Value of Travel Time Reliability in Transportation Decision Making: Proof of Concept—Maryland (2014)

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73 P A R T 2 DEScRiPtion of thE MEthoD While the data-driven method shows promise, additional validation is required for the under- lying theories, method, and test applicant results. Suggestions for further research are presented in this section.

74 C H A P T e R 5 The objectives of the L35B project are to • Select and defend a value or range of values for travel time reliability for the Maryland State Highway network. • Use the value of travel time reliability (VOR) in the Maryland State Highway Administration (SHA) project development process to prioritize operational and capital improvements and determine if (and how) the ranking of projects changes due to the addition of VOR. • Report for the benefit of others the step-by-step process used to develop, justify, apply, and assess the use of VOR in the Maryland SHA project evaluation and decision process. This technical memorandum is the deliverable for L35B Task 2, which was to develop and apply a methodology to select a travel time reliability performance measure and a value or range of values for travel time reliability. Currently, the Maryland SHA is using planning time index (PTI) to measure travel time reliability on highway facilities. Also, the Maryland SHA has adopted a 0.75 reliability ratio (RR) to measure the economic benefits of improvements in travel time reliability when conducting benefit–cost analysis of congestion relief projects. Conventionally, RR is defined as the ratio of value of travel time reliability (VOR) and value of time (VOT). This value is adopted by the Maryland SHA based on a comprehensive literature search of existing national and inter- national resources as well as existing federal recommendations for RR value. This task report seeks to validate this number or provide a basis for changing it based on local data. A data-driven methodology is proposed for estimating a reliability ratio (RR) and ultimately a value of travel time reli- ability (VOR). The methodology has been implemented in MATLAB to automate the process. A year’s worth of archived probe-based travel time data is used to estimate the local RR and VOR values on five different corridors (in two directions) in Maryland. The initial results indicate that the current num- ber is conservative and may have to be revised. However, the estimated number (0.87) is believed to be at the upper range of values for travel time reliability. Further analysis is needed to justify any decision to increase the current value of travel time reliability. This analysis will be facilitated by some of the data that are currently being used to complete tasks 3, 4, and 5 of this project. In particular, Maryland Statewide Transporta- tion Model (MSTM) results will be crucial in aggregating the results for all origin–destination (O-D) pairs in the state. It is recommended, at this point, to consider 0.75 to 0.87 as the local range of viable values for the travel time reliability ratio. This report includes specific details of an approach to esti- mate VOR and RR. The proposed method is data driven and requires access to fine granularity and long-term archived travel time data. This method is based on the analogy of an insurance policy designed to cover travelers against the negative impacts of unexpected variations in travel time. The proposed method has been designed to provide maximum flexibility for valuing travel time reliability based on existing local information and experiences. A review of the previous attempts to apply Real Options concepts to the problem of travel time reliability valu- ation is provided. Reasons as to why the previous attempts have received a cautious review are explained. Also, this report sets out to unravel some of the less clear parts of previous works by venturing further into the nuts and bolts of the approach. This report clearly identifies the distinctions between the proposed method and the earlier works. Also, included in the report is a brief background on clas- sical utility theory and its application in travel time reliability valuation. Strengths and limitations of utility-based estima- tion methods are discussed. A travel time insurance analogy is adopted to illustrate different aspects of the proposed approach. Setting a premium on the proposed travel time insurance is presented and discussed in the context of option- theoretic valuation and asset pricing. Examples are provided throughout the text to facilitate the discussions and to dem- onstrate application of the concepts. Applications of the pro- posed methodology using a year’s worth of travel time data in introduction

75 the state of Maryland are reported. Analysis performed on the results of this application are presented and models to relate the travel time reliability ratio and average travel time, as well as 95th percentile travel time and average travel time, are cali- brated. The next steps to finalize the range of values for travel time reliability in the state, based on the statewide model’s results, are discussed. Finally, this technical memorandum includes two appendi- ces. Appendix D provides a brief review of stochastic processes, and in particular, the geometric Brownian motion process, including its properties and relationships with random walks. Appendix E presents more details about the application of the proposed methodology to the 10 directional corridor cases in Maryland and their various results. Real Options and Applicability to Travel Time Reliability Valuation In this section an analogy is used to develop a methodology to select a value or range of values for travel time reliability. The analogy relates to an insurance premium that one would be willing to pay in order to keep one’s travel time below a certain threshold. For instance, if a traveler, based on experience, knows that their morning commute to work takes 10 minutes on average, they might be willing to add 5 minutes to their trip time so that they could be certain the trip time would be less than 15 minutes; otherwise they will be compensated for any additional time spent on the trip. In other words, this approach strives to find a certainty equivalent for travel time unreliabil- ity in terms of additional expected travel time. From this brief description it is clear that this valuation approach relies on knowledge of the following factors from a traveler’s perspective: • Expected travel time; • Level of travel time variations; • Acceptable level of travel time variations (traveler’s toler- ance); and • A sense of how longer than expected travel times (or even shorter) will negatively (or maybe sometimes even posi- tively) affect the traveler’s experience. In addition, from a rational decision making perspective, the valuation approach should take into account conditions under which the traveler would consider existing travel time variations as reliable. These certainty-equivalent conditions would, in fact, determine the state of the world in which a traveler would be indifferent between experiencing an unreli- able travel time scenario or incurring an additional (but fixed) travel time up front that guarantees a certain level of travel time reliability. The approach stems from asset pricing efforts in finance in which risky assets are valued according to their expected future payoffs. The specific type of assets that is commonly used to model insurance policies are referred to as options. Options are common in stock trading and are meant to protect share holders against excessive increase or decrease in share prices.

76 Background Before details of the proposed approach are presented, some background is provided on the relationship between travel time and travel cost, the role of random utility in classic discrete choice analysis, and definitions of travel time and travel time reliability values based on the existence of a utility concept. In addition, two different approaches to travel time reliability val- uation based on the travel time insurance analogy are presented. These approaches include the Real Options concept and option pricing theory in the context of general consumption-based asset pricing. Applications of Real Options in the field of trans- portation decision making, planning, and reliability valuation are also briefly reviewed. Travel Time and Cost Travel time and travel cost are usually directly linked to each other. It is common practice to assume travel cost (TC) is equal to travel time (TT) times a constant factor. This factor is commonly referred to as value of time (VOT), which reflects the perceived rate at which travelers would value the time they spend in their trips. The linear relationship between travel time and travel cost is shown in Equation 6.1. TC TT VOT (6.1)= × It should be noted that this is the simplest expression of travel cost as a function of travel time. If a more general form is known to exist that better specifies this relationship (such as multivalued, piecewise, or nonlinear functions), the proposed methodology, in its general form, will be capable of estimating appropriate travel time reliability values. Discrete Choice Analysis and Random Utility/Consumer Theory Discrete choice analysis, in the context of trip decisions, is based on consumer theory in microeconomics. Consumer theory allows for modeling the action of consumers under given circumstances (e.g., budget, prices). A discrete choice model can be presented by a set of general assumptions about (1) 1. Decision maker (individual, household, socioeconomic attributes) 2. Alternatives (set of options available to the decision maker) 3. Attributes (the measures of benefit/cost of an alternative available to the decision maker) 4. Decision rule (the process by which the decision maker chooses an alternative) The decision rule commonly used for travel behavior applications is based on utility theory, which assumes a deci- sion maker’s preference for an alternative is captured by util- ity, a single value that is a function of decision maker and alternatives attributes. The decision maker selects the alterna- tive with the highest utility. Random utility theory assumes that the decision maker has perfect discrimination capability but, at the same time, the utility cannot be exactly specified. In fact, the uncertainty in utility may be explained by (2) • Unobserved alternative attributes; • Unobserved individual characteristics (taste variations); • Measurement errors; and • Proxy, or instrumental variables. Classic VOT and VOR Estimation Value of travel time reliability (VOR) is usually derived from utility function calibration performed in the context of discrete choice analysis for travel demand modeling. This approach is basically known as a risk-return model in finance, in which a decision maker looks to maximize the asset’s return while minimizing its associated risk. The asset’s return is rep- resented by the expected value and the risk by the variance (3). In the current context, therefore, both expected travel time C h A p T E R 6

77 and its variability (a measure of unreliability) are regarded as sources of disutility. In its most general form, the deterministic part of the utility of an alternative (mode, route, or both) can be stated as a func- tion of expected travel time (TT), associated out-of-pocket cost (OPC) of travel, and some measure of travel time (un) reliability/variation (RM) as shown in Equation 6.2. TT, OPC, RM (6.2)u U ( )= + ε In the classic approach, VOT is specified as a ratio of the marginal utility with respect to travel time and the marginal utility with respect to out-of-pocket cost. In essence, based on this definition VOT is the rate of substitution between the marginal utility of average travel time and the marginal utility of the trip’s out-of-pocket cost (10) as shown in Equation 6.3. VOT TT OPC (6.3) U U = ∂ ∂ ∂ ∂ VOT is known to vary with trip purpose, income level, and other socioeconomic attributes of the subject population. Practical evidence shows that the magnitude of VOT esti- mated based on this approach is normally comparable to the relevant wage rate of the individual decision maker or the subject population. Similarly, VOR may be estimated in an identical manner to VOT from a utility function calibration. VOR is the rate of substitution between the marginal utility of travel time un- reliability and the marginal utility of the trip’s out-of-pocket cost as shown in Equation 6.4. VOR RM OPC (6.4) U U = ∂ ∂ ∂ ∂ However, apart from the practical difficulties in conducting regular preference surveys (stated or revealed), other theoreti- cal obstacles still exist that render applying this approach dif- ficult to implement if not impractical in most cases. First, the fact that average travel time and travel time variability mea- sures are naturally correlated makes it difficult to find un- biased VOT and VOR estimates using this approach. Second, stated preference respondents are known to have a subjective bias toward shorter average travel times and alternatives with lower costs. Nevertheless, random utility-based models are state-of-the-practice in estimating the above measures. Reliability ratio (RR) is classically defined as the ratio of value of reliability (VOR) to the value of time (VOT). In other words, reliability ratio is the fraction of VOT that speci- fies the value travelers assign to a unit variation in their travel time as shown in Equation 6.5. RR VOR VOT RM TT (6.5) U U = = ∂ ∂ ∂ ∂ It should be noted that the RR definition in Equation 6.5 is based again d on the rate of substitution between two rele- vant marginal utilities. Note that in the special (and widely used) case where utility can be expressed as an additive linear function, VOT, VOR, and RR will be equal to the ratios of the relevant parameters in the utility function. Equations 6.6 through 6.9 follow. TT OPC RM (6.6)U a b c= × + × + × + ε VOT (6.7)a b= VOR (6.8)c b= RR (6.9)c a= Utility-Based Reliability Valuation To estimate a value for travel time reliability, a simple approach based on the concept of utility in discrete choice analysis is presented. For a given individual and a given trip, substitution between different attributes while utility is maintained at a constant level (indifferent decision maker) can be used to esti- mate a reliability value. To explain this approach further, let’s assume that a certainty-equivalent addition to the average travel time (X) is known; the utility function in Equation 6.6 can be written as shown in Equation 6.10. TT , OPC, RM (6.10)u U X r( )= + where (RMr) is a known parameter referring to the level of travel time variability measure that is perceived as tolerable by decision makers. Using the first-order Taylor’s expansion around the current point (TT, OPC, RM), Equation 6.10 can be approximated by the expression shown in Equation 6.11: TT, OPC, RM TT RM RM RM (6.11) u U U X U r ( ) ( ) ( )( ) ≅ + ∂ ∂ + ∂ ∂ − Comparing Equations 6.2 and 6.10, it is clear that to main- tain the indifference condition, the second and third terms in Equation 6.10 must add up to zero. By referring to the reliability ratio definition in Equation 6.5, the following expression for reliability ratio (RR) can be derived as shown in Equation 6.12: = ∂ ∂ ∂ ∂ ≅ − RR RM TT RM RM (6.12) U U X r This statement suggests that the reliability ratio can be esti- mated by dividing the certainty-equivalent additional travel time X by how much the current reliability measure RM devi- ates from its reliable norm, RMr. It should be noted that in general the second equality in Equation 6.12 is approximate.

78 However, in the case of the additive linear utility function Equation 6.6, this equality will be exact. Note that the certainty-equivalent addition to average travel time (X) can be interpreted as an insurance premium. The policy ensures a reliable trip time with no (or tolerable) varia- tion at the cost of adding X units to the average travel time. This is a very interesting result as it suggests that conceptu- ally, RR can be stated as the ratio of two variables as opposed to being the ratio of two unknown model parameters that requires model calibration to determine their values. Fur- thermore, it should be noted that multiplying both sides of Equation 6.12 by VOT, the following expression for the value of reliability VOR is obtained as shown in Equation 6.13. VOR RM RM VOT (6.13) X r ≅ − × However, note that the usefulness of this alternative approach hinges on access to good estimates of the additional certainty-equivalent travel time (Xˆ). In general, finding a good estimate for X based on the utility theory is not a straight forward problem. The next section provides an alternative framework to approach the travel time reliability valuation problem. In transportation project evaluation and in travel behavior mod- eling in general, time can be viewed as the asset (with a cor- responding capital value) that travelers invest into their trips going from Point A to Point B. An insurance policy that offers to compensate the traveler for variations in travel time is therefore an asset that would help travelers (investors) to con- trol the risk associated with their travel times (capital invest- ments). In this context, establishing the relationship between expected return and the risk measure is an essential part of any asset pricing theory. A brief background on consumer theory in finance asset pricing models that are potentially useful for the problem at hand is provided. Asset pricing is a mature topic in econom- ics and finance. This exposition of asset pricing is mainly from the perspective of consumption-based models (4). Consumption-Based Asset pricing Model The decision to take a trip can be viewed as an investment problem in which time is the essential asset travelers have. If the decision is to take a trip, then it means the individual has decided to invest a portion of their available time budget into the trip; in other words they have decided to consume their time in moving from Point A to Point B. By doing so, the trav- eler knowingly has reduced their available budget for initial consumption at Point A in the hope of gaining more utility by consuming their remaining time in a desirable activity later on at Point B. So, in a way, the trip decision involves a trade-off between consuming available time at the current Point A and moving to the new location (and losing some of the available time in the process) and consuming the remaining time (or portions of it) at Point B. In general, the decision is compli- cated by the fact that travel time from A to B is not determin- istic. Expected and unexpected components of the travel time between A and B are both important to the decision makers. A traveler may be willing to build an extra amount of time into their trip (on top of what they expect the trip to take) to safeguard against variability in travel time. In fact, the payoff they will get at the end of their trip as a result of this decision is determined by the amount of time the actual trip time devi- ates from the guaranteed level. At the decision point, the decision maker (traveler) must decide how much time they want to spend at the current location and, by moving to the new location, how much time they will have at the new location considering that moving from A to B introduces uncertainty in the amount of time that needs to be budgeted for the trip itself. Figure 6.1 illustrates the consumption-based asset pricing model for a particular traveler. To formalize the rest of the discussion, let the traveler set aside (at time t0) a budget of B0 in equivalent monetary units for the trip. The traveler expects to spend E(t) time units on this trip (with a value rate equal to the VOT) and is willing to spend an extra X time units (with a value rate equal to the VOR) in order to buy the afore- mentioned insurance policy that guaranties them a reliable travel time. Therefore, the amount of budget left to be con- sumed initially (c0) can be stated as shown in Equation 6.14. VOT VOR (6.14)0 0c B E X( )= − × τ − × However, in general, as the trip takes place, the actual/ realized travel time t will include a Dt time units deviation from the expected travel time E(t) plus the additional allotted time X. See Equation 6.15. (6.15)E X( )τ = τ + + ∆τ At the end of the trip, depending on how long it actually took the traveler to get to their destination, they may be left with a terminal budget Be and a payoff at a rate of fe(Dt) from the initial investment in the aforementioned insurance policy of the size X. See Equation 6.16. ( )= + ∆τ × (6.16)c B f Xe e e The traveler’s decision problem can be stated in terms of a utility maximization problem in which the objective is to max- imize the utility of consumptions (of the traveler’s time in activities other than the trip). Of course there is a distinction between consuming now and in the future and also between spending time at A or at B. So, in general the objective function

79 is the sum of the utility of the initial consumption at Point A and the traveler’s expectation of the discounted (with factor b) utility of consumption at the end of the trip at Point B as shown in Equation 6.17. max (6.17),0 0 ,u c E u cA B e e[ ]( ) ( )+ β To solve Equation 6.17, subject to Equations 6.14 through 6.16, assuming regularity conditions on utility functions (mainly concavity due to a decision maker’s risk averseness), it is possible to take the derivative of the objective function with respect to the additional time X and then set the deriva- tive equal to zero as shown in Equation 6.18. 0 (6.18),0 0 0 ,u c c E u c cA B e e e[ ]( ) ( )′ ′ + β ′ ′ = Substituting the derivatives from Equations 6.14 and 6.16 into Equation 6.18, the following equality is derived as shown in Equation 6.19. VOR (6.19),0 0 ,u c E u c fA B e e e[ ]( ) ( ) ( )× ′ = β ′ ∆τ Therefore, it is shown that under optimality conditions, the marginal utility loss of spending a little less time at the origin and buying a little more of the asset (insurance) should equal the marginal utility gain of spending a little more time at the destination in the future. Then value of reliability (VOR) may be stated as the expected value of the discounted payoffs scaled by the relative marginal utilities at the trip’s origin and destination as shown in Equation 6.20. VOR (6.20) , ,0 0 E u c u c f B e e A e ( ) ( ) ( )= β ′ ′ ∆τ   Setting , ,0 0 u c u c m B e e A ( ) ( )β ′ ′ = , the following expression for VOR is obtained as shown in Equation 6.21: VOR (6.21)E mfe[ ]( )= ∆τ Different asset pricing models are proposed for application in the case of nonlife insurance policy valuation. Each model makes certain assumptions about how the asset in question evolves over time to produce a distribution of different out- comes (Dt) and how the payoffs fe(Dt) are determined. Also, they make assumptions on the discount factor m used to trans- form the value of payoffs at a future time to the current time. The most widely used asset pricing methods in practice that are also extensively studied in the literature include the following: • Capital asset pricing model (CAPM) (5,6) • Arbitrage pricing theory (APT) (7) • Option pricing theory (OPT) CAPM and APT are essentially linear-factor pricing mod- els in which the discount and marginal utility growth expres- sions in the consumption-based model are replaced with a linear model of the form shown in Equation 6.22. (6.22) , ,0 0 m u c u c a b F B e e A ( ) ( )= β ′ ′ ≅ + × where a and b are parameters. Factor pricing models look for variables (F) that are good proxies for aggregate marginal utility growth. In the classic CAPM, the adopted factor is the return on the “wealth portfolio” (11). Figure 6.1. Consumption-based asset pricing model depiction.

