Value of Traveler Information for Adaptive Routing in Stochastic Time-Dependent Networks

Transcription

1 Univesity of Massachusetts Amhest Amhest Mastes Theses Febuay Value of Tavele Infomation fo Adaptive Routing in Stochastic Time-Dependent Netwoks He Huang Univesity of Massachusetts Amhest Follow this and additional woks at: Huang, He, "Value of Tavele Infomation fo Adaptive Routing in Stochastic Time-Dependent Netwoks" (2009). Mastes Theses Febuay Retieved fom This thesis is bought to you fo fee and open access by ScholaWoks@UMass Amhest. It has been accepted fo inclusion in Mastes Theses Febuay 2014 by an authoized administato of ScholaWoks@UMass Amhest. Fo moe infomation, please contact scholawoks@libay.umass.edu.

2 VALUE OF TRAVELER INFORMATION FOR ADAPTIVE ROUTING IN STOCHASTIC TIME-DEPENDENT NETWORKS A Thesis Pesented by HE HUANG Submitted to the Gaduate School of the Univesity of Massachusetts Amhest in patial fulfillment of the equiements fo the degee of MASTER OF SCIENCE IN CIVIL ENGINEERING Febuay 2009 Civil & Envionmental Engineeing

3 Copyight by He HUANG 2009 All Rights Reseved

4 VALUE OF TRAVELER INFORMATION FOR ADAPTIVE ROUTING IN STOCHASTIC TIME-DEPENDENT NETWORKS A Thesis Pesented by HE HUANG Appoved as to style and content by: Song Gao, Chai John Collua, Membe Michael Knodle, Membe Richad N. Palme, Depatment Head Depatment of Civil & envionmental Engineeing

5 ACKNOWLEDGMENTS The autho is indebted to the Univesity of Massachusetts (UMass) Amhest, the Depatment of Civil and Envionmental Engineeing and the School of Engineeing at UMass Amhest, and U.S. Dept. of Tanspotation (USDOT DTRT06-G-0032) fo funding the eseach. iv

6 ABSTRACT VALUE OF TRAVELER INFORMATION FOR ADAPTIVE ROUTING IN STOCHASTIC TIME-DEPENDENT NETWORKS FEBRUARY 2009 HE HUANG, M.S., Civil & Envionmental Engineeing UNIVERSITY OF MASSACHUSETTS AMHERST Diected by: Pofesso Song Gao Real-time infomation plays an impotant ole in taveles outing choices in an uncetain netwok by enabling online adaptation to evealed taffic conditions. The quality of the infomation affects its effectiveness. Usually thee ae some limitations in the infomation povided to the taveles, spatially, tempoally o both. In this thesis, thee vaiants of an optimal adaptive outing poblem with patial online infomation poblem ae intoduced: global infomation with time lag, global pe-tip infomation and adio infomation on a subset of links without time lag. A geneic desciption of online infomation is povided. An algoithm is designed fo the optimal outing poblem in stochastic time-dependent netwoks with patial online infomation and specializations equied fo each of the thee vaiants ae given. A test example is conducted and computationally veifies the non-negative value of infomation. The wok in this thesis is potentially of inteest to tavele infomation systems evaluation and design. v

7 TABLE OF CONTENTS Page ACKNOWLEDGMENTS... iv ABSTRACT...v LIST OF TABLES... viii LIST OF FIGURES... ix CHAPTER 1. INTRODUCTION Motivation Liteatue Review Contibutions PROBLEM DEFINITION The Netwok Online Infomation Event Collection The Decisions and the Optimal Routing Policy Poblem The Value of Infomation ALGORITHM DESIGN Patial Online Infomation Poblem Vaiants The Optimality Conditions Algoithm DOT-PART COMPUTATIONAL TESTS Objectives The Test Netwok Test Results CONCLUSION AND FUTERE DIRECTION Conclusion Futue Diection...38 vi

8 APPENDIX: AN ILLUSTRATIVE EXAMPLE FOR ALGORITHM DOT-PART...40 BIBLIOGRAPHY...60 vii

9 LIST OF TABLES Table Page Table 1-1 Taxonomy of the optimal outing policy poblem... 6 Table 2-1 Suppot points fo the Small Netwok Table 3-1 Relationship between CPU time (sec) and input vaiables in LAG vaiant Table A-1 Suppot points fo the Small Netwok Table A-2 Results in the Static Peiod and at Time 0 in POI vaiant Table A-3 Results in the Static Peiod and at Time 0 in NOI vaiant Table A-4 Results in the Static Peiod and at Time 1 and 0 in LAG vaiant Table A-5 Results in the Static Peiod and at Time 0 in PRE vaiant Table A-6 Results in the Static Peiod and at Time 0 in RADIO vaiant Table A-7. The expected tavel time fom node a to node c in all vaiants viii

10 LIST OF FIGURES Figue Page Figue 2-1 Algoithm DOT-PART: A Small Netwok Figue 2-2 A schematic view of S 1 containing S Figue 4-1 The test netwok Figue 4-2 The test esults Figue A-1 Algoithm DOT-PART: A Small Netwok ix

11 CHAPTER 1 INTRODUCTION 1.1 Motivation Congestion is an impotant woldwide tanspotation issue. In developed counties whee building moe infastuctues is usually politically, financially and envionmentally constained, a lot of effots have been devoted to making full use of cuent infastuctue system with the help of Intelligent Tanspotation Systems (ITS). Advanced Tavele Infomation System (ATIS), a sub-system of ITS, aims to povide taveles with eal-time infomation about netwok conditions, in the hope that bette infomed taveles can make bette decisions, and thus collectively the congestion would be elieved. In ode to assess an ATIS, a compehensive model is needed to take into consideation the demand-supply inteaction unde the influence of ATIS (Gao, 2008). This thesis deals with the demand side of the poblem, which descibes the optimal outing decisions a tavele can make with the help of ATIS and how much benefit can be obtained fom tavele infomation. Note that no demand-supply inteaction is modeled in this thesis, i.e., tavel times ae not affected by taveles choices. This is the study of the value of tavele infomation fo a single tavele in an uncetain netwok. The value of infomation povided by ATIS is most evident when taffic conditions ae stochastic. In a netwok whee taffic quantities ae almost cetain, taveles ae aleady quite well infomed and ATIS has little to povide. Stochasticity is a basic featue of congested taffic netwoks. One significant souce of the andomness is the distubances to taffic netwoks, such as incidents, vehicle beakdown, bad weathe, 1

