Integrating MTS Commerce Data with Multimodal Freight Transportation Performance Measures to Support MTS Maintenance Investment Decision Making (2014)

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65 A P P E N D I X D9 In an attempt to estimate the capacity of the inland water- way system on a link-by-link basis, we developed a process using monthly historical traffic and delay data to estimate the delay as a function of traffic and then to define a capacity for each lock. The process was designed to be easily replicated across all of the locks in the system using a small dataset and limited computation. We used one year (2008) of monthly data for the 193 locks reporting through the Army Corps of Engineers Lock Perfor- mance Monitoring System (LPMS) and the Operations and Maintenance of Navigation Information (OMNI) system. Of these locks, we had no lockage data for 18 locks, leaving 175 locks reporting at least some data. Another 14 had lim- ited data (less than 1 lockage per day for each active month). Locks are built in different sizes based on anticipated traffic and tow sizes. Many of the smaller rivers or upper reaches of rivers have chambers 600 ft long or shorter. Busier sections have 1200 ft long chambers. Many locks have a main cham- ber and an auxiliary chamber. If the auxiliary is smaller than the main chamber, it is often used primarily for recreational traffic allowing the main chamber to process the large com- mercial tows. The auxiliary also acts as a backup if the main is closed for maintenance or repair. A few locks have twin chambers and use both for commercial traffic. We also gath- ered data on the physical characteristics of each of the locks: location, length, width, number of chambers. A theoretical bound on capacity (in terms or tows pro- cessed) could be estimated by dividing the available process- ing time by the average processing time for a vessel. This number is almost meaningless, however, since we are dealing with a system where tows arrive at random times and shippers do not have limitless patience for delay. Instead, we need to estimate the traffic that can reasonably be expected to endure the delay transiting the lock. Delay can be directly translated to cost for waterway shippers. At some level of traffic, adding additional tows will cause the average delay to exceed the time and cost threshold of some shippers, and these shippers will no longer find the waterway a cost effective transportation alternative. They will move their goods by other modes (rail, highway) or possibly they will not have a profitable trans- portation alternative and not make the movement at all. A full analysis of this economic tradeoff requires an economic equilibrium model with detailed costs for each shipment. As an alternative, we attempted to estimate the delay shippers would endure by observing their shipping patterns at vari- ous locks. We calculated the hours per Kiloton transiting each lock (delay and processing time) assuming the movements were in average tow sizes for that lock. Thus, for each lock we calculated the current Level of Service (LOS) as: Average Delay Average Processing Average Tow Size( )+ Plotting the distribution of these values (see Figure D-1) shows that they cluster around 0.2 and drop off sharply at about 0.8. In fact, 90 percent of the locks have an LOS less than or equal to 0.8. The 0.8 value appeared to represent the maximum delay most tows were willing to endure at a lock. Assuming a 0.8 LOS value, we calculated a MAXDELAY value for each lock using the lock’s processing time and average tow size. Given the MAXDELAY value, we estimated each lock’s capacity as the traffic that can be accommodated with the expected delay equal to MAXDELAY using the transit delay functions derived below. The 17 locks that currently have an LOS above 0.8 are assumed to be at capacity currently. Any additional demand for traffic through those locks is assumed to be offset by the loss of existing traffic. Lock Capacity Calculation 9The methodology employed to calculate the theoretical capacity of locks was borrowed from an unpublished paper titled “Using Historic Data to Estimate Capacity of the U.S. Inland Waterway System,” prepared by M. R. Hilliard, D. P. Vogt, and M. S. Schultze for the Center for Transportation Analysis at Oak Ridge National Laboratory. An abridged version of that paper with minimal editing is presented in Appendix D. This abridged version is used with permission. This material has not been edited by TRB.