80 In this project, option pricing theory is adopted to deter- mine the value of travel time reliability (VOR). First, however, a brief overview of options and their applications in the inter- disciplinary area of transportation economics is provided. What Is a Real Option? Trigeorgis (8) gives the following concise definition of an option as a financial instrument: “An option is the right (not the commitment/obligation) to buy (if a call) or to sell (if a put) a specified asset (e.g. common stock) by paying a speci- fied price (the exercise or strike price) on or before a specified date (the expiration or maturity date). If the option can be exercised before maturity, it is called an American option; if only at maturity, a European option.” However, the concept of insurance on travel time variabil- ity introduced here actually falls in the real options category. When the asset in question is tangible or real (as opposed to intangible financial instruments), the choices and decisions that come to existence in regard to operating and managing (such as altering, abandoning, expanding, shrinking, or deferring) that asset are commonly referred to as real options. Real Options in Transportation projects Garvin and Cheah (9) introduced the real options valuation techniques in the context of infrastructure investment deci- sions. They bring to light the fact that traditional project evaluation methods fundamentally fall short in taking into account the inherent uncertainty in cash flow and interest rates in their assumptions. Frequently, this leads to flawed evaluations and inappropriate investment decisions. They provide an interesting and somewhat detailed account of the Dulles Greenway (an early toll road project in Virginia that went into operation in 1995), whose forecast demand and income levels were not met. Project sponsors therefore had to renegotiate a plan for deferring debt payments and had to restructure the loan contracts with their creditors. Pichayapan et al. (10) and Zhao et al. (11) use real options approaches to plan for highway investments under stochastic demand conditions. Saphores and Boarnet (12) analyzed the impact of uncertainty in population levels on optimal tim- ing for investment in a congestion relief project. They con- sidered the case of a linear city with fixed boundaries and a single CBD. It is shown that under certainty conditions, maximizing the utility of living in the city for its population is approximately equivalent to a standard benefit–cost analy- sis (BCA). However, when the urban population levels evolve as a stochastic process it is shown that, depending on the length of project implementation, optimal timings would vary considerably. Vergara-Alert (13) proposes an extension of the real options theory for application in decisions regarding transportation projects. It is assumed both construction costs (outflows) and operating revenues (inflows) follow standard stochastic pat- terns. Then providing a different perspective, it is argued that the ratio of social operating revenues over construction costs can be modeled as a mean-reverting process that provides for improved modeling and description of real transportation finance cases. Chow and Regan (14, 15) propose a mathematical optimi- zation framework to incorporate deferral options in network level investment decision making. In their study, the source of stochasticity is random variations in O-D demand, which is exogenous to the problem. A variation of the Monte Carlo method originally proposed by Longstaff and Schwartz (16) (Least Squares Monte Carlo Simulation—LSM) was adopted to solve the resulting dynamic and stochastic network design problem. The method is applied to a small-size network but it does not scale up efficiently if the number of investment projects considered increases significantly. Another limita- tion of this method is that in the long run, despite evidence suggesting otherwise, O-D demands are not affected by con- gestion levels, travel time uncertainties, or infrastructure investments. Later they applied the model with some modifi- cations to a larger network (17). Real Options in Trip and Route Choice Decision Making Friesz et al. (18) introduce the idea of a European type conges- tion call option to value commuting to work along a given path for a given departure time selected by automobile drivers. Their treatment is based on the dynamic user equilibrium (DUE) concept in which drivers are modeled as Cournot-Nash non- cooperative agents competing for limited roadway capacity when telecommuting from home is offered as an alternative to driving in a congested and unreliable network. Using a small net- work example, they show that offering a congestion call option to travelers may lower the net social costs of congestion. Real Options in Travel Time Reliability Valuation To the best of our knowledge, there is only one reported case of applying Real Options methodology to the problem of travel time reliability valuation in the literature. A brief account of this application has been first reported by Puget Sound Regional Council (PSRC) (19) in the context of travel time reliability benefits estimation in a more general benefit–cost analysis setting. A more detailed account of this unique appli- cation is given in SHRP 2 Project L11 (20) and then summa- rized as a guidebook in Project L17 (21).

81 The reported application is based on the option-theoretic concept, which is a well-documented and comprehensively studied topic in the field of finance. While this is an innova- tive and bold application of a mature concept from finance into transportation economics, it has been met with criti- cism by some. These criticisms mainly stem from the fact that L11 failed to convey the option valuation process and its main ideas in a way that is accessible by other experts. Of course, using a closed-form solution built on a very specific set of assumptions about the underlying travel speed process and envisioned reimbursement policy did not provide a great deal of transparency. Besides, the fact that option char- acterization in Project L11 is based on speeds, and not travel times, raised serious questions about the justification of its application. At the time the traveler decides to take a trip, they only have an idea about the trip time in terms of its expected travel time and some measure of its variation. If the traveler considers paying a premium in terms of leaving earlier (adding time to the expected travel time) in order to obtain insurance on the trip time (e.g., it will not deviate from a certain level or they will be compensated), then it is presumable that they will be willing to obtain the policy for the maximum possible dura- tion of their trip to protect against the worst odds. In general, the proposed method is very flexible and can be applied to a wide range of all possible conditions. In the next chapter, components of such a “hypothetical” travel time insurance policy are introduced and discussed. The compo- nents of the methodology that lead to the design of the travel time insurance policy and its valuation are discussed next.

82 Methodology The methodology proposed to value travel time reliability based on option theory is described in this chapter. The pro- posed method is based on an analogy of a travel time insur- ance policy. The method applies historical travel time data over an extended period of time as input and performs the necessary analysis to identify variations that are experienced by travelers and, based on these variations, estimates a ratio- nal value for reliability that would be offered as the travel time insurance policy. In summary, to describe the method, the following questions need to be answered: 1. How can travel time evolutions over time be modeled? 2. How can a penalty/reward (payoff) of early/late arrivals at the destination be determined? 3. What is the guaranteed level of travel time? 4. What is the duration of time for which the travel time insurance policy is issued? 5. How do future payoffs get valued at the outset of trip? Figure 7.1 illustrates the above-mentioned components of an option-theoretic valuation method. Note that this is a generic graphic. The methodology will be fully described by specifying each component in the following sections. Travel Time evolution Current research supports the notion that, at any given time, travel times are lognormally distributed (22) and that, over time, they represent a memory less Markov process (43). By definition, a continuous lognormally dis tributed process, which is also a Markov process, can be modeled using the geo- metric Brownian motion with drift (GBM) stochastic process (see Appendix E). This implies that changes in the continu- ous travel time process {t} can be expressed as (7.1)d a dt dzτ = τ + στ where a and s are instantaneous drift (trend) and standard deviation parameters of the process, respectively. Recalling Ito’s lemma, the GBM process suggests that ran- dom variable t is lognormally distributed with the follow- ing mean and variance at time t when the initial time is denoted by t0: exp (7.2)0 0E t t a t t[ ]( ) ( ) ( ){ }τ = τ − exp 2 exp 1 (7.3)2 0 0 2 0V t t a t t t t( )[ ] { }( ) ( ) ( ){ } ( )τ = τ − σ − − For more information on the GBM stochastic process and relevant derivations, interested readers are referred to Appendix D. Random Walk Representation of Geometric Brownian Motion The GBM process can be approximated by a discrete random walk. Set discrete time intervals equal to Dt and increments of the log-travel time equal to (7.4)h t∆ = σ ∆ Then, the multiplicative step-up and step-down factors will be calculated as exp (7.5)u t= σ ∆  exp (7.6)d t= −σ ∆  Respectively, probabilities of taking a step up or a step down in the random walk are given by 1 2 1 2 (7.7) 2 q a t= + − σ    σ     ∆     C H A P T e R 7

83 1 1 2 1 2 (7.8) 2 q q a t′ = − = − − σ    σ     ∆     Then, it can be shown that over any T time units (T = nDt), the expected and variance of log-travel time changes (a bino- mial distribution) can be expressed as log log 0 2 (7.9) 2 E T n q q h a T[ ] ( )( )( ) ( )( )τ − τ = − ′ ∆ = − σ  log log 0 1 1 2 (7.10) 2 2 2 2 2 V T n q q h a t T [ ] ( )( )( ) ( )( ) ( )τ − τ = − − ′  ∆ = − − σ    σ     ∆     σ In the limit, as time steps become smaller (Dt → 0), the mean and variance of the travel time displacement from its initial value as described by the random walk will be equal to the following: log log 0 2 (7.11) 2 E T a T[ ]( )( ) ( )( )τ = τ + − σ  log 0 (7.12)2V T T[ ]( )( ) ( )τ τ = σ Note that both mean and variance of the random walk in the limit are independent of the adopted discretization stencil (Dt &Dh). Example 1 To illustrate the above concepts, a simple example is pre- sented in which travel time variations are modeled as a GBM stochastic process with instantaneous trend and standard deviation parameters equal to 5 and 10%, respectively. Figure 7.2 illustrates a realization of such process over the next 20 minutes when initial travel time is equal to 10 min- utes. The travel time realization shown in blue is only one instance (out of an infinite number of instances) that travel time could have evolved under the assumption of this parti- cular GBM process. The solid red line represents the travel time trend and at any time T can be expressed by 10exp 0.05 (7.13)E T T[ ]( ) ( )τ = Similarly, the pair of dashed red lines in Figure 7.2 repre- sents the lower and upper 95% confidence intervals around the mean of the process. The variance of travel time at any time T is given by 10 exp 2 0.05 exp 0.1 1 100 exp 0.11 exp 0.1 (7.14) 2 2V T T T T T [ ] [ ] [ ] ( )( ) ( ) ( ) ( ) τ = × − = − The random walk representation of this process at 2-minute increments (Dt = 2) can be built as a binomial tree with 0.1 2 0.14h( )∆ = ≅ increments in the y-axis with a loga- rithmic scale. However, in the normal travel time scale, the multiplicative step-up and step-down factors will be equal to exp 0.14 1.15 (7.15)u ( )= = 1 exp 0.14 0.87 (7.16)d u ( )= = − = With the following probabilities associated with step-up and step-down moves, respectively: 1 2 1 0.05 0.1 2 0.1 2 0.82 (7.17) 2 q = + −             ≅ Time Travel Time Guaranteed Travel Time Policy Duration ETA Penalty/ Claim Travel Time PDF Time 1 2 3 4 Figure 7.1. Various components of a travel time insurance pricing method.

84 1 0.82 0.18 (7.18)q′ = − ≅ Also, the expectation and variance of log-travel times at any time step (T = nDt) will be given by the following expressions: log log 10 0.05 0.1 2 2.30 0.045 (7.19) 2 E T T T[ ]( )( ) ( )τ = + −  ≅ + log 10 1 0.05 0.1 2 0.1 2 0.1 0.00595 (7.20) 2 2 2V T T T ( )[ ]( ) ( )( )τ = − −   = Similarly, at 1-minute time increments (Dt = 1), the relevant parameters of the corresponding binomial tree representation will be the following: ∆ = =0.1 1 0.1 (7.21)h ( )= =exp 0.1 1.11 (7.22)u ( )= = − =1 exp 0.1 0.90 (7.23)d u 1 2 1 0.05 0.1 2 0.1 1 0.725 (7.24) 2 q = + −           = ′ = − =1 0.725 0.275 (7.25)q log log 10 0.05 0.1 2 2.30 0.045 (7.26) 2 E T T T[ ]( )( ) ( )τ = + −  ≅ + ( )( ) ( )[ ]( )( )τ = − −  = log 10 1 0.05 0.1 2 0.1 1 0.1 0.007975 (7.27) 2 2 2V T T T Payoff Characterization It is conceivable that travelers would normally incur a penalty as a result of arriving later or earlier than scheduled at their destination. Under the current analogy with an insurance policy, the travelers who obtain the option at the start of their trip will be reimbursed for any penalty they incur at the ter- mination of their trip due to deviation of their actual arrival time at the destination from their expected arrival time. In this section a general framework for characterization of such hypothetical payoffs is presented. It should be noted that in the scheduling approach (23, 24) to activity decision making, time spent in travel between ori- gin and destination, the magnitudes of lateness and earliness 0 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 50 60 70 80 ALPHA = 0.05 SIGMA = 0.1 Time (Minute) Tr av el T im e (M inu te) Travel Time Trend Lower 95% CI Upper 95% CI Figure 7.2. Sample travel time evolution path simulated as geometric Brownian motion (GBM) process.

85 at the destination compared with the preferred time of arrival PTA are introduced in a linear-additive form to specify trip utility. Of course, this is a simple specification of the utility in which trip costs and additional terms are ignored. , PTA PTA PTA (7.28) 0 0 0 U t a t t DL i i i i ( ) ( ) ( ) τ = τ + β + τ − + γ + τ − + θ − + The scheduling preference expressed by Equation 7.28 is often referred to as (a, b, g) model. Extending this activity scheduling approach to explicitly include the uncertainty of travel time, and then minimizing the expected disutility for a traveler, estimates of optimal departure time t*0, VOT, and VOR may be obtained (25, 26). The scheduling approach pro- vides insight into the relationships between the value of travel time reliability and theoretical or empirical travel time distri- butions (27). One important result of this analysis is the first- order condition for optimal departure time in the special case q = 0 (28, 29): = β β+ γ * (7.29)PL where P*L is the optimal probability of being late. In the United States, a choice of P*L = 0.05 is consistent with selection of 95th percentile travel time to determine buffer time travelers add to their average travel times (30). This leads to the inter- esting result that being 1 minute late is almost 19 times as negatively perceived as being 1 minute early (g @ 19b) by an average traveler. Similar to the scheduling approach, in this study the pen- alty associated with arriving late or early is assumed to mainly depend on the departure time t0, actual travel time t, and pre- ferred time of arrival PTA. In general, additional factors such as trip purpose TP and traveler’s socioeconomics SE can also be introduced in the payoff model. ( ) ( )τ = + τ −PTA, TP, SE (7.30)0 0C t f t For a given trip purpose and an individual traveler (constant last two arguments), the simple linear-additive expression for payoff conditioned on departing at t0 with a fixed PTA can be written as ( ) ( ) ( )τ = β + τ − + γ + τ −− +PTA PTA (7.31)0 0 0C t t t For brevity purposes, by explicitly noting that cost of travel depends on departure time t0, let us drop the conditional and set PTA = t0 + E(t), which suggests the traveler ignores the unreliability of travel time and budgets only the expected travel time for their trip. This makes sense in this context since the traveler is assumed to have obtained an insurance policy that provides full protection against potentially negative impacts of travel time variations. Finally, let’s express unit earliness and lateness costs as coefficients of VOT: VOT (7.32)C a E b Ei i i[ ]( ) ( )( ) ( )( )τ = τ − τ + τ − τ− + Note that the above payoffs are, in fact, retroactively calcu- lated meaning that initially the realized travel time (t) and therefore its corresponding payoff (C(t)) is not known to the traveler. This implies that the cost statements discussed in this section are only applicable at the end of the insurance policy validity period. Provided that the insurance policy covers a period longer than the actual travel time experienced by the traveler, the cost associated with travel time variability around its expected value after its realization can be obtained by the following expression: VOT (7.33)C a T E b T EN i i i( ) ( )( ) ( ) ( ) ( ) ( )τ = τ − τ + τ − τ − + where CN denotes the cost associated with travel time variabil- ity as calculated at the termination of the insurance policy period (time step N). In most practical cases, it is expected that b is positive while a may assume negative values (for instance when arriving early is incentivized). Also, it is reasonable to expect b to be larger than a (b >> a), since the cost of being late is usually much larger than the cost of being early. Figure 7.3 illustrates the general bilinear form of the above cost function in which normalized costs are depicted versus deviation of travel time from its expected value. It should be noted that, in general, the cost function can take any form. Figure 7.4 depicts the common sense constraints on magni- tudes of cost parameters a and b as well as their feasible region. As noted earlier, the left half of the feasible region (a < 0) is indicative of situations in which early arrivals are rewarded as, for instance, in the case of work trips when travelers start getting paid as soon as they get to their workplace no matter how early they are. In this research it is assumed that b = 1, which indicates the cost of being late is equal to the value of extra time (compared with the expected time) the traveler has spent on the road. This is a conservative assumption since it does not account for Figure 7.3. Bilinear payoff function.

