Global and Local Information in Traffic Congestion
aa r X i v : . [ n li n . AO ] J un Global and Lo al Information in Tra(cid:30) CongestionGiovanni Petri , , Henrik Jeldtoft Jensen , , John W. Polak Centre for Transport Studies, Department of Civil and Environmental Engineering,Imperial College London, South Kensington Campus, London Sw7 2AZ, UK Institute for Mathemati al S ien es, Imperial College London, 53 Prin es Gate, London SW7 2PG, UK Department of Mathemati s, Imperial College London,South Kensington Campus, London Sw7 2AZ, UKA generi network (cid:29)ow model of transport (of relevan e to information transport as well as physi altransport) is studied under two di(cid:27)erent ontrol proto ols. The (cid:28)rst involves information on erningthe global state of the network, the se ond only information about nodes' nearest neighbors. Theglobal proto ol allows for a larger external drive before jamming sets in, at the pri e of signi(cid:28) antlarger (cid:29)ow (cid:29)u tuations. By triggering jams in neighboring nodes, the jamming perturbation growsas a pulsating ore. This feature explains the di(cid:27)erent results for the two information proto ols.The interplay between information networks and trans-port networks determine the (cid:29)ow on the latter. In par-ti ular ongestion formation, persisten e and eliminationdepend on how information is spread a ross the trans-port system. In line with a number of re ent studies[1, 2, 3, 4, 5℄ we model here these issues from a networktheoreti al perspe tive [6, 7℄. Our aim is to study simplemodels to obtain qualitative as well as semi-quantitativeinsight into fundamental aspe ts of the dynami s of net-work ongestion. Transport networks have indeed beenobje t of many studies in re ent years. In analogy to equi-librium (cid:29)uid dynami s, steady-state des riptions [8℄ wereproposed and studied in depth. Although they produ einteresting results about lo al phenomena [9, 10, 11, 12℄they are not able to apture the inherent "intelligent"behaviour involved in route hoi e. Mi ro-level agent-based models [13, 14, 15℄ solve the issue of individual hoi e, but require an enormous amount of detailed in-formations about the network (e.g. position and signalingof jun tions, tra(cid:30) lights syn hronizing et .).We propose here a minimal network (cid:29)ow model withunjammed/jammed nodes evolving under two di(cid:27)erenttypes of information distribution. In the (cid:28)rst ase welet ea h node have omplete information (global model).In the se ond ase ea h node only re eive information on erning its neighborhood (lo al model). Nodes arethought of as stations or jun tions in a tra(cid:30) network,while links as roads or rail-tra ks between two nodes. Anode i is hara terised by its threshold T i = Ω k i , where k i is the node's degree and Ω a positive onstant, rep-resenting the maximum load it an support before jam-ming, the load L i representing the load (people, trains, ars..) present on the node, and the state S i , whi h isset to 1 or 0, respe tively when the station is unjammedor jammed.When a node i jams, no in oming (cid:29)ow from neighbor-ing nodes is permitted until L i be omes smaller then T i again. This mimi ks the behavior of real systems, wherea station an blo k due to too mu h in oming (cid:29)ow, ef-fe tively utting o(cid:27) its neighbors. Normal operation isrestablished after su(cid:30) ient o(cid:27)-loading. The equation forthe state of node i is S i ( t + 1) = Θ ( T i − L i ( t )) , where Θ is the Heaviside step fun tion, and the outgoing (cid:29)ow F outij from node i to node j is F glob/locij ( t ) = min ( L i ( t ) , T i ) k i S j ( t ) (1 − J ( t )) (1)where J ∈ [0 , represents the fra tion of jammedstations that a station 'sees' or is informed of andthe min fun tion is used to put a superior limit onthe outgoing (cid:29)ow from a station. Note that, forgrowing J , the outgoing (cid:29)ow from stations de reasesin general. Also, the term S j on the r.h.s. of (1)a ounts for the impossibility of sending (cid:29)ow to ajammed station ( F ij = 0 if S j = 0 ). The idea behindthis hoi e is that a station, if informed that manystations are jammed, will try to gradually redu e itsoutgoing (cid:29)ux to avoid ongesting the system any further.a. Initial Conditions We drive the system with twodi(cid:27)erent me hanisms. In the (cid:28)rst ase, we introdu e inthe network a total load, L tot = β P Ni =0 T i = βT tot , ontrolled by the parameter β , that represents the (cid:28)lledfra tion of the total network apa ity. The total load isdistributed in the network randomly, the only onditionbeing avoiding to jam nodes from the beginning. Oper-atively, one an a omplish this by assigning to node i a load equal to a random fra tion of the node's thresh-old and then iterating over the nodes until the load in-trodu ed in the network rea hes the desired value. Theinitial onditions are thus: S i (0) = 1 ∀ iL i (0) = X i , < X i < T i ∀ i (2)where X i a random variable ∈ [0 , T i ] , so that P i L i = βT tot . The system is then driven through random redis-tribution of load among nodes.In the se ond ase, we pla e the entire load on one singlenode, denoted as the seed , and study how the systemrelax without any further drive. So the initial onditionsare: S i (0) = 1 , L i (0) = 0 ∀ i = seedS seed = 0 , L seed = L tot (3)b. Dynami s The parameter J in (1) represents thefra tion of jammed nodes that a station sees in the sys-tem. In the lo al dynami s, ea h station has informa-tion only about its nearest neighbors, thus J is node-dependent, J = J loci ( t ) : J loci ( t ) = 1 k i X { j ∈ I i } (1 − S j ( t )) (4)where the sum is restri ted to the neighborhood of node i ,while in the global dynami s ea h station has informationabout the whole network, therefore produ ing an uniquevalue for J for all the nodes at ea h time: J glob ( t ) = 1 N X m (1 − S m ( t )) , (5)We note that the ompletely jammed and unjammedstates, N J = N and N J = 0 , are absorbing states. In-deed, if the system rea hes the state N J = N we have J loci = J glob = 1 ∀ i , implying F ij = 0 ∀ i, j . Similarly,if N J = 0 , onsider the load L i ( t ) on node i at time t . Sin e all the neighbors of i are unjammed, the to-tal outgoing (cid:29)ow from i is P j ∈ I i F ij = L i ( t ) while themaximum in oming (cid:29)ow is P j ∈ I i F ji = k i α = T i . Thus, L i ( t + 1) = L ( t ) − X j ∈ I i F ij + X j ∈ I i F ji ≤ k i Ω = T i (6)and node i does not jam, sin e the ondition for jammingis L i > T i .We have seen that starting from N J = 0 the system an-not in itself trigger jamming perturbations. However, itis a very unstable state be ause, in presen e of a jammedneighbor, a node keeps some fra tion of its previous loadon itself. This will move the node loser to the jam-ming threshold and thus making itself more sus eptibleto jamming. So if we produ e in some way even a singlejammed node, we expe t the jamming perturbation toexpand through a sort of hain rea tion up to a station-ary state that depends on the amount of load introdu ed.This is a (cid:28)rst indi ation that inhomogeneity in the sys-tem is the driving for e behind jam propagation. . Simulations We performed extensive simulationson random graph (RG) and s alefree networks(SF) withthe number of nodes N varying between and × under both driving me hanisms. Realisations initialisedas in (2) were driven by redistributing a fra tion of arandomly hosen node's load to another randomly ho-sen node ( L i → L i + cL j , L j → L j (1 − c ) , c ∈ (0 , ).When a jam appeared, the driving was suspended forthe duration of the a tive phase ( N J = 0 ). Realizationsinitialized as in Eq. (3) relaxed to their stationary statewithout a tive driving.