Dynamic Equilibria in Fluid Queueing Networks

Transcription

1 OPERATIONS RESEARCH Vol. 63, No. 1, January Fbruary 215, pp ISSN print) ó ISSN onlin) INFORMS Dynamic Equilibria in Fluid Quuing Ntworks Downloadd from informs.org by [ ] on 1 April 215, at 13:33. For prsonal us only, all rights rsrvd. Robrto Comintti, José Corra, and Omar Larré Dpartamnto d Ingniría Industrial, Univrsidad d Chil, Santiago, Chil {rccc@dii.uchil.cl, jcorra@dii.uchil.cl, olarr@dii.uchil.cl} Flows ovr tim provid a natural and convnint dscription for th dynamics of a continuous stram of particls travling from a sourc to a sink in a ntwork, allowing to track th progrss of ach infinitsimal particl along tim. A basic modl for th propagation of flow is th so-calld fluid quu modl in which th tim to travrs an dg is composd of a flow-dpndnt waiting tim in a quu at th ntranc of th dg plus a constant travl tim aftr laving th quu. In a dynamic ntwork routing gam ach infinitsimal particl is intrprtd as a playr that sks to complt its journy in th last possibl tim. Playrs ar forward looking and anticipat th congstion and quuing dlays inducd by othrs upon arrival to any dg in th ntwork. Equilibrium occurs whn ach particl travls along a shortst path. This papr is concrnd with th study of quilibria in th fluid quu modl and provids a constructiv proof of xistnc and uniqunss of quilibria in singl origin-dstination ntworks with picwis constant inflow rat. This is don through a dtaild analysis of th undrlying static flows obtaind as drivativs of a dynamic quilibrium. Furthrmor, for multicommodity ntworks, w giv a gnral nonconstructiv proof of xistnc of quilibria whn th inflow rats blong to L p. Subjct classifications: dynamic quilibrium; flows ovr tim; fluid quus; congstion; ntworks. Ara of rviw: Gams, Information, and Ntworks. History: Rcivd January 214; rvision rcivd August 214; accptd Dcmbr Introduction Undrstanding tim varying flows on ntworks is rlvant in contxts whr a stady stat is rarly obsrvd such as urban traffic or th Intrnt. In ordr to dscrib th tmporal volution of such systms, on has to considr th propagation of flow across th ntwork by tracking th position of ach particl along tim. In th most basic modl, th so-calld fluid quu modl, a continuous stram of particls is injctd at a sourc s and travls towards a sink t through dgs charactrizd by a latncy and a pr-timunit capacity: flow propagats according to dg dynamics in which particls arriving to an dg join a quu with srvic rat ç and, aftr laving th quu, mov along th dg to rach its had aftr í tim units. Flows ovr tim wr initially studid in th framwork of optimization. Ford and Fulkrson 1958, 1962) considrd a fluid quu modl in a discrt tim stting and dsignd an algorithm to comput a flow ovr tim carrying th maximum possibl flow from th sourc s to th sink t in a givn timspan. Gal 1959) thn showd th xistnc of a flow pattrn that achivs this optimum simultanously for all tim horizons. Ths rsults wr xtndd to continuous tim by Andrson and Philpott 1994) and Flischr and Tardos 1998). W rfr to Skutlla 29) for an xcllnt and up-to-dat survy. Whn ntwork flows suffr from a lack of coordination among th participating agnts, it is natural to considr thm from a gam thortic prspctiv. In this stting, ach infinitsimal inflow particl is intrprtd as a playr that sks to complt its journy in th last possibl tim, so that quilibrium occurs whn ach particl travls along an s-t shortst path. Th travl tim for a particl ntring th ntwork at any givn tim must tak into account th quuing dlays inducd by othr particls on th dgs along its path. This rquirs to anticipat th quu lngths by th tim whn an dg will b rachd. Th following xampl provids som intuition on flow propagation and dynamic quilibrium in th fluid quu modl prcis dfinitions will b givn in th nxt sction). Exampl 1. Considr th ntwork in th figur blow with a constant inflow at th sourc s qual to u4à5 = 2 for all tims à æ. í í í Th inflow u4à5 is smallr than th srvic rat ç a so that no quu will form on th first dg and a particl dparting at tim à rachs th intrmdiat nod r at tim l r 4à5 = à + 1. Thus, a constant flow of 2 arrivs at r starting from tim à = 1. This flow must b routd along a shortst path 21

2 Comintti, Corra, and Larré: Dynamic Equilibria in Fluid Quuing Ntworks 22 Oprations Rsarch 631), pp , 215 INFORMS Downloadd from informs.org by [ ] on 1 April 215, at 13:33. For prsonal us only, all rights rsrvd. and, sinc í b <í c, at last initially it gos on dg b. Sinc th quu rat for this dg is ç b = 1, a quu starts to grow linarly as z b 4à5 = à É 1 for à æ 1, and consquntly th travl tim z b 4à5/ç b + í b quuing plus latncy) will incras until it qualizs í c. This occurs xactly at tim à = 4 whn th quu lngth rachs z b 445 = 3. From that point on th flow splits qually btwn b and c, kping a constant quu on dg b and an mpty quu on c, so that both dgs hav a constant travl tim of 4. To comput th tim l t 4à5 at which a particl dparting at tim à rachs th sink t, w distinguish two cass. For a dpartur tim à th particl arrivs at r at tim à + 1 and thn follows th dg b whr it facs a quu of lngth à and an additional latncy í b = 1, raching th sink at tim l t 4à5 = 2à + 2. For à æ 3 th particl arrivs to r at à + 1 as bfor but spnds a constant tim 4 travrsing ithr b or c to rach th sink at tim l t 4à5 = à + 5. Both xprssions for l t 4à5 coincid at th brakpoint à = 3 so this arrival tim function is continuous and incrasing, and no particl ovrtaks th flow that ntrd arlir. Th study of flows ovr tim whn particls bhav slfishly has mostly bn considrd in th transportation litratur. Probably, th first to considr th fluid quu modl as a gam was Vickry 1969), who usd it as a tool for valuating transport invstmnts to mitigat congstion. Th sminal papr by Frisz t al. 1993) s also th book by Ran and Boyc 1996) proposd a gnral framwork to modl dynamic quilibrium using an appropriat variational inquality. Th modl supports vry gnral flow propagation ruls and dg dynamics that includ th fluid quu modl as a particular cas. Unfortunatly, in this gnral framwork littl is known in trms of xistnc, uniqunss, and charactrization of solutions. Undr suitabl assumptions, an xistnc rsult was vntually obtaind by Zhu and Marcott 2), which howvr dos not apply to th fluid quu modl. Also, Munir and Wagnr 21) stablishd th xistnc of dynamic quilibria using an altrnativ spcification of th modl and xploiting gnral rsults for gams with a continuum of playrs. A mor dtaild discussion of ths and rlatd works is postpond until 6. Rcntly, Koch and Skutlla 211) obtaind a mor spcific and vry usful charactrization of dynamic quilibria in th fluid quu modl by introducing th concpt of thin flows with rstting. Ths thin flows charactriz th drivativs of a dynamic quilibrium and can b usd to rconstruct quilibria by intgration. Whil th authors did not prov th xistnc of thin flows, th concpt was usd to analyz th pric-of-anarchy for this class of dynamic routing gams. Th fluid quu modl has also bn considrd rcntly by Bhaskar t al. 214) to invstigat th pric-of-anarchy in Stacklbrg routing gams. Our Contribution. This papr considrs flows ovr tim for th fluid quu modl as in Koch and Skutlla 211), and is an outgrowth of our prvious work in Larré 21) and Comintti t al. 211). W provid a constructiv proof for th xistnc and uniqunss of quilibria, xploiting th ky concpt of thin flow with rstting introducd