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48 If the sketch planning method was used to estimate seg- 6.4 Estimating Reliability ment travel times, the analyst should enter field measured and estimated travel times for each segment into a spread- All of the reliability metrics can be computed from the sheet and use the linear regression function to find the ap- travel-time variance data. This section provides a method to propriate parameters for ADT/lane, signals per mile, access predict the variance in the travel time given the variance in points per mile, and constant. the volume and the variance in the capacity. If the planning method was used to estimate travel times, Traffic operations improvements generally affect the prob- the field data and the estimated trip times should be entered ability of the facility being able to deliver a given capacity, and into a spreadsheet and the optimization function used in the have minor effects on the variability of the volume of traffic. spreadsheet to find the values of the parameters a and b in Thus this method predicts how changes in the variability of Equation 6.16 that minimize the squared error between the the delivered capacity for the facility affect the travel-time field data and the estimates. The search should be limited to variance and ultimately reliability. positive values for a and to values greater than 1.00 for b. 6.4.1 Predicting Changes in Capacity Variance 6.3 Estimating Delay The expected (mean) value of the inverse of capacity and Once travel time is known, delay can be estimated by sub- the square of the inverse of capacity are needed to predict the tracting the ideal travel time (often the travel time during un- travel-time variance. If the expected value of capacity can be congested periods of the day) from the actual travel time. considered as the ideal capacity (C0) minus a random variable (x), then the expected values of the inverse values can be com- 6.3.1 Definition of Ideal Travel Time puted using the following formulae. The ideal travel time against which delay is measured 1 N 1 E = P( x i ) * 0 <= xi C0 (Eq. 6.19) should be set by agency policy. Several definitions of the ideal C i =1 C0 - x i travel time are possible; two are provided here: One perspective is to take the "no other cars on the road" 1 N 1 E 2 = P( x i ) * 0 <= xi < C0 (Eq. 6.20) travel time as the ideal travel time. This method would as- C i =1 ( C 0 - x i )2 sume that all signals are green, so that all travel is at the PSL. This is often called the FFS or zero-flow travel time. For each study segment Exhibit 6.5 would be constructed. FFS however is not readily measurable in the field. So an The probability of a given capacity reduction (ai*C0) is approximation of the FFS would be the mean travel time and computed as a function of the frequency of that event type speed measured under low flow conditions. This method of occurring each year and the average number of hours that the measuring speed and travel time includes nominal delays at capacity reduction endures for each event. signals due to modest amounts of traffic on the main street and the side street. This speed would be defined as the mean Events/Year * Hours/Event P(x = ai * C0 ) = (Eq. 6.21) speed measured over the length of the trip during a nonpeak Hours/Yeear hour, say 10:00 a.m. to 11:00 a.m. or 2:00 p.m. to 3:00 p.m. This speed would generally be lower than the posted speed It also is possible that an ITS project might cause an inci- limit for signalized streets, but could be higher than the PSL dent to have a lesser impact on capacity. Then one would cre- for freeways, highway, and rural roads. ate two incident types, one before ITS, and the same one after ITS. Each event type would have a different capacity reduc- tion. The probability of after ITS incident happening before 6.3.2 Computation of Delay would be set to zero; the same for the before ITS incident hap- pening after. Delay is the difference between the actual and ideal travel time. d = Ta - T0 (Eq. 6.18) 6.4.2 Computation of Travel-Time Variance The travel-time variance is a function of the variance in where the volume/capacity ratio (A simple linear travel-time d = Delay (hr:min:sec); function with a breakpoint at v/c = 1.0 has been assumed to Ta = Actual Travel Time (hr:min:sec); and facilitate the computation of the travel-time variance from T0 = Ideal Travel Time (hr:min:sec). the v/c variance).