86 the wages lost, impacts of schedule disruptions, and negative reactions by other parties involved (e.g., boss at work, teacher at school, friends and relatives). Therefore, from what we know about the average traveler’s relative perceived costs of being late or early, a is estimated to be negligible (a @ 1/19 @ 0.05) for practical purposes. It is widely believed that VOT for local personal and busi- ness travel in the United States is about 50 and 100% of the wage rate, respectively (31). Therefore, in the case of work trips in which arriving late would reduce travelers’ income proportional to the amount of time they are late, b = 2 would be more realistic (a @ 2/19 @ 0.1). Guaranteed Travel Time Travel time unreliability is measured as the variability around the mean travel time. Therefore, it is common sense to assume that a guaranteed level of any travel time insurance policy designed to protect travelers from unreliability of their trip time must be the expected travel time (its mean or average). This is also in line with the previous definition of payoff function at the termination of the travel time insurance policy. Note that the proposed methodology is able to deal with any other level of travel time as the guaranteed value. The choice to proceed with expected travel time as the guaran- teed level of insurance policy is based on the current under- standing and interpretation of reliability as perceived by travelers. This selection is not a limitation for this method and can be relaxed as soon as another desirable level or range of levels for guaranteed travel times are deemed as more reasonable. Duration of Travel Time Insurance Policy It is customary to assume that the threshold between recur- ring and nonrecurring traffic congestion falls somewhere between 80th and 95th percentiles of a travel time distribution. This means that on average, 5–20% of the trip times are subject to nonrecurrent disturbances (e.g., incidents, weather events). While this probability may be different in any particular case, on average for well-designed, well- maintained, and carefully operated surface facilities with traffic incident management practices in place, the 5% per- cent risk level in encountering nonrecurrent congestion seems more acceptable. The insurance policy duration adopted in this research reflects the maximum conceivable duration of travel time as a result of recurrent congestion. The 95th percentile travel time is again a conservative choice as it, in effect, creates a policy long enough for any trip impacted by recurrent congestion to be compensated after the trip is terminated. Again, it should be noted that the proposed method is able to deal with any other policy duration and by no means is restricted to the particular duration selected in this research. Certainty-equivalent Payoff Valuation So far, we know how payoffs at the termination of the insur- ance policy duration are calculated. But, we still need to answer the question of how the payoffs will be valued at the start of the trip. To answer this question, first we need to define under what conditions travel time and its variations would be con- sidered as reliable. Figure 7.5 shows a branch of the binomial tree that is used to represent the random walk approximating a GBM process that models travel time (t) variations. The top branch illustrates the random walk in terms of travel time loga- rithms where in one time step the current logarithm gets incremented up and down by complementary probabilities Early arrival is incentivized Figure 7.4. Space of normalized early and late arrival perceived costs. Figure 7.5. A binary branch of the binomial tree representing travel time variations as a random walk.

87 p and 1 - p, respectively. The middle branch is a rescaled version of the top branch where all travel times are expressed in unit time scale. From a traveler’s perspective, at the start node of this branch, travel times in the next time step will be con sidered as reliable only if they expect them to vary within a certain range around its current value. The expected next-step travel time can be simply calculated as the weighted average of the binary next-step travel times: 1 (7.34)1E p u p dn n n ni i( ) ( )( ) ( )τ τ = τ + − τ+ Note that at this point, unlike process probabilities (q and 1 - q), the certainty-equivalent probabilities (weights, p and 1 - p) used in calculation of the above expectation are not known. Also, the certainty condition can be expressed as (7.35)1E r tn n n n i i( )τ τ − τ ≤ τ ∆+ where certainty threshold r is defined as a percentage rate of the current travel time and the size of time step. In practice this certainty threshold can be assumed to be small (r ≤ 5%). For instance, when the current trip time is 40 minutes, the travel time is deemed as reliable when, in the next 5 minutes, it is still within the range of 40 ± 10 minutes. Expanding Equation 7.35 as inequality pairs and substitut- ing for expected future travel time from Equation 7.34, the following expressions are obtained: 1 1 (7.36)1r t E r tn n n ni i i i( ) ( )( )τ − ∆ ≤ τ τ ≤ τ + ∆+ 1 1 1 (7.37)r t p u p d r ti i i i( )− ∆ ≤ + − ≤ + ∆ Note that Equation 7.37 is not dependent on travel times as they cancel out from both sides of inequalities. This gives a range for binary probabilities that would ensure certainty conditions in travel time: 1 1 (7.38) d r t u d p d r t u d i i− − ∆ − ≤ ≤ − + ∆ − For small tolerance rates (r @ 0), or in general when the time steps are small (Dt → 0), or simply if the midrange probability is targeted, the certainty-equivalent probability is expressed as 1 (7.39)p d u d = − − And, thus: 1 1 (7.40)p u u d − = − − Now that certainty-equivalent probabilities are specified, they can be used to value previous step payoffs by taking their certainty-equivalent expectation: 1 , 0, 1, 2, . . . , 1 (7.41) 1 1C p C u p C d n Nn n ni i( )( ) ( ) ( )τ = τ + − τ = −+ + In Figure 7.5, the bottom branch depicts the end point pay- offs and the certainty-equivalent probabilities that are speci- fied by Equations 7.41 and 7.39, respectively. In the case where the binomial tree has multiple time steps, the same process can be repeated recursively to calculate intermediate and ini- tial certainty-equivalent values for the terminal payoffs. The proposed valuation process, in the context of binomial trees, is designed to reflect the certainty that the insurance policy creates for its holders (travelers).

88 Summary Table 8.1 summarizes the reliability valuation method described in previous chapters. For data input, the method uses field mea- surements of travel time in the form of an ordered data set (time series) as well as an estimate of VOT. In the proposed method, average and 95th percentile travel times are used as the guaranteed travel time level and policy duration, respectively. In order to run hypothesis testing for a GBM stochastic process, travel time series need to be transformed to a loga- rithmic scale and then get differenced once. Then, based on the transformed series, trend and standard deviation param- eters can be estimated. The GBM hypothesis testing is carried out on the transformed series using a chi-square statistic to verify whether the series is normally distributed. After establishing the validity of using a GBM process to model travel time variations, a binomial tree can be formed to represent its approximate random walk process. The bino- mial tree is specified by the number and length of time steps as well as the size of log-travel time increments and up and down move probabilities. Once the binomial tree is specified, terminal payoffs at all nodes on the last time step are estimated. Then, certainty- equivalent probabilities are calculated. These probabilities are then used to carry out expectation calculations at the binary ends of each branch and to evaluate policy values at all inter- mediate and initial nodes of the binomial tree. The estimated value at the initial node is the VOR and by dividing it over VOT, the reliability ratio (RR) can be estimated. Table 8.2 summarizes all the parameters used in the proposed method. As was discussed, lateness and earliness parameters are set equal to 1 and 0.05, respectively. This choice indicates the relative cost perceptions of being late and being early based on experience in U.S. urban areas. The travel time insurance policy is designed to provide guaranteed travel times at the average travel time level and to have a lifetime equal to 95th percentile of travel time distribution to cover all recurrent congestion cases. The threshold to define certainty (limit on variations) in travel time is set strictly at 0%. Comparison Between the Proposed Approach and the SHRP 2 Project L11 Methodology Table 8.3 provides a side-by-side comparison between the proposed approach in this study and the L11 methodology. Both approaches take advantage of the analogy between value of travel time reliability and an insurance policy that guaran- tees a specific level of travel time for a specific duration of time. While the L11 method was criticized for discounting speeds, the proposed approach in this study directly works with travel times, which, in the transportation literature, are commonly associated with cost. The stochastic process adopted in both methods are essentially the same, but this study uses the binomial tree representation of a discrete random walk. This choice gives the proposed method a tremendous level of flexibility in dealing with any conceivable scenario in terms of payoffs and, more importantly, provides insight into the evalu- ation process. Whereas the L11 approach was more like a black box, the proposed method can be tweaked carefully to fit new circumstances and any theoretical/empirical evidence that may become available. Example 2 Building on the GBM process described in Example 1 in Chapter 7, we would like to build the binomial tree structure and to estimate the terminal payoffs. Then based on the reli- ability and variation threshold arguments provided earlier, certainty-equivalent probabilities as well as intermediate and current values of reliability will be calculated. First, let us assume the 95th percentile travel time is 20 minutes. The only case presented here is when 2-minute time intervals (Dt = 2) are considered. In that case, the forward GBM factors and probabilities (Equations 7.15 through 7.17) are used to build the binomial tree presented in Table 8.4. Also, Table 8.5 summarizes the results of recursive valuation C H A P t e R 8

89 Table 8.1. Summary of the Proposed Travel Time Data-Driven Method for Estimating VOR/RR Step Description Input Travel time series (time ordered data set) Value of time (VOT) Primary Calculations Travel time distribution (frequency of obser- vations, unordered data set) • Average travel time • 95th percentile travel time Stochastic process One-difference lognormal transform of travel time series • Average • Standard deviation Trend and standard deviation parameter estimates Hypothesis testing for GBM Binomial tree Number and length of time steps Increment size Up and down probabilities Payoff Terminal step calculations Valuation Certainty-equivalent probabilities Intermediate and initial values Output Value of reliability (VOR) Reliability ratio (RR) Table 8.2. Parameters Used in the Proposed Approach Description Parameter Value Lateness parameter b 1 Earliness parameter a = b/19 0.05 Guaranteed travel time Average Policy duration 95th percentile Certainty threshold rate r 0% Note: empty cells = to be determined based on data. Table 8.3. Comparison Between Project L11 Methodology and Proposed Approach Method L11 Proposed Approach Analogy used? Insurance premium Insurance premium What is being insured? Average speed Average travel time Policy duration? 95th percentile trip time 95th percentile trip time Stochastic process GBM (continuous) Binomial tree (discrete) Payoff? Speeds lower than average Lateness/earliness penalty Valuation? Discounted value Certainty-equivalent value Solution type? Closed form (Black-Scholes) Numerical simulation of reliability. In this case, the set of certainty-equivalent prob- abilities used are calculated as follows: = − − = 1 0.87 1.15 0.87 0.46 (8.1)p And, thus − = − =1 1 0.46 0.54 (8.2)p Table 8.4 shows the binary tree constructed to represent the aforementioned GBM process as time step (columns) increments from left to the right. Travel time has time units and therefore note should be taken in interpreting the GBM process values. The reported values indicate travel time over the same link as time passes. In this representation, each cell is identified by the time elapsed since the start of process (col- umn header) and the travel time level (row entries). For instance, at initial time (T = 0), link travel time is 10 minutes, and 2 minutes later (T = 2), travel time on the same link can be either 11.5 or 8.7 minutes. Note that travel time increments between two adjacent rows are not uniform as the travel times are reported in their normal scale (minutes). Had travel times been reported in loga- rithmic scale then rows would have been uniformly spaced from each other. The modeled travel times at the termination of the simulation (rightmost column) can be used to calculate payoffs. Payoff calculation at the termination of the period for which the travel time insurance policy has been valid (T = 20) is performed using [ ]( ) ( )τ = × τ − + × τ − ×− +( ) 0.05 10 1 10 VOT (8.3)CN N N N For instance, in the case of highest possible travel time at ter- mination (40.5 minutes) the payoff is [ ]( ) ( )= × − + × − × = × − +(40.5) 0.05 40.5 10 1 40.5 10 VOT 30.5 VOT (8.4) 10C And in the case of smallest possible travel time at termination (2.5 minutes) the payoff is [ ]( ) ( )= × − + × − × = × − +(2.5) 0.05 2.5 10 1 2.5 10 VOT 0.38 VOT (8.5) 10C

90 Table 8.4. Forward Time Binary Tree Construction (a 5 5%, s 5 10%, Dt 5 2, t(0) 5 10, t95 5 20) n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 T 5 0 T = 2 T = 4 T = 6 T = 8 T = 10 T = 12 T = 14 T = 16 T = 18 T  20 40.5 35.2 30.6 30.6 26.6 26.6 23.1 23.1 23.1 20.1 20.1 20.1 17.5 17.5 17.5 17.5 15.2 15.2 15.2 15.2 13.2 13.2 13.2 13.2 13.2 11.5 11.5 11.5 11.5 11.5 10 10 10 10 10 10 8.7 8.7 8.7 8.7 8.7 7.6 7.6 7.6 7.6 7.6 6.6 6.6 6.6 6.6 5.7 5.7 5.7 5.7 5.0 5.0 5.0 4.3 4.3 4.3 3.8 3.8 3.3 3.3 2.9 2.5 Note: Dh = 0.14, u = 1.15, d = 0.87, q = 0.82. At the intermediate and initial time steps (n = 0, 1, 2, . . . , 9), by taking the expectations recursively and using certainty- equivalent probabilities, reliability values can be estimated using the following expression: ( ) 0.46 0.54 , 0,1,2, . . . ,9 (8.6) 1 1C C u C d n n n n( ) ( )τ = × τ + × τ = + + For instance, after 18 minutes (T = 18), if travel time is 35.2 minutes, the insurance premium the traveler is willing to pay in order to guarantee travel time at 10 minutes in the next 2 minutes is equal to 35.2 0.46 30.5 0.54 20.6 VOT 25.15 VOT (8.7) 9C [ ]( ) = × + × × = × Table 8.5 summarizes the recursive reliability values obtained at all terminal, intermediate, and initial steps along the binary tree. The reported numbers are normalized val- ues by VOT amount. In other words, these numbers are, in fact, the reliability ratio (RR) times average travel time (TT) at all nodes on the tree. Of course, in the case of travel times, the only important value is the initial value (1.71 in this case).

91 Table 8.5. Recursive Reliability Valuation (a 5 0.05, b 5 1, r 5 0%, Dt 5 2, E(t) 5 10, t95 5 20) n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 T 5 0 T = 2 T = 4 T = 6 T = 8 T = 10 T = 12 T = 14 T = 16 T = 18 T 5 20 30.5 25.15 20.51 20.6 16.48 16.55 12.99 13.05 13.1 9.96 10.02 10.08 7.39 7.38 7.43 7.5 5.31 5.21 5.13 5.18 3.71 3.54 3.36 3.18 3.2 2.53 2.34 2.12 1.85 1.47 1.71 1.53 1.32 1.06 0.71 0 1.01 0.84 0.64 0.39 0.06 0.56 0.43 0.28 0.12 0.12 0.33 0.25 0.18 0.17 0.25 0.22 0.22 0.22 0.26 0.26 0.26 0.29 0.29 0.29 0.32 0.32 0.34 0.34 0.36 0.38 Note: p = 0.46.