+ Figure 1 shows the hara teris-ti behaviors observed for the asymptoti jammed pop-ulation N J ( t → ∞ ) . For β < . the system does notshow a stationary population of jammed nodes[16℄. Forhigher values, we observe N J = 0 for di(cid:27)erent valuesof β depending on the topology and dynami s, but, in FIG. 1: N J vs β plot for global and lo al models on SF (fullline) and RG (dashed line) networks ( N = 1000 ) with seededand uniform (inset) initial onditions. Error bars are stan-dard deviations of (cid:29)u tuations of N J around its mean value,averaged over the realizations at a (cid:28)xed β . Data shown areobtained over 100 realizations of the RG and SF networks.striking ontrast to the naive expe tation N J → N , as β → N J remains in he vi inity of N/ for a broadrange of β values. Indeed the applied load is su(cid:30) ientto jam the entire network. However the load is trappedin the jammed nodes and being redire ted from there.This prevents a uniform redistribution of the load a rossthe network keeping N J far below its maximal possiblevalue. We noti e that in the lo al model the in rease in N J sets in for lower values of β than in the global model.Also, the SF networks appear to be more sus eptible ofdeveloping a stationary jammed population than the RGnetwork. There reason for this is the higher abundan eof low degree nodes in SF networks than in RG networks.The low degree nodes are more easily jammed and pro-vide the ore needed for seeding a larger jamming pertur-bation. Indeed, N J ( t ) under the seeded driving ((cid:28)g.2) in- reases through a sequen e of jumps and quasi-plateaus,these are the signature of an os illating ore me hanism((cid:28)g.3): the initial jammed seed makes the neighboringnodes more vulnerable to jamming by sending (cid:29)ow to-ward them, while not a epting (cid:29)ow from them. Thisbrings the seed's neighbors loser to their threshold. Ifthere is enough load on the seed node, it will eventu-ally jam its own neighborhood ((cid:28)g. 3a); the neighborswill relax onto their unjammed neighbors, but this willmake the se ond neighbors vulnerable or even jam ((cid:28)g.3b); this me hanism an ontinue as long as the ore has(cid:29)ow to send. The growth of the ore an end in twos enarios, a) the seed node's neighbors manage to dis-tribute the load without jamming their own unjammedneighbors and so the load is distributed freely outward((cid:28)g. 3d), or b) the se ond neighbors jam and while theyrelax, they jam the seed's (cid:28)rst neighbors again thus re-ating a bigger, more stable jammed ore ((cid:28)g.3 ). Thequasi-plateaus in N J ( t ) orrespond to periods of growthof the vulnerable population, and are followed by sharpjumps, where the jammed ore grows by rapidly invad-ing the vulnerable nodes. Another interesting feature FIG. 2: Time series plot for global (green) and lo al (bla k)models in a RG network with β = 0 . , averaged over realiza-tions.is that the lo al model exhibits mu h narrower (cid:29)u tu-ations ( σ l ≃ for N = 10 ) than the global model( σ g ≃ ). These (cid:29)u tuations are another signature ofthe ore expansion me hanism: when the systems rea hesits stationary N J value, it keeps (cid:29)u tuating around thevalue due to the onstant overshooting between jammedand unjammed nodes. The more violent (cid:29)u tuations ob-served under the global dynami s are indeed onsistentwith the ore me hanism: the global dynami s, throughits global (cid:29)ow suppression, produ es smaller "pa kets"of (cid:29)ow, that allow nodes to stay nearer to their thresh-olds than under the lo al dynami s, and therefore pro-du es a larger population of nodes sus eptible of jam-ming, thereby generating bigger (cid:29)u tuations. For thissame reason, the global dynami s develops a stationaryjammed population for β values larger than for the lo aldynami s but exhibits also a steeper in rease with β .d. Fokker-Plan k Rate Model Our deterministi (cid:29)ow model is omputationally heavy to simulate and an-alyti ally hard to des ribe. Although the model is deter-ministi , the (cid:29)u tuations due to spatial inhomogeneitysuggest it may be possible to understand observed be-havior by use of a set of simple transition pro esses. Weseparate the population of nodes in to three types: un-jammed (U), vulnerable (V) and jammed (J), with rateFIG. 