12 wok zones, special events, etc. Taffic netwoks ae teated as stochastic to bette eflect eality and enable the modeling of infomation. Thee ae vaious implementations of ATIS, and they diffe in the spatial and tempoal availability, the quality, the fomat, and limitation of infomation povided. Fo example, a vaiable message sign (VMS) is usually fixed in location and thus only taveles passing it can obtain the infomation. It is also limited in the amount of infomation it can povide. Radio-based systems can povide infomation to taveles anywhee in the adio coveage. Relatively moe detailed infomation is available compaed to VMS, yet still the coveage is usually limited to majo highways and ateials. Besides the limitation on the spatial side, thee is also limitation on the tempoal side. Usually adio boadcast povides taffic condition infomation evey 15 minutes fo example, and so fo taveles thee is a time lag with the infomation. Intenet can also be an access to ATIS, poviding taveles with infomation such as camea images, tavel time estimations, wok zone and event schedule, and tavel advisoies. Howeve, once taveles ae en oute, they can hadly have access to intenet, and so intenet-based ATIS implementation is usually viewed as a pe-tip planne. Moe advanced in-vehicle systems ae also emeging, possibly with a database of oad map, tavel times unde nomal conditions, ecods of past incidents, etc., and can communicate with infomation centes to obtain vey detailed and updated infomation. Taveles' outing decisions in a stochastic netwok with online infomation is conceivably diffeent fom those in a deteministic netwok. It is geneally believed that adaptive outing will save tavel time and enhance tavel time eliability. Fo example, in a netwok with andom incidents, if one does not adapt to an incident scenaio, he/she 2

13 could be stuck in the incident link fo a vey long time. Howeve, if adequate online infomation is available about the incident, the tavele can avoid it by switching to an altenative oute. The adaptation also ensues that the tavel time is not pohibitively high in incident scenaios, and thus povides a moe eliable tavel time. It is theefoe a vey inteesting eseach question how an individual tavele makes adaptive outing decisions based on povided infomation in a stochastic and timedependent (STD) netwok, whose link tavel times ae andom vaiables with timedependent distibutions. In pevious wok Gao and Chabini (2006) and Gao (2005), the optimal outing poblem with pefect online infomation is studied and the value of pefect online infomation evaluated. This study is an extension of Gao and Chabini (2006) and Gao (2005) whee a numbe of patial online infomation situations ae consideed. A diffeent algoithm is equied, and it is a geneic one which can also solve the pefect online infomation poblem in Gao and Chabini (2006) and Gao (2005). The thesis is oganized as follows. Fist a liteatue eview is given and the contibutions of this thesis ae summaized. Next the optimal outing policy poblem in a stochastic time-dependent netwok is defined fo patial online infomation situations. Fou vaiants which ae paticulaly petinent in taffic netwoks ae then studied. A geneic algoithm which povides an exact optimal solution to the vaiants is pesented. Computational tests ae caied out and the esults ae given. Finally some conclusions ae made and futue eseach diections ae given. 3

14 1.2 Liteatue Review Two of the impotant chaacteistics of a netwok ae time-dependency (whethe a link cost is dependent on the aival time at the beginning of the link) and stochasticity (whethe link costs ae andom vaiables). Routing poblems in deteministic netwoks, both static and dynamic, have been impotant and well eseached topics fo a long time (see e.g., Ahuja et al. 1993, Chabini, 1998). Routing poblems in stochastic netwoks ae elatively less studied compaed to thei deteministic countepat. Two possible types of outing poblems exist in a stochastic netwok: non-adaptive and adaptive. Non-adaptive outing does not conside the fact that infomation on aival times on intemediate nodes and/o link tavel time ealizations will be available duing a tip, and thus a fixed path is detemined at the oigin node and followed egadless of the actual ealizations of the stochastic netwok. On the othe hand, adaptive outing consides intemediate decision nodes, and a next link (o sub-path) is chosen based on infomation collected thus fa. The adaptive outing poblem is the focus of the eview. Vaious assumptions ae made to define a stochastic netwok and how the ealizations of the stochastic netwok ae evealed to the taveles. Studies in both static and time-dependent (and stochastic) netwoks ae eviewed. In Andeatta and Romeo (1988), the topology of the static netwok is stochastic; in Polychonopoulos and Tsitsiklis (1996), the whole static netwok is descibed by a joint distibution of link tavel costs in the dependent case, and by maginal distibutions of link tavel times in the independent case; in Polychonopoulos (1992), Psaaftis and Tsitsiklis (1993) and Kim et al. (2005), the link costs evolve as Makov pocesses; in Hall (1988), Chabini (2000), Mille-Hooks and Mahmassani (2000), Petolani (2000), Mille-Hooks (2001), Yang and 4

15 Mille-Hooks (2004), Bande and White (2002), Fan et al. (2005b) and Opasanon and Mille-Hooks (2006), time-dependent netwoks ae descibed by maginal distibutions of link tavel times; in Gao and Chabini (2006), time-dependent netwoks ae descibed by joint distibution of tavel times of all links at all times; and in Walle and Ziliaskopoulos (2002), Fan et al. (2005a) and Boyles (2006), conditional pobabilities of adjacent link tavel costs ae utilized. As fo the evealing of netwok conditions, in Andeatta and Romeo (1988), Polychonopoulos and Tsitsiklis (1996), Cheung (1998), Fu (2001), Walle and Ziliaskopoulos (2002) and Povan (2003) it is assumed that one leans the ealization of a link tavel cost once he/she aives at the node fom which the link emanates; in Chabini (2000), Mille-Hooks and Mahmassani (2000), Mille-Hooks (2001), Yang and Mille-Hooks (2004), Bande and White (2002), Petolani (2000), Fan et al. (2005b), Opasanon and Mille-Hooks (2006) it is not stated explicitly how taveles lean about the netwok conditions othe than the aival times at decision nodes, hence the tem time-adaptive ; in Walle and Ziliaskopoulos (2002), Fan et al. (2005a) and Boyles (2006) it is assumed that taveles emembe only the tavel time on the last link they tavese; in Gao and Chabini (2006) it is assumed that taveles have knowledge about all link tavel time ealizations up to the cuent time; and in Psaaftis and Tsitsiklis (1993) and Kim et al. (2005) it is assumed that Makovian tavel times and thus taveles lean the cuent state of the Makovian chain at any time. The optimal adaptive outing poblem studies in stochastic time-dependent (STD) netwoks ae summaized in Table 1 1 with a taxonomy developed by Gao and Chabini (2006). A moe detailed eview follows. 5