92 Value of Reliability Savings Quantification In classic utility-based reliability valuation, using the reliability ratio concept (Equation 6.5), value of travel time reliability (VOR) is equal to value of time (VOT) multiplied by the reli- ability ratio (RR). = ×VOR RR VOT (9.1) This simple relationship gives an estimate of the value of reliability (VOR) for each unit of the reliability measure (RM). Therefore, cost of reliability (RC) can be linearly estimated by multiplying the unit value of reliability (VOR), the RM, and the number of users affected (V) of the road segment under consideration. = × × = × × ×RC VOR RM RR VOT RM (9.2)V V Reliability savings of an improvement can be estimated as the difference between reliability costs in the before and after scenarios (DRC = RCb - RCa). Assuming the value of reliability (VOR) remains unchanged before and after the improvement, the reliability savings can be estimated as the following: ( ) ( )∆ = × × × − ×RC RR VOT RM RM (9.3)V Vb b a a Note that b and a subscripts used in this chapter indicate the before and after scenarios of the improvement under consid- eration, respectively. Plugging the certainty-equivalent estimate of the reliability ratio according to utility theory (Equation 6.12) into the reli- ability cost estimate (Equation 9.2) results in the following expression for reliability cost estimation: = × × − ×RC VOT RM RM RM (9.4)X V r Note that in case the reliability measure used takes a value equal to or near zero under reliable conditions (RMr @ 0), then the reliability cost estimate expression can be further simplified as = × ×RC VOT (9.5)X V For instance, standard deviation and buffer index are the reli- ability measures that meet this condition. In these cases, the reliability savings of an improvement can be estimated as the following: ( ) ( )∆ = × × −RC VOT (9.6)X V Vb a Note that the certainty-equivalent based reliability cost expres- sion (Equation 9.5) can be rewritten as = × × ×RC TT VOT TT (9.7) X V Comparing Equations 9.2 and 9.7 implies an analogy between utility-based and certainty-equivalent based reliability cost esti- mates. In fact, in the certainty-equivalent approach, the ratio of certainty-equivalent addition to the average travel time (X) and average travel time (TT) is analogous to the reliability ratio (RR): =RR TT (9.8) X This may become more evident if both nominator and denomi- nator in Equation 9.8 are both multiplied by VOT and V. The nominator in this case will represent the reliability cost (RC), and denominator will be equal to the cost of average travel time (TC): = × × × × =RR VOT TT VOT RC TC (9.9) X V V Note that the option-based reliability value Equation 7.41 can be expressed as the product of a certainty-equivalent addi- tional travel time (X) and value of time (VOT): = × VOT (9.10)0C X C h a p t e r 9

93 Therefore, substituting the option-based reliability value into the reliability cost estimate (Equation 9.5) will result in the following simple expression: = ×RC (9.11)0C V And, substituting the certainty-equivalent additional travel time (X) from Equation 9.10 into Equation 9.8, the reliability ratio (RR) can be expressed as the following: = =RR VOT TT TC (9.12) 0 0C C Note that the option-based reliability value (C0) is dependent on the average travel time (TT). Therefore, in the option- based approach, reliability savings due to an improvement in general are calculated as ∆ = × − ×RC (9.13)0, 0,C V C Vb b a a Using the more expansive expression of certainty-based reli- ability cost (Equation 9.7), the reliability savings due to an improvement can be expressed as ( )∆ = × × − × × ×RC RR TT RR TT VOT (9.14)V Vb b b a a a

94 Corridor Methodology Example Applications In this chapter the application of the proposed methodology using real-world cases is presented. Five major corridors in the state of Maryland are selected for the analysis. These cases include three major north-south corridors running parallel to each other between Washington, D.C., and Baltimore and used heavily on a daily basis by both commuters and other travelers alike. The other two corridors also carry significant traffic levels and are among the most congested urban high- ways in the country. The following provide more details on the selected corridors and their geographical extent: • I-95 between Capital Beltway (I-495) and Baltimore Belt- way (I-695) • MD-295 (Baltimore-Washington Parkway) between D.C. border and Baltimore Beltway (I-695) • US-29 (Columbia Pike) between Capital Beltway (I-495) and I-70 • Capital Beltway (I-495) between American Legion Bridge (Virginia border) and I-95 • I-270 between Capital Beltway (I-495) and I-70 Figure 10.1 shows the geographic location and extent of the selected corridors on a map. Note that in the map, I-95 is red, MD-295 is green, US-29 is blue, I-495 is gray, and I-270 is yellow. Details on the number of segments in each corridor and their lengths are provided in Appendix E. Also included in Appendix E are further details on each corridor such as average travel times, spatial correlations of travel times along the corridor, and graphs showing the calculated reliability ratio’s relationship with trip length and average travel times for each direction of travel as well as morning and afternoon peak periods. In this chapter, however, the emphasis of reporting (and modeling) is on the first three parallel corridors since they are, by and large, of the same length and virtually are stretched between the same origin and destination pair. However, it should be noted that while I-95 (red) is a four-lane (northbound and southbound) access-controlled freeway facility, MD-295 (green) for the most part is a two-lane (northbound and southbound) access-controlled highway that is under the National Park Service’s jurisdiction. Trucks are not allowed on MD-295. Columbia Pike (blue) is mostly a multilane access-controlled highway between I-70 and the newly built MD-200 (Intercounty Connector, or ICC) that runs east-west from I-95 to I-270. However, between the ICC and the Capital Beltway (I-495), US-29 turns into a high- level multilane arterial highway with widely spaced signal- ized intersections. Therefore, the three selected corridors provide a representative mix of geometry and traffic for fur- ther analysis. Travel time data used as input in this study are provided by INRIX (32) through the Vehicle Probe Project (33) of the I-95 Corridor Coalition. Data have been pulled and archived since 2009 in the Regional Integrated Transportation Information System (RITIS) (34) housed at the Center for Advanced Trans- portation Technology (CATT) Lab of the University of Mary- land. In this study, data archived during calendar year 2011 are used at 1-minute resolution on all segments considered. Analy- sis is focused on 2-hour peak periods in the morning (7:00 a.m.– 9:00 a.m.) and in the afternoon (4:00 p.m.–6:00 p.m.). Path travel times are constructed using segment travel times at their original 1-minute granularity using an instantaneous path travel time estimation algorithm. Figures 10.2 and 10.3 demonstrate sample histograms of one-time differenced log-travel times on the northbound I-95 corridor and southbound I-270 corridor during AM peak periods, respectively. The histograms show a close match to the hypothesized normal distribution (see Appendix E for details). The chi-square hypothesis testing for all paths formed on all studied corridors at all time periods indicate travel times follow a GBM stochastic process. Figure 10.4 summarizes the results of analysis on the northbound direction of the three parallel corridors. Each dot on the graphs represents the average of reliability ratios C h a p t e r 1 0

95 Source: Map data ©2014 Google. Figure 10.1. Corridor examples in Maryland. obtained by applying a binary tree for each minute of the cor- responding peak period over a given path segment. There- fore, each dot is the average of 120 (2 hours, every minute) binary tree applications for the corresponding set of seg- ments that comprise the same path. Average reliability ratios during AM peak periods are shown on the left graphs, while PM peak reliability ratios are shown on the right graphs. The top graphs depict reliability ratios versus average travel times experienced by travelers. The bottom graphs show the reli- ability ratios versus trip length. From these graphs it can be seen that reliability ratios are uniformly increasing with both trip length and average travel times. However, the rate of increase diminishes as trips become longer both in space and time. This is due to the fact that over a given corridor as trips become longer by incrementally adding new segments, the trips would inherit both the risks of the currently included segments as well as the risks associated with the newly added segment. The concave form of the reliability ratio curves is mainly due to the fact that, as trip length becomes longer, the risk impact of any newly added segment, while still positive, becomes marginal compared with the rest of the path. Similarly, Figure 10.5 demonstrates the calculated reliability ratios on the southbound direction of the same three corri- dors. Both Figure 10.4 and Figure 10.5 indicate more stability in reliability ratio estimates when they are drawn versus aver- age travel times. This fact implies that reliability ratios are strongly correlated with average travel times. Note that while reported reliability ratios are between zero and one, theoretically there is no real constraint on the maxi- mum possible ratio that can be obtained from applying the proposed method. Example 2 presented in the previous chap- ter is a case in point. Parameters a and b, defined earlier in the

96 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0 10 20 30 40 50 60 70 80 ALPHA: 6.4028e-08 SIGMA: 9.9581e-05 TRAVEL TIME IS LOG-NORMALLY DISTRIBUTED TRAVEL TIME LOGARITHM (LOG-MINUTE) FR EQ UE NC Y -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0 20 40 60 80 100 120 140 ALPHA: 1.148e-07 SIGMA: 0.00016289 TRAVEL TIME IS LOG-NORMALLY DISTRIBUTED TRAVEL TIME LOGARITHM (LOG-MINUTE) FR EQ UE NC Y -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0 10 20 30 40 50 60 70 80 ALPHA: 4.5363e-08 SIGMA: 0.00013584 TRAVEL TIME IS LOG-NORMALLY DISTRIBUTED TRAVEL TIME LOGARITHM (LOG-MINUTE) FR EQ UE NC Y -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0 20 40 60 80 100 120 ALPHA: 7.6576e-08 SIGMA: 0.00016771 TRAVEL TIME IS LOG-NORMALLY DISTRIBUTED TRAVEL TIME LOGARITHM (LOG-MINUTE) FR EQ UE NC Y HISTOGRAM OF DIFFERENCES IN TRAVEL TIME LOGARITHMS CORRIDOR: I95 NB PATH SEGMENT: 110P04424 (LENGTH: 22.0677 MILES) AM PEAK PERIOD Figure 10.2. Sample histograms of one-time differenced log-travel times on northbound I-95 corridor during AM peak period. payoff characterization section, play an important role in determining the magnitude of reliability ratios. Figure 10.6 shows the relationship between the 95th per- centile travel times and average travel times as is estimated on all incremental paths formed on both directions of the five subject corridors during AM and PM peak periods. The linear relationships between the two measures are clearly visible. However, dispersions around the mean increase as average travel time increases. The linear model fitted to the data by regression displays a high goodness-of-fit measure. 0.291 1.320 , 0.97 (10.1)95 2E R( )( )τ = − + × τ = Figure 10.7 shows the ensemble of all estimated reliability ratios along the five studied corridors in both directions and both peak periods. While larger dispersions are visible in the 10 to 20 minutes average travel time range, for both shorter and longer trip times, further convergence in reliability ratios is evident. The general trend is increasing at a diminishing rate. The fitted Gompertz function provides the best estimate of the trend compared with other alternatives. The mean square error reported for this model is just 0.1%. ( )( )( )( )= − × − × τ − = RR 1 exp 17.355 exp 0.004 1 , (MSE 0.001) (10.2) E Figure 10.8 further illustrates scatter of the observed (pro- posed method) versus model estimated (Equation 10.2) reli- ability ratios. The linear model fitted to this scattergram indicates a very good fit of the model to the data. The intercept is very close to zero (0.6) and the slope is close to one (0.987) with a strong goodness-of-fit measure (0.98). ( )= + × =RR 0.006 0.987 RR, 0.98 (10.3)2R In the case of Maryland, a good estimate of the average statewide travel time based on U.S. Census Bureau data during the 5-year period (2006–2010) is approximately 31 minutes

97 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 20 40 60 80 100 120 140 160 ALPHA: 9.733e-07 SIGMA: 0.0007289 TRAVEL TIME IS LOG-NORMALLY DISTRIBUTED TRAVEL TIME LOGARITHM (LOG-MINUTE) FR EQ UE NC Y -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0 20 40 60 80 100 120 ALPHA: -4.5665e-08 SIGMA: 0.00069247 TRAVEL TIME IS LOG-NORMALLY DISTRIBUTED TRAVEL TIME LOGARITHM (LOG-MINUTE) FR EQ UE NC Y -0.1 -0.05 0 0.05 0.1 0.15 0 10 20 30 40 50 60 70 80 ALPHA: 3.589e-07 SIGMA: 0.0005864 TRAVEL TIME IS LOG-NORMALLY DISTRIBUTED TRAVEL TIME LOGARITHM (LOG-MINUTE) FR EQ UE NC Y -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0 20 40 60 80 100 120 140 ALPHA: 8.6881e-07 SIGMA: 0.00070041 TRAVEL TIME IS LOG-NORMALLY DISTRIBUTED TRAVEL TIME LOGARITHM (LOG-MINUTE) FR EQ UE NC Y HISTOGRAM OF DIFFERENCES IN TRAVEL TIME LOGARITHMS CORRIDOR: I270 SB PATH SEGMENT: 110-10531 (LENGTH: 41.1712 MILES) AM PEAK PERIOD Figure 10.3. Sample histograms of one-time differenced log-travel times on southbound I-270 corridor during AM peak period. (35, 36). Plugging this value into Equation 10.2 would result in a statewide reliability ratio equal to 0.87, which is larger than the current 0.75 value adopted by the state. However, it should be noted that due to the nonlinear (in fact, concavity) form of the model, this is an overestimation. As discussed earlier, in case of the concave function f(x), [ ] [ ]( ) ( )< (10.4)E f x f E x The results of applying the methodology indicate that SHA’s use of the current RR of 0.75 is conservative for com- mute trips. According to the U.S. Census Bureau statistics, average commute trips in Maryland during the 5-year period (2006–2010) has been approximately 31 minutes long (35, 36). However, the corresponding RR value (0.87) is believed to be at the upper range of values for travel time reliability. Further analysis was conducted to justify any decision to increase the current value of travel time reliability. Maryland Statewide Transportation Model (MSTM) long- term demand and travel time estimates are used in aggregating the results for all origin–destination (O-D) pairs in the state. Based on MSTM, for all trip purposes currently an average reli- ability ratio value of 0.52 is obtained. This value is expected to increase to 0.55 over the next 15 years until 2030. Similarly, the current average reliability ratio for commute trips in Maryland is estimated to be 0.68 and would remain relatively unchanged until 2030. However, it should be noted that in comparison with Census Bureau estimates, MSTM travel times are on aver- age about 6 minutes smaller. Note that due to bias in self- reporting, Census Bureau estimates tend to be an overestimate. At the same time, it may be argued that MSTM travel times are underestimates caused by spatial aggregations in zone defini- tions as well as the use of long-term performance functions. In summary, it can be concluded that during peak hours in congested urban areas, the average reliability ratio ranges between 0.68 and 0.87, derived from MSTM and Census Bureau travel times, respectively. In non-urban areas and at off-peak hours, the average reliability ratio can be taken as 0.52. There- fore, it seems the current value (0.75) is reasonable when reli- ability of commute travel times during peak hours in congested urban areas is considered.

98 0 5 10 15 20 25 30 35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Average Travel Time (Minute) R el ia bi lit y Ra tio - RR (U nit les s) AM Peak Period I-95 MD-295 US-29 0 10 20 30 40 50 600 0.2 0.4 0.6 0.8 1 Average Travel Time (Minute) R el ia bi lit y Ra tio - RR (U nit les s) PM Peak Period I-95 MD-295 US-29 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Length (Mile) R el ia bi lit y Ra tio - RR (U nit les s) AM Peak Period I-95 MD-295 US-29 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 Length (Mile) R el ia bi lit y Ra tio - RR (U nit les s) PM Peak Period I-95 MD-295 US-29 Figure 10.4. Reliability ratios on the northbound direction of parallel corridors between Capital Beltway and Baltimore Beltway: left, AM peak period; right, PM peak period; top, versus average travel time; and bottom, versus length. 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 Average Travel Time (Minute) R el ia bi lit y Ra tio - RR (U nit les s) AM Peak Period I-95 MD-295 US-29 0 5 10 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Average Travel Time (Minute) R el ia bi lit y Ra tio - RR (U nit les s) PM Peak Period I-95 MD-295 US-29 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 Length (Mile) R el ia bi lit y Ra tio - RR (U nit les s) AM Peak Period I-95 MD-295 US-29 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Length (Mile) R el ia bi lit y Ra tio - RR (U nit les s) PM Peak Period I-95 MD-295 US-29 Figure 10.5. Reliability ratios on the southbound direction of parallel corridors between Capital Beltway and Baltimore Beltway: left, AM peak period; right, PM peak period; top, versus average travel time; bottom, versus length.

99 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 80 90 100 Y = –0.291+1.320X R2= 0.971 Average Travel Time (Minute) 95 P er ce nt ile T ra ve l T im e (M inu te) Path Level / Across Days AM Peak PM Peak Figure 10.6. 95th percentile travel time versus average travel time. 0 10 20 30 40 50 60 70 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Gompertz Function: Y = 1 – e17.355(e–0.004X–1) MSE = 0.001 Average Travel Time (Minute) R e li a b il it y R a ti o - R R ( U n it le ss ) Path Level / Across Days AM Peak PM Peak Figure 10.7. Reliability ratio versus average travel time.

