Algorithms for distributing traffic flows

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ISSN 0005-1179, Automation and Remote Control, 2009, Vol. 70, No. 10, pp. 17281736. c Pleiades Publishing, Ltd., 2009.Original Russian Text c V.I. Shvetsov, 2009, published in Avtomatika i Telemekhanika, 2009, No. 10, pp. 148157.AUTOMATIC CONTROL SYSTEMSAlgorithms for Distributing Trac FlowsV. I. ShvetsovInstitute for Systems Analysis, Russian Academy of Sciences, Moscow, RussiaReceived January 16, 2009AbstractWe discuss models and algorithms of distributing trac ows in the network ofa large city or an algomeration. We provide comparative analysis of dierent algorithms fornding an equilibrium distribution in a transportation network. We discuss the problem ofambiguity in distributing correspondences by the paths in equilibrium and the use of entropymodels to avoid this ambiguity.PACS number: 89.40.-aDOI: 10.1134/S00051179091001051. INTRODUCTIONThe transportation infrastructure is one of the most important infrastructures that providefor the life of cities and regions. Over the recent decades, many large cities have exhausted ornearly exhausted opportunities for extensive development of their transport networks. Thus, itis especially important to plan the networks optimally, improve the trac ow, and optimize thepublic transportation routes. Solving these problems is impossible without mathematical modelingof the networks. The primary problem of said mathematical models is to determine and foresee allparameters of a transportation network, such as the trac intensity on all elements of the network,transportation capacities in the public transportation network, average movement speeds, delaysand downtimes, etc.The general outline of modeling a transportation system includes the following interrelatedtasks [1]: computing the total capacities of trac ows between all regions of the city or aglomeration(computing the interregional correspondence matrix); distributing the correspondences to particular paths in the transportation network; computing the load of all network elements by trac ows.The present paper is devoted to the second task of this list, namely the task of distributing thecorrespondences to particular paths in the transportation network. If such a distribution is known,the load of all elements can be computed by simply summing up all correspondences that use theseelements for transportation.Modeling correspondence distributions is based on the equilibrium principle for transportationnetworks [2,3], which is discussed in Section 2. Sections 34 are devoted to a comparative analysisof dierent algorithms for nding the equilibrium. The equilibrium principle uniquely determinesthe values of ows on the networks arcs but is very ambiguous with respect to distributing cor-respondences by paths. Using the entropy maximization principle to remove this ambiguity isdiscussed in Section 5.A transportation network is modeled by a graph consisting of nodes and arcs. Arrival anddeparture regions are special nodes that the whole city territory is subdivided into. In what followswe use the following notation: the index a runs over the arcs of the graph; ua is the trac ow1728ALGORITHMS FOR DISTRIBUTING TRAFFIC FLOWS 1729on the arc a; p, q are indices of arrival and departure regions; Kpq is the set of paths in the graphfrom p to q; the index k runs over paths in Kpq; ukpq is the part of correspondence from p to q thatuses the path k (we also call it the ow on the path k).2. THE EQUILIBRIUM PRINCIPLETrac ow management algorithms work on the assumption that each trac participant aimsto minimize the generalized cost of his movement. The generalized path cost is an aggregateassessment criterion of a path. The main ingredient of the generalized cost is the time spent formovement; however, other ingredients may step in, such as the regular cost of the path, i.e., moneyspent. In what follows we call the generalized path cost simply the path cost. It is the sumof movement costs of the arcs the path consists of and the costs of all transfers from arc to arc(intersection maneuvers). To simplify the formulas, in what follows we omit maneuver costs andonly specify arc costs, which we denote by ca, where a is the arc index.A ow distribution is said to be in equilibrium if no trac participant can change his pathand thus reduce his path cost. According to the modern assumptions, a ow distribution in atransportation network in a stable situation is in equilibrium in this sense. This condition is knownas the Wardrop condition (or Wardrop conditions, as it is often formulated in two