16 Table 1-1 Taxonomy of the optimal outing policy poblem Infomation Netwok No link-wise and time-wise dependency Complete dependency Patial dependency Pefect online Gao and Chabini (2002, 2006) Patial online This thesis Psaaftis and Tsitsiklis (1993), Kim et al. (2005), Boyles (2006) No online infomation (time-adaptive) Hall (1987), Mille-Hooks and Mahmassani (2000), Chabini (2000), Petolani (2000), Mille-Hooks (2001), Bande and White (2002), Yang and Mille- Hooks (2004), Fan et al. (2005b), Opasanon and Mille-Hooks (2006) Hall (1986) studies fo the fist time the time-dependent vesion of the ORP poblem. It is shown that in an STD netwok, outing policies ae moe effective than paths. Chabini (2000) gives a dynamic pogamming algoithm based on the concept of deceasing ode of time (DOT). The algoithm is optimal in the sense that no algoithms with bette theoetical complexity exist. Mille-Hooks and Mahmassani (2000) develop a label-coecting algoithm. Insight into the diffeence between an optimal outing policy poblem and a least expected time path poblem is povided. Late Mille-Hooks (2001) compaes the said label-coecting algoithm and the dynamic pogamming algoithm woking in deceasing ode of time (Chabini, 2000) in both spase tanspotation netwoks and dense telecommunication data netwoks. Yang and Mille-Hooks (2004) also extend the study of the time-adaptive outing policies to a signalized netwok. Petolani (2000) uses a hype-path epesentation of the adaptive outing poblem based on aival times. Bande and White (2002) design a heuistic appoach with a pomising featue: it will teminate with an optimal solution if one exists, given that the heuistic function undeestimates the tue cost-to-go. The poposed heuistic has a 6

17 significant computational advantage compaed to dynamic pogamming, shown though computational tests. Fan et al. (2005b) maximize the pobability of aiving on time with continuous pobability density functions on link tavel times. Late in Fan and Nie (2006), algoithmic issues ae exploed fo the same poblem. Opasanon and Mille-Hooks (2006) study the time-adaptive poblem with multiple optimization citeia. Psaaftis and Tsitsiklis (1993) study the poblem in acyclic netwoks, implying that no link would be visited twice, so it is not helpful to keep infomation of any aleady tavesed links. This assumption along with the infinite hoizon assumption makes a polynomial unning time algoithm possible. Kim et al. (2005) study a simila poblem in a geneal netwok with a wide infomation ange. Boyles (2006) studies the poblem with minimum expected disutility, which is a geneal piece-wise polynomial function of aival time at the destination. 1.3 Contibutions Gao and Chabini (2006) establish a fomal famewok fo the poblem and design both exact and appoximation algoithms fo the poblem with pefect online infomation. This thesis builds on Gao and Chabini (2006) and the contibutions to the state of at ae theefold: A geneic epesentation of online infomation is povided fom which thee specializations of patial online infomation ae deived. The geneic epesentation povides a unified view towads outing poblems with online infomation. 7

18 A theoetical poof of the value of infomation is given which shows that in an adaptive outing context in an STD netwok, moe infomation is always bette (at least not wose) in a flow-independent netwok. A geneic algoithm fo a numbe of patial online infomation vaiants is designed. This enables the systematic study of the value of tavele infomation fo adaptive outing in an uncetain netwok whee a wide vaiety of infomation access situations can be modeled. 8

19 CHAPTER 2 PROBLEM DEFINITION 2.1 The Netwok Let G = (N, A, T,C ~ ) denote a stochastic time-dependent netwok. N is the set of nodes and A is the set of links, with N = n and A = m. It is assumed that thee is at most one diectional link fom node j to k, and thus a link can be denoted as (j, k). T is the set of time peiods {0, 1,, K-1}. A suppot point is defined as a distinct value (vecto of values) that a discete andom vaiable (vecto) can take. Theefoe a pobability mass function (PMF) of a andom vaiable (vecto) is a combination of suppot points and the associated pobabilities. Thoughout this thesis, a symbol with a ove it is a andom vaiable (vecto), while the same symbol without the is its suppot point. The tavel time on each link (j, k) at each time peiod t is a andom vaiable ~ C jk, t with finite numbe of discete, positive and integal suppot points. Beyond time peiod K-1 tavel times ae static, i.e., tavel times on link (j, k) at any time t > K-1 is equal to that at time K 1 fo any given suppot point. The time peiod fom 0 to K-1 is denoted as the dynamic peiod, while that beyond K-1 static peiod. It is geneally possible to model the peak peiod as dynamic, while off-peak as static when taffic is moe stable. {C 1,,C R } is the set of suppot points of the joint pobability distibution of all link tavel times at all times, whee C is a vecto of time-dependent link tavel times with a dimension K m, = 1, 2,, R. C, is the tavel time of link (j, k) at time t in the -th suppot point, which has jk t a pobability p, and p = 1. R = 1 9

20 An example netwok is shown in Figue 2-1 and Table 2-1 with 3 nodes, 3 links and 2 time peiods. Thee ae 3 suppot points, each with a pobability of 1/3, fo the joint distibution of 6 tavel time andom vaiables (links (a, b), (b, c) and (a, c) ove time peiods 0 and 1). A suppot point can be conveniently viewed as a day. Tavel times beyond time 1 ae the same as those at time 1 fo each of the 3 suppot points. b a c Figue 2-1 Algoithm DOT-PART: A Small Netwok Table 2-1 Suppot points fo the Small Netwok Time Link C 1 C 2 C (a, b) (b, c) (a, c) (a, b) (b, c) (a, c) p 1 = p2 = p3 = 1/ Online Infomation Let H be a tajectoy of (node, time) pais a tavele could expeience in the netwok to the cuent node j and time t: H = {(j 0, t 0 ),, (j, t)}, whee j 0 is the oigin, t 0 is the depatue time, j is the cuent node and t is the cuent time. Denote the infomation coveage on links and time peiods as Q A T. Infomation is epesented as the tavel time ealizations on time-dependent links in Q. It is assumed thee is no eo in evealing the tue tavel times, i.e., a 1 minute tavel time will be evealed as 1 10