100 Figure 10.8. Estimated versus observed reliability ratios. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y = 0.006 + 0.987X R2 = 0.98 Observed Reliability Ratio - RR (Unitless) Es tim at ed R el ia bi lit y Ra tio - RR (U nit les s) Path Level / Across Days

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103 Background on Stochastic Processes Brownian Motion and Wiener Process Robert Brown (1773–1858) was an early nineteenth-century botanist who studied the jittery motion of small grains of pollen in water under a microscope. He then observed the same motion in particles of inorganic matter suspended in water, which enabled him to rule out the existence of a bio- logic cause for the observed motion (37). Thus, he concluded the motions have to have a physical source. In 1905 Albert Einstein published a paper (38) that showed movements of small particles in liquids can be explained by the thermal motions of liquid molecules and their kinetic impacts on the floating particles. While he mentions Brown- ian motion in this work, he states he does not have enough data to give a verdict on whether the type of motions that he discusses here are the same as the ones that were reported by Brown. This work made it possible to determine the real size of atoms and molecules. Norbert Wiener, a renowned mathematician and MIT pro- fessor (1894–1964), built on earlier work and argued that Einstein’s assumptions on the independence of an interval from previous intervals and applicability of Stokes’ law are approximations. He showed that mean square displacement in a given direction of a spherical particle in a fluid over any given time interval Dt is effectively proportionate to the length of the time interval 2z c t( )∆ ≅ ∆ . In other words, he showed that floating particles displacement is proportional to the square root of the time interval over which displacement is taking place (39). Louis Bachelier (1870–1946), a French mathematician and the so-called founder of mathematical finance, as part of his PhD dissertation (40) modeled a stochastic process that today is known as Brownian motion. Interestingly, his dissertation was published in 1900 (5 years before Einstein’s work). Some even have suggested that what is known today as Brownian motion should be renamed as the Wiener-Bachelier process (41). Bachelier’s work later inspired A. Kolmogorov to develop the formal foundations of Markov processes. A Wiener process (also called a Brownian motion) is a continuous-time stochastic process with three important properties (42). First, it is a Markov process, which means that the probability distribution for all future values of the process depends only on its current value, and is unaffected by past values of the process or by any other current informa- tion. In other words, Markov property suggests that process is memoryless, which in the case of process {z} can be written as the following: , , . . . , 0 , , 0 (D.1) P z s t z s z s t z P z s t z s s t ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ∆ − ∆ = + ∆ ∀ ∆ > In the field of transportation, travel time variations over time are usually assumed to resemble a Markov process (43). Second, the Wiener process has independent increments. This property suggests that the probability distribution for the change in the process over any time interval is indepen- dent of any other (non-overlapping) time interval. Third, changes in the process (Dz) over any finite interval of time (Dt) are normally distributed with a variance that increases linearly with the time interval. 0, (D.2)z s t z s z N t∼( ) ( ) ( )+ ∆ − = ∆ ∆ Thus, any increments in a Wiener process over a finite time interval are linearly related to the square root of the time step. , 0, 1 (D.3)z t Nt t ∼ ( )∆ = ε ∆ ε where et is a standard normal random variable. And, in the limit as the time interval is reduced, the process is defined as , 0, 1 (D.4)dz dt Nt t ∼ ( )= ε ε A P P e n d i x d

104 It should be noted that the latter property holds for every time step size. For instance, over a time step T (equal in size to n smaller time steps), (D.5)T n t= ∆ The process increment may be written as the sum of incre- ments in all smaller time intervals: (D.6) 1 1 z s T z s t tii n ii n∑ ∑( ) ( )+ − = ε ∆ = ∆ ε = = In the above equation, it should be noted that the sum in the rightmost term is, in fact, the sum of n independent and identically distributed (iid) standard normal random vari- ables. Using central limit theorem (CLT), it can be shown that this sum is normally distributed with mean zero and vari- ance n. Thus, the process increment is also normally distrib- uted with mean zero and variance equal to the time interval T, 0, (D.7) 2 z s T z s N T∼ ( )( )( ) ( )+ − Therefore, in the limit as T increases (T → ∞), the expected increment is zero while the variance of increment will increase unboundedly. Also, note that the Wiener process has no time derivatives in a conventional sense: (D.8)z t tt∆ ∆ = ε ∆ and, as time interval Dt is reduced, lim (D.9)0 z t dz dt t ( )∆∆ = = ∞∆ → Brownian Motion with drift The Wiener process can be easily extended to represent more complex processes. The following process is called “Brownian motion with drift”: (D.10)dx adt dz= + σ where dz is the increment of a Wiener process as defined above; a is the drift parameter; and s is the standard deviation parameter. From the previous discussion it is straightforward to see that over a time interval Dt, the process increment is normally distributed: , (D.11)2x N a t t∼ ( )∆ ∆ σ ∆ This leads to the following difference equation in discrete time to represent the trajectory of process x: (D.12)1x x a t tn n n= + ∆ + σ ∆ ε+ Random Walk Representation of Brownian Motion From the preceding discussion, it is clear that a continuous Wiener process can be simply described in terms of a random walk. A random walk is a succession of random steps (44). In the case of one-dimensional movements, a simple metaphor is the position of a person on a ladder (Figure D.1), where at Figure D.1. Random walk representation of Wiener process in one dimension: left, man on the ladder metaphor; right, Manhattan grid metaphor.

105 the end of each time interval that person takes a step up or down from his current position with certain probabilities. Expanding on the same idea, and starting from the man’s initial position on the ladder (x0), over time we can draw his possible positions on a rectangular grid. In the grid shown in Figure D.1, the x-axis represents time steps and the y-axis rep- resents the man’s position on the ladder. In this grid represen- tation, the uniform step size on the ladder is denoted by Dh and the probability of taking one step up is denoted by p, while the probability of taking one step down is denoted by q. It should be noted that p + q = 1. By connecting his possible positions from one time step to the next, a tree form emerges that has its root at his initial position (x0) at the initial time (t = 0), and which spreads (diffuses) in both directions as time goes by. The probability of the man being at any of the nodes on this tree depends on the number of paths along the tree (starting from his initial position) he can take to reach that particular node and the number of up and down steps he has to take along each path. Since the man’s decisions at each time step are independent of other time steps, the up and down probabilities can be multiplied to obtain the path probability. At each time step, probability of the man being positioned at any of the possible steps on the ladder is determined by a binomial distribution. , 1; , 2, . . . (D.13) 0P X x k h n k p q p q k n nk n k( )( )= ± ∆ = + = = −− (D.14)E x p q h[ ] ( )∆ = − ∆ (D.15)2E x p q h h( )( )∆  = + ∆ = ∆ ( ) ( ) ( ) ( ) ( ) [ ] [ ]∆ = ∆ − ∆ = − −  ∆ = ∆1 4 (D.16) 2 2 2 2 2 V x E x E x p q h pq h (D.17)T n t= ∆ 0 Binomial (D.18)x T x ∼[ ]( ) ( )− 0 (D.19)E x T x n p q h T p q h t[ ] ( ) ( )( ) ( ) ( )− = − ∆ = − ∆ ∆ 0 1 4 (D.20) 2 2 2V x T x n p q h pqT h t( )[ ] ( )( ) ( ) ( ) ( )− = − −  ∆ = ∆ ∆ We would like the mean and variance of [x(T) - x(0)] to remain unchanged and to be independent of the particular choice of probabilities (p, q), and discretization stencil (Dh, Dt). To achieve this goal, it is common to set (D.21)h t∆ = σ ∆ 1 2 1 (D.22)p a t= + σ ∆   1 1 2 1 (D.23)q p a t= − = − σ ∆   Thus, (D.24) 2 p q a t a h− = σ ∆ = σ ∆ 1 4 1 (D.25) 2 2 pq a t= − σ ∆   Then, 0 (D.26) 2 E x T x T a h h t aT( )[ ]( ) ( ) ( )− = σ ∆ ∆ ∆ = 0 1 1 (D.27) 2 2 2 2 2 2V x T x a t T h t a t T( )[ ]( ) ( ) ( )− = − σ ∆   ∆ ∆ = − σ ∆     σ In the limit, as time steps become smaller the mean and variance of the displacement from initial position as described by random walk will be equal to the following: 0 (D.28)E x T x aT[ ]( ) ( )− = 0 (D.29)2V x T x T[ ]( ) ( )− = σ Note that both mean and variance of the random walk in the limit are independent of the adopted discretization stencil. Besides, they are equal to the mean and variance of the process described by Brownian motion with drift. Generalized Brownian Motion (ito Processes) The Wiener process provides a very basic and natural descrip- tion of variability in many physical and social phenomena. As such it can be further generalized to model a wide range of stochastic processes: ( ) ( )= +, , (D.30)dx a x t dt b x t dz where dz is the increment of a Wiener process; a(x,t) is the expected instantaneous drift rate; and b(x,t) is the instantaneous standard deviation rate. Note that in the general definition the drift and standard deviation coefficients are both known (nonrandom) functions of the current state and time. The generalized continuous-time stochastic process x(t) presented here is called an Ito process.

106 The mean and variance of the increments of this process are, respectively, ( ) ( )= , (D.31)E dx a x t dt ( ) ( ) ( )[ ] [ ]=  − = , (D.32)2 2 2V dx E dx E dx b x t dt Note that in calculating the variance, terms in which dt orders are higher than one are dropped: ( ) ( )( ) ( ) ( ) ( )( )= + +, , 2 , , (D.33) 2 2 2 2 3 2dx a x t dt b x t dt a x t b x t dt , (D.34)2 2E dx b x t dt( ) ( )  = ito’s Lemma It was discussed earlier that the Ito process is continuous in time, but it is not necessarily smooth enough to be differen- tiable. However, in most practical cases we deal with functions of Ito processes. Therefore, computationally it is desirable to be able to differentiate or to integrate such functions. This possi- bility is provided through use of the so-called Ito’s lemma. Ito’s lemma is very similar to a Taylor series expansion of a function around a given point. Suppose that x(t) follows an Ito process, and consider a function F(x,t) that is at least twice differentiable in x and once in t. According to the rules of cal- culus, the total differential of this function can be written as 1 2 1 6 . . . (D.35) 2 2 2 3 3 3dF F t dt F x dx F x dx F x dx( ) ( )= ∂∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + Substituting Equation D.30 for dx and dropping higher-order terms would result in the following expression for the total differential of the function: , 1 2 , , (D.36) 2 2 2 dF F t a x t F x b x t F x dt b x t F x dz( ) ( ) ( )= ∂∂ + ∂ ∂ + ∂ ∂     + ∂ ∂ which illustrates that any function of an Ito process is itself an Ito process. See the following example. ( ) ( ) ( )= = = σ, log ; , ; ,F x t x a x t ax b x t x Note that in this case, 0, 1 , 12 2 2 F t F x x F x x ∂ ∂ = ∂ ∂ = ∂ ∂ = − Substituting these partial derivatives in Equation D.36 would result in 1 2 (D.37)2dF a dt dz( )= − σ + σ This suggests that over a time interval Dt, the change in log x is normally distributed with mean 1 2 2a( )− σ Dt and vari- ance s2 Dt. So, in this case compared with the instantaneous rate of change in process x that is, dx x( ), log x is expected to change over time with a rate less than half its variance. Note that log x is a strictly concave function of x, 0 2 2 F x ∂ ∂ <  , so applying Jensen’s inequality, it can be shown that with x uncertain, the expected value of log x changes by less than the logarithm of the expected value of x. In the case of two random points x1 and x2, the following illustrates the above reasoning: log 1 log log 1 (D.38)1 2 1 2w x w x wx w x( )( ) ( )+ − ≤ + − log log (D.39)E x E x[ ] [ ]( )≤ These equations can be easily extended to the general case where an infinite number of points on the x axis (random variable) are considered. Geometric Brownian Motion A very important special case of Ito processes is the geometric Brownian motion with drift (GBM), in which a(x,t) = ax, and b(x,t) = sx, where a and s are constants. The following is the expression for a GBM process: (D.40)dx axdt xdz= + σ Note that dx/x = d log x = dF, therefore GBM suggests that natural logarithm of random variable x is following a simple Brownian motion with drift stochastic process, and therefore F = log x is normally distributed. In other words, recalling the Ito’s lemma and the example previously discussed, the GBM process suggests that random variable x is lognormally dis- tributed and can be expressed using the following Brownian motion with drift process: 1 2 (D.41)2dF a dt dz( )= − σ + σ 1 2 (D.42)2E F a t( )[ ]∆ = − σ ∆ (D.43)2V F t[ ]∆ = σ ∆ As for x itself, starting from initial time t0, its expected posi- tion at time t is given by (45) exp (D.44)0 0E x t x t a t t[ ] { }( ) ( ) ( )= −

107 And the variance of x(t) is given by ( )( ) ( ) ( ) ( )[ ] { }{ }= − σ − −exp 2 exp 1 (D.45) 2 0 0 2 0V x t x t a t t t t Random Walk Representation of Geometric Brownian Motion As shown previously, a simple Brownian motion process can be represented by a random walk. In this section we show that geometric Brownian motion can also be represented by a ran- dom walk. This argument is supported by the fact when a random variable follows GBM process, its natural logarithm would follow a simple Brownian motion. Building on this fact we can write the size of increments in terms of the loga- rithm of x at three neighboring points as log log (D.46)1x x hn n− = ∆+ log log (D.47)1x x hn n− = ∆− Thus, starting from the middle point (xn) it is possible to find the other two neighboring points based on the following equations: exp (D.48)1x x h uxn n n( )= ∆ =+ exp (D.49)1x x h dxn n n( )= −∆ =− where u and d are multiplicative factors by which xn gets transformed into the upper and lower neighboring points, respectively. Also, note that u and d are inverse of each other. 1 (D.50)u d= As before, setting the step size Dh equal to the standard deviation of increments in the logarithm of random vari- able x would lead to the following expressions for u and d factors: (D.51)h t∆ = σ ∆ exp (D.52)u t= σ ∆  exp (D.53)d t= −σ ∆  Similarly, probabilities of taking a step up or a step down in the random walk is given by 1 2 1 2 (D.54) 2 p a t( )= + − σ σ  ∆  1 1 2 1 2 (D.55) 2 q p a t( )= − = − − σ σ  ∆  Thus, ( )( ) ( )( )− = − σ σ ∆ = − σ σ ∆2 2 (D.56)2 2 2p q a t a h 1 4 1 2 (D.57) 2 2 pq a t( )= − − σ σ  ∆  Then, E x T x T a h t a( ) − ( )[ ] = −       ( ) = −0 2 2 2 2 σ σ ∆ ∆ σ 2 2 58   T ( . )D 0 1 2 1 2 (D.59) 2 2 2 2 2 2 V x T x a t T h t a t T ( )[ ]( ) ( ) ( )− = − − σ   σ     ∆     ∆ ∆ = − − σ    σ     ∆     σ In the limit, as time steps become smaller the mean and vari- ance of the displacement from initial position as described by random walk will be equal to the following: E x T x a T( ) − ( )[ ] = − 0 2 60 2σ ( . )D 0 (D.61)2V x T x T[ ]( ) ( )− = σ Note that both mean and variance of the random walk in the limit are independent of the adopted discretization stencil. Besides, they are equal to the mean and variance of the stochas- tic process described as geometric Brownian motion with drift. GBM Calibration and Hypothesis Testing Given a series {x} of random variables sampled at Dt time intervals, the following hypothesis test needs to be performed in order to determine whether {x} is a GBM process: : : (D.62) 0 1 H x is a GBM process H x is not a GBM process  Recall that a GBM process is equivalent to asserting that increments in the natural logarithm of {x} are normally dis- tributed with specific mean and variance, log log 2 , (D.63)1 2 2x x N a t tn n ∼( )− − σ  ∆ σ ∆ +

108 Therefore, the first step to test the hypothesis is to form a series of increments of the natural logarithm of the series {x}: log log log , 1,2, . . . (D.64)1 1 y x x x x nn n n n n ( )= − = =− − The second step is to verify whether series {y} is normally distributed. For this purpose, initially we need to estimate the mean and variance of the transformed series {y}: 1 2 (D.65) 2 E y a t( )[ ] = − σ ∆ (D.66) 2 V y t[ ] = σ ∆ Solving for the instantaneous trend and standard deviation of the GBM process, the following estimates of the pair of param- eters are obtained: (D.67) V y t SD y t  [ ] [ ]σ = ∆ = ∆ 2 2 (D.68)a E y V y t  [ ] [ ] = + ∆ Now, the original hypothesis test can be written in an equiva- lent form: : 1 2 , : (D.69) 0 2 2 1 H y N a t t H Otherwise ∼   − σ   ∆ σ ∆        This hypothesis can be tested using a chi-square goodness-of- fit test. The chi-square test statistic is of the form (D.70)2 2 1 O E Ei i ii N∑ ( )χ = − = where N is the number of bins; Oi are the observed counts; and Ei are the expected counts based on the hypothesized distribution. Usually bins are defined in such a way that the expected count in a given bin based on the hypothesized distribution does not fall below 5. As a result, bin sizes do not have to be uniform. In most cases the square root of the length of the series {y} is a good starting point for the number of bins (N) to be considered in performing the hypothesis test. The test statistic has an approximate chi-square distribution when the counts are sufficiently large. At a given significance level (a), if the test statistic is smaller than the corresponding value of the chi-square distribution (c2 ≤ c2a), then the chi- square test does not reject the null hypothesis at the a signifi- cance level. Otherwise, the null hypothesis is rejected. Note that if the null hypothesis is not rejected it is not auto- matically accepted. In fact, failure to reject the null hypothesis at a significance level merely means that evidence against the hypothesis is not overwhelming. It does not mean that there is evidence in favor of the hypothesis. Therefore, when the null hypothesis is not rejected, other evidence, including the nature of the process and visuals such as graphs and histo- grams, may be sought to confirm the nature of the process that data are hypothesized to follow.