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49 Capacity Reducing Event Capacity Reduction Probability Before ITS Probability After ITS No Incident x=0 P b (x=0) P a (x=0) Shoulder Incident x = a 1*C0 P b (x=a 1*C0) P a (x=a 1*C0) Bad Weather Type 1 x = a 2*C0 P b (x=a 2*C0) P a (x=a 2*C0) Bad Weather Type 2 x = a 3*C0 P b (x=a 3*C0) P a (x=a 3*C0) Shoulder Work Zone x = a 4*C0 P b (x=a 4*C0) P a (x=a 4*C0) Single-Lane Closure x = a 5*C0 P b (x=a 5*C0) P a (x=a 5*C0) Two-Lane Closure x = a 6*C0 P b (x=a 6*C0) P a (x=a 6*C0) Three-Lane Closure x = a 7*C0 P b (x=a 7*C0) P a (x=a 7*C0) Total Closure x = C0 1 P b (x= C 0 1) P a (x= C 0 1) Note: "x" is never allowed to exceed C0-1. This avoids divide by zero problems when computing the expected value of 1/C. Exhibit 6.5. Capacity reductions. For v/c < = 1.00 TC = Travel time at capacity; Var (T) = a2 * Var (v/c) (Eq. 6.22) a = Calibration parameter = TC T0; and b = 0.25 (average delay per deterministic queuing theory). For v/c > 1.00 Var (T) = b2 * Var (v/c) (Eq. 6.23) The variance of the travel time is equal to the variance in the v/c ratio times the square of the slope of the linear travel- where time function for the facility. T = predicted travel time (hours); According to the HCM, the following free-flow and capac- T0 = Free-flow travel time (hours); ity travel-time rates (hours/mile) are appropriate (Exhibit 6.6). Calibration Free-Flow Travel- Capacity Travel- Parameter Free-Flow Speed Speed at Time Rate (T0) Time Rate (TC) a=TC-T0 HCM Facility Type (MPH) Capacity (MPH) (Hours/Miles) (Hours/Mile) (Hours/Mile) Freeway 75 53.3 0.0133 0.0188 0.0054 70 53.3 0.0143 0.0188 0.0045 65 52.2 0.0154 0.0192 0.0038 60 51.1 0.0167 0.0196 0.0029 55 50.0 0.0182 0.0200 0.0018 Multilane Highway 60 55.0 0.0167 0.0182 0.0015 55 51.2 0.0182 0.0195 0.0013 50 47.5 0.0200 0.0211 0.0011 45 42.2 0.0222 0.0237 0.0015 Arterial 50 20.0 0.0200 0.0500 0.0300 40 17.0 0.0250 0.0588 0.0338 35 9.0 0.0286 0.1111 0.0825 30 7.0 0.0333 0.1429 0.1095 Two-Lane Highways 55 40.0 0.0182 0.0250 0.0068 Sources: 1. Freeways: Exhibit 23-2 HCM. 2. Multilane Highways: Exhibit 21-2 HCM. 3. Arterials: HCM Exhibits 15-8, 15-9, 15-10, 15-11, Middle Curve. 4. Two-lane Highways: Exhibit 20-2, HCM. Exhibit 6.6. Free-flow and capacity travel-time rates per HCM.
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50 The variance in the volume/capacity ratio can be com- 3.0 Var (T ) puted from the expected value (the mean) of the volume (v), BI = * - 1 * 100% (Eq. 6.26) Mean (T )2 the volume squared, the inverse of the capacity, and the in- verse of the capacity squared. Var (v/c) = E(v2) * E(1/c2) [E(v)]2 * [E(1/c)]2 (Eq. 6.24) On-Time Arrival The Percent On-Time Arrival is computed using the fol- 6.4.3 Computation of Reliability Metrics lowing equation, which assumes a Gamma distribution for The following equations are for use with forecasted travel the travel times. times, where only the mean and variance are known. The dis- %OnTime = [1.10 * Mean (T )] (Eq. 6.27) tribution of times in this case must be assumed. For these equa- tions we have assumed that travel time is Gamma distributed where Gamma is the cumulative Gamma probability distri- with mean equal to mean (T) and variance equal to Var (T). bution with Mean = Mean (T) and variance = Var (T). Percent Variation Misery Index The following equation is used to compute percent varia- The Misery Index is computed according to the following tion based on the forecasted mean and variance in travel equation, which assumes a Gamma distribution for the travel times. times. Since it is inconvenient to compute the mean of the top 20 percent of the values of a function, we have approximated Var (T ) %V = * 100% (Eq. 6.25) this value with the 85 percentile highest value for the distribu- Mean (T ) tion. For Gamma (T): -1 [ 85% ] MI = -1 (Eq. 6.28) Buffer Index Mean (T ) BI is computed according to the following equation, which where Gamma1 is the inverse of the cumulative Gamma assumes a Gamma distribution for the travel times. distribution with Mean = Mean (T) and variance = Var (T).