66 We approached the development of delay curves (the func- tional relationship between traffic level and delay) on two parallel tracks: a queuing model and a regression model. In both cases the goal was to use the monthly traffic data at the locks to estimate delay at any given level of traffic. Combining a delay curve with a MAXDELAY then defines a maximum endurable traffic level, which we call capacity. The queuing approach assumed that the equations for estimating the delay in an M/G/1 queuing system could be applied to the lock operations. The M/G/1 queue refers to a system with exponentially distributed inter-arrival times, a random service time governed by any reasonable distribu- tion, and a single server. The standard form of the delay (wait time) as a function of traffic level is the Pollaczek-Khinchin formula. It appears in several versions, but the one most use- ful for this analysis was: 1 1 1 2 (1) 2 W c ( ) ( ) = µ ρ − ρ + Where µ = service rate (tows locked per hour when busy) c = conditional variance of the service time r = ratio of arrival rate to service rate (between 0 and 1) Attempting to fit the monthly data to this form was diffi- cult. We did not have the individual lockage times to generate the c value, and attempting to estimate it from the monthly means or working backwards from the average waiting time and service times to estimate a value for c did not produce reasonable values for many locks. The M/G/1 queuing model has fundamental assumptions about the system. In particular, the service time is assumed to be independent of the queue length. This assumption is prob- ably not true for the lockage process. There are efficiencies in processing several tows in a row. Also, lockage times are not independent. Since weather and flow conditions affect lock- ages, the processing time for a tow is likely to be correlated to the previous tow’s processing time. The loss of these assump- tions means that the theoretical model can be far from the actual system’s performance. In many cases, estimating the parameters for the queuing model produced delay estimates considerably higher than the observed delay. This overesti- mation of delay created some estimates of capacity less than the current traffic levels. We needed another approach to pro- vide an estimate using a limited dataset. The second approach was to consider each month at each lock as an observation and to develop a formula for the rela- tionship between traffic levels and delay using a regression approach. The lock data were aggregated based on lock size and the presence of an auxiliary chamber. The main cham- bers of the majority of locks fall into one of two standard lengths, 640 ft or 1200 ft. These two classes had sufficient data and produced reasonable results for the regression using a log-log form. For the 640 ft locks, the equation was: 2.1483 (2)Log Delay Log TowsPerDay Intercept( ) ( )= ∗ + For each individual lock, the intercept was chosen such that the curve goes through the annual average point for that lock. The other large cluster of locks has 1200 ft main chambers. The regression for this set of locks produced a slope of 1.682. Once again, the intercept for each lock was calculated from the annual averages. All locks under 1200 ft were modeled with the 640 ft equation, and the rest of the locks were mod- eled with the 1200 ft equation. For this effort, the goal of the development of transit curves is not to estimate the delay under all traffic levels. The final goal was to estimate the traffic level that gives the delay identified as the MAXDELAY. By substituting MAXDELAY Figure D-1. The distribution of the level of service (hrs/kton) for 174 locks calculated for 2008 traffic.

67 for each lock into its transit equation and solving for the Tow- PerDay variable, we were able to generate an estimate of the capacity in terms of annual traffic. Comparing the capacity traffic and the current traffic pro- vides a first level check of the estimates. Figure D-2 shows the distribution of the ratio of the estimated capacity to the cur- rent traffic level based on the queuing-based approach and the regression approach. The queuing approach produced a large number of estimates showing locks already at or above capacity (ratio less than or equal to 1). While many locks may be congested, it does not seem reasonable to argue that many are already at capacity. The regression approach seemed to pro- duce a more reasonable estimate relative to current traffic with a median of 3.7 and only a few ratios less than 1. A few locks have less than one tow per day with the current shipping patterns. The capacity estimates are based on the physical limitations, and though they are not large values, the ratio can exceed 20. There are a number of opportunities to improve the regression approach. Analyzing several years data on individ- ual lockages would allow the data to be averaged in smaller time units (days or weeks) providing a larger dataset and a wider range of delays and traffic intensities. Also, other func- tional forms could be tested. Clearly, if the arrival rate equals or exceeds the processing rate, delay eventually becomes infinite. In queuing terms, the queue “explodes.” While the log-log form provides a reasonable fit for the monthly data and moderate traffic levels, it does not have an asymptote to model the extreme high values of the relationship. Thus, at extremely high traffic levels we know the model is under- estimating delay. Conclusions The typical approach to establishing the relationship between traffic and delay at a lock (and estimating capac- ity) is to develop and run detailed simulations of the lock operations at various levels of demand. The timing for lock operations (vessels entering the chamber, opening and clos- ing gates, filling and emptying the chamber, etc.) is estimated using distributions based on historical data. These results are then used to estimate delay at any level of traffic. This process requires a great deal of data and a large number of simulation runs. The effort often consumes weeks or months of time. The resulting curve is not necessarily guaranteed to coincide with any one historical year. This approach is a first attempt at a simplified analysis for the system as a whole. While we would still consider the results preliminary, the overall approach seems promising for further study. Figure D-2. Ratio of estimated capacity to current traffic under two models (175 locks).