109 Details on Corridor Examples This appendix provides further details on corridor examples presented in this report. For each corridor, a comprehensive description of standard traffic message channel (TMC) seg- ments included therein is provided (Tables E.1 through E.11). Average travel times over the length of each corridor in AM and PM peak periods are presented (Figures E.1, E.4, E.7, E.10, E.13, E.16, E.19, E.22, E.25, and E.28). These graphs can be used conveniently to identify the peak direction of flow in each corridor and also to identify mileposts along the corri- dor in which congestion builds up frequently during each peak period. Also, to illustrate the correlations between travel time at different segments along the corridor, travel time correlation heat maps are presented (Figures E.2, E.5, E.8, E.11, E.14, E.17, E.20, E.23, E.26, and E.29). Note that in the heat maps, red colors represent higher correlations while blue colors represent lower correlations. Naturally, segments next to each other are more likely to show a simultaneous increase or decrease in travel time (travel speed). The correlation heat maps can be used as a tool to segment the corridors into sub- corridors with homogeneous traffic patterns during AM and PM peak periods. Finally, for each corridor and peak period combination, reliability ratios on paths formed by incrementally adding single TMC segments are reported for every minute of the 2-hour-long peak period Figures E.3, E.6, E.9, E.12, E.15, E.18, E.21, E.24, E.27, and E.30. This is potentially very informative as the information makes it abundantly clear which TMC seg- ments and exactly at what times would contribute the most to the corridor unreliability. Also, average reliability ratios of incremental subpaths in each peak period are depicted versus the length of the corresponding corridor subpaths. A p p e n d i x e

110 Table E.1. Summary Details of Corridors Reported Highway Direction Number of TMC Segments Length (mi) I-95 Northbound 20 22.1 I-95 Southbound 20 21.8 I-270 Northbound 49 41.0 I-270 Southbound 49 41.2 I-495 Clockwise (inner loop) 23 15.0 I-495 Counterclockwise (outer loop) 23 16.0 MD-295 Northbound 38 29.5 MD-295 Southbound 38 29.3 US-29 Northbound 34 20.7 US-29 Southbound 34 20.8 Table E.2. TMC Segment Definitions on Northbound I-95 Corridor TMC Roadnumber Firstname County Direction Miles 110+04261 I-95 MD-212/Exit 29 PRINCE GEORGE’S NORTHBOUND 1.238947 110P04261 I-95 MD-212/Exit 29 PRINCE GEORGE’S NORTHBOUND 1.147974 110+04262 I-95 MD-198/Exit 33 PRINCE GEORGE’S NORTHBOUND 2.922258 110P04262 I-95 MD-198/Exit 33 PRINCE GEORGE’S NORTHBOUND 1.319046 110+04263 I-95 Howard/Prince George’s Co Line (Laurel) (West) PRINCE GEORGE’S NORTHBOUND 0.641658 110+04417 I-95 Howard/Prince George’s Co Line (Laurel) (East) HOWARD NORTHBOUND 0.011807 110+04418 I-95 MD-216/Exit 35 HOWARD NORTHBOUND 0.597041 110P04418 I-95 MD-216/Exit 35 HOWARD NORTHBOUND 1.088444 110+04419 I-95 MD-32/Exit 38 HOWARD NORTHBOUND 1.966358 110P04419 I-95 MD-32/Exit 38 HOWARD NORTHBOUND 0.870892 110+04420 I-95 MD-175/Exit 41 HOWARD NORTHBOUND 1.339863 110P04420 I-95 MD-175/Exit 41 HOWARD NORTHBOUND 0.923773 110+04421 I-95 MD-100/Exit 43 HOWARD NORTHBOUND 1.053211 110P04421 I-95 MD-100/Exit 43 HOWARD NORTHBOUND 0.912837 110+04422 I-95 I-895/Exit 46 HOWARD NORTHBOUND 2.336029 110P04422 I-95 I-895/Exit 46 HOWARD NORTHBOUND 0.247628 110+04423 I-95 I-195/MD-166/Exit 47 BALTIMORE NORTHBOUND 0.583805 110P04423 I-95 I-195/MD-166/Exit 47 BALTIMORE NORTHBOUND 0.830501 110+04424 I-95 I-695/Exit 49 BALTIMORE NORTHBOUND 1.223226 110P04424 I-95 I-695/Exit 49 BALTIMORE NORTHBOUND 0.812418

111 0 5 10 15 20 25 0 5 10 15 20 25 30 LENGTH (MILES) A VE R A G E TR AV EL T IM E (M IN UT ES ) AVERAGE TRAVEL TIMES CORRIDOR NAME: I95 NB (LENGTH = 22.0677 MILES) AM PEAK PM PEAK Figure E.1. Average travel time versus length on northbound I-95 corridor. Figure E.2. Travel time correlations on northbound I-95 corridor: left, AM peak period; right, PM peak period.

112 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) 0 20 40 60 80 100 120 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) AM PEAK PERIOD 0 0.2 0.4 0.6 0.8 1 R EL IA BI LI TY R AT IO (U NI TL ES S) PM PEAK PERIOD 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) RELIABILITY RATIOS (RR) PATH LEVEL & INTERDAY (ACROSS DAYS) CORRIDOR NAME: I95 NB Figure E.3. Analysis results on northbound I-95 corridor: left, AM peak period; right, PM peak period; top, reliability ratio over time; and bottom, average reliability ratio versus length.

113 Table E.3. TMC Segment Definitions on Southbound I-95 Corridor TMC Roadnumber Firstname County Direction Miles 110N04424 I-95 I-695/Exit 49 BALTIMORE SOUTHBOUND 0.924333 110-04423 I-95 I-195/MD-166/Exit 47 BALTIMORE SOUTHBOUND 1.249014 110N04423 I-95 I-195/MD-166/Exit 47 BALTIMORE SOUTHBOUND 0.663158 110-04422 I-95 I-895/Exit 46 HOWARD SOUTHBOUND 0.727535 110N04422 I-95 I-895/Exit 46 HOWARD SOUTHBOUND 0.449024 110-04421 I-95 MD-100/Exit 43 HOWARD SOUTHBOUND 2.200626 110N04421 I-95 MD-100/Exit 43 HOWARD SOUTHBOUND 0.804464 110-04420 I-95 MD-175/Exit 41 HOWARD SOUTHBOUND 0.971372 110N04420 I-95 MD-175/Exit 41 HOWARD SOUTHBOUND 0.754504 110-04419 I-95 MD-32/Exit 38 HOWARD SOUTHBOUND 1.902851 110N04419 I-95 MD-32/Exit 38 HOWARD SOUTHBOUND 0.658187 110-04418 I-95 MD-216/Exit 35 HOWARD SOUTHBOUND 1.950575 110N04418 I-95 MD-216/Exit 35 HOWARD SOUTHBOUND 1.035439 110-04417 I-95 Howard/Prince George’s Co Line (Laurel) (East) HOWARD SOUTHBOUND 0.582314 110-04263 I-95 Howard/Prince George’s Co Line (Laurel) (West) PRINCE GEORGE’S SOUTHBOUND 0.040764 110-04262 I-95 MD-198/Exit 33 PRINCE GEORGE’S SOUTHBOUND 1.090495 110N04262 I-95 MD-198/Exit 33 PRINCE GEORGE’S SOUTHBOUND 1.261877 110-04261 I-95 MD-212/Exit 29 PRINCE GEORGE’S SOUTHBOUND 2.855954 110N04261 I-95 MD-212/Exit 29 PRINCE GEORGE’S SOUTHBOUND 0.888291 110-04260 I-95 I-495/Exit 27-25 PRINCE GEORGE’S SOUTHBOUND 0.786195 AVERAGE TRAVEL TIMES CORRIDOR NAME: I95 SB (LENGTH = 21.797 MILES) 0 5 10 15 20 25 0 5 10 15 20 25 30 LENGTH (MILES) A VE R A G E TR AV EL T IM E (M IN UT ES ) AM PEAK PM PEAK Figure E.4. Average travel time versus length on southbound I-95 corridor.

114 Figure E.5. Travel time correlations on southbound I-95 corridor: left, AM peak period; right, PM peak period. 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) AM PEAK PERIOD 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) PM PEAK PERIOD 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) RELIABILITY RATIOS (RR) PATH LEVEL & INTERDAY (ACROSS DAYS) CORRIDOR NAME: I95 SB Figure E.6. Analysis results on southbound I-95 corridor: left, AM peak period; right, PM peak period; top, reliability ratio over time; and bottom, average reliability ratio versus length.

115 Table E.4. TMC Segment Definitions on Northbound I-270 Corridor TMC Roadnumber Firstname County Direction Miles 110+04103 I-270 MD-187/Old Georgetown Rd/Exit 1 MONTGOMERY NORTHBOUND 1.080863 110P04103 I-270 MD-187/Old Georgetown Rd/Exit 1 MONTGOMERY NORTHBOUND 0.767429 110+04104 I-270 I-270 MONTGOMERY NORTHBOUND 0.08246 110P04104 I-270 I-270 MONTGOMERY NORTHBOUND 0.519118 110+04105 I-270 Montrose Rd/Exit 4 MONTGOMERY NORTHBOUND 1.138529 110P04105 I-270 Montrose Rd/Exit 4 MONTGOMERY NORTHBOUND 0.523902 110+04106 I-270 MD-189/Falls Rd/Exit 5 MONTGOMERY NORTHBOUND 0.956645 110P04106 I-270 MD-189/Falls Rd/Exit 5 MONTGOMERY NORTHBOUND 0.3496 110+04107 I-270 MD-28/Montgomery Ave/Exit 6 MONTGOMERY NORTHBOUND 0.530738 110P04107 I-270 MD-28/Montgomery Ave/Exit 6 MONTGOMERY NORTHBOUND 0.44281 110+04108 I-270 Shady Grove Rd/Exit 8 MONTGOMERY NORTHBOUND 1.468555 110P04108 I-270 Shady Grove Rd/Exit 8 MONTGOMERY NORTHBOUND 0.490285 110+04109 I-270 I-370/Sam Eig Hwy/Exit 9 MONTGOMERY NORTHBOUND 0.393781 110P04109 I-270 I-370/Sam Eig Hwy/Exit 9 MONTGOMERY NORTHBOUND 0.574049 110+04110 I-270 MD-117/Exit 10 MONTGOMERY NORTHBOUND 1.228819 110P04110 I-270 MD-117/Exit 10 MONTGOMERY NORTHBOUND 0.019512 110+04111 I-270 MD-124/Quince Orchard Rd/Exit 11 MONTGOMERY NORTHBOUND 0.419818 110P04111 I-270 MD-124/Quince Orchard Rd/Exit 11 MONTGOMERY NORTHBOUND 0.403289 110+04112 I-270 Middlebrook Rd/Exit 13 MONTGOMERY NORTHBOUND 2.074047 110P04112 I-270 Middlebrook Rd/Exit 13 MONTGOMERY NORTHBOUND 0.212208 110+04113 I-270 MD-118/Exit 15 MONTGOMERY NORTHBOUND 0.477794 110P04113 I-270 MD-118/Exit 15 MONTGOMERY NORTHBOUND 0.648307 110+04114 I-270 Father Hurley Blvd/Exit 16 MONTGOMERY NORTHBOUND 0.28137 110P04114 I-270 Father Hurley Blvd/Exit 16 MONTGOMERY NORTHBOUND 0.635444 110+04115 I-270 MD-121 MONTGOMERY NORTHBOUND 2.170053 110P04115 I-270 MD-121 MONTGOMERY NORTHBOUND 0.220597 110+04116 I-270 MD-109/Exit 22 MONTGOMERY NORTHBOUND 3.841557 110P04116 I-270 MD-109/Exit 22 MONTGOMERY NORTHBOUND 0.216185 110+04117 I-270 MD-80/Exit 26 FREDERICK NORTHBOUND 3.499849 110P04117 I-270 MD-80/Exit 26 FREDERICK NORTHBOUND 0.175235 110+04118 I-270 MD-85/Exit 31 FREDERICK NORTHBOUND 4.713754 110P04118 I-270 MD-85/Exit 31 FREDERICK NORTHBOUND 0.526637 110+04119 I-270 I-70/US-40 FREDERICK NORTHBOUND 0.386697 110P04119 I-270 I-70/US-40 FREDERICK NORTHBOUND 1.014187 110+10532 I-270 Montrose Rd MONTGOMERY NORTHBOUND 0.325862 110P10532 I-270 Montrose Rd MONTGOMERY NORTHBOUND 0.523902 110+10533 I-270 MD-189/Great Falls Rd MONTGOMERY NORTHBOUND 0.956645 110P10533 I-270 MD-189/Great Falls Rd MONTGOMERY NORTHBOUND 0.3496 (continued on next page)

116 0 5 10 15 20 25 30 35 40 45 0 10 20 30 40 50 60 70 LENGTH (MILES) A VE R A G E TR AV EL T IM E (M IN UT ES ) AVERAGE TRAVEL TIMES CORRIDOR NAME: I270 NB (LENGTH = 40.9652 MILES) AM PEAK PM PEAK Figure E.7. Average travel time versus length on northbound I-270 corridor. Table E.4. TMC Segment Definitions on Northbound I-270 Corridor TMC Roadnumber Firstname County Direction Miles 110+10534 I-270 MD-28/W Montgomery Ave MONTGOMERY NORTHBOUND 0.756679 110P10534 I-270 MD-28/W Montgomery Ave MONTGOMERY NORTHBOUND 0.004661 110+10535 I-270 Shady Grove Rd MONTGOMERY NORTHBOUND 1.680763 110P10535 I-270 Shady Grove Rd MONTGOMERY NORTHBOUND 0.490285 110+10536 I-270 I-370 MONTGOMERY NORTHBOUND 0.393781 110P10536 I-270 I-370 MONTGOMERY NORTHBOUND 0.574049 110+10537 I-270 MD-117/W Diamond Ave MONTGOMERY NORTHBOUND 1.228819 110P10537 I-270 MD-117/W Diamond Ave MONTGOMERY NORTHBOUND 0.019512 110+10538 I-270 MD-124/Montgomery Village Ave MONTGOMERY NORTHBOUND 0.419818 110P10538 I-270 MD-124/Montgomery Village Ave MONTGOMERY NORTHBOUND 0.403289 110+10539 I-270 I-270/Washington National Pike MONTGOMERY NORTHBOUND 0.35339 (continued)

117 Figure E.8. Travel time correlations on northbound I-270 corridor: left, AM peak period; right, PM peak period. 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) AM PEAK PERIOD 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) PM PEAK PERIOD 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) RELIABILITY RATIOS (RR) PATH LEVEL & INTERDAY (ACROSS DAYS) CORRIDOR NAME: I270 NB Figure E.9. Analysis results on northbound I-270 corridor: left, AM peak period; right, PM peak period; top, reliability ratio over time; and bottom, average reliability ratio versus length.