statements [2]).The key assumption about the cost is the following: the movement cost of an arc is a nonde-creasing function of the total ow in this arc: ca = ca(ua). In other words, the more cars movevia a given arc, the slower and less comfortable the trac is, and, correspondingly, the higherthe movement cost. Under this assumption it can be shown that every transportation system hasan equilibrium. This equilibrium is unique in the sense that the equilibrium condition uniquelydetermines the ows on every arc of the network. In other words, there exists a unique load of allnetwork elements that satises the equilibrium condition. However, the equilibrium distribution isnot unique in terms of distributing the correspondences by the paths in the network. That is, dier-ent correspondence distributions may yield equal load on all network elements (in fact, for a givenload distribution there exists an innite number of correpondence distributions that generate it).The equilibrium distribution is given by the solution of the following optimization problem withconstraints [3]:aua0ca(v) dv min . (1)The arc ows ua in this expression are not the independent variables of the problem. They aresums of ows ukpq over all paths k that use this arc a. The constraints in the problem state that theows on paths ukpq should add up to the correspondences Fpq between all pairs of regions p and q:ua =p,qkKpq,akukpq,Fpq =kKpqukpq, (2)ukpq 0.3. THE FRANKWOLFE ALGORITHMAlthough the equilibrium principle in a transportation network was rst formulated back in1950s, rst practical implementations of this principle appeared much later, in mid-1970s [4]. Thisis because the numerical implementation of a model requires considerable memory size. Storing allAUTOMATION AND REMOTE CONTROL Vol. 70 No. 10 20091730 SHVETSOVDistribution un = {unkpq} on step nPivotal distribution unL at the prices cna = ca(una )Selection : un+1 = (1 )un + unLFig. 1. The FrankWolfe algorithm.information about distributing correspondences by paths in a large city requires up to several giga-bytes of memory. Computers with such specications have become widely available only recently.However, equilibrium distributions became possible to model much earlier due to the developmentof the FrankWolfe algorithm [5]. This algorithm allows to compute the nal equilibrium load ofthe transportation network without storing the correpondence distribution itself in memory.The FrankWolfe algorithm is depicted on Fig. 1. On the initialization step, all correspondencesare distributed to optimal (by cost) paths computed with an empty network. On each subsequentstep, n+ 1, each correpondence is already distributed to a number of paths (at most n) computedon the previous n steps. As a result, some load of the network arcs una arises. Arc costs cna arecomputed with this load, and a new system of optimal paths is constructed. This path system ispivotal for redistributing correspondences. For some correspondences the computed paths may benew, while for others the computed optimal path is already among the paths obtained on previoussteps. Then, the fraction of correspondences , which will be redistributed from all previous pathsto new optimal paths, is computed (the precise method for computing is irrelevant for furtheranalysis).The algorithm always redistributes the same fraction of correspondences from all previouspaths. This means that the total load of all arcs reduces in precisely (1 ) times. Therefore, tocompute the new load of all arcs one does not have to store the path distribution, it suces tostore arc loads from the previous step. Since arc costs depend only on arc loads, ca = ca(ua), tocompute itself one also does not need to know the path distribution.The algorithm is illustrated on Fig. 2 with a simple example, where the equilibrium distributionof two correspondences between two pairs of regions is computed. Suppose that on a certain stepof the algorithm we have to redistribute from one path to another 15% of the rst correspondenceand only 5% of the second. The algorithm will select a certain common fraction of redistributedcorrespondences, say 10%. As a result, for the rst correspondence the redistribution will turn outto be insucient, while for the second correspondence it will be excessive. Due to the new arcloads, on the next algorithm step the rst correspondence will have the same optimal path as onthe previous step, while for the second correspondence the optimal path would be the one thatlost the correspondence on the previous step. Thus, on this step the remaining 5% of the rstcorrespondence will be redistributed, and 