by Koch and Skutlla: a static flow togthr with an associatd labling that charactriz th tim drivativs of an quilibrium. W actually considr a slightly mor rstrictiv dfinition by adding a normalization condition. Using a fixd point formulation w show that normalizd thin flows xist, and thn w prov that th labling is uniqu. As a by-product, this yilds an xponntial tim algorithm to comput a normalizd thin flow and shows that this problm blongs to th complxity class Polynomial Parity argumnts on Dirctd graphs PPAD), though w conjctur that it might b solvabl in polynomial tim. By intgrating ths thin flows w dduc th xistnc of an quilibrium for th cas of a picwis constant inflow rat, and w show that th quilibrium is uniqu within a natural family of flows ovr tim. Finally, w giv a nonconstructiv xistnc proof whn th inflow rat blongs to th spac of p-intgrabl functions L p with 1 <p<à, and w discuss how th rsult xtnds to multipl origindstination pairs. Organization of th Papr. Sction 2 dscribs th fluid quu modl for flows ovr tim. Sction 3 charactrizs th tim drivativs of a dynamic quilibrium using th notion of normalizd thin flows with rstting and provs th xistnc and uniqunss of th lattr. In 4 w xploit th prvious rsults to giv a constructiv proof for th xistnc of an quilibrium in th cas of a picwis constant inflow rat, and w discuss th uniqunss of this quilibrium. In 5 w prsnt a nonconstructiv xistnc rsult for mor gnral inflow rats, including th cas of multipl origin-dstination pairs. Finally, in 6 w compar our findings with prvious rsults in th litratur and stat som opn qustions. Th appndix at th nd summarizs som tchnical facts usd in th papr. 2. A Fluid Quu Modl for Dynamic Routing Gams Throughout this papr w considr a ntwork N = 4G1 ç1 í1 s1 t1 u5 consisting of a dirctd graph G with nod st V and dg st E, a vctor ç = 4ç 5 2E of positiv numbrs rprsnting quu srvic rats, a vctor í = 4í 5 2E of nonngativ numbrs rprsnting dg travl tims, a sourc s 2 V, a sink t 2 V, and an inflow rat function u2! + takn from th st F 4 5 of nonngativ and locally intgrabl functions which vanish on th ngativ axis; R that is, u4à5 = for a.. à<. W dnot U4à5= à u4é5 dé th cumulativ inflow so that U 2 AC loc4 5, th spac of locally absolutly continuous functions. For th prcis dfinition of ths functional spacs and som of thir basic proprtis, w rfr to th appndix as wll as Loni 29, Chaptr 3). A continuous stram of particls is injctd at th sourc s at a tim-dpndnt rat u4à5 and flows through th ntwork towards th sink t. Particls arriving to an dg join

3 Comintti, Corra, and Larré: Dynamic Equilibria in Fluid Quuing Ntworks Oprations Rsarch 631), pp , 215 INFORMS 23 Downloadd from informs.org by [ ] on 1 April 215, at 13:33. For prsonal us only, all rights rsrvd. a quu with srvic rat ç and, aftr laving th quu, travl along th dg to rach its had aftr í tim units. Each infinitsimal inflow particl is intrprtd as a playr that sks to complt its journy in th last possibl tim, so that quilibrium occurs whn ach particl travls along an s-t shortst path. Th rlvant dg costs for a particl ntring th ntwork at tim à must considr th quuing dlays inducd by othr particls along its path by th tim whn ach dg is rachd. This introducs intricat spatial and tmporal dpndncis among th flows that ntr th ntwork at diffrnt tims, possibly at futur dats if ovrtaking occurs. Th rst of this sction maks ths notions mor prcis. For simplicity, and without loss of gnrality, w assum that thr is at most on dg btwn any pair of nods in G, that thr ar no loops, and that for ach nod v 2 V thr is a path from s to v. An dg 2 E from nod v to nod w is writtn vw, whil th forward and backward stars of a nod v 2 V ar dnotd Ñ + 4v5 and Ñ É 4v5. W also suppos that th sum of latncis along any cycl is positiv, namly P 2C í > for vry cycl C in G Flows-Ovr-Tim Th modl is formulatd in trms of th flow rats on vry dg. A flow-ovr-tim is a pair f = of arrays of functions f + 1fÉ 2 F 4 5 for ach 2 E, rprsnting th rat at which flow ntrs th tail of and th rat of flow laving th had of, rspctivly. W say that f is fasibl if th following flow consrvation constraints hold at vry nod v 6= t and for almost all tims à 2 4à5É 4à5= u4à5 for v =s 1) for v 2V \8s1t9 2Ñ + 4v5 2Ñ É 4v5 Th cumulativ inflow and cumulativ outflow of an dg ar dfind as th AC loc 4 5 functions F + 4à5 = Z à F É 4à5 = Z à 4é5 dé1 4é5 dé 2.2. Quu Dynamics and Quuing Dlays An dg is modld as a fluid quu with srvic rat ç followd by a link with constant travl tim í. Th quu lngth z 4à5 at any tim à is th nt flow that has ntrd th dg and has not yt lft th quu. Accounting for th tim í rquird to rach th had of th dg aftr laving th quu, w hav z 4à5 = F + 4à5 É F É 4à + í 5 Throughout th papr w assum that quus oprat at capacity. By this w man that for almost all à 2 ç if z 4à5 > 1 4à + í 5 = min8 4à51 ç 9 othrwis. 2) This condition can b quivalntly statd in trms of th quu lngth dynamics z 4à5 = 4à5 É ç if z 4à5 > 6f othrwis, whos uniqu solution is givn by s.g., Prabhu 22, 1.3) z 4à5 = max á261à7 Z à á 3) 6 4é5 É ç 7dé 4) This formula shows that in a quu that oprats at capacity, th inflow f + compltly dtrmins th quu lngth z, and thrfor th outflow f É is also uniquly dtrmind by 2). Th quuing dlay xprincd by a particl ntring at tim à bfor it starts travrsing th dg is dfind as Z à+q q 4à5 = min q æ 2 4é + í 5dé= z 4à5 5) à W dnot W à = 6à1 à + q 4à55 th intrval on which th particl waits in th quu and Q = 8à2 z 4à5 > 9 th instants at which th quu is nonmpty. Not that for all à 2 W à th quu rmains nonmpty sinc z 4à 5 = z 4à5 + æ z 4à5 É Z à à Z à à 6 4é5 É 4é + í 57 dé 4é + í 5dé>1 and thrfor Q = S à W à. Th nxt rsult shows that in a quu that oprats at capacity th quuing dlay is xactly q 4à5 = z 4à5/ç, providing in fact an quivalnt charactrization of opration at capacity. Proposition 1. Lt f + 1fÉ b th inflow and outflow on dg with corrsponding quu lngth z.th quu oprats at capacity if and only if th nxt thr conditions hold simultanously: a) Capacity constraint: 4à5 ç for almost all à, b) Nondficit constraint: z 4à5 æ for all à, c) Quuing dlay: q 4à5 = z 4à5/ç for all à. Proof. Suppos th quu oprats at capacity. From 2) w clarly hav a) whil 4) implis b). To prov c) w obsrv that R à+q à 4é + í 5dé ç q from which it follows that q 4à5 æ z 4à5/ç. On th othr hand, sinc th quu is nonmpty on W à, condition 2) implis 4é + í 5 = ç a.. é 2 W à and thn Z à+z 4à5/ç à 4é + í 5dé= z 4à51 which yilds q 4à5 = z 4à5/ç. R Convrsly, suppos a) c) hold. From c) w gt à+q 4à5 6 à 4é + í 5 É ç 7dé = so that a) givs 4é + í 5 = ç for almost all é 2 W à, and Lmma 8 implis that this quality holds a.. on S à W à = Q proving th first cas of 2). For th scond cas, b) and Lmma 9: a) ) c) giv that almost vrywhr z 4à5 = implis = z 4à5 = f + 4à5Éf É4à +í 5 and thrfor 4à +í 5 = min8f + 4à51 ç 9. É