118 Table E.5. TMC Segment Definitions on Southbound I-270 Corridor TMC Roadnumber Firstname County Direction Miles 110N04119 I-270 I-70/US-40 FREDERICK SOUTHBOUND 0.828202 110-04118 I-270 MD-85/Exit 31 FREDERICK SOUTHBOUND 0.85225 110N04118 I-270 MD-85/Exit 31 FREDERICK SOUTHBOUND 0.512904 110-04117 I-270 MD-80/Exit 26 FREDERICK SOUTHBOUND 4.835362 110N04117 I-270 MD-80/Exit 26 FREDERICK SOUTHBOUND 0.162993 110-04116 I-270 MD-109/Exit 22 MONTGOMERY SOUTHBOUND 3.554346 110N04116 I-270 MD-109/Exit 22 MONTGOMERY SOUTHBOUND 0.173619 110-04115 I-270 MD-121 MONTGOMERY SOUTHBOUND 3.446906 110N04115 I-270 MD-121 MONTGOMERY SOUTHBOUND 0.219727 110-04114 I-270 Father Hurley Blvd/Exit 16 MONTGOMERY SOUTHBOUND 2.257981 110N04114 I-270 Father Hurley Blvd/Exit 16 MONTGOMERY SOUTHBOUND 0.720016 110-04113 I-270 MD-118/Exit 15 MONTGOMERY SOUTHBOUND 0.350407 110N04113 I-270 MD-118/Exit 15 MONTGOMERY SOUTHBOUND 0.622332 110-04112 I-270 Middlebrook Rd/Exit 13 MONTGOMERY SOUTHBOUND 0.487799 110N04112 I-270 Middlebrook Rd/Exit 13 MONTGOMERY SOUTHBOUND 0.276896 110-04111 I-270 MD-124/Quince Orchard Rd/Exit 11 MONTGOMERY SOUTHBOUND 1.934977 110N04111 I-270 MD-124/Quince Orchard Rd/Exit 11 MONTGOMERY SOUTHBOUND 0.256141 110-04110 I-270 MD-117/Exit 10 MONTGOMERY SOUTHBOUND 0.624072 110N04110 I-270 MD-117/Exit 10 MONTGOMERY SOUTHBOUND 0.277579 110-04109 I-270 I-370/Sam Eig Hwy/Exit 9 MONTGOMERY SOUTHBOUND 0.70591 110N04109 I-270 I-370/Sam Eig Hwy/Exit 9 MONTGOMERY SOUTHBOUND 0.920231 110-04108 I-270 Shady Grove Rd/Exit 8 MONTGOMERY SOUTHBOUND 0.404345 110N04108 I-270 Shady Grove Rd/Exit 8 MONTGOMERY SOUTHBOUND 0.419134 110-04107 I-270 MD-28/Montgomery Ave/Exit 6 MONTGOMERY SOUTHBOUND 1.353658 110N04107 I-270 MD-28/Montgomery Ave/Exit 6 MONTGOMERY SOUTHBOUND 0.456915 110-04106 I-270 MD-189/Falls Rd/Exit 5 MONTGOMERY SOUTHBOUND 0.62836 110N04106 I-270 MD-189/Falls Rd/Exit 5 MONTGOMERY SOUTHBOUND 0.574422 110-04105 I-270 Montrose Rd/Exit 4 MONTGOMERY SOUTHBOUND 0.662475 110N04105 I-270 Montrose Rd/Exit 4 MONTGOMERY SOUTHBOUND 0.538567 110-04104 I-270 I-270 MONTGOMERY SOUTHBOUND 1.321096 110N04104 I-270 I-270 MONTGOMERY SOUTHBOUND 0.191081 110-04103 I-270 MD-187/Old Georgetown Rd/Exit 1 MONTGOMERY SOUTHBOUND 0.241165 110N04103 I-270 MD-187/Old Georgetown Rd/Exit 1 MONTGOMERY SOUTHBOUND 0.808069 110-04102 I-270 I-495/MD-355 MONTGOMERY SOUTHBOUND 1.091365 110-10538 I-270 MD-124/Montgomery Village Ave MONTGOMERY SOUTHBOUND 0.290132 110N10538 I-270 MD-124/Montgomery Village Ave MONTGOMERY SOUTHBOUND 0.302498 110-10537 I-270 MD-117/W Diamond Ave MONTGOMERY SOUTHBOUND 0.577716 110N10537 I-270 MD-117/W Diamond Ave MONTGOMERY SOUTHBOUND 0.277579 (continued on next page)

119 0 5 10 15 20 25 30 35 40 45 0 10 20 30 40 50 60 70 LENGTH (MILES) A VE R A G E TR AV EL T IM E (M IN UT ES ) AVERAGE TRAVEL TIMES CORRIDOR NAME: I270 SB (LENGTH = 41.1712 MILES) AM PEAK PM PEAK Figure E.10. Average travel time versus length on southbound I-270 corridor. Table E.5. TMC Segment Definitions on Southbound I-270 Corridor TMC Roadnumber Firstname County Direction Miles 110-10536 I-270 I-370 MONTGOMERY SOUTHBOUND 0.70591 110N10536 I-270 I-370 MONTGOMERY SOUTHBOUND 0.920231 110-10535 I-270 Shady Grove Rd MONTGOMERY SOUTHBOUND 0.404345 110N10535 I-270 Shady Grove Rd MONTGOMERY SOUTHBOUND 0.419134 110-10534 I-270 MD-28/W Montgomery Ave MONTGOMERY SOUTHBOUND 1.634779 110N10534 I-270 MD-28/W Montgomery Ave MONTGOMERY SOUTHBOUND 0.002175 110-10533 I-270 MD-189/Great Falls Rd MONTGOMERY SOUTHBOUND 0.801979 110N10533 I-270 MD-189/Great Falls Rd MONTGOMERY SOUTHBOUND 0.574422 110-10532 I-270 Montrose Rd MONTGOMERY SOUTHBOUND 0.662475 110N10532 I-270 Montrose Rd MONTGOMERY SOUTHBOUND 0.538567 110-10531 I-270 I-270 MONTGOMERY SOUTHBOUND 0.347984 (continued)

120 Figure E.11. Travel time correlations on southbound I-270 corridor: left, AM peak period; right, PM peak period. 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) AM PEAK PERIOD 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) PM PEAK PERIOD 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) RELIABILITY RATIOS (RR) PATH LEVEL & INTERDAY (ACROSS DAYS) CORRIDOR NAME: I270 SB Figure E.12. Analysis results on southbound I-270 corridor: left, AM peak period; right, PM peak period; top, reliability ratio over time; and bottom, average reliability ratio versus length.

121 Table E.6. TMC Segment Definitions on Clockwise (Inner Loop) I-495 Corridor TMC Roadnumber Firstname County Direction Miles 110+04615 I-495 Clara Barton Pkwy/Exit 41 MONTGOMERY CLOCKWISE 0.213389 110P04615 I-495 Clara Barton Pkwy/Exit 41 MONTGOMERY CLOCKWISE 0.35252 110+04616 I-495 Cabin John Pkwy/Exit 40 MONTGOMERY CLOCKWISE 1.236897 110P04616 I-495 Cabin John Pkwy/Exit 40 MONTGOMERY CLOCKWISE 0.444177 110+04617 I-495 MD-190/River Rd/Exit 39 MONTGOMERY CLOCKWISE 0.090103 110P04617 I-495 MD-190/River Rd/Exit 39 MONTGOMERY CLOCKWISE 0.00814 110+04618 I-495 I-270 Spur MONTGOMERY CLOCKWISE 1.131072 110+04619 I-495 MD-187/Old Georgetown Rd/Exit 36 MONTGOMERY CLOCKWISE 1.895954 110P04619 I-495 MD-187/Old Georgetown Rd/Exit 36 MONTGOMERY CLOCKWISE 0.440262 110+04620 I-495 I-270/Exit 35 MONTGOMERY CLOCKWISE 0.700753 110+04621 I-495 MD-355/Wisconsin Ave/Exit 34 MONTGOMERY CLOCKWISE 0.046481 110P04621 I-495 MD-355/Wisconsin Ave/Exit 34 MONTGOMERY CLOCKWISE 0.362587 110+04622 I-495 MD-185/Connecticut Ave/Exit 33 MONTGOMERY CLOCKWISE 1.117899 110P04622 I-495 MD-185/Connecticut Ave/Exit 33 MONTGOMERY CLOCKWISE 0.588466 110+04623 I-495 MD-97/Georgia Ave/Exit 31 MONTGOMERY CLOCKWISE 1.609737 110P04623 I-495 MD-97/Georgia Ave/Exit 31 MONTGOMERY CLOCKWISE 0.390177 110+04624 I-495 US-29/Colesville Rd/Exit 30 MONTGOMERY CLOCKWISE 1.067503 110P04624 I-495 US-29/Colesville Rd/Exit 30 MONTGOMERY CLOCKWISE 0.422055 110+04625 I-495 MD-193/University Blvd/Exit 29 MONTGOMERY CLOCKWISE 0.240668 110P04625 I-495 MD-193/University Blvd/Exit 29 MONTGOMERY CLOCKWISE 0.435104 110+04626 I-495 MD-650/New Hampshire Ave/Exit 28 MONTGOMERY CLOCKWISE 1.091241 110P04626 I-495 MD-650/New Hampshire Ave/Exit 28 MONTGOMERY CLOCKWISE 0.627241 110+04627 I-495 Exit 27 PRINCE GEORGE’S CLOCKWISE 0.499916

122 0 2 4 6 8 10 12 14 16 0 5 10 15 20 25 30 35 40 45 LENGTH (MILES) A VE R A G E TR AV EL T IM E (M IN UT ES ) AVERAGE TRAVEL TIMES CORRIDOR NAME: I495 CW (LENGTH = 15.0123 MILES) AM PEAK PM PEAK Figure E.13. Average travel time versus length on clockwise (inner loop) I-495 corridor. Figure E.14. Travel time correlations on clockwise (inner loop) I-495 corridor: left, AM peak period; right, PM peak period.

123 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) AM PEAK PERIOD 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) PM PEAK PERIOD 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) RELIABILITY RATIOS (RR) PATH LEVEL & INTERDAY (ACROSS DAYS) CORRIDOR NAME: I495 CW Figure E.15. Analysis results on clockwise (inner loop) I-495 corridor: left, AM peak period; right, PM peak period; top, reliability ratio over time; and bottom, average reliability ratio versus length.

124 Table E.7. TMC Segment Definitions on Counterclockwise (Outer Loop) I-495 Corridor TMC Roadnumber Firstname County Direction Miles 110N04627 I-495 Exit 27 PRINCE GEORGE’S COUNTERCLOCKWISE 0.942975 110-04626 I-495 MD-650/New Hampshire Ave/Exit 28 MONTGOMERY COUNTERCLOCKWISE 0.675089 110N04626 I-495 MD-650/New Hampshire Ave/Exit 28 MONTGOMERY COUNTERCLOCKWISE 0.556153 110-04625 I-495 MD-193/University Blvd/Exit 29 MONTGOMERY COUNTERCLOCKWISE 1.139523 110N04625 I-495 MD-193/University Blvd/Exit 29 MONTGOMERY COUNTERCLOCKWISE 0.22389 110-04624 I-495 US-29/Colesville Rd/Exit 30 MONTGOMERY COUNTERCLOCKWISE 0.598843 110N04624 I-495 US-29/Colesville Rd/Exit 30 MONTGOMERY COUNTERCLOCKWISE 0.258067 110-04623 I-495 MD-97/Georgia Ave/Exit 31 MONTGOMERY COUNTERCLOCKWISE 1.023011 110N04623 I-495 MD-97/Georgia Ave/Exit 31 MONTGOMERY COUNTERCLOCKWISE 0.365756 110-04622 I-495 MD-185/Connecticut Ave/Exit 33 MONTGOMERY COUNTERCLOCKWISE 1.60781 110N04622 I-495 MD-185/Connecticut Ave/Exit 33 MONTGOMERY COUNTERCLOCKWISE 0.701871 110-04621 I-495 MD-355/Wisconsin Ave/Exit 34 MONTGOMERY COUNTERCLOCKWISE 1.118706 110N04621 I-495 MD-355/Wisconsin Ave/Exit 34 MONTGOMERY COUNTERCLOCKWISE 0.424975 110-04620 I-495 I-270/Exit 35 MONTGOMERY COUNTERCLOCKWISE 0.01423 110-04619 I-495 MD-187/Old Georgetown Rd/Exit 36 MONTGOMERY COUNTERCLOCKWISE 0.785325 110N04619 I-495 MD-187/Old Georgetown Rd/Exit 36 MONTGOMERY COUNTERCLOCKWISE 0.42802 110-04618 I-495 I-270 Spur MONTGOMERY COUNTERCLOCKWISE 1.780062 110-04617 I-495 MD-190/River Rd/Exit 39 MONTGOMERY COUNTERCLOCKWISE 1.251686 110N04617 I-495 MD-190/River Rd/Exit 39 MONTGOMERY COUNTERCLOCKWISE 0.008575 110-04616 I-495 Cabin John Pkwy/Exit 40 MONTGOMERY COUNTERCLOCKWISE 0.032499 110N04616 I-495 Cabin John Pkwy/Exit 40 MONTGOMERY COUNTERCLOCKWISE 0.554413 110-04615 I-495 Clara Barton Pkwy/Exit 41 MONTGOMERY COUNTERCLOCKWISE 1.143376 110N04615 I-495 Clara Barton Pkwy/Exit 41 MONTGOMERY COUNTERCLOCKWISE 0.389618

125 0 2 4 6 8 10 12 14 16 18 0 5 10 15 20 25 30 35 LENGTH (MILES) A VE R A G E TR AV EL T IM E (M IN UT ES ) AVERAGE TRAVEL TIMES CORRIDOR NAME: I495 CCW (LENGTH = 16.0245 MILES) AM PEAK PM PEAK Figure E.16. Average travel time versus length on counterclockwise (outer loop) I-495 corridor. Figure E.17. Travel time correlations on counterclockwise (outer loop) I-495 corridor: left, AM peak period; right, PM peak period.

126 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) AM PEAK PERIOD 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) PM PEAK PERIOD 0 2 4 6 8 10 12 14 16 18 0 0.2 0.4 0.6 0.8 1 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) 0 2 4 6 8 10 12 14 16 18 0 0.2 0.4 0.6 0.8 1 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) RELIABILITY RATIOS (RR) PATH LEVEL & INTERDAY (ACROSS DAYS) CORRIDOR NAME: I495 CCW Figure E.18. Analysis results on counterclockwise (outer loop) I-495 corridor: left, AM peak period; right, PM peak period; top, reliability ratio over time; and bottom, average reliability ratio versus length.

127 Table E.8. TMC Segment Definitions on Northbound MD-295 Corridor TMC Roadnumber Firstname County Direction Miles 110+04265 MD-295 US-50/MD-201/Kenilworth Ave PRINCE GEORGE’S NORTHBOUND 0.319213 110+04266 MD-295 MD-202 PRINCE GEORGE’S NORTHBOUND 0.902521 110P04266 MD-295 MD-202 PRINCE GEORGE’S NORTHBOUND 0.186731 110+04267 MD-295 MD-450 PRINCE GEORGE’S NORTHBOUND 0.09756 110P04267 MD-295 MD-450 PRINCE GEORGE’S NORTHBOUND 0.456294 110+04268 MD-295 Riverdale Rd PRINCE GEORGE’S NORTHBOUND 1.176062 110P04268 MD-295 Riverdale Rd PRINCE GEORGE’S NORTHBOUND 0.48842 110+04269 MD-295 I-495/I-95 PRINCE GEORGE’S NORTHBOUND 1.818714 110P04269 MD-295 I-495/I-95 PRINCE GEORGE’S NORTHBOUND 0.555034 110+04270 MD-295 MD-193 PRINCE GEORGE’S NORTHBOUND 0.242284 110P04270 MD-295 MD-193 PRINCE GEORGE’S NORTHBOUND 0.18555 110+04271 MD-295 Goddard Rd PRINCE GEORGE’S NORTHBOUND 0.828948 110P04271 MD-295 Goddard Rd PRINCE GEORGE’S NORTHBOUND 0.29212 110+04272 MD-295 Powder Mill Rd PRINCE GEORGE’S NORTHBOUND 1.605635 110P04272 MD-295 Powder Mill Rd PRINCE GEORGE’S NORTHBOUND 0.471518 110+04273 MD-295 MD-197/Exit 11 PRINCE GEORGE’S NORTHBOUND 1.121006 110P04273 MD-295 MD-197/Exit 11 PRINCE GEORGE’S NORTHBOUND 0.711814 110+04274 MD-295 Arundel/Prince George’s Co Line (Laurel) (South) PRINCE GEORGE’S NORTHBOUND 0.631032 110+04494 MD-295 Arundel/Prince George’s Co Line (Laurel) (North) ANNE ARUNDEL NORTHBOUND 0.016902 110+04495 MD-295 MD-198 ANNE ARUNDEL NORTHBOUND 2.148428 110P04495 MD-295 MD-198 ANNE ARUNDEL NORTHBOUND 0.680682 110+04496 MD-295 MD-32 ANNE ARUNDEL NORTHBOUND 0.944217 110P04496 MD-295 MD-32 ANNE ARUNDEL NORTHBOUND 0.724304 110+04497 MD-295 Canine Rd ANNE ARUNDEL NORTHBOUND 0.167902 110P04497 MD-295 Canine Rd ANNE ARUNDEL NORTHBOUND 0.2654 110+04498 MD-295 MD-175 ANNE ARUNDEL NORTHBOUND 1.275796 110P04498 MD-295 MD-175 ANNE ARUNDEL NORTHBOUND 0.592008 110+04499 MD-295 MD-100 ANNE ARUNDEL NORTHBOUND 1.678588 110P04499 MD-295 MD-100 ANNE ARUNDEL NORTHBOUND 0.790234 110+04500 MD-295 I-195 ANNE ARUNDEL NORTHBOUND 2.180058 110P04500 MD-295 I-195 ANNE ARUNDEL NORTHBOUND 0.805024 110+04501 MD-295 Nursery Rd ANNE ARUNDEL NORTHBOUND 0.528811 110P04501 MD-295 Nursery Rd ANNE ARUNDEL NORTHBOUND 0.529806 110+04502 MD-295 I-695 ANNE ARUNDEL NORTHBOUND 0.714921 110P04502 MD-295 I-695 ANNE ARUNDEL NORTHBOUND 0.438708 110+04503 MD-295 I-895/Harbor Tunnel Trwy BALTIMORE NORTHBOUND 0.734743 110P04503 MD-295 I-895/Harbor Tunnel Trwy BALTIMORE NORTHBOUND 0.110671 110+04504 MD-295 MD-648/Waterview Ave/Annapolis Rd BALTIMORE NORTHBOUND 2.065471

128 0 5 10 15 20 25 30 0 10 20 30 40 50 60 LENGTH (MILES) A VE R A G E TR AV EL T IM E (M IN UT ES ) AVERAGE TRAVEL TIMES CORRIDOR NAME: MD295 NB (LENGTH = 29.4831 MILES) AM PEAK PM PEAK Figure E.19. Average travel time versus length on northbound MD-295 corridor. Figure E.20. Travel time correlations on northbound MD-295 corridor: left, AM peak period; right, PM peak period.