21 minute, not any othe value. An infomation scheme is defined as a mapping fom tajectoy H to coveage Q, that is, infomation depends on tavesed locations and times. Hee ae examples of online infomation schemes with tajectoy H = {(j 0, t 0 ),, (j, t)}: Pefect online infomation (Gao and Chabini, 2006): Q POI (H) = A {0,1,,t} (all links up to the cuent time) Global infomation with time lag : Q LAG (H) = A {0,1,,t - } (all links up to time ago) Global pe-tip infomation with depatue time t 0 : Q PRE (H) = A {0,1,,t 0 } (all links up to the depatue time t 0 ) Radio infomation on B A with no time lag: Q RADIO (H) = B {0,1,,t} (a subset of links up to the cuent time) No online infomation (see e.g., Gao and Chabini, 2006): Q NOI (H) = (no infomation on any link at any time) The example in Figue 2-1 and Table 2-1 is used to illustate the diffeent infomation schemes. At time 0 and any node, a tavele with POI knows the tavel time ~ ~ ~ ealizations of { C ab, 0, C bc, 0, C ac, 0 } which could be eithe {1,2,3} o {1,1,2}; a tavele with global infomation with a lag of 1 minute does not know any tavel time ealization yet; a tavele with global pe-tip infomation with depatue time 0 has the same knowledge as with POI; a tavele with adio infomation on link (a, b) with no time lag ~ knows the tavel time ealization of C, 0 which is always 1; and a tavele with NOI simply does not know any tavel time ealization. ab 11

22 As the time moves fom 0 to 1, moe infomation could be obtained while that fom time 0 is kept. A tavele with POI knows the tavel time ealizations of ~ ~ ~ ~ ~ ~ C, C, 0, C, 0, C, 1, C, 1, C, 1 } which could be each of the 3 suppot points; a { ab, 0 bc ac ab bc ac tavele with global infomation with a lag of 1 minute knows what happened at time 0: ~ ~ ~ the tavel time ealizations of { C ab, 0, C bc, 0, C ac, 0 } which could be eithe {1,2,3} o {1,1,2}; a tavele with global pe-tip infomation with depatue time 0 does not gain any moe infomation en oute and thus his/he infomation emains unchanged ; a tavele with adio infomation on link (a, b) with no time lag knows the tavel time ~ ~ ealization of { C, 0, C, 1} which could be {1,1} o {1,2}; and a tavele with NOI still ab does not know any tavel time ealization. ab As the time moves fom 1 to 2, only the tavele with global infomation with a lag of 1 minute will gain moe useful infomation, as he/she now knows what happened in time 1. A tavele with POI, pe-tip o adio infomation does not gain any moe useful infomation because his/he infomation is always up-to-date and the infomation he/she had at time 1 is enough fo any time peiods beyond 1 due to the static peiod assumption. A tavele with NOI does not gain any moe infomation by definition. 2.3 Event Collection The concept of event collection is genealized fom that defined in Gao and Chabini (2006) to the case of a geneal infomation scheme. Let C ~ Q be the vecto of andom tavel times of all time-dependent links in Q. Fo a given suppot point C Q, thee exists one o moe suppot points C of the netwok, such that the tavel time on any 12

23 time-dependent link in Q is the same in both C Q and C. In othe wods, fo any possible evealed link tavel times in Q, a set of suppot points of the netwok that ae compatible with the infomation can be identified. Such a set is defined as an event collection, EV. As moe infomation is collected, infomation coveage Q gows and the size of EV deceases o emains unchanged. When EV becomes a singleton, a deteministic netwok (not necessaily static) is evealed to the tavele. If a tavele has pefect online infomation with Q POI = A {0, 1,, t}, the netwok becomes deteministic no late than the stat of the static peiod, i.e., K 1. When taveles have less than pefect online infomation, it is possible that the netwok emains stochastic beyond the dynamic peiod. In the example of Figue 2-1 and Table 2-1, it is assumed that a tavele has POI. At time 0 he/she eceived the infomation that tavel times on links (a, b), (b, c) and (a, c) ae 1, 2 and 3 espectively. By utilizing his/he a pioi knowledge of the joint distibution of link tavel times, he/she can infe that suppot points C 1 o C 2 ae possible as both povide compatible tavel times with what is evealed, while suppot point C 3 is not. Theefoe his/he event collection is {C 1, C 2 }. As the time moves fom 0 to 1, the tavele obtains moe infomation. If the newly evealed tavel times on links (a, b), (b, c) and (a, c) ae 1, 1 and 3 espectively, the tavele knows fo sue that suppot point C 1 will be ealized and his/he event collection is {C 1 }. Similaly, If the newly evealed tavel times on links (a, b), (b, c) and (a, c) ae 1, 2 and 2 espectively, the tavele knows fo sue that suppot point C 2 will be ealized and his/he event collection is {C 2 }. Similaly a tavele with global infomation with a lag of 1 minute has no idea which suppot point will be ealized at time 0 and his/he event collection is {C 1, C 2, C 3 }. At time 1, he/she knows link tavel times ealized at time 0, and is faced with the same 13

24 situation as a tavele with POI did at time 0. If the evealed tavel times on links (a, b), (b, c) and (a, c) at time 0 ae 1, 2 and 3 espectively, his/he event collection is {C 1, C 2 }. At time 2, he/she will have an event collection {C 1 } o {C 2 }. The same logic can be applied to othe infomation schemes. Note that fo NOI, the event collection emains as {C 1, C 2, C 3 } fo any time peiod. All the possible event collections with infomation coveage Q, denoted as EV(Q), can be geneated by pefoming a patition of {C 1,,C R } based on C ~ Q. EV(Q) = {EV 1, EV 2, }, whee C, is invaiant ove EV i, ((j, k), t) Q, i, and ((j, k), t) Q jk t ' such that C C, fo EV i, EV j, j i, i, j. In othe wods, suppot points in jk, t jk, t an EV ae undistinguishable in tems of evealed tavel times on links in Q, but ae distinctive fom those in anothe EV. All the possible event collections fo a given infomation scheme can be geneated in pepocessing. Hee ae some impotant facts about event collections: Thee is no ovelapping among elements of EV(Q), so thee ae at most R event collections at any cetain time and location ( EV(t) R); Any element EV of EV(Q) is a subset of one and only one element EV of a late EV (Q ): EV EV = o EV ; EV(Q) EV(Q ) ; The geneation of event collection can be caied out in inceasing ode of time, as the infomation coveage can only gow and late patitions can be done based on ealie ones. An example fom Figue 2-1 and Table 2-1 is shown hee fo a tavele with 14