4 Comintti, Corra, and Larré: Dynamic Equilibria in Fluid Quuing Ntworks 24 Oprations Rsarch 631), pp , 215 INFORMS Downloadd from informs.org by [ ] on 1 April 215, at 13:33. For prsonal us only, all rights rsrvd Edg Travl Tims Th tim at which a particl xits from an dg can b computd as th sum of th ntranc tim à, plus quuing dlay, plus latncy, namly T 4à5 = à + z 4à5 ç + í 6) For notational convninc, w omit th dpndnc of T on th flow f. Clarly T 2 AC loc 4 5 and using 3) w can comput its drivativ almost vrywhr as 8 1 >< T 4à5 = ç 4à5 if z 4à5 > 1 >: max ) ç 4à5 othrwis. Hnc T 4à5 æ so that T is nondcrasing and thus particls travrsing rspct FIFO without ovrtaking. Morovr, all th flow that ntrs up to tim à xits by tim T 4à5. Indd, sinc th quu is nonmpty ovr th intrval W à É, srvic at capacity implis f 4é + í 5 = ç for almost all é 2 W à and thn F É 4T 4à55 = Z à+í Z 4é5 dé + T 4à5 ç dé = F É 4à + í 5 + z 4à5 à+í = F + 4à5 8) 2.4. Dynamic Shortst Paths A flow particl ntring a path P = k 5 at tim à will rach th ndpoint of th path at th tim l P 4à5 = T k û ût 1 4à51 9) Thus, dnoting P w th st of all s-w paths in G, th arlist tim at which a particl starting from s at tim à can rach w is givn by l w 4à5 = min P2P w l P 4à5 1) Ths functions corrspond to shortst paths with dg costs that considr th quuing dlays along th path at th appropriat tims, taking into account th tim it taks to rach vry dg. W rfr to thm as dynamic shortst paths. Sinc th T s ar absolutly continuous and nondcrasing, th sam holds for thir compositions l P and thrfor also for th l w s s appndix or Loni 29, Chaptr 3). Not also that l w 4 5 is surjctiv with l w 4à5! ±à whn à! ±à. Indd, for à!à this is a consqunc of th inquality l w 4à5 æ à, whil for à! Éà this follows sinc all th quus ar mpty and l w 4à5 = à + d sw with d sw th minimum tim from s to w considring only th travl tims í and no quuing. Th monotonicity of T togthr with th nondficit constraints and th fact that th sum of latncis on any cycl is positiv, imply that dynamic shortst paths do not contain cycls, and thrfor 1) can also b computd by solving 8 < à for w = s1 l w 4à5 = 11) : min T 4l v 4à55 for w 6= s =vw2ñ É 4w5 Th à-shortst-path graph is dfind as th acyclic graph G à = 4V 1 Eà 5 containing all th shortst paths at tim à. An dg = vw is in Eà if and only if T 4l v 4à55 l w 4à5, or quivalntly T 4l v 4à55 = l w 4à5, in which cas it is said to b activ. Not that an inactiv dg has T 4l v 4à55 > l w 4à5, so by continuity it rmains inactiv narby. W also dnot th st of all tims à at which is activ. Not that Eà and dpnd on th givn flow-ovr-tim f Dynamic Equilibrium A fasibl s-t flow-ovr-tim can b intrprtd as a dynamic quilibrium by looking at ach infinitsimal inflow particl as a playr that travls from th sourc to th sink along an s-t path that yilds th last possibl travl tim. Th following dfinition maks this notion prcis. Dfinition 1 Dynamic Equilibrium). A fasibl flowovr-tim f is calld a dynamic quilibrium if for ach = vw 2 E w hav f + 4é5 = for almost all é 2 l v4 c5. Th nxt lmma provids an altrnativ charactrization of dynamic quilibrium. Lmma 1. A fasibl flow-ovr-tim f is a dynamic quilibrium iff for all = vw 2 E and almost all é 2 w hav f + 4é5 > ) é 2 l v4 5. Proof. Th condition in th lmma is quivalnt to f + 4é5 = for almost all é 2 l v4 5 c. Hnc, to stablish th rsult it suffics to show that th sts l v 4 5 c and l v 4 c5 diffr on a st of null masur. W not that th first st is includd in th scond. Indd, tak any é 2 l v 4 5 c. Sinc l v 4 5 is surjctiv w may find à 2 with é = l v 4à5, and sinc é y l v 4 5 it must b th cas that à y so that é 2 l v 4 c5. Now, for ach é 2 l v4 c5\l v4 5 c = l v 4 c5 \ l v 4 5 w may find à 2 c and à 2 such that é = l v 4à5 = l v 4à 5. Sinc l v 4 5 is nondcrasing, it follows that l v 4á5 = é for all á btwn à and à and sinc à 6= à w may tak á 2 in ordr to dduc that l v 4 c5\l v4 5 c l v 4 5. This shows that th sts l v 4 5 c and l v 4 c5 diffr on a countabl st, hnc a st of masur zro. É Rmark. A slightly diffrnt notion, which w call strong dynamic quilibrium, was considrd in Koch and Skutlla 211) rquiring that y Eà ) f + 4l v4à55 = for ach = vw 2 E and almost all à. This condition implis dynamic quilibrium sinc l v is absolutly continuous and maps null sts into null sts), and it is in fact strictly strongr as

5 Comintti, Corra, and Larré: Dynamic Equilibria in Fluid Quuing Ntworks Oprations Rsarch 631), pp , 215 INFORMS 25 Downloadd from informs.org by [ ] on 1 April 215, at 13:33. For prsonal us only, all rights rsrvd. illustratd in th xampl blow. Th point is that whil th concpt of dynamic quilibrium is insnsitiv to modifications on sts of masur zro, this is not th cas for strong quilibrium: th composition f + 4l v4à55 is not wll dfind undr th standard idntification of functions that coincid almost vrywhr and dpnds on th rprsntativ function f + that is chosn. Indd, sinc l v 4 5 may b constant ovr a nontrivial intrval, a simpl modification of f + at a singl point may spoil th almost vrywhr condition with rspct to à. Dfinition 1 avoids this difficulty. Exampl 2. Considr th sam ntwork as in Exampl 1 in th introduction with inflow function 8 >< 4 if à<11 u4à5 = if 1 à 21 >: 2 if 2 <à Th inflow of link a is f a + 4à5 = u4à5 so that a quu builds up in th intrval and is mptid during , aftr which it stays mpty s th lft plot blow). Thus, th quu on dg a has a constant throughput qual to 2 and th outflow, which is also th inflow at th intrmdiat nod r, is givn by a 4à5 = for à 11 2 for à æ 1 z a ) z b ) Th outflow of dg a is xactly th sam as in Exampl 1 in th introduction, so that th quilibrium flows in dgs b and c ar th sam as dscribd thr. Mor xplicitly, th inflow and outflow functions ar 8 >< for à<11 b 4à5 = 2 for 1 à<41 b >: 4à5 = for à<21 1 for à æ 21 1 for à æ 41 c 4à5 = for à<41 c 1 for à æ 41 4à5 = for à<81 1 for à æ 8 For ths flows th quu on dg c rmains mpty at all tims and th quu on dg b volvs as in th right plot abov. A routin calculation shows that ths flows yild a strong dynamic quilibrium with corrsponding arlist tim functions 8 >< 1 + 2à for à<11 l r 4à5 = 3 for 1 à<21 >: 1 + à for à æ à for à<11 >< 6 for 1 à<21 l t 4à5 = 2 + 2à for 2 à<31 >: 5 + à for à æ 3 W obsrv that th dg c is not in Eà for any à<3. If w modify f c + + 4à5 at just on point by taking f c 435>, w still hav a dynamic quilibrium. Howvr, sinc l r 4à5 = 3 for all à w now hav f c + 4l r4à55 > throughout this intrval, and strong quilibrium fails Quus and Cumulativ Flows in a Dynamic Equilibrium It is worth noting that at quilibrium all dgs with positiv quu must b activ. Namly, lt Eà dnot th st of links with positiv quu E à = 8 = vw 2 E2 z 4l v 4à55 > 9 12) Proposition 2. If f is a dynamic quilibrium, thn E à E à and w hav E à = 8 = vw 2 E2 l w4à5 æ l v 4à5 + í 91 13) E à = 8 = vw 2 E2 l w4à5 > l v 4à5 + í 9 14) Proof. Lt = vw 2 Eà and considr th largst à à at which was activ. Equilibrium implis f + 4é5 = for almost all é 2 4l v 4à 51 l v 4à57, so th quu must b nonmpty throughout this intrval and 7) givs T 4é5 = almost vrywhr. Hnc T is constant in this intrval so that T 4l v 4à55 = T 4l v 4à 55 = l w 4à 5 l w 4à51 which yilds 2 Eà proving th inclusion E à E à. To show 13), w not that for 2 Eà w hav l w 4à5 = T 4l v 4à55 æ l v 4à5 + í whr th inquality follows from dfinition of T and th nondficit constraints. Convrsly, suppos that l w 4à5 æ l v 4à5 + í. If z 4l v 4à55 = this yilds l w 4à5 æ T 4l v 4à55 so that 2 Eà, whras in th cas z 4l v 4à55 > th sam conclusion follows sinc w alrady provd that Eà E à. A similar argumnt provs 14). For 2 Eà w hav z 4l v 4à55 > and thn 2 Eà, so that l w4à5 = T 4l v 4à55 > l v 4à5 + í. Convrsly, if l v 4à5 + í <l w 4à5 th inquality l w 4à5 T 4l v 4à55 and th dfinition of T yild z 4l v 4à55 >. É Intuitivly, at quilibrium any flow routd through an dg = vw up to tim l v 4à5 should rach w bfor th optimal tim l w 4à5. This is, in fact, an quivalnt charactrization of dynamic quilibrium.

6 Comintti, Corra, and Larré: Dynamic Equilibria in Fluid Quuing Ntworks 26 Oprations Rsarch 631), pp , 215 INFORMS Downloadd from informs.org by [ ] on 1 April 215, at 13:33. For prsonal us only, all rights rsrvd. Thorm 1. Lt f b a fasibl s-t flow-ovr-tim. Th following ar quivalnt a) f is a dynamic quilibrium; b) for ach = vw and all à w hav F + 4l v4à55 = F É4l w4à55; c) for ach = vw and almost all à w hav y Eà ) f + 4l v4à55lv 4à5 =. Proof. For ach à considr th intrval I à = 4à 1à7 with à à th largst tim such that T 4l v 4à 55 = l w 4à5. Not that à is wll dfind sinc l w 4à5 T 4l v 4à55 and T 4l v 4à 55! Éà whn à! Éà. Not also that I à =ô for à 2 sinc in this cas à = à. W claim that c coincids with th union of th I à s. Indd, for ach à 2 c w hav à <à and thrfor à 2 I à so that c Sà I à. Convrsly, for à 2 I à w hav by dfinition of à that T 4l v 4à 55 > l w 4à5 æ l w 4à 5 so that à 2 S c à I à c. Now, invoking 8), for ach à w hav F + 4l v4à55 É F É 4l w4à55 = Z lv 4à5 l v 4à 5 and thn 4é5 dé æ 1 15) with quality iff f + vanishs almost vrywhr on 4l v 4à 51 l v 4à57 = l v 4I à 5. Lmma 8 thn shows that b) holds iff f + 4é5 = for almost all é 2 S à l v 4I à 5 = l v 4 c 5, proving b), a). Similarly, a chang of variabls cf. appndix) allows to rwrit 15) as Z à F + 4l v4à55 É F É 4l w4à55 = à 4l v4z55l v4z5 dz æ 1 with quality iff f + 4l v4z55lv 4z5 = for almost all z 2 I à. By Lmma 8, b) holds iff this map vanishs almost vrywhr on S à I à = c, proving b), c). É Thorm 1b) abov provids a way to synchroniz th flow ovr tim on th diffrnt dgs by using th dpartur tim as a common clock. This proprty motivats th dfinition of cumulativ flow on an dg at a givn tim, consisting of all flow that dpartd up to that tim and which uss th dg. Dfinition 2 Cumulativ Flow). Th cumulativ flow inducd by a dynamic quilibrium f is dfind as x4à5 = 4x 4à55 2E with x 4à5 = F + 4l v4à55 = F É4l w4à55 for all = vw 2 E and à 2. Intgrating th flow consrvation constraints 1) ovr th intrval 61l v 4à57, it follows that for ach à 2 th cumulativ flow x4à5 is a static s-t flow of valu U4à5, 2Ñ + 4v5 x 4à5É 2Ñ É 4v5 x 4à5= U4à5 for v =s1 for v 2V \8s1t9 16) Diffrntiating, for almost all à w gt that x 4à5 is a static s-t flow of valu u4à5 with x 4à5 = for y E à Path Formulation of Dynamic Equilibrium Sinc th à-shortst path graph G à is acyclic, x 4à5 dos not rout flow on cycls. Hnc, dnoting P th st of simpl s-t paths w may find a dcomposition u4à5 = P P2P h P 4à5 into nonngativ path-flows h P 4à5 æ such that x 4à5 = P3 h P 4à5 Indd, start with y = x 4à5 and considr th paths P 2 P in a fixd ordr stting h P 4à5 = min 2P y and updating y Ñ y É h P 4à5 for 2 P. This yilds a masurabl dcomposition h P 2 F 4 5 such that h P 4à5 > only for paths that blong to th à-shortst-path graph G à. It is appaling to tak th lattr as th dfinition of dynamic quilibrium. Th difficulty is to proprly dfin shortst path sinc this rquirs th xit-tim functions T, which in turn rquir an appropriat flow-ovr-tim f to b associatd with h = 4h P 5 P2P. Sinc f dpnds on how th flow h propagats along th paths, both f and T must b dtrmind simultanously. This ntwork loading procss typically rquirs additional conditions to b wll dfind, such as an acyclic ntwork structur or whn link travl tims ar boundd away from zro, which is a natural and mild assumption s.g., Munir and Wagnr 21, u t al. 1999, Zhu and Marcott 2). Sinc w will not rquir ntwork loading until 5, w dfr its discussion to that sction. 3. Drivativs of Dynamic Equilibria: Normalizd Thin Flows Th functions x and l w ar absolutly continuous, and thrfor thy can b rcovrd by intgrating thir drivativs. In this sction w charactriz ths drivativs, yilding a constructiv mthod to find an quilibrium. Our charactrization is closly rlatd to th notion of thin-flow with rstting introducd by Koch and Skutlla 211). Rcall that for almost all à th drivativ x 4à5 is an s-t flow of valu u4à5 with x 4à5 = for y E à. On th othr hand, clarly ls 4à5 = 1 whil for w 6= s w may us 11) and th rul of diffrntiation of a pointwis) minimum function, which combind with 7) yilds almost vrywhr l w 4à5 = min =vw2e à T 4l v4à55l v 4à5 = whr for ach = vw 2 Eà w st min =vw2e à ê 4l v 1x 5 = x /ç if 2 E à 1 max8l v 1x /ç 9 if y E à ê 4l v 4à51 x 4à551 Sinc Eà is acyclic, this allows to comput l w4à5 by scanning th nods w in topological ordr.