5% of the second correspondence will be put back intoplace. Of course, all numbers are approximate. In fact, due to nonlinearity of the ca(ua) function,the value of on the rst step will dier from 10%, and even in this simple case the iterations willnot converge in exactly two steps.The above-mentioned advantage of the FrankWolfe algorithm has determined its longevity. TheFrankWolfe algorithm is still the most widely used algorithm in the practice of load modeling forAUTOMATION AND REMOTE CONTROL Vol. 70 No. 10 2009ALGORITHMS FOR DISTRIBUTING TRAFFIC FLOWS 1731 = 10% = 5%5% 5%15% 5%Fig. 2. How the FrankWolfe algorithm works.transportation networks. However, the algorithm has an important disadvantage as well. Althoughin theory the algorithm always converges, in practice the convergence rate substantially decreasesduring the iterations. On early iterations, the algorithm approaches equilibrium very rapidly, butafter the rst few dozen iterations the approximation may virtually halt. The eect is especiallystrong in large networks. The convergence deteriorates because of the so-called residual ows eect.This eect manifests itself in a strong non-uniformity of the ows convergence to equilibrium valueson dierent arcs. That is, while the network may converge quite rapidly as a whole, there mayremain a small number of arcs on which the ow diers substantially from the equilibrium, andthese outliers are not corrected by subsequent iterations.The reason for the residual ows eect is clear from the algorithms operation as shown onFig. 2. As a result of the algorithm, each correspondence should be distributed by a numberof alternative paths in such proportions that all costs for all paths in the set are equal. If theminimal and maximal path costs in a set do not dier much, should be chosen small to approachthe equilibrium. A larger value of would only move the correspondences distribution o theequilibrium. Suppose that due to random eects on early stages of the algorithm a small numberof correspondences were left behind from the bulk of the correspondences on their way to theequilibrium. These correspondences require a substantially larger to be redistributed. Since ischosen as the average (in some sense) for all correspondences, it will be chosen small. Thus, thelarger part of correspondences will move even closer to the equilibrium, while this small part willremain behind and will not catch up. On subsequent iterations will become smaller and smalleras required by the majority of correspondences, and several stuck correspondences will remainunredistributed.The author has experienced the residual ow eect while modeling the transportation net-work of the Moscow aglomeration after increasing the number of arcs of the network graph to3050 thousand with about 1500 departurearrival regions. Similar problems were reported inforeign publications [6, 7].4. OTHER ALGORITHMS FOR COMPUTING THE EQUILIBRIUM DISTRIBUTIONRecently, new algorithms for computing the equilibrium distribution, free from the Frank-Wolfealgorithms disadvantages, have been developed [810]. The basic idea is to work with correspon-dences individually; it allows for timely redistribution of individual residual ows. Of course,redistributing a correspondence among alternative paths changes the arc costs and indirectly inu-ences the distributions of other correspondences that use the same arcs. Thus, the algorithm hasto traverse the whole correspondence array several times to ensure that the system has got into theequilibrium state.AUTOMATION AND REMOTE CONTROL Vol. 70 No. 10 20091732 SHVETSOVCreation of the initial sheafs of pathsSearch of restricted equilibriumBig iterationsCycle on each pairs of regions p, qAlignment of the prices of sheafs of paths Kpqkmax, kmin Kpq , ukmaxpq ukminpqEnd of Big iterations: all the sheafs are balancedAdding of the new optimal paths in sheafsEnd of the algorithm: there are no new paths addedFig. 