25 up-to-date adio infomation on link (a, b). Since the infomation coveage depends only on the cuent time t, not the tajectoy, Q (H) can be simplified as Q (t) and EV (Q) as EV (t). At time 0, infomation coveage Q (0) = {(a, b)} {0}. The tavel time on link (a, b) at time 0 is 0 fo all 3 suppot points, so the patition yields only one event collection and EV (0) = {{C 1, C 2, C 3 }}. At time 1, infomation coveage Q (1) = {(a,b)} {0, 1} whee the incemental infomation is on {(a, b)} {1}. The patition can then be caied out on EV(0) based on tavel time ealizations of link (a, b) at time 1, which can be eithe 1 o 2. Theefoe EV(1) = {{C 1, C 2 }, {C 3 }}. Duing the static peiod, no moe useful infomation will be available, so EV (t) = {{C 1, C 2 }, {C 3 }} fo all t > 1. Anothe example is shown fo a tavele with global infomation with a lag of 1 minute. At time 0, Q (0) =, and thus EV (0) = {{C 1, C 2, C 3 }}. At time 1, Q (1) = {(a,b), (b,c), (a,c)} {0}. Fist check link-time pai ((a,b), 0) with only 1 possible value, and {{C 1, C 2, C 3 }} emains unchanged. Next check ((b,c),0) with 2 possible values and {{C 1, C 2, C 3 }} is patitioned as {{C 1, C 2 }, {C 3 }}. Lastly check ((a,c),0) and {{C 1, C 2 }, {C 3 }} emains unchanged because C 1 ac,0 and C 2 ac,0 ae the same, while {C 3 } is aleady a singleton. Theefoe EV (1) = {{C 1, C 2 }, {C 3 }}. Similaly EV (t 2) = {{C 1 }, {C 2 }, {C 3 }}. 2.4 The Decisions and the Optimal Routing Policy Poblem It is assumed that taveles can make decisions only at nodes. The decision is what node k to take next at each node, based on the cuent state x = {j, t, EV}, whee j is the cuent node, t is the cuent time, and EV is the cuent event collection. A outing 15

26 policy µ is defined as a mapping fom state to decision on the next node, fo all possible states and all possible next nodes out a given state, µ : x= { j, t, EV} a k. The next state ~ y= k, ~ t ', EV ' of the tavele is uncetain. The tavel time on link ~ (j, k) at time t given EV could be uncetain, esulting in an uncetain aival time t ' at node k. The next event collection the next infomation coveage ~ ~ EV ' is uncetain because: 1) t ' is uncetain and thus ~ Q ' is uncetain, e.g., at 8:00 with a possible tavel time of ~ 1 o 2 minute(s) on the next link, Q' could cove eithe 8:01 o both 8:01 and 8:02; 2) Even with a given Q and a given t, tavel times of links in Q between t and t' ae uncetain. Fo a tavele with up-to-date adio infomation on link (a,b) in Figue 2-1 and Table 2-1, let µ { a,0,{ C, C, C }} = c. The tavel time on link (a, c) could be eithe 3 o 2 given the event collection {C 1, C 2, C 3 }, with a pobability of 2/3 o 1/3. If the tavel time is 3, the event collection at node c will be an element of EV(3); if the tavel time is 2, the event collection at node c will be an element of EV(2). In this specific example, EV(3) = EV(2), but geneally they ae not equal. The tavele makes anothe decision at state y, and continues the pocess until the destination node is eached. The tavel time of a outing policy fom any initial state to a destination is a andom vaiable; a outing policy can be manifested as diffeent paths in diffeent suppot points. Definition 1: (Optimal outing policy poblem). The optimal outing policy (ORP) poblem in a stochastic time-dependent netwok is to find the outing policy that 16

27 optimizes an objective function to a given destination d, fo all possible states, i.e., all possible combinations of oigins, depatue times and event collections. The objective function can be expected tavel time, tavel time vaiance and expected tavel time schedule delay, o a combination of some of the citeia. 2.5 The Value of Infomation Let e(µ,x) be the objective function of following outing policy µ fom an initial state x, and e*(x) = min µ e(µ,x). The value of infomation is to be investigated theoetically in obtaining e*(x) and a theoem is to be poved that moe infomation is always bette (o at least not wose) in flow-independent netwoks. Two infomation schemes 1 and 2 in the same netwok ae to be studied. It is assumed that fo any tajectoy H, infomation scheme 2 has a lage coveage Q 2 than that of infomation scheme 1, Q 1 : Q 1 Q 2. Definition 1 (S 1 contains S 2 ). A patition of set S is a set of subsets which ae mutually exclusive and collectively exhaustive of S. Let S 1 and S 2 be two patitions of S. S 1 contains S 2 if fo any y S 2, thee exists z S 1, such that y z and y z =, z z. In othe wods, any element of S 2 is a subset of one and only one element of S 1. See Figue 2 fo a schematic epesentation. S a b c d e f g h S 1 a b c d e f g h S 2 a b c d e f g h Figue 2-2 A schematic view of S 1 containing S 2 Lemma 1. EV(Q 1 ) contains EV(Q 2 ), fo any tajectoy H. 17