7 Comintti, Corra, and Larré: Dynamic Equilibria in Fluid Quuing Ntworks Oprations Rsarch 631), pp , 215 INFORMS 27 Downloadd from informs.org by [ ] on 1 April 215, at 13:33. For prsonal us only, all rights rsrvd. This discussion motivats th nxt dfinition. Lt u æ and 4E 1E 5 b a pair of dg sts such that 4H5 E E E with E acyclic and for all v 2 V thr is an s-v path in E. W dnot by K4E 1u 5 th nonmpty, compact, and convx st of all static s-t flows x = 4x 5 2E æ of valu u with x = for y E. With ach x 2 K4E 1u 5 w associat th uniqu labls givn as abov by l s = 1 and l w = min =vw2e ê 4l v 1x 5 for w 6= s. Not that th map x 7! l is continuous. Dfinition 3 Normalizd Thin Flow). A flow x 2 K4E 1u 5 is calld a normalizd thin flow 4NTF5 of valu u with rstting on E E iff x = for vry dg = vw 2 E such that lw <ê 4lv 1x 5. Thorm 2. Lt f b a dynamic quilibrium and à 2 such that th right drivativs u = 4dU /dà + 54à5, lv = 4dl v /dà + 54à5 and x = 4dx /dà + 54à5 xist. Thn x is an NTF of valu u with rstting on Eà E à, with corrsponding labls l. Proof. Diffrntiating 16), it follows that x is an s-t flow of valu u. Morovr, if y Eà thn rmains inactiv on som intrval 6à1 à + Ö5, so th chain rul s appndix) and quilibrium imply that on this intrval x 4é5 = f + 4l v4é55lv 4é5 = a.., so x 4 5 is constant and x =. This provs that x 2 K4Eà 1u 5. Lt us show that l ar th corrsponding labls. Clarly ls = 1. For th rst of th argumnt w distinguish two mor substs of Eà : E + contains th links = vw, which hav a quu or ar about to build on with z 4é5 > for all é on a small intrval 4l v 4à51 l v 4à5 + Ö5, whras E+ includs th links without quu at tim à but which ar activ along a strictly dcrasing squnc à n # à. For 2 E+ w hav l w 4à n 5 = T 4l v 4à n 55 æ l v 4à n 5 + í and l w 4à5 = l v 4à5 + í so that l w 4à n 5 É l w 4à5 æ l v 4à n 5 É l v 4à5 and dividing by à n É à with n!àw gt lw æ l v. Similarly, for 2 E à \E + w may tak à n # à with z 4l v 4à n 55 = so that l w 4à n 5 T 4l v 4à n 55 = l v 4à n 5 + í and w gt lw l v. Also, for = vw 2 Eà th capacity constraint givs for à æ à x 4à 5Éx 4à5= Z lw 4à 5 l w 4à5 4é5dé ç 4l w 4à 5Él w 4à551 17) which implis lw æ x /ç. Finally, whn 2 E+ w hav z 4l v 4à 55 > for à clos to à and as obsrvd aftr Equation 5) th quu rmains nonmpty ovr 6l v 4à 51 l v 4à 5 + z 4l v 4à 55/ç 5 so that 2) implis that 4é5 = ç for almost all é 2 6l v 4à 5 + í 1l w 4à 55. This radily givs 4é5 = ç almost vrywhr on a small intrval to th right of l w 4à5 and thn quality holds in 17) for à sufficintly clos to à, so that lw = x /ç for 2 E+. In summary a) lw æ l v, for = vw 2 E + 1 b) lw l v, for = vw 2 E à \E + 1 c) lw æ x /ç, for = vw 2 Eà 1 d) lw = x /ç, for = vw 2 E+ Combining b) and d) w gt lw min =vw2eà ê 4lv 1x 5 with quality if thr is som = vw 2 Eà. To prov th quality whn no dg from Eà is incidnt on w, choos any à n # à and a squnc of activ dgs n 2 Eà n, and tak a subsqunc with n = vw constant so that = vw 2 E+. Thn a) and c) combind giv lw æ ê 4lv 1x 5. Altogthr this provs lw = min =vw2eà ê 4lv 1x 5 for w 6= s. Lt us finally show that x is an NTF. Suppos x > on som = vw 2 Eà with l w <ê 4lv 1x 5. Th lattr and d) imply y Eà, whil x > givs x 4à 5>x 4à5 for all à >àso must b activ on a squnc à n # à and 2 E+. Thn a) and c) yild th contradiction lw æ ê 4lv 1x 5. É Thorm 2 drivs th xistnc of NTF s from a dynamic quilibrium. To procd in th othr dirction, w study th xistnc of NTF s, and thn by intgration w rconstruct a dynamic quilibrium. Thorm 3. Lt u æ and 4E 1E 5 satisfying H). Thn thr is an NTF of valu u with rstting on E E. Proof. Lt K = K4E 1u 5 and obsrv that th NTF s ar prcisly th fixd-points of th st-valud map 2 K! 2 K with nonmpty convx compact valus givn by 4x 5 = y 2 K2 y = for all 2 E such that l w <ê 4l v 1x 5 with l th labls corrsponding to x and E. Sinc x 7! l is continuous, it follows that is uppr-smicontinuous, and a fixd point x 2 4x 5 xists by virtu of Kakutani s fixd point thorm. É This rsult shows that finding an NTF blongs to th complxity class PPAD. It also suggsts a finit xponntial tim) algorithm to comput an NTF: w guss th st E of links 2 E that satisfy lw = ê 4lv 1x 5, and thn solv max 4x 1l 5 l w 2x 2 K4E 1u 53 l s = 13 l w min ê 4l =vw2e w2v v 1x 5 Th lattr can b rstatd as a mixd intgr linar program and solvd in finit tim. By considring all possibl substs E E, th mthod vntually finds an NTF. In gnral thr may xist diffrnt NTF s, ach on with its corrsponding labls. W show nxt that th labls in all of thm coincid. Thorm 4. Lt u æ and 4E 1E 5 satisfying H). Thn th labls l ar th sam for all NTF s of valu u with rstting on E E. Proof. Lt x and y b two NTF s with diffrnt labls l 6= h, and suppos without loss of gnrality that S = 8v 2 V : lv >h v9 is nonmpty. Considr th nt flow across th boundary of S: sinc x and y satisfy flow consrvation, stting b s = u, b t =Éu and b v = for v 2 V \8s1 t9, w gt x 4Ñ + 4S55 É x 4Ñ É 4S55 = v2s b v = y 4Ñ + 4S55 É y 4Ñ É 4S55 18)

8 Comintti, Corra, and Larré: Dynamic Equilibria in Fluid Quuing Ntworks 28 Oprations Rsarch 631), pp , 215 INFORMS Downloadd from informs.org by [ ] on 1 April 215, at 13:33. For prsonal us only, all rights rsrvd. For = vw 2 Ñ + 4S5 w hav x y sinc othrwis x >y implis x > and l w = ê 4lv 1x 5>ê 4h v 1y 5 æ h w contradicting w y S. Similarly, x æ y for all = vw 2 Ñ É 4S5 sinc y >x implis y > and h w = ê 4h v 1y 5 æ ê 4lv 1x 5 æ l w contradicting w 2 S. Ths inqualitis and 18) imply x = y for all 2 Ñ4S5, with y = for 2 Ñ É 4S5 sinc y > yilds a contradiction as bfor. Sinc E is acyclic, w may find w 2 S with all dgs = vw 2 E blonging to Ñ É 4S5. Now, lw >h w æ and x = implis that y E for all ths dgs, and thn ê 4lv 1x 5 = l v as wll as ê 4h v 1y 5 = h v, from which w gt th contradiction h w = min vw2e h v æ min vw2e l v = l w. É 4. Existnc and Uniqunss of Dynamic Equilibria Koch and Skutlla 211) dscrib a mthod to xtnd an quilibrium for th cas of a constant inflow rat u4à5 u. Givn a fasibl flow-ovr-tim f that satisfis th quilibrium conditions in 61à k 7, th quilibrium is xtndd as follows: 1) Find x an NTF of valu u with rstting on E à k E à k, and lt l dnot th corrsponding labls. 2) Comput à k+1 = à k + Å with Å> th largst valu with l w 4à k 5+Ål w Él v4à k 5ÉÅl v í 1 for all =vwye à k 1 19) l w 4à k 5+Ål w Él v4à k 5ÉÅl v æí 1 for all =vw2e à k 2) 3) Extnd th arlist-tim functions and th flow-ovrtim as l v 4à5=l v 4à k 5+4àÉà k 5l v 1 for v 2V and à 26à k1à k é5=x /l v 1 for =vw2e and é 26l v4à k 51l v 4à k é5=x /l w 1 for =vw2e and é 26l w4à k 51l w 4à k+1 55 Thorms 3 and 4 imply that x in stp 1) xists and l is uniqu. Morovr thr ar finitly many l, ach on corrsponding to a diffrnt pair 4Eà 1E à5. Th Å computd in 2) is strictly positiv so that ach itration xtnds th arlist-tim functions to a strictly largr intrval. Th conditions 19) and 2) corrspond, rspctivly, to th maximum rangs on which th inactiv dgs rmain inactiv, and th positiv quus rmain positiv. Hnc, for à 2 6à k 1à k+1 5 th pair 4Eà 1E à5 rmains constant, whras at à k+1 this pair changs and w must rcomput th NTF. Not that whn lv + = th updat of f dos not xtnd its domain of dfinition and similarly for f É whn lw =. As shown in Koch and Skutlla 211), th xtnsion maintains at all tims th conditions for dynamic quilibrium in th strong sns s Rmark aftr Dfinition 1). This xtnsion procdur can b usd to stablish th xistnc of a dynamic quilibrium. Starting from th intrval 4Éà1à 7 with à = and zro flows, th xtnsion can b itratd as long as rquird to find a nw intrval 6à k 1à k+1 7 with à k+1 >à k at vry stp k. Evntually, à k may hav a finit