3: Jammed omponent expansion me hanism. FIG. 4: An example of (cid:29)u tuations PDF for RG for β ≃ . ( N = 1000 ).equations, U ( n, t ) + J ( m, t ) α → V ( n, t ) + J ( m, t ) (7) V ( n, t ) + J ( m, t ) ν → J ( n, t ) + J ( m, t ) (8) J ( n, t ) + U ( m, t ) γ → U ( n, t ) + U ( m, t ) (9)Instead of treating individual nodes, we onsider degree lasses and introdu e U k , V k , J k as the probabilities of hoosing randomly a member of a k -degree node belong-ing to one of the three types. The rate terms are veryintuitive and an be written down dire tly from the rateequations, onsidering the ontributions from the di(cid:27)er-ent degree lasses. For example, the ontribution of (9)to J k is of the form Γ Jν ( k, t ) = ν P k max k ′ =1 P ( V k ( t ) , J k ′ ( t )) .Later on, we will set ν = 1 as de(cid:28)nition of vulnerablepopulation. For the moment we negle t degree-degreeFIG. 5: SF simulated data (dots) and (cid:28)t (line) obtained usingeq. 14 for lo al (green and blue) and global (violet and red)models. orrelations and write the probabilities P ( a, b ) as theprobability of randomly hoosing a node belonging to a and a se ond node belonging to b times the proba-bility of having a link between the two. For example, P ( V k , J k ′ ) = V k N kk ′ M J k ′ ( N − , in other words the pro essrate times the number of relevant ouples V J . If wenow substitute the Γ terms inside the FP equations forthe populations we obtain, for ea h degree lass k , theset of equations: J k ( t + 1) = J k ( t ) + kN ( N − M × (10) k max X k ′ =1 k ′ [ νV k ( t ) J k ′ ( t ) − γJ k ( t ) U k ′ ( t )] U k ( t + 1) = U k ( t ) + kN ( N − M × (11) k max X k ′ =1 k ′ [ γJ k ( t ) U k ′ ( t ) − αU k ( t ) J k ′ ( t )] V k ( t + 1) = V k ( t ) + kN ( N − M × (12) k max X k ′ =1 k ′ [ αU k ( t ) J k ′ ( t ) − νV k ( t ) J k ′ ( t )] with J k + U k + V k = p k and P k p k = 1 . Performing thesum over k ′ we obtain the average degree for the respe -tive populations. The (10-11) annot be solved easily.We restri t to the ase h k J i = h k U i = h k V i , a strong as-sumption that is however on(cid:28)rmed by the simulations.So for the stationary state we (cid:28)nd: J k = νγ V k U k = να V k V k = p k να + νγ (13)Assuming a simple proportionality between β and the"jamming rate" α ∝ ( β − β ) , we an ompare theFokker-Plan k predi tion J ( β ) ∝ ( β − β ) γ + ( γ + 1)( β − β ) (14)with the simulated data. Figure 5 shows this ompari-son and we an easily see that the Fokker-Plan k solu-tion reprodu es well the behavior of the simulated data up to β ≃ . In parti ular, the (cid:28)t orre tly identi(cid:28)esthe value of β and reprodu es the steep transition fromno stationary non-zero N J to a signi(cid:28) ant non-zero N J .As expe ted, eq. 14 does not predi t the se ond jumpin N J → N : the approximation of un orrelated degreesand onstant rates break down. Indeed, eqs. (8-9) arenot valid anymore as the system is better des ribed byonly two ompeting population V and J. Moreover, ourmean-(cid:28)eld-like rate model does not a ount for random(cid:29)u tuations that ould bring the system in the absorbing N J = N state.e. In on lusion we have identi(cid:28)ed how a jammed ore is able to push neighboring nodes to their thresh-olds and eventually invade them. We also demonstratedthat the network topology and the di(cid:27)erent informationregimes (lo al and global) play a signi(cid:28) ant role only atthe onset of jams (by shifting threshold load level β ),but do not signi(cid:28) antly in(cid:29)uen e the rest of the evolu-tion, whi h is qualitatively well des ribed by a model of3 ompeting populations, mimi king the unjammed (U),vulnerable (V) and jammed (J) populations of the (cid:29)owmodel.Of relevan e to tra(cid:30) ontrol we have shown that un-der global dynami s the jam pi ks up at a larger drivingload than when only lo al information is used. 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Daganzo, Transportation Resear h Part B: Method-ologi al, 41, (2007) 1.[16℄ A transient jammed population is indeed observed but itde ays into N J = 0= 0