28 Poof: By definition, EV(Q 1 ) and EV(Q 2 ) ae two patitions of the set of netwok suppot points {C 1,,C R }. Assume by contadiction that EV(Q 1 ) does not contain EV(Q 2 ), then thee exists EV 2 EV(Q 2 ), such that fo any EV 1 EV(Q 1 ), EV 2 EV 1 EV 2. As EV(Q 1 ) and EV(Q 2 ) ae patitions of the same set, thee must exist EV 1, such that EV 2 EV 1. By definition tavel times on all time-dependent links in Q 2 is invaiant acoss suppot points in EV 2. As Q 1 Q 2, tavel times on all time-dependent links in Q 1 is also invaiant acoss suppot points in EV 2, specifically fom EV 2 EV 1 to EV 2 \(EV 1 EV 2 ). Since EV 1 EV 2 and EV 2 \(EV 1 EV 2 ) ae subsets of two distinctive elements of EV(Q 1 ), by definition tavel times on all time-dependent links in Q 1 vay fom EV 1 EV 2 to EV 2 \(EV 1 EV 2 ). Thee is a contadiction and this completes the poof. Theoem 1. The optimal objective function value unde infomation scheme 2 is no wose than that unde infomation scheme 1, fo the same oigin j 0, depatue time t 0, and event collections EV 2 and EV 1, whee EV 2 is a subset of EV 1. Mathematically * * e2 ( j0, t0, EV2 ) e1 ( j0, t 0, EV1 ), j N t,. 0, 0 T EV1, EV2 EV2 EV1 Poof: It is to be shown that any feasible outing policy µ 1 unde infomation scheme 1 is equivalent to at least one feasible outing policy µ 2 unde infomation scheme 2. It is poved by constuction. At the initial state, set µ 2 ( j 0, t0, EV2 ) = µ 1 ( j0, t0, EV1 ). Upon aival at the next node j 1 at time t 1, the infomation coveage Q 1 is a subset of Q 2 fom the tajectoy {(j 0, t 0 ), (j 1, t 1 )}. By Lemma 1, EV(Q 1 ) contains EV(Q 2 ), theefoe set ' ' ' ' ' µ 2 ( j1, t1, EV2 ) =µ 1 ( j1, t1, EV1 ), EV2 EV ( Q2 ), EV2 EV1. The pocess continues and a outing policy µ 2 is obtained defined ove infomation scheme 2 which poduces exactly the same tajectoy as µ 1, and thus the same objective function value. Theefoe 18

29 thee exists a feasible outing policy unde infomation scheme 2 with the same objective function value as the optimal outing policy unde infomation scheme 1 and the optimal objective function unde scheme 2 is at least as good as that unde scheme 1. This completes the poof. 19

30 CHAPTER 3 ALGORITHM DESIGN 3.1 Patial Online Infomation Poblem Vaiants In ode to study the value of infomation in the context of optimal adaptive outing in a flow-independent netwok, algoithms ae designed to solve the optimal outing policy (ORP) poblem with patial online infomation. The vaiants consideed ae petinent in a taffic netwok: Global infomation with time lag (LAG). Fo example, at 7:00 taveles only have infomation about taffic conditions up to 6:45. Global pe-tip infomation with depatue time t 0 (PRE). Fo example, taveles get pe-tip infomation fom intenet befoe they stat the jouney. Once depated, they can no longe get online fo moe infomation. Theefoe EV(t) = EV(t 0 ), t t 0. Infomation on a subset of links with no time lag (RADIO). Fo example, only taffic conditions on seveal majo highways and ateials will be epoted in a adio boadcast. No online infomation (NOI). This can be viewed as a special case of patial online infomation. A geneic algoithm is pesented based on geneic optimality conditions fo the fou patial online infomation poblem vaiants and pefect online infomation (POI) vaiant. It can be shown that the geneic algoithm is equivalent to Algoithm DOT-SPI in Gao and Chabini (2006) which is designed to solve the POI vaiant only. 20

31 Some chaacteistics of the five vaiants ae: In all vaiants, infomation coveage Q is detemined by the cuent time, instead of the whole tajectoy, theefoe EV(t) is used instead of EV(Q). Note that time lag in LAG, depatue time t 0 in PRE and adio coveage B in RADIO ae teated as exogenous system paametes. With the exception of LAG, in all othe vaiants taveles eceive no moe useful infomation duing the static peiod, i.e., Q does not gow beyond time K 1, eithe because no infomation is povided by definition (PRE and NOI), o additional infomation will not enlage Q (RADIO and POI); In the case of LAG with a time lag, a tavele continues eceiving infomation beyond the static peiod until K 1 +, at which time Q = A T. Let T* denote the time beyond which a tavele eceives no moe infomation, and thus T* = K 1 + fo LAG, and T* = K 1 fo all othe fou vaiants (PRE, RADIO, POI and NOI). Conside the outing decision making beyond T*. The event collection will emain the same duing all futue time peiods as that at time T*, EV EV(T*), since no moe infomation will be eceived. The tavel times ae also static by definition. It is like taveling in a static and stochastic netwok defined by EV with no infomation. An optimal outing poblem in such a netwok can be solved by a classical static shotest path algoithm in a conveted deteministic netwok by taking link tavel time means. 3.2 The Optimality Conditions Since link tavel times ae andom vaiables, thee exist multiple optimization citeia. The expected tavel time is used in the emaining of the thesis, as geneally it is 21

32 the pimay citeion in outing choices. Othe citeia egading tavel eliability, such as tavel time vaiance and expected tavel time schedule delay, and a combination of some of the citeia, will be exploed in futue eseaches. Let e µ (j, t, EV) be the expected tavel time to the destination node d if the depatue fom node j happens at time t with the event collection EV by following outing policy µ. S µ (j, t, ) is the tavel time to the destination node d if suppot point is ealized with a depatue fom node j at time t by following outing policy µ. The elationship between e µ (j, t, EV) and S µ (j, t, ) is as follows: e µ ( j, t, )P( EV ) (1) ( j, t, EV ) = Sµ EV The outing policy is defined on event collection, not suppot point. Howeve, fo each suppot point, a outing policy is manifested as a path with a cetain tavel time. Fo example, fo a tavele with up-to-date adio infomation on link (a,b) in Figue 2-1 and Table 2-1, the outing decision at node a at time 0 can only be made based on the event collection {C 1, C 2, C 3 }. Let µ { a,0,{ C, C, C }} = c. The tavel time by following outing policy µ stating fom node a at time 0 is a andom vaiable with possible 1 2 diffeent outcomes in diffeent suppot points: S µ ( a,0, C ) = 3, ( a,0, C ) S µ =3, and 3 S µ ( a,0, C ) =2. The elationship between S µ at node j and the succeeding node k by following µ is citical to solving the ORP poblem. S µ (j, t, ) is defined fo a tip depating at time t. Fo the vaiants POI, LAG, RADIO and NOI, the infomation coveage is not a function of depatue time, and thus the event collections at time t is the same no matte whethe t is the depatue time o not. In this case, 22