129 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) AM PEAK PERIOD 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) PM PEAK PERIOD 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) RELIABILITY RATIOS (RR) PATH LEVEL & INTERDAY (ACROSS DAYS) CORRIDOR NAME: MD295 NB Figure E.21. Analysis results on northbound MD-295 corridor: left, AM peak period; right, PM peak period; top, reliability ratio over time; and bottom, average reliability ratio versus length.

130 Table E.9. TMC Segment Definitions on Southbound MD-295 Corridor TMC Roadnumber Firstname County Direction Miles 110-04503 MD-295 I-895/Harbor Tunnel Trwy BALTIMORE SOUTHBOUND 1.873956 110N04503 MD-295 I-895/Harbor Tunnel Trwy BALTIMORE SOUTHBOUND 0.051638 110-04502 MD-295 I-695 ANNE ARUNDEL SOUTHBOUND 0.813164 110N04502 MD-295 I-695 ANNE ARUNDEL SOUTHBOUND 0.408819 110-04501 MD-295 Nursery Rd ANNE ARUNDEL SOUTHBOUND 0.669745 110N04501 MD-295 Nursery Rd ANNE ARUNDEL SOUTHBOUND 0.555532 110-04500 MD-295 I-195 ANNE ARUNDEL SOUTHBOUND 0.568705 110N04500 MD-295 I-195 ANNE ARUNDEL SOUTHBOUND 0.752888 110-04499 MD-295 MD-100 ANNE ARUNDEL SOUTHBOUND 2.189689 110N04499 MD-295 MD-100 ANNE ARUNDEL SOUTHBOUND 0.830688 110-04498 MD-295 MD-175 ANNE ARUNDEL SOUTHBOUND 1.666906 110N04498 MD-295 MD-175 ANNE ARUNDEL SOUTHBOUND 0.609345 110-04497 MD-295 Canine Rd ANNE ARUNDEL SOUTHBOUND 1.222791 110N04497 MD-295 Canine Rd ANNE ARUNDEL SOUTHBOUND 0.30113 110-04496 MD-295 MD-32 ANNE ARUNDEL SOUTHBOUND 0.058474 110N04496 MD-295 MD-32 ANNE ARUNDEL SOUTHBOUND 0.791104 110-04495 MD-295 MD-198 ANNE ARUNDEL SOUTHBOUND 1.138902 110N04495 MD-295 MD-198 ANNE ARUNDEL SOUTHBOUND 0.463937 110-04494 MD-295 Arundel/Prince George’s Co Line (Laurel) (North) ANNE ARUNDEL SOUTHBOUND 2.234741 110-04274 MD-295 Arundel/Prince George’s Co Line (Laurel) (South) PRINCE GEORGE’S SOUTHBOUND 0.014976 110-04273 MD-295 MD-197/Exit 11 PRINCE GEORGE’S SOUTHBOUND 0.437714 110N04273 MD-295 MD-197/Exit 11 PRINCE GEORGE’S SOUTHBOUND 0.799183 110-04272 MD-295 Powder Mill Rd PRINCE GEORGE’S SOUTHBOUND 1.232733 110N04272 MD-295 Powder Mill Rd PRINCE GEORGE’S SOUTHBOUND 0.497182 110-04271 MD-295 Goddard Rd PRINCE GEORGE’S SOUTHBOUND 1.698659 110N04271 MD-295 Goddard Rd PRINCE GEORGE’S SOUTHBOUND 0.172936 110-04270 MD-295 MD-193 PRINCE GEORGE’S SOUTHBOUND 0.836653 110N04270 MD-295 MD-193 PRINCE GEORGE’S SOUTHBOUND 0.312626 110-04269 MD-295 I-495/I-95 PRINCE GEORGE’S SOUTHBOUND 0.059095 110N04269 MD-295 I-495/I-95 PRINCE GEORGE’S SOUTHBOUND 0.538132 110-04268 MD-295 Riverdale Rd PRINCE GEORGE’S SOUTHBOUND 1.896761 110N04268 MD-295 Riverdale Rd PRINCE GEORGE’S SOUTHBOUND 0.467293 110-04267 MD-295 MD-450 PRINCE GEORGE’S SOUTHBOUND 1.183581 110N04267 MD-295 MD-450 PRINCE GEORGE’S SOUTHBOUND 0.254463 110-04266 MD-295 MD-202 PRINCE GEORGE’S SOUTHBOUND 0.088984 110N04266 MD-295 MD-202 PRINCE GEORGE’S SOUTHBOUND 0.232217 110-04265 MD-295 US-50/MD-201/Kenilworth Ave PRINCE GEORGE’S SOUTHBOUND 1.046313 110-04264 MD-295 Eastern Ave PRINCE GEORGE’S SOUTHBOUND 0.329591

131 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 LENGTH (MILES) A VE R A G E TR AV EL T IM E (M IN UT ES ) AVERAGE TRAVEL TIMES CORRIDOR NAME: MD295 SB (LENGTH = 29.3012 MILES) AM PEAK PM PEAK Figure E.22. Average travel time versus length on southbound MD-295 corridor. Figure E.23. Travel time correlations on southbound MD-295 corridor: left, AM peak period; right, PM peak period.

132 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) 0 20 40 60 80 100 120 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) 0 0.2 0.4 0.6 0.8 1 R EL IA BI LI TY R AT IO (U NI TL ES S) 0 0.2 0.4 0.6 0.8 1 R EL IA BI LI TY R AT IO (U NI TL ES S) AM PEAK PERIOD PM PEAK PERIOD 0 5 10 15 20 25 30 LENGTH (MILES) 0 0.2 0.4 0.6 0.8 1 R EL IA BI LI TY R AT IO (U NI TL ES S) 0 5 10 15 20 25 30 LENGTH (MILES) RELIABILITY RATIOS (RR) PATH LEVEL & INTERDAY (ACROSS DAYS) CORRIDOR NAME: MD295 SB Figure E.24. Analysis results on southbound MD-295 corridor: left, AM peak period; right, PM peak period; top, reliability ratio over time; and bottom, average reliability ratio versus length.

133 Table E.10. TMC Segment Definitions on Northbound US-29 Corridor TMC Roadnumber Firstname County Direction Miles 110+05898 US-29 Cherry Hill Rd/Randolph Rd MONTGOMERY NORTHBOUND 1.58022 110P05898 US-29 Cherry Hill Rd/Randolph Rd MONTGOMERY NORTHBOUND 0.508678 110+05899 US-29 Fairland Rd MONTGOMERY NORTHBOUND 0.539624 110P05899 US-29 Fairland Rd MONTGOMERY NORTHBOUND 0.059841 110+05900 US-29 Briggs Chaney Rd MONTGOMERY NORTHBOUND 0.453622 110P05900 US-29 Briggs Chaney Rd MONTGOMERY NORTHBOUND 0.598781 110+05901 US-29 Greencastle Rd MONTGOMERY NORTHBOUND 0.754193 110+05902 US-29 MD-198/Sandy Spring Rd MONTGOMERY NORTHBOUND 0.93123 110P05902 US-29 MD-198/Sandy Spring Rd MONTGOMERY NORTHBOUND 0.648679 110+06887 US-29 Dustin Rd MONTGOMERY NORTHBOUND 0.373896 110P06887 US-29 Dustin Rd MONTGOMERY NORTHBOUND 0.306909 110+05241 US-29 Howard/Montgomery County Line HOWARD NORTHBOUND 0.474066 110+05242 US-29 Old Columbia Rd HOWARD NORTHBOUND 0.569762 110P05242 US-29 Old Columbia Rd HOWARD NORTHBOUND 0.050209 110+05243 US-29 MD-216 HOWARD NORTHBOUND 0.778987 110P05243 US-29 MD-216 HOWARD NORTHBOUND 0.392103 110+05244 US-29 Johns Hopkins Rd/Exit 15 HOWARD NORTHBOUND 0.416027 110P05244 US-29 Johns Hopkins Rd/Exit 15 HOWARD NORTHBOUND 0.616864 110+05245 US-29 MD-32/Exit 16 HOWARD NORTHBOUND 1.020028 110P05245 US-29 MD-32/Exit 16 HOWARD NORTHBOUND 0.840444 110+05246 US-29 Brokenland Pkwy/Exit 18 HOWARD NORTHBOUND 0.839449 110P05246 US-29 Brokenland Pkwy/Exit 18 HOWARD NORTHBOUND 0.673908 110+05247 US-29 MD-175 HOWARD NORTHBOUND 1.152759 110P05247 US-29 MD-175 HOWARD NORTHBOUND 0.609531 110+05248 US-29 MD-108 HOWARD NORTHBOUND 0.480466 110P05248 US-29 MD-108 HOWARD NORTHBOUND 0.654334 110+05249 US-29 MD-100/Exit 22 HOWARD NORTHBOUND 0.450826 110P05249 US-29 MD-100/Exit 22 HOWARD NORTHBOUND 0.615745 110+05250 US-29 MD-103 HOWARD NORTHBOUND 0.045052 110P05250 US-29 MD-103 HOWARD NORTHBOUND 0.44573 110+05251 US-29 US-40 HOWARD NORTHBOUND 0.628484 110P05251 US-29 US-40 HOWARD NORTHBOUND 0.822858 110+05252 US-29 I-70 HOWARD NORTHBOUND 0.457599 110P05252 US-29 I-70 HOWARD NORTHBOUND 0.859458

134 0 5 10 15 20 25 0 5 10 15 20 25 30 LENGTH (MILES) A VE R A G E TR AV EL T IM E (M IN UT ES ) AVERAGE TRAVEL TIMES CORRIDOR NAME: US29 NB (LENGTH = 20.6504 MILES) AM PEAK PM PEAK Figure E.25. Average travel time versus length on northbound US-29 corridor. Figure E.26. Travel time correlations on northbound US-29 corridor: left, AM peak period; right, PM peak period.

135 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) AM PEAK PERIOD 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) PM PEAK PERIOD 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) RELIABILITY RATIOS (RR) PATH LEVEL & INTERDAY (ACROSS DAYS) CORRIDOR NAME: US29 NB Figure E.27. Analysis results on northbound US-29 corridor: left, AM peak period; right, PM peak period; top, reliability ratio over time; and bottom, average reliability ratio versus length.

136 Table E.11. TMC Segment Definitions on Southbound US-29 Corridor TMC Roadnumber Firstname County Direction Miles 110N05252 US-29 I-70 HOWARD SOUTHBOUND 0.868904 110-05251 US-29 US-40 HOWARD SOUTHBOUND 0.629913 110N05251 US-29 US-40 HOWARD SOUTHBOUND 0.59207 110-05250 US-29 MD-103 HOWARD SOUTHBOUND 0.729959 110N05250 US-29 MD-103 HOWARD SOUTHBOUND 0.529495 110-05249 US-29 MD-100/Exit 22 HOWARD SOUTHBOUND 0.027342 110N05249 US-29 MD-100/Exit 22 HOWARD SOUTHBOUND 0.651041 110-05248 US-29 MD-108 HOWARD SOUTHBOUND 0.478789 110N05248 US-29 MD-108 HOWARD SOUTHBOUND 0.563983 110-05247 US-29 MD-175 HOWARD SOUTHBOUND 0.355254 110N05247 US-29 MD-175 HOWARD SOUTHBOUND 0.764136 110-05246 US-29 Brokenland Pkwy/Exit 18 HOWARD SOUTHBOUND 1.297297 110N05246 US-29 Brokenland Pkwy/Exit 18 HOWARD SOUTHBOUND 0.462197 110-05245 US-29 MD-32/Exit 16 HOWARD SOUTHBOUND 0.866977 110N05245 US-29 MD-32/Exit 16 HOWARD SOUTHBOUND 0.905442 110-05244 US-29 Johns Hopkins Rd/Exit 15 HOWARD SOUTHBOUND 0.941483 110N05244 US-29 Johns Hopkins Rd/Exit 15 HOWARD SOUTHBOUND 0.724801 110-05243 US-29 MD-216 HOWARD SOUTHBOUND 0.76165 110N05243 US-29 MD-216 HOWARD SOUTHBOUND 0.383777 110-05242 US-29 Old Columbia Rd HOWARD SOUTHBOUND 0.250114 110N05242 US-29 Old Columbia Rd HOWARD SOUTHBOUND 0.095447 110-05241 US-29 Howard/Montgomery County Line HOWARD SOUTHBOUND 0.645697 110-06887 US-29 Dustin Rd MONTGOMERY SOUTHBOUND 0.419569 110N06887 US-29 Dustin Rd MONTGOMERY SOUTHBOUND 0.360847 110-05902 US-29 MD-198/Sandy Spring Rd MONTGOMERY SOUTHBOUND 0.422428 110N05902 US-29 MD-198/Sandy Spring Rd MONTGOMERY SOUTHBOUND 0.296843 110-05901 US-29 Greencastle Rd MONTGOMERY SOUTHBOUND 1.23379 110-05900 US-29 Briggs Chaney Rd MONTGOMERY SOUTHBOUND 0.788743 110N05900 US-29 Briggs Chaney Rd MONTGOMERY SOUTHBOUND 0.273789 110-05899 US-29 Fairland Rd MONTGOMERY SOUTHBOUND 0.705289 110N05899 US-29 Fairland Rd MONTGOMERY SOUTHBOUND 0.032313 110-05898 US-29 Cherry Hill Rd/Randolph Rd MONTGOMERY SOUTHBOUND 0.596233 110N05898 US-29 Cherry Hill Rd/Randolph Rd MONTGOMERY SOUTHBOUND 0.564728 110-05897 US-29 MD-650/New Hampshire Ave MONTGOMERY SOUTHBOUND 1.555923

137 0 5 10 15 20 25 0 5 10 15 20 25 30 LENGTH (MILES) A VE R A G E TR AV EL T IM E (M IN UT ES ) AVERAGE TRAVEL TIMES CORRIDOR NAME: US29 SB (LENGTH = 20.7763 MILES) AM PEAK PM PEAK Figure E.28. Average travel time versus length on southbound US-29 corridor. Figure E.29. Travel time correlations on southbound US-29 corridor: left, AM peak period; right, PM peak period.

138 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) AM PEAK PERIOD 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 TIME (MINUTES) R EL IA BI LI TY R AT IO (U NI TL ES S) PM PEAK PERIOD 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 LENGTH (MILES) R EL IA BI LI TY R AT IO (U NI TL ES S) RELIABILITY RATIOS (RR) PATH LEVEL & INTERDAY (ACROSS DAYS) CORRIDOR NAME: US29 SB Figure E.30. Analysis results on southbound US-29 corridor: left, AM peak period; right, PM peak period; top, reliability ratio over time; and bottom, average reliability ratio versus length.