3. The path balancing algorithm.Figure 3 depicts the layout of a simple algorithm of this type, used to model the Moscowalgomeration. The algorithm can be called path balancing. The basic idea of the algorithm is asfollows. In equilibrium, each correspondence is distributed by a certain number of alternative paths;this number may vary for dierent correspondences and is not known in advance. If we x certainsubsets of paths between all region pairs, we can formulate the restricted problem of computingequilibrium: nd the equilibrium of the system with an additional restriction that only paths fromthese xed subsets are used, and none other. Obviously, if these subsets include all paths used inthe real equilibrium, the restricted equilibrium coincides with the general equilibrium.We call a xed set of paths between a pair of regions a sheaf of paths. On the rst step of thealgorithm we create sheaves consisting of only one paththe optimal path between the correspond-ing pair of regions computed on the empty network. On each subsequent step of the algorithm,sheaves are extended with new paths as follows. First, we solve the restricted equilibrium prob-lem for current sheaves (on the rst step this problem is trivial). Then, each sheaf is extended withthe optimal path computed with the current arc loads in the network. Of course, the new pathmay in fact coincide with one of the old ones. The algorithm halts when no actually new path isfound.Restricted equilibrium can be computed as follows. Denote by Kpq the sheaf of paths fromregion p to region q. In equilibrium, costs of all used paths in the sheaf are equal, and costs of allunused paths are higher. We call a sheaf balanced if this condition is satised. Fix a distribution ofall other correspondences and vary only the distribution of the correspondence fpq between paths ofthe sheaf Kpq. Denote the shortest and longest (by cost) paths in the sheaf by kmin and kmax. If wetransfer some part of the correspondence from kmax to kmin, their costs will get closer due to loadbalancing. There are two possible scenarios here: either there exists a ratio of the correspondenceswhen the costs are equal, or all correspondence from kmax is transferred to kmin, and the longestroute is not used anymore. By repeatedly computing kmin and kmax we can balance the sheaf Kpqwith any given precision.Iterating over all region pairs, we can balance all sheaves. However, since paths from dierentsheaves may use the same arcs, balancing another sheaf will mess with the balancing of theprevious sheaves. Thus, we need to iterate over region pairs several times (big iterations onFig. 3).Note that this algorithm arrives at the equilibrium after a nite number of steps, although itwill not be exact, as the sheaf balancing on each step is only carried out up to some nite precision.AUTOMATION AND REMOTE CONTROL Vol. 70 No. 10 2009ALGORITHMS FOR DISTRIBUTING TRAFFIC FLOWS 1733According to the current terminology, equilibrium nding algorithms for a transportation net-work are divided into link-based, which are based on arc ows, and route-based, which work withows over whole paths (routes) in the network. The FrankWolfe algorithm is a typical exam-ple of a link-based algorithm, while the above-mentioned path balancing algorithm represents theroute-based kind. The primary advantage of route-based algorithms is the substantially increaseduniformity of the arc ow convergence which, in particular, solves the residual ow problem. How-ever, these algorithms require a lot more memory to run. And its not even the physical memorycapacity; modeling large networks may hit the general virtual address space constraint of a 32-bitsystem, so that one would have to migrate to a 64-bit system.In this situation a combined approach suggested by Bar-Gera [10] may help. This approachis a cross between the link-based and route-based approaches. The basic idea is to distinguishthe representatives of correspondences moving from a common departure region on every arc,without distinguishing them by arrival regions. Thus, instead of arc ows ua or path ows ukpq theindependent variables are ows with a common source (origin) uap, where a runs over the arcs, andp over departure regions. Obviously, this array is considerably (by a factor of approximately thenumber of regions) smaller than the path ows array and may t the RAM. At the same time, itallows to work with correspondences from dierent regions independently. Numerical experimentshave shown that the algorithm provides the necessary convergence uniformity and solves the residualows problem.5. NON-UNIQUENESS OF THE EQUILIBRIUM DISTRIBUTION AND ENTROPYAs we have already noted in Section 2, the Wardrop condition uniquely determines total owvalues on all arcs ua in the network. This follows from the form of the objective function (1).Indeed, if arc costs ca(ua) are increasing functions of the ow, then the integrals in (1) are convexfunctions of ua, and the equilibrium distribution exists and is unique since it is the minimum ofa convex function under linear constraints. However, the equilibrium distribution is not unique interms of distributing correspondences over paths in the network. The cause of this non-uniquenessis shown on a simple example on Fig. 4.Suppose that two regions A and B