33 Sµ j, t, ) = C + Sµ ( k, t C, ), whee k = µ ( j, t, EV ), EV (2) ( jk, t + jk, t Fo the PRE vaiant, howeve, the infomation coveage does depend on depatue time, and thus in geneal (2) does not hold. A diffeent vaiable S µ j, t, ; t ) can then ( 0 be defined as the tavel time fom node j and time t to the destination node if suppot point is ealized by following outing policy µ, with a depatue time t 0. Similaly e µ j, t, EV ; t ) and µ j, t, EV; t ) can be defined. In this case, ( 0 ( 0 Sµ ( j, t, ; t0 ) = C, + Sµ ( k, t C,, ; t0 ), whee k = µ j, t, EV; t ), EV jk t + jk t µ ( j, t, EV; t0 ) = Sµ ( j, t, ; t0 EV ( 0 e )P( EV ) Poposition 1: Fo the POI, LAG, RADIO and NOI vaiants, the minimum expected tavel time *( j, t, EV ), j N\{d}, t, EV EV(t) and optimal outing policy e µ µ* ae solutions to the following system of equations: ( C + S ( k, t+ C, ) ) e µ ( j, t, EV ) = min jk t jk t P( EV ) *, *, k A( j) µ (3) EV ( C + S ( k, t+ C, ) ) µ *( j, t, EV ) = ag min jk, t µ * jk, t P( EV ) (4) k A( j) EV whee S j, t, ) C + S ( k*, t C, ), * µ *( = jk *, t µ * + jk*, t k µ *( j, t, EV ) set of downsteam nodes out of node j. The bounday conditions ae: =, EV. A (j) is the 1) At the destination: e µ ( d, t, EV ) 0, µ * ( d, t, EV ) = d, t, EV EV(t). * = 23

34 * * 2) Beyond T*: µ ( j, t T*, EV ) = µ ( j, T*, EV ), j, EV EV(T*), whee T*=K 1+ fo LAG, and T*= K 1 fo all othe 3 vaiants (RADIO, POI and NOI). Poof: (Necessity). The necessity can be poved by showing that if *( j, t, EV ) is the minimum expected tavel time and µ* the optimal outing policy, they must satisfy the system of equations (3) ~ (4). e µ not optimal. Tivially, if the bounday conditions at the destination node ae not satisfied, µ* is The optimal outing policy beyond T* is not a function of time t, because both the tavel times and event collections do not change ove time. Thus * * µ ( j, t T*, EV ) = µ ( j, T*, EV ), j, EV EV (T*). Futhe making use of (1) in (3) and (5) and the following ae obtained: e ( j, T*, EV ) = min C, * P( EV ) + e *( k, T*, ) * ( ) EV µ jk T µ (5) k A j EV µ *( j, T*, EV ) = ag min C jk, T* P( EV ) + eµ *( k, T*, EV ) (6) k A( j) EV These ae the optimality conditions of a static shotest path poblem in a conveted deteministic netwok whee link tavel times ae eplaced by thei means EV C, P( EV ) at T* given an event collection EV EV(T*). In a static stochastic jk T* netwok, the expected path tavel time is equal to the sum of the expected link tavel times along the path, and theefoe the minimum expected time path is the same as the shotest path in a conveted deteministic netwok whee link times ae eplaced by thei 24

35 means. If µ* is optimal, it must manifest as the shotest path in each of the conveted deteministic netwok defined by EV, and thus (5) and (6) must be satisfied. Assume by contadiction that (3) and (4) ae not satisfied fo some state with a depatue time ealie than T*. Let (j, t, EV) be such a state. Theefoe thee must exist an outgoing node k A (j), such that ( C jk ', t + Sµ *( k', t+ C jk ', t, ) ) P( EV ) < ( C jk*, t + Sµ *( k*, t+ C jk*, t, ) ) EV EV P( EV ) Theefoe a diffeent outing policy µ can be constucted such that µ (j, t, EV) = k, and µ = µ* fo all othe states. Then the following is obtained: e = < µ '( j, t, EV ) = Sµ '( j, t, )P( EV ) = ( C jk ', t + Sµ '( k', t+ C jk ', t, ) ) EV EV ( C jk ', t + Sµ *( k', t+ C jk*, t, ) ) P( EV ) EV ( C jk*, t + Sµ *( k*, t+ C jk*, t, ) ) P( EV ) = eµ *( j, t, EV ) EV P( EV ) which is contadicted with fact that µ* is optimal. (Sufficiency) The sufficiency can be poved by showing that if e µ *( j, t, EV ) and µ* satisfy the system of equations (3) ~ (4), they must be the minimum expected tavel time and the optimal outing policy espectively. Assume by contadiction that µ* is not optimal, theefoe thee must exist a outing policy µ such that e µ (j, t, EV) < e µ* (j, t, EV) fo some (j, t, EV). Fom the poof of necessity, (5) ~ (6) ae also the sufficient condition fo µ* to be optimal at o beyond T*. Theefoe t < T*. Assume t is the latest time when the inequality occus, and thus e µ (j, t, EV) = e µ* (j, t, EV), t >t. Assume µ(j, t, EV) = k: 25

36 e = = µ = = = e ( j, t, EV ) = EV EV C jk, t P( EV ) + eµ *( k, t+ C EV ( C jk, t + Sµ *( k, t+ C jk, t, ) ) P( EV ( C jk*, t + Sµ *( k*, t+ C jk*, t, ) ) EV µ * C C jk, t jk, t ( j, t, EV ) EV S µ P( EV ) + P( EV ) + e ( j, t, ) P( EV ) = EV µ S µ ( k, t+ C ( k, t+ C jk, t jk, t jk, t, EV ), EV ) EV ) P( EV ) ( C jk, t + Sµ ( k, t+ C jk, t, ) ) EV, ) P( EV ) P( EV ) Theefoe the assumption is not valid and µ* is optimal. Poposition 2: Fo the PRE vaiant with depatue time t 0, the minimum expected tavel time j, t, EV ; ), j N\{d}, EV EV(t) and optimal outing policy µ* ae e µ *( t0 solutions to the following system of equations: ( C + S ( k, t+ C, ; t )) eµ *( j, t, EV; t0 ) = min jk, t µ * jk, t 0 P( EV ) (7) k A( j) EV ( C + S ( k, t+ C, ; t )) µ *( j, t, EV; t ) = ag min jk, t * jk, t 0 P( EV ) (8) 0 k A( j) µ EV whee S j, t, ; t ) C + S ( k*, t C, ; ), * µ *( 0 = jk *, t µ * + jk*, t t0 k µ *( j, t, EV ; t0 ) =, EV. A(j) is the set of downsteam nodes out of node j. The bounday conditions ae: 1) At the destination: e µ d, t, EV ; t ) 0, µ ( d, t, EV; t ) = d, t, EV EV(t). *( 0 = * 0 * * 2) Beyond T*: µ j, t T*, EV ; t ) = µ ( j, T*, EV ; t ) ( 0 0, j, EV EV(T*), whee T*=K 1. 26