limit à à : in this cas, sinc th labl functions ar nondcrasing and hav boundd drivativs, w can dfin th quilibrium at à à as th limit point of th labl functions l, and rstart th xtnsion procss. A standard argumnt using Zorn s lmma shows that a maximal solution is dfind ovr all +. Not that th f constructd abov is right-constant. Dfinition 4. A function g2! is calld rightconstant if for ach à 2 thr is an Ö> such that g is constant on 6à1 à + Ö5. Similarly, g is right-linar if for ach à it is affin on 6à1 à + Ö5 for som Ö>. Th xtnsion mthod works vn if th inflow rat function is picwis constant, so w hav th following xistnc rsult. Thorm 5. Suppos that th inflow u is picwis constant, i.., thr is an incrasing squnc 4é k 5 k2 with é = such that u4 5 is constant on ach intrval 6é k 1é k+1 5. Thn thr xists a strong dynamic quilibrium f that is right-constant and whos labl functions l ar right-linar. Dynamic quilibria in gnral ar not uniqu. Considr, for instanc, a constant inflow u4à5 = 8àæ9 in th ntwork in Exampl 1 but with í c = í b. Thn all quus rmain mpty at all tims, and any splitting of th outflow fa É4à5 = 8àæ19 among th dgs b and c yilds a dynamic quilibrium. Nvrthlss, using Thorm 4 on can prov that th arlist-tim functions in all sufficintly rgular dynamic quilibria ar th sam and coincid with thos givn by th constructiv procdur. Thorm 6. Suppos that th inflow u is picwis constant. Thn, th arlist-tim functions 4l v 5 v2v ar th sam for all dynamic quilibria f which ar right-continuous. Proof. Whn f is right continuous, it follows that th quu lngths z 4à5, th xit-tim functions T 4 5, and th arlist-tim functions l v 4à5 ar right-diffrntiabl vrywhr with right-continuous drivativs. Thorm 2 implis that 4dl v /dà ar an NTF, and Thorm 4 shows that ths drivativs ar uniqu. Sinc thy can tak only finitly many valus, continuity from th right imply that 4dl v /dà is right-constant and l v 4 5 is right-linar. It follows that any two right-continuous dynamic quilibria must hav th sam arlist-tim functions. Indd, if ths functions coincid up to tim à, thir right drivativs at à coincid, and sinc thy ar right-linar thy will also coincid on a nontrivial intrval 6à1 à + Ö7. This implis that in fact th functions must coincid throughout. É 5. Existnc of Equilibria for Inflow Rats in L p Th prvious sctions studid dynamic quilibria for a singl origin-dstination with picwis constant inflows.

9 Comintti, Corra, and Larré: Dynamic Equilibria in Fluid Quuing Ntworks Oprations Rsarch 631), pp , 215 INFORMS 29 Downloadd from informs.org by [ ] on 1 April 215, at 13:33. For prsonal us only, all rights rsrvd. W considr nxt mor gnral inflow rats and thn xtnd th rsults to multipl origin-dstination pairs. W procd as in Frisz t al. 1993) using a variational inquality for a path-flow formulation of dynamic quilibrium. Th analysis is nonconstructiv and xploits th following particular cas of th xistnc rsult Brézis 1968, Thorm 24). Lt 41 ò ò5 b a rflxiv Banach spac and ì 1 î th canonical pairing btwn and its dual. If A2 K! is a wak-strong continuous map dfind on a nonmpty, closd, boundd, and convx subst K, thn th following variational inquality problm has a solution: VI4K1 A5 Find x 2 K such that ìax1y É xî æ 1 for all y 2 K Variational Inquality Formulation Lt us considr first th cas of a singl origin-dstination pair st and an inflow rat u 2 L p 41T5 whr T is a finit horizon and 1 <p<à. W xtnd u4à5 outsid 61T7 so that u may b sn as a function in F 4 5. As bfor, lt P b th st of paths conncting s to t and dnot by K th nonmpty, boundd, closd, and convx st of fasibl path-flows givn by K = h 2 L p 41T5 P 2 h P = u and h P æ P2P for all P 2 P 21) Th spac = L p 41T5 P is rflxiv with dual = L q 41T5 P whr 1/p + 1/q = 1. W will show that a dynamic quilibrium can b obtaind by solving th variational inquality VI4K1 A5 with A2 K! such that A P 4h5 2 L q 41T5 is th continuous function à 7! lh P 4à5É à giving th tim rquird to travl from s to t using path P undr th path-flow pattrn givn by h, namly, th problm is to find h 2 K as dfind by 21) such that P2P Z T 4l P h 4à5 É à54h P 4à5 É h P 4à55 dà æ 8 h 2 K 22) To proprly dfin th map A, our first task is to show that vry h 2 K dtrmins a uniqu fasibl flow-ovr-tim f, which in turn inducs link travl tims T and path travl tims lh P. This is achivd by th ntwork loading procdur dscribd in th nxt subsction. In 5.3 w stablish th wak-strong continuity of A, and thn in 5.4 w conclud th xistnc of a dynamic quilibrium. Finally, 5.5 xtnds th xistnc rsult to multipl origin-dstinations Ntwork Loading Th following ntwork loading procdur rquirs í > on vry link, which w assum from now on. Lt h = 4h P 5 P2P b a givn family of path-flows with h P 2 F 4 5 for all P 2 P. Antwork loading is a flow-ovrtim f = togthr with nonngativ and masurabl link-path dcompositions 4à5 = P3 P1 4à51 4à5 = P1 4à51 23) P3 such that for all links = vw and almost all à 2 on has 8 >< h P 4à5 P1 4à5 = fp1 >: É 4à5 if is th first link of P1 if is th link in P just bfor, togthr with th link transfr quations Z T 4à5 P1 4é5 dé = Z à 24) P14é5 dé 25) whr T is th link travl tim inducd by f + through Equations 4) and 6). W dnot by ó th tupl comprising all th flows f +, f P1, +, f P1 É for 2 E and P 2 P. In ordr to prov th xistnc and uniqunss of a ntwork loading, w first stablish th following tchnical lmma. Lmma 2. Lt a link-path dcomposition of th inflow 4à5 = P3 P1 4à5 b givn ovr an initial intrval 4Éà1 à7. Thn thr ar uniqu outflows P1 2 Là 44Éà1T 4 à575 satisfying 25), with P1 4é5 ç for all é T 4 à5. Proof. Sinc T maps 4Éà1 à7 surjctivly onto 4Éà1 T 4 à57, it is clar that thr is at most on fp1 É satisfying 25) undr th usual idntification of functions diffring on a ngligibl subst of ). To stablish th xistnc lt A 4Éà1 à7 b th st of tims à at which th drivativ T 4à5 xists and is strictly positiv, and st P1 4T 4à55 = P14à5/T 4à5 for à 2 A1 othrwis. This unambiguously dfins fp1 É 4é5 for all é T 4 à5 as a nonngativ masurabl function. Morovr, for à 2 A w hav P1 4T 4à55 = 4à5/T 4à5 = 4T 4à55 ç 1 P3 which implis fp1 É 4é5 ç for all é T 4 à5 so that th fp1 É s ar ssntially boundd. Finally, a chang of variabls in th intgral s appndix) givs Z T 4à5 Z à Z à P1 4é5dé = P1 4T 4é55T 4é5dé = P1 4é5dé whr w usd th quality fp1 É 4T 4é55T 4é5 = f P14é5, + which follows from th dfinition of fp1 É 4é5 whn é 2 A and from th fact that, almost vrywhr, 7) implis that if T + 4é5 = thn f 4é5 = and thrfor f P14é5 + =. É

10 Comintti, Corra, and Larré: Dynamic Equilibria in Fluid Quuing Ntworks 3 Oprations Rsarch 631), pp , 215 INFORMS Downloadd from informs.org by [ ] on 1 April 215, at 13:33. For prsonal us only, all rights rsrvd. Proposition 3. Suppos that í > on all links. Thn to ach path-flow tupl h it corrsponds a uniqu ntwork loading ó. Proof. Lt h = 4h P 5 P2P b a givn family of path-flows and suppos that w hav a link-path dcomposition satisfying 23) 25) ovr an intrval 4Éà1 à7. For à = this is asy sinc all flows vanish on th ngativ axis. By Lmma 2, th inflow dcompositions f + 4à5 = P P3 f P14à5 + ovr 4Éà1 à7, togthr with condition 25), dtrmin uniqu link-path dcompositions for th outflows P 4à5 = P3 fp1 É 4à5 ovr th intrval 4Éà1T 4 à57. Ths intrvals includ 4Éà1 à + ò7 with ò = min í >, and thn using 24) it follows that th link inflows and thir link-path dcompositions hav uniqu xtnsions to 4Éà1 à + ò7. Procding inductivly it follows that th inflows and outflows, togthr with thir link-path dcompositions, ar uniquly dfind on all of. É 5.3. Continuity of Path Travl Tims W prov nxt that th ntwork loading procdur dfins path travl tim maps h 7! lh P that ar wak-strong continuous from K L p 41T5 P to th spac of continuous functions C461T 71 5 ndowd with th uniform norm. Th proof is split into svral lmmas. Lmma 3. Thr xists a constant M æ such that all th flows in th ntwork loading corrsponding to any h 2 K ar supportd on 61M7. Proof. W claim that th quu lngths ar boundd by z 4à5 z = R T u4é5 dé. Indd, an inductiv argumnt basd on 24) and 25) shows that for ach path P and ach link 2 P w hav R P14é5 dé = R h P 4é5 dé. Sinc z 4à5 F + 4à5, using 23) w gt Z à z 4à5 F + 4à5 = P1 4é5 dé P3 P Z T = u4é5 dé Z h P 4é5 dé This bound implis that th tim to travrs a link is at most z/ç + í. Dnoting by Ñ th maximum of ths quantitis ovr all 2 E and stting M = T + mñ whr m is th maximum numbr of links in all paths P 2 P, thn lh P 4à5 M for all P 2 P and à 2 61T7. This, togthr with 24) and 25), implis in turn that all th flows in a ntwork loading ar supportd on th intrval 61M7. É Lmma 4. Th maps 7! z and 7! T dfind by 4) and 6) ar wak-strong continuous from L p 41M5 to C461 M71 5. Proof. Th continuity of f + 7! T is immdiat from that of f + 7! z. To show th lattr, w rcall that Arzla-Ascoli s thorm implis that th intgration map I: L p 41M5! C461M71 5 dfind by Ix4à5 = R à x4é5dé is a compact oprator, and hnc it is wak-strong continuous. R It follows that th map f + 7! y givn by y 4à5 = à 6 4é5 É ç 7dé is wak-strong continuous, and thn th sam holds for f + 7! z sinc 4) givs z 4à5 = max á261à7 y 4à5 É y 4á5 = y 4à5 É min y 4á5 á261à7 and th map y 7! Hy oprating on C461 M71 5 as Hy4à5 = min á261à7 y 4á5 is nonxpansiv. É Lmma 5. Lt Ï dnot th st of all th rstrictions to 61M7 of th pairs 4h1 ó5 whr h 2 K and ó is th corrsponding ntwork loading. Thn Ï is a boundd and wakly closd subst of L p 41M5 k whr k is th dimnsion of th tupl 4h1 ó5, namly k =ópó+2óeó+2ópóóeó. Proof. From Lmma 3 w know that all flows 4h1 ó5 2 Ï ar supportd on 61M7, whil 24) and Lmma 2 imply that thy ar uniformly boundd in L p 41M5. Lt us tak a wakly convrgnt nt 4h Å 1ó Å 5 * 4h1 ó5 with 4h Å 1ó Å 5 2 Ï. It is clar that conditions 23) and 24) ar stabl undr wak limits so that ó satisfis ths quations. In ordr to show 25) it suffics to pass to th limit in Z T Å 4à5 f ÅÉ P1 4é5 dé = Z à f Å+ P14é5 dé 26) Th right-hand sid convrgs to R à P14é5 dé whil th intgral on th lft can b writtn as th sum Z T Å 4à5 f ÅÉ P1 4é5 dé = Z T 4à5 Z T Å f ÅÉ P1 4é5 dé + 4à5 f ÅÉ P14é5 dé T 4à5 Th first trm on th right convrgs to R T 4à5 P14é5 dé whil th scond convrgs to zro. Indd, ltting q = p/4p É 15, by Höldr s inquality w hav Z T Å 4à5 T 4à5 f ÅÉ ÅÉ P14é5 dé òfp1 ò qp p ót Å 4à5 É T 4à5ó so th conclusion follows sinc fp1 ÅÉ4é5 ç implis òfp1 ÅÉ ò p ç p M, and according to Lmma 4 w hav T Å4à5! T 4à5. Hnc, w may pass to th limit in 26), which provs that w satisfis 25) and thrfor 4h1 ó5 2 Ï as was to b provd. É Lmma 6. Th maps h 7! T dfind by th ntwork loading procdur ar wak-strong continuous from K L p 41T5 P to C461M71 5. Proof. Tak a wakly convrgnt nt h Å *hin K and lt ó Å b th corrsponding ntwork loading. From Lmma 3 w know that th nt ó Å is boundd in L p 41M5, whil Lmma 5 implis that any wak accumulation point of w Å is a ntwork loading for h. Sinc th lattr is uniqu, it follows that w Å *w. In particular f Å+ *f + wakly in L p 41M5 so that th conclusion T Å! T strongly in C461M71 5 is a consqunc of Lmma 4. É

11 Comintti, Corra, and Larré: Dynamic Equilibria in Fluid Quuing Ntworks Oprations Rsarch 631), pp , 215 INFORMS 31 Downloadd from informs.org by [ ] on 1 April 215, at 13:33. For prsonal us only, all rights rsrvd. Lmma 7. For ach P 2 P th map h 7! lh P dfind by th ntwork loading procdur is wak-strong continuous from K L p 41T5 P to C461T Proof. Lt P = k 5. St M i = T + iñ with Ñ as in th proof of Lmma 3, and considr th rstrictions T i 261M ié1 7! 61M i 7 so that for all à 2 61T7 l P h 4à5 = T k û ût 1 4à5 27) By Lmma 6 th maps h 7! T i ar wak-strong continuous, so th conclusion follows by noting that composition is a continuous opration. Mor prcisly, th map 4f 1 g5 7! g û f dfind on th spacs û2 C461M ié M i 75 C461M i 71 61M i+1 75! C461M ié M i+1 75 is a continuous map with rspct to uniform convrgnc). Indd, considr a strongly convrgnt nt 4f Å 1g Å 5! 4f 1 g5. Thn for ach à 2 61M ié1 7 w hav óg Å û f Å 4à5 É g û f4à5ó óg Å 4f Å 4à55 É g4f Å 4à55ó +óg4f Å 4à55 É g4f 4à55ó Th first trm on th right can b boundd by òg Å É gò à, which tnds to, whil th scond trm also tnds to zro uniformly in à sinc g is uniformly continuous and òf Å É f ò à tnds to zro. É 5.4. Existnc of Dynamic Equilibrium for a Singl Origin-Dstination With ths prliminary rsults w may now prov that th variational inquality VI4K1A5 has a solution, and th corrsponding ntwork loading givs a dynamic quilibrium. Thorm 7. Lt u 2 L p 41T5 with 1 <p<à and assum that í > on vry link. Thn thr xists a dynamic quilibrium. Proof. According to Lmma 7 th map h 7! A4h5 is wak-strong continuous from K to so that th variational inquality VI4K1 A5 has a solution h 2 K. W claim that th corrsponding flow-ovr-tim f givn by Proposition 3 is a dynamic quilibrium. If not, by Thorm 1 w may find à> and a link = vw y Eà such that for all Ö> w hav f 4l v 4é55lv 4é5 > on a subst of positiv masur in 6à1 à + Ö7. Choos Ö small nough so that Eé dcrass on 6à1 à + Ö7 and choos P 2 P with 2 P and h P 4é5 > on a subst I 6à1 à + Ö7 with positiv masur. Tak also P 2 P with all links in Eà+Ö so that P is optimal for ach é 2 6à1 à + Ö7 that is, P is an s-t path in th é-shortst-path graph G é ), and lt h 2 K b idntical to h xcpt for é 2 I whr w transfr flow from P to P, that is h P 4é5 = and h P 4é5 = h P 4é5 + h P 4é5. A dirct calculation thn givs Z ìah1 h É hî= ìah4é51 h 4é5 É h4é5î dé Z = 61T 7 I 4l P h 4é5 É lp h 4é55h P 4é5 dé Sinc P is optimal for all é 2 I whil P is not sinc y Eé ), it follows that lp h 4é5 < lp h 4é5, which yilds a contradiction. É 5.5. Extnsion to Multipl Origin-Dstination Pairs Th xtnsion to multipl origin-dstinations is straightforward. For ach pair st 2 N N lt u st 2 L p 41T5 b th corrsponding inflow possibly zro) and lt P st b th st of s-t paths which is assumd nonmpty if u st is nonzro. A fasibl flow-ovr-tim is now a family of inflows f + = Pst f 1st + and outflows f É = P st f1st É satisfying flow consrvation for ach st pair, namly, for all nods v 6= t and almost all à 2 2Ñ + 4v5 1st 4à5 É = 2Ñ É 4v5 1st 4à5 u st 4à5 for v = s1 for v 2 V \8s1 t9 28) Th dfinitions of quu lngths, link travl tims, and path travl tims rmain unchangd, and w only nd to introduc th origin-dstination optimal tims l st 4à5 = min P2P st l P 4à5 Dynamic quilibrium holds whn for ach pair st and all = vw 2 E w hav f 1st + 4é5 = for almost all é 2 l sv 4 \ s5 whr s dnots th st of all tims à at which link = vw is activ for origin s, namly l sw 4à5 = T 4l sv 4à55. Dnoting P th union of all th P st s, th ntwork loading procdur in 5.2 rmains unchangd as it did not dpnd on having a singl origin-dstination pair. Also th rsults in 5.3 ar asily xtndd by considring K as th st of path-flows h = 4h P 5 P2P 2 L p 41T5 P, which ar nonngativ and that satisfy flow consrvation for ach pair st, that is h P = u st P2P st For th bound z = R T u4é5 dé of th quu lngths in Lmma 3 it suffics to tak u as th sum of all th u st s. With ths prliminaris, th proof of Thorm 7 is radily adaptd to stablish th xistnc of a dynamic quilibrium for multipl origin-dstinations. Thorm 8. Lt u st 2 L p 41T5 with 1 <p<à th inflows for multipl origin-dstination pairs st 2 N N, and assum that í > on vry link. Thn thr xists a dynamic quilibrium.

12 Comintti, Corra, and Larré: Dynamic Equilibria in Fluid Quuing Ntworks 32 Oprations Rsarch 631), pp , 215 INFORMS Downloadd from informs.org by [ ] on 1 April 215, at 13:33. For prsonal us only, all rights rsrvd. 6. Concluding Rmarks Although dynamic traffic assignmnt has rcivd considrabl attntion sinc th sminal work by Mrchant and Nmhausr 1978a, b), th xistnc and charactrization of dynamic quilibria still poss many challnging qustions. For a rviw of th litratur and opn problms w rfr to Pta and Ziliaskopoulos 21). Svral of th prvious studis hav rlid on a strict FIFO condition that rquirs th xit tim functions T 4 5 to b strictly incrasing. For instanc, Frisz t al. 1993) considr a situation in which usrs choos simultanously rout and dpartur tim, with link travl tims spcifid as D 4y 5 = Å y + Ç whr y = F + 4à5 É F É 4à5 is th total flow on link at tim à and Å 1Ç ar strictly positiv constants. Strict FIFO was shown to hold for such linar volum-dlay functions, which allowd to charactriz th quilibrium by a variational inquality, though no xistnc rsult was givn. Strict FIFO was also usd by u t al. 1999) to invstigat th ntwork loading problm, namly, to dtrmin th link volums and travl tims that rsult from a givn st of path-flow dpartur rats. Shortly aftr, th xistnc of quilibria was stablishd by Zhu and Marcott 2) undr a strong FIFO condition that holds for linar volumdlay functions vn in th cas Å = ), assuming in addition that inflows ar uniformly boundd. Unfortunatly, as illustratd by th xampl in 2.5, strict FIFO dos not hold in our framwork and ths prvious rsults do not apply. This is somwhat surprising sinc w also considr linar travl tims. Th subtl diffrnc is that w considr th quu lngth z instad of th total volum y on th link. Not that th fluid quu modl could b cast into th linar volum-dlay framwork by dcomposing ach link into a pur quuing sgmnt with travl tim z /ç that is, Å = 1/ç, Ç = ), followd by a link with constant travl tim í that is, Å =, Ç = í ). Strict FIFO fails prcisly bcaus th quuing sgmnt has Ç =. In this rspct it is worth noting that our xistnc rsults do not rquir strict FIFO, as long as í >, and Thorm 5 holds vn if í =. A gnral xistnc rsult for dynamic ntwork quilibrium byond strict FIFO was rcntly prsntd by Munir and Wagnr 21). Thir modl considrs both rout choic and dpartur tim choic and is basd on a wak form of strict FIFO: th travl tim T 4 5 strictly incrass on any intrval on which thr is som inflow into th link. This wakr proprty dos hold in our contxt, and th rsult applis providd that th inflow u4 5 blongs to L à loc 4 5. An intrsting fatur of th approach in 4, compard with prvious xistnc rsults, is that it provids a way to construct th quilibrium. In this rspct, our work ows much to Koch and Skutlla 211). Thr ar, howvr, svral diffrncs. On th modling sid, w distinguish th notion of dynamic quilibrium from th strongr dynamic quilibrium condition s Rmark 1). Both concpts wr usd intrchangably in Koch and Skutlla 211), although thy might diffr as shown in Exampl 2. In particular, our Thorm 1 prciss Koch and Skutlla 211, Thorm 1) which charactrizs dynamic quilibria, not strong quilibria. Also, Thorm 2 is an xtnsion of Koch and Skutlla 211, Thorm 2), which applis to th largr class of dynamic quilibria and provids a sharpr conclusion by including th normalization condition. Th xistnc and uniqunss for NTF s in Thorms 3 and 4 ar nw, and so is th subsqunt xistnc and uniqunss of a dynamic quilibrium in Thorms 5 and 6. To th bst of our knowldg, th lattr uniqunss rsult has not bn obsrvd prviously in th litratur. Th constructiv approach in 4 raiss a numbr of qustions. On th on hand, it would b rlvant to know if th stp sizs computd in stp 2) of th xtnsion mthod ar boundd away from. In this cas th à k s would not accumulat, and th quilibrium would b computd in finitly many stps for any givn horizon T. A rlatd qustion is whthr a stady stat could vntually b attaind with Å =àat som itration, in which cas th algorithm would b finit. A wakr but still difficult conjctur is whthr th quu lngths z 4 5 rmain boundd as long as th capacity of any s-t cut is larg nough, for instanc largr than th inflow at any point in tim. Th difficulty for proving such a claim is that th flow across a cut can b arbitrarily largr than th inflow: th quuing procsss might introduc dlay offsts in such a way that th flow ntring th ntwork at diffrnt tims rachs th cut simultanously at a latr dat, causing a suprposition of flows that xcds th capacity of th cut. On th othr hand, whil it is asy to giv a finit algorithm to comput thin flows with rstting, th computational complxity of th problm rmains opn. A polynomial tim algorithm for this would imply that for picwis constant inflows on could comput a dynamic quilibrium in polynomial tim in input plus output). Anothr intrsting qustion is whthr th constructiv approach in 4 can b adaptd to dal with mor gnral inflows u4à5. Mor prcisly, lt N4l1u 5 dnot th uniqu labls in a normalizd thin flow of valu u with rstting on th st E of all links = vw with l w >l v + í, and E th st of links with l w æ l v +í s Proposition 2). Rcalling Thorms 2 and 4, an quilibrium could b computd by solving th systm of ordinary diffrntial quations l 4à5 = N 4l4à51 u4à55 with initial condition l v 45 qual to th minimum s-v travl tim with mpty quus. Th cumulativ flows x 4à5 could thn b rcovrd by intgrating a masurabl slction of th corrsponding thin flows. Th main difficulty hr is that th map N is discontinuous in l so that th standard thory and algorithms for ordinary diffrntial quations do not apply dirctly. A final opn problm is to xtnd th constructiv approach to multipl origin-dstinations.