are connected by two pairs of identical paths with a crossingpoint in the middle, as shown on Fig. 4. We distribute a single correspondence from region A toregion B. Obviously, in equilibrium the loads of all arcs should be equal, that is, in both fork pointsthe correspondence should be divided in half. There are exactly 4 dierent paths from A to B, andthere is an innite number of ways to distribute the correspondence along these paths for whichthe correspondence on both fork points is divided in half. Let x be the share of drivers who havechosen the left path on both forking points. As shown on the gure, for every 0 x 0.5 weobtain a distribution over the four paths that generates the equilibrium load. The only dierenceFig. 4. Non-uniqueness of the equilibrium distribution.AUTOMATION AND REMOTE CONTROL Vol. 70 No. 10 20091734 SHVETSOVis in how many drivers who have chosen the left path on the rst fork make the same decision onthe second.In a real-world network with many regions the situation is more complicated, but the non-uniqueness of path distributions remains and is of the same nature. There are a lot of such forksin the network, and from the equilibrium principles point of view only the overall arc ows areimportant. It is absolutely irrelevant which particular drivers comprise these arc ows.Thus, the Wardrop condition does not uniquely determine the correspondences path distribu-tion. In fact, any possible set of values of arc ows ua corresponds to a multidimensional linearspace of path distributions ukpq. Therefore, to correctly formulate the equilibrium distributionproblem one has to add some assumptions to the Wardrop condition.The probabilistic approach looks natural. Consider a pair of regions p, q linked by a sheaf ofpathsKpq. We can assume that drivers (representatives of the correspondence fpq) choose one of theways k Kpq randomly and independently of each other. If we set the probability values of choosingone way or the other, we can easily calculate the probability of realizing a given correspondencepath distribution fpq. In particular, in equilibrium the costs of all used paths are equal, so we canassume that each driver chooses one of these paths with equal probabilities.In terms of the macrosystems theory approach [11], this situation can be described as follows.The drivers are elements of a macrosystem, and the choice of a path determines the state of anelement. The microstate of the system is determined by the element states, that is, by specifying apath for each individual driver. The macrostate is determined by the occupation numbers, i.e., theoverall number of elements in a given state. In our case, the occupation numbers are path ows ukpq.Thus, dierent correspondence distributions over paths correspond to dierent macrostates of thesystem. Each macrostate can be realized by a dierent number of microstates, and that determinesits statistical weight. In our case, when the number of elements and the number of states are nite,statistical weights of macrostates are simply proportional to the probabilities of their realization.If the network is in equilibrium, and we know a set of equilibrium arc ows ua, this imposeslinear constraints on the possible values of ukpq (see constraints in (2)). Assuming that in realitythe state with maximal statistical weight should take place, we arrive to the entropy (logarithm ofthe statistical weight) maximizing model. The entropy model of the ows equilibrium distributionmay be formulated as follows [12]:p,qkKpq(ukpq ln(ukpqfpq) ukpq) max (3)under the following constraints:ua =p,qkKpq,akukpq,Fpq =kKpqukpq, (4)ukpq 0.In this regard it appears interesting to compare path distributions arising from the FrankWolfealgorithm and route-based algorithms. These algorithms are outlined on Fig. 5. The space of possi-ble values for the arc ows ua is represented as a one-dimensional line. There exists a unique pointua in this space that corresponds to equilibrium ows. Thus, the space of possible path distributionsukpq turns out to be bered into subspaces, each of which corresponds to a certain set of ua. Eachof these subspaces has a most probable (entropy maximizing) distribution ukpq. The algorithmsthat search for equilibrium usually begin with distributing the correspondences by optimal pathsAUTOMATION AND REMOTE CONTROL Vol. 70 No. 10 2009ALGORITHMS FOR DISTRIBUTING TRAFFIC FLOWS 1735Fig. 5. Comparing dierent algorithms for nding equilibrium.computed with an empty network. Then, correspondences are iteratively redistributed among thepaths. The logic of this redistribution aims to get the point ua to the equilibrium point ua, withno regard for how close ukpq is to ukpq. Therefore, the nal path distribution appears randomlydierent from the most probable path distribution for all these algorithms, and it is hard to makeany theoretically founded conclusions.However, numerical experiments evidence that the Frank-Wolfe algorithm often yields a pathdistribution closer to ukpq than a route-based algorithm. In particular, in a route-based algorithmcorrespondences are much less likely to split into dierent paths, that is, the algorithm aims tominimize the number of dierent paths. The same results were obtained in [13] when comparingthe FrankWolfe algorithm with an origin-based algorithm. The conclusion we draw from thismay sound a little paradoxical: route-based algorithms as compared with link-based provide betterconvergence on the arcs, but tend to result in worse distributions on paths (routes).In the authors opinion, these observations correspond to an objective tendency and have thefollowing (informal) explanation. The path distribution corresponding to maximal entropy is themost random of all possible distributions. While a link-based algorithm by denition does notuse the path ow information, a route-based algorithm works with individual paths. Thus, thedistribution becomes more specic and less random.6. CONCLUSIONApplying the equilibrium principle to transportation networks is a basis for transportation net-work load computer modeling. Two levels can be distinguished in the trac ows modeling problem:computing ows on the arcs of the network, i.e., computing the network load, and computing thepath distribution of all correspondences in the network that generates this load. The equilibriumprinciple uniquely determines arc ows but leaves a lot of slack in specifying the path distribution.To remove this ambiguity the equilibrium principle may be augmented with the entropy maximiza-tion principle that allows computing the most probable path distribution for given values of arcows.The most widely used equilibrium nding algorithm is, since the mid-1970s, the FrankWolfealgorithm. However, applying this algorithm to very large networks encounters convergence non-uniformity problems and the residual ows problem. Recently, algorithms that distribute corre-spondences to paths on a more individual basis have been developed. These algorithms improve theconvergence of arc ows, but the resulting distributions are far from maximal entropy distributions.A correct prospective solution of this problem appears to be a combination of computing theequilibrium arc ows by one of the known approaches with computing the path distribution ofcorrespondences with the entropy maximization approach, although this solution meets signicantcomputational problems due to large dimensions of the problems.AUTOMATION AND REMOTE CONTROL Vol. 70 No. 10 20091736 SHVETSOVREFERENCES1. Shvetsov, V.I., Mathematical Modeling of Trac Flows, Autom. Remote Control , 2003, no. 11, pp. 16511689.2. Wardrop, J.G., Some Theoretical Aspects of Road Trac Research, Proc. Inst. Civil Engin. II , 1952,pp. 325378.3. Shey, Y., Urban Transportation Networks , Englewood Clis: Prentice Hall, 1985.4. LeBlanc, L.J., Morlok, E.K., and Pierskalla,W., An Ecient Approach to Solving the Road NetworkEquilibrium Trac Assignment Problem, Transpn. Res. B , 1975, vol. 9, pp. 309318.5. Frank, M. and Wolfe, P., An Algorithm for Quadratic Programming, Naval Res. Logistics Quart ., 1956,vol. 3, pp. 95110.6. Boyce, D., Ralevic-Dekic, B., and Bar-Gera, H., Convergence of Trac Assignments: How Much isEnough?, J. Transport. Engin., 2004, vol. 130, no. 1, pp. 4955.7. Janson, B. and Zozaya-Gorostiza, C., The Problem of Cyclic Flows in Trac Assignment, Transpn.Res. B , 1987, vol. 21, pp. 299310.8. Patriksson, M., The Trac Assignment ProblemModels and Methods , Utrecht: VSP, 1994.9. Kupsizewska, D. and Vliet, D. Van, 101 Uses for Path-based Assignment, in Transport Planning Methods:Proc., PTRC Planning and Transport Summer Ann. Meeting , Univ. Sussex, 1999.10. Bar-Gera, H., Origin-based Algorithm for the Trac Assignment Problem, Transpn. Sci., 2002, vol. 36,no. 4, pp. 398417.11. Popkov, Yu.S., Macrosystems Theory and Its Applications , Berlin: Springer-Verlag, 1995.12. Rossi, T.F., McNeil, S., and Hendrickson, C., Entropy Model for Consistent Impact Fee Assessment,J. Urban Planning Development/ASCE , 1989, vol. 115, pp. 5163.13. Bar-Gera, H., Origin-based Algorithms for Transportation Network Modeling, Chicago: Univ. of Illinois,1999.This paper was recommended for publication by Yu.S. 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