37 The poof of Poposition 2 is simila to that of Poposition 1 with notation change only. Poposition 3: Optimality conditions (3) ~ (4) ae equivalent to the optimality conditions fo the pefect online infomation (POI) vaiant in Gao and Chabini (2006). The poof of Poposition 3 is done by making use of the fact that with POI, the tavel time on an outgoing link is a deteministic value given an event collection. 3.3 Algoithm DOT-PART The evaluation of e µ *( j, t, EV ) only depends on S µ *( j, t', ) fom a late time t > t, due to the positive and integal link tavel time assumption. Theefoe the labels can be optimally set in a deceasing ode of time, making use of the acyclic popety of the netwok along the time dimension. At time T* and beyond, any deteministic static shotest path algoithm can be used to compute *( j, t, EV ) e µ, j N, t T*, EV EV(T*). The pocedue to geneate event collections cay out patitions of the netwok suppot points in an inceasing ode of time. At time t, a patition is made on EV(t-1) based on each (link, time) pai in the incemental infomation coveage, Q(t)\Q(t- 1). Note that Q is witten as a function of t, because in all the 5 vaiants, Q only depends on t, not the tajectoy. The algoithm solves the ORP poblem fom all initial states fo POI, LAG, RADIO and NOI, but only fom depatue time 0 fo PRE. In ode to solve PRE vaiant with all depatue times, an oute loop ove all depatue times t 0 has to be added to the main loop, and the main loop ove time t will be fom T*-1 down to t 0. This is because the event collection at time t when t is the depatue time is diffeent fom when t is not. 27

38 Fo any othe vaiant, the event collection at t is the same egadless of the depatue time. Adding an oute loop is not the most efficient implementation to solve the PRE vaiant. Howeve since the focus of this thesis is the study of the value of tavele infomation, a coect implementation is enough. Moe efficient implementations will be exploed in futue eseach. Algoithm DOT-PART (Geneic fo the 5 vaiants: POI, LAG, PRE with depatue time 0, RADIO and NOI) Initialization Step 1: If infomation scheme = LAG with a lag of then T* = K 1 + else T* = K 1 Constuct EV(t), t = 0,, T* by calling Geneate_Event_Collection (see the statement below) Step 2: Compute e µ *( j, T*, EV ) and µ * ( j, T*, EV ), j N, EV EV ( T*) with a static deteministic shotest path algoithm in a conveted static deteministic netwok whee link tavel times ae eplaced by thei means at time T*. Compute S µ ( j, T*, ) by executing µ* in the oiginal static stochastic netwok, * j N, EV ; j, t > T*, ) S ( j, T*, ). S µ *( = µ * 28

39 Step 3: e µ* (j, t, EV) +, j N \{ d}, t< T*, EV EV ( t) e µ* (d, t, EV) 0, t< T*, EV EV ( t) Main Loop Fo t = T*-1 down to 0 Fo each EV EV(t) Fo each link (j, k) A ( C jk, t + Sµ*( k, t+ C jk, t, ) ) temp = P( EV ) EV If temp_e < e µ* (j, t, EV) then e µ* (j, t, EV) = temp_e µ*(j, t, EV) = k Fo each EV and each j N k* = µ*(j, t, EV) S *( j, t, ) C *, + S *( k*, t C *,, ) µ = jk t µ + jk t Geneate_Event_Collection D = {C 1,,C R } Fo t = 0 to T* If infomation scheme = POI Q(t) = A {0,1,,t } If infomation scheme = LAG with a lag 29

40 Q(t) = A {0,1,,t - } If infomation scheme = PRE with depatue time 0 Q(t) = A {0} If infomation scheme = RADIO with link set B Q(t) = B {0,1,,t} Q(-1) = //a poxy fo the convenience of epesentation Fo t = 0 to T* Fo each (link, time) pai ((, k), t' ) j Q(t) \ Q(t-1) Fo each disjoint subset S D D A patition of S based on C ~ jk, t D Union of all D EV(t) D; Following a simila analysis as in Gao and Chabini (2006), it can be deived that Algoithm DOT-PART has a complexity of O(mKRlnR + R SSP) and Ω(mKR + SSP), whee SSP is the complexity of the static deteministic shotest path algoithm. The algoithm is stongly polynomial in R, the numbe of suppot points of the link tavel time joint distibution. Fo eal applications, time-dependent tavel time obsevations on all links fom each day can be viewed as one suppot point. A unning time test is conducted with andomly geneated netwoks on a Dell Optiplex with 2.40GHz Intel Coe 2 CPU and 2.00GB of RAM. Details of the andom netwok geneato can be found in Gao (2005). The numbe of nodes (n), the numbe of time peiods (K), and the numbe of suppot points (R) ae chosen as input vaiables; the 30

41 numbe of links (m) is thee times as geat as the numbe of nodes. Random numbes fom multivaiate nomal distibutions ae geneated fo link tavel times. The elationship between unning time of the algoithm and the input vaiables fo the LAG vaiant is shown in Table 3-1. It can be seen that the elationship between unning time and each of the 3 input vaiables is close to linea. Simila tests ae conducted fo othe vaiants and the elationships ae simila. Table 3-1 Relationship between CPU time (sec) and input vaiables in LAG vaiant Running time of Geneate_Event_Collection m K R Running time of DOT-PART fo LAG vaiant (excluding Geneate_Event_Collection) m K R Fo applications in eal life size netwoks, the computational time is not the constaint, but the memoy is. One possible solution is to change the epesentation of link tavel times fom discete time based to continuous time based. A piece-wise linea epesentation of link tavel times has been implemented and computational tests have been conducted in a Swedish city netwok with about 7500 diectional links, time peiods and 30 suppot points. The esults in eal-life netwoks will be epoted in succeeding eseaches. 31

42 CHAPTER 4 COMPUTATIONAL TESTS 4.1 Objectives The objectives of the computational tests ