Comparison of Methods for Increasing the Performance of a DUA Computation

Transcription

1 Comparison of Methos for Increasing the Performance of a DUA Computation Michael Behrisch, Daniel Krajzewicz, Peter Wagner an Yun-Pang Wang Institute of Transportation Systems, German Aerospace Center, Germany Abstract Computing realistic routes for a given roa network an a known eman of vehicles is one of the most important steps when preparing a roa traffic simulation. The approach evelope by Christian Gawron in 1998 which we use for this purpose computes a ynamic user equilibrium by iteratively performing the simulation an computing new vehicle routes. The results are vali, but the computation is very time consuming ue to the nee to perform both the complete simulation an rerouting of all vehicles within each iteration step. Herein, we want to iscuss some approaches to reuce the neee time an memory consumption. The results show that this can be achieve without reucing the algorithm s quality. Keywo: microscopic simulation, traffic assignment, SUE, DUA 1 Motivation The ynamic user assignment metho evelope by Gawron [1] was implemente as the efault assignment metho for the open source microscopic traffic simulation package SUMO [2-5] from the very begin on. Within the past few yea, SUMO has been use in several large-scale projects where whole cities ha to be simulate [6] an Gawron s algorithm has proven to be a vali tool, which allows using a eman base on OD-matrices in the microscopically simulate roa network. Because of using travel time information from the simulation an iteratively ajusting vehicles routes to this, jams are solve automatically an the generate route set results in realistic simulation. Nonetheless, the preparation of routes for the simulate scenarios was often painful, ue to the large computation time of up to several ays the algorithm nees. Since the algorithm has to be rerun even after small changes in the network, waiting such a long time for the results is often unacceptable. Furthermore, if ealing with even larger scenarios, which inclue major cities with their surrouning area an over one million vehicle journeys, one further issue arises: as the algorithm retrieves the ege travel times from the simulation in each iteration step, an initially all vehicles are using the shortest route in the empty network, jams an network ealocks are quite common within the fit iteration steps. The simulation is then fille with several hunres of thousans vehicles which causes severe memory problems.

2 Due to the above mentione problems, two approaches for speeing up the computation were implemente an evaluate. In this publication, Gawron s DUA algorithm will be presente fit, followe by a escription of the observe reasons for large computation times an simulation failures. After this, two moifications are introuce, followe by a comparison of the original an the moifie algorithms. The publication is close with a conclusion an ieas for future improvements. 2 Original Algorithm an Propose Moifications The ynamic user assignment algorithm evelope by C. Gawron is a microscopic approach meaning that each vehicle has its own route an knows its own travel time through the network when using it. The basic proceure is as follows: Step 1: Initialize the process by computing the fastest route through the empty network for each vehicle to simulate. Then a this route to the river s list of known routes, set its probability to be chosen to 1, an choose it as the one to use. Step 2: Perform the simulation using the currently chosen routes in orer to obtain the eges travel times over simulation time. Step 3: Compare the mean travel times to the last run (if any) an quit if the algorithm converges, i.e. if the mean travel time reuction falls below a given threshol. Step 4: Compute new routes for vehicles using the current travel times within the network for all rive. Then a new fastest routes to the respective river s route list an upate all routes travel times as estimate by the river an the route choice probabilities for all routes the river knows. After that, choose one route regaring the route choice probabilities an continue with step 2 In the following, we focus on the algorithm that generates new vehicular routes (step 4) an o not put emphasis on the use of a particular simulation moel neee to compute the vehicle s an eges travel times. At fit, the travel times for the routes known by a river are aapte to the travel times obtaine from the simulation: τ s x = last _ chosen _ route τ ' = (1) βτ r + (1 β ) τ otherwise where τ, τ s, τ r are route x s prior travel times as estimate by the river, retrieve from the simulation, an reconstructe from the ege travel times that were etermine by the simulation, respectively τ ' is river s new estimation of the uration of route x β is a factor affecting the spee of aapting remembere travel times to the current The choice probability of each route is then recompute. The probability for each unuse route known by the river is recompute by a function that compares its travel time with the travel time of the route use in the last simulation step. The later route s probability is also ajuste, herein. This is one using the following equations:

3 p' ( r) = p αδ ( r)( p ( r) + p ( s))exp( 1 δ αδ p ( r)exp( ) + p ( s) 2 1 δ 2 ) (2a) p' ( s) = p ( r) + p ( s) p' ( r) (2b) where p (x) is the prior probability to use route x by river p' is the new probability to use route x by river r is the route use in the last simulation run, s another route from the list of known routes δ is the relative costs ifference between routes r an s, compute as: δ τ ( s) τ ( r) = (3) τ ( s) + τ ( r) where τ (x) is the travel time for river to complete route x. As escribe before, the list of routes known by a river is not compute initially. Instea, a new fastest route is compute in each iteration step an ae to the list of known routes, if it was not known before. Within the investigations escribe herein, ege travel times were collecte an aggregate within the simulation for intervals of 900 secons. These time series were rea by the router an use for computing new routes an the travel times for alreay known routes. Herein, each ege s travel time was etermine by looking up in the corresponing time-series generate by the simulation for the interval that matche interval begin <= entry time < interval en. If the travel time for an ege is aske for which no matching interval exists, which can only be the case if asking for times later than the simulation s en, the last interval s value was use. α was set to 0.5, an β to 0.9. These values were foun to be proper within previous projects an generate a vali route set, where vali means in this case that using this route set, all vehicles are simulate, manage to en their route at the en of the simulate ay, an neither large jams nor network ealocks occur. 3 Observe Bottlenecks Two processes are responsible for computing the DUA, the fit is the routing application which computes new vehicle routes, an the secon is the simulation which etermines the ege travel times using the new vehicle routes. The major cause of computation time is easily ientifie if the execution times of both applications are plotte along the iterations as seen in Figure 1.

4 uration [s] iteration [#] routing simulation Figure 1: Duration of the route computation (re) an the simulation (blue) over iterations for the test scenario (escribe in chapter 5) Obviously, most of the computation time is use by the simulation uring the fit runs. Along the DUA iterations, this time (almost monotonously) ecreases uring the fit steps. The reason why the simulation nees much time in the fit steps can be foun by observing the number of vehicles within the simulation over the simulation time an iteration step (see Figure 2a). a) b) Figure 2: Development of the maximum number of vehicles (a) an the vehicles mean travel time (b) over simulation time an iterations (shown iterations: 0, 5, 10, 15, 20, 25) Accoring to Figure 2b, one can see that, in the fit steps of the assignment, many of the vehicles are staying within the simulation for a long time. Because they o not leave the simulation, the absolute number of vehicle movements the simulation has to compute is increase which results in a large uration. Also, because each vehicle has besies other paramete its own route store within the simulation, the large number of vehicles that are within the network causes memory consumption problems.

5 Figures 2a also shows that the maximum number of vehicles that are within the network simultaneously is shrinking over iteration steps. As a conclusion, the major problem is the large number of vehicles kept in the simulation uring the fit DUA steps. 4 Propose Moifications Both propose moifications o not change the route assignment itself, but only try to reuce the number of simulate vehicles, especially within the fit iteration steps. The following methos that achieve this were evaluate: a) Increasing vehicle amounts in the simulation This approach resembles the macroscopic incremental assignment [7]. While route computation is still one for all vehicles in each iteration step with reference to the ege travel times compute by the simulation, the simulation only uses a fraction of the vehicles in the analyze scenario. In each iteration step, this number of vehicles emitte into the simulation is increase. The motivation is - to keep the simulation fast uring the fit steps by reucing the number of vehicles; an - to reuce the number of jams that occur by reucing the number of vehicles. The ege travel times generate by the simulation are use for fining new routes an computing the uration of known routes as in the original algorithm. This approach will be name inc_sim in the evaluation section, meaning that the number of vehicles is incremente in the simulation. b) Increasing simulation en time Instea of increasing the amount of vehicles within the whole simulation, this approach uses the complete number of vehicles, but the time the simulation ens is increase in regular intervals. The reason for trying this metho is that, from our observations, simulate networks are jamme by falsely route vehicles when the eman is high enough. Also, in most cases, we are intereste in having simulations of whole ays. This means that uring the fit simulate hou, with only minor traffic, no large jams occur even if no proper assignment was yet compute. The iea of the metho is to benefit from the travel times collecte uring the fit simulation runs, where only minor traffic is within the network an no jams occur, an incrementally increase the eman by following the aily eman time-series. Another benefit of this approach is that the number of vehicles inserte into the network is boune by the simulation en time. The ege travel times generate by the simulation are use as escribe earlier. In the next sections, it will be name inc_en for increasing the simulation en time. However, both methos loose information. As travel times over eges are not proportional to the number of vehicles that use the ege, both moifications lea the router to operate on ege travel times that o not correspon to the real eman. Nonetheless, because the original algorithm also changes the assignment in a probabilistic way an, instea of operating on global measures, only takes into account a river s knowlege about the network. One can thus be optimistic that the information loss can be compensate incrementally, while the number of vehicle (inc_sim) or the simulation en time (inc_time) is increase. The lack of possibility to evaluate the effects of this information loss makes it necessary to evaluate both methos using a simulation.

6 5 Evaluation The scenario we use for evaluating the ifferent methos was evelope for the INVENT project an contains a roa network an the respective eman for a normal weekay for the city of Mageburg using 1h-matrices for the hou between 5am an 9pm. Both the network an the eman were given in VISUM format, so that several moifications were neee for making it useable in a microscopic simulation. The roa network was extene by highway on an off ramps, lane-to-lane connections, an traffic lights. The eman was converte from the original O/D-matrices that use about 250 istricts into single vehicle trips. The resultant scenario consists of a network mae of about 4800 eges (roas) an 2500 noes (junctions) an a traffic eman of about 600,000 vehicles. The evaluation was one by executing 50 iterations of the original algorithm, which was known to compute a vali route set for the scenario. Then, the following moification settings were run for also 50 iterations, each: - inc_sim10 : inc_sim increasing the number of simulate vehicles in steps of 1/10 th of the eman - inc_sim20 : inc_sim increasing the number of simulate vehicles in steps of 1/20 th of the eman - inc_sim30 : inc_sim increasing the number of simulate vehicles in steps of 1/30 th of the eman - inc_sim40 : inc_sim increasing the number of simulate vehicles in steps of 1/40 th of the eman - inc_sim50 : inc_sim increasing the number of simulate vehicles in steps of 1/50 th of the eman - inc_time1800 : inc_time using steps of 1800s - inc_time3600 : inc_time using steps of 3600s - inc_time5400 : inc_time using steps of 5400s - inc_time7200 : inc_time using steps of 7200s We chose three main inicato for evaluating the quality of the assignment process among the ifferent methos. The fit one is the execution time which we try to reuce; the secon one is the memory consumption, which shall be reuce, too. While we measure the uration irectly, we ecie to use the maximum number of vehicles, which are simultaneously within the simulate area, as a measurement for memory consumption. There are three reasons for using this value: a) the number of currently simulate vehicles is irectly available within the simulation an can be save an evaluate, b) because the network an other infrastructure information stay the same over the escribe simulation runs, the maximum number of simultaneously running vehicles is the only reason for ifferent memory footprints of the application, an c) this value oes not epen on the use architecture or a single vehicle's memory size. The thir inicator is the mean travel time of all vehicles to simulate after the simulation s en. In the case that all vehicles have left the network, this gives a goo assessment of the assignment's quality, because it prefe faster routes an penalizes jams, retrieving better scores for assignments that simulate rive who want to reach their estination fast. Figure 3 shows the maximum number of vehicles that were within the simulation simultaneously. All iteration steps of each of the name methos were taken into account. As expecte, the maximum number of simulate vehicles is reuce within the propose methos, when

7 compare to the original ( plain ) metho. Still, in almost all cases, except inc_sim50, this value is higher than that for the simulation in the equilibrate state, as obtaine from the 50 th iteration of the plain metho, which is shown as a re line in Figure 3. This means that, uring the process, jams occur which may fill the simulation uner circumstances max. vehicle number [#] plain inc_sim10 inc_sim20 inc_sim30 inc_sim40 inc_sim50 inc_time1800 inc_time3600 inc_time5400 inc_time7200 Metho Figure 3: Maximum number of vehicles within the simulation over all iteration steps of each of the evaluate methos; re line: maximum number of vehicles in the final step of the original algorithm The reuce number of the simulate vehicle movements results in a lower computation times, as inicate in Figure 4. Each of the ba shows the time in hou that was neee to perform 50 iterations for each metho. In most cases, except inc_sim50, where the number of really simulate vehicles was reache in the last iteration step, the respective metho neee less time to compute an assignment for all vehicles which resulting mean vehicle travel times were lower than the one of the original algorithm. Each bar shown in Figure 4 pictures the complete time neee by the respective metho to perform 50 iteration steps. The ark component of each of these ba shows the time, the accoring metho neee to compute an assignment which results in a mean travel time that lies below the one obtaine using after performing the original metho for 50 iterations. The light component is the aitional time of the 50 iterations, that the respective metho use to further improve the assignment beyon the quality, reache by the original metho.

8 execution time [h] plain inc_sim10 inc_sim20 inc_sim30 inc_sim40 inc_sim50 inc_time1800 inc_time3600 inc_time5400 inc_time7200 Metho Figure 4: Execution time of 50 iterations of each of the methos (colo: see text) Because most of the moifications were able to reach the original algorithm s quality in less than 50 iterations, the following iterations coul achieve further improvement of the mean travel times, shown in Figure mean travel time [s] plain inc_sim10 inc_sim20 inc_sim30 inc_sim40 inc_sim50 inc_time1800 inc_time3600 inc_time5400 inc_time7200 Metho Figure 5: Mean vehicle travel time after 50 iterations of the methos The figures show that using any of the moifications is of benefit. Now, one shoul ask the question which metho an parameter set is the best. Though some of the presente methos nee a lower execution time to achieve the same quality (see Figure 4), inc_time1800 seems to combine both of the escribe qualities execution spee an reuction of use memory at best. 6 Conclusions an Future Work We have shown that both approaches, increasing the number of simulate vehicles after staring with only a fraction of the complete eman, an to slowly increase the simulation en

9 time, result in a faster computation of the DUA without loosing the original algorithm s quality. However, both methos have the rawback that their parameter, the percentage with which the number of the simulate vehicles is increase, an the amount of time by which the simulation s en is extene, respectively, have to be given explicitly. The next evelopment steps are to establish a system which recognizes starting jams within the simulation an uses this information as a feeback to the DUA system. Such a system shoul then aapt the parameter to the current DUA progress. Also, for the inc_sim approach, we have only consiere changing the fraction of the processe vehicles within the simulation. Furthermore, it shoul be also evaluate whether all routes must be compute by the router moule an whether only computing the use ones can reuce the computational effort without affecting the results quality. 7 References [1] Gawron, C. (1998). Simulation-base traffic assignment computing user equilibria in large street networks. Ph.D. Dissertation, Univeity of Köln, Germany. [2] Krajzewicz, D., Hertkorn, G., Rössel, C., Wagner, P. (2002). SUMO (Simulation of Urban MObility): an open-source Traffic Simulation. Proceeings of the 4th Mile East Symposium on Simulation an Moeling (MESM2002), , SCS European Publishing House. [3] Krajzewicz, D., Hartinger, M., Hertkorn, G., Nicolay, E., Rössel, C., Ringel, J., Wagner, P. (2004). Recent Extensions to the open source Traffic Simulation SUMO. Proceeings of the 10th Worl Conference on Transport Research (on CD), WCTR04-10th Worl Conference on Transport Research, Istanbul (Turkey), 4-8 Jul [4] Krajzewicz, D., Bonert, M. an Wagner, P. (2006). The Open Source Traffic Simulation Package SUMO. In: RoboCup 2006, RoboCup 2006, Bremen, Germany, Jun [5] ITS/DLR. SUMO Website: [6] Behrisch, M., Bonert M., Brockfel, E., Krajzewicz, D. an Wagner, P. (2008). Event traffic forecast for metropolitan areas base on microscopic simulation. International Symposium on Transport Simulation [7] Thomas, R. (1991). Traffic assignment techniques, Vermont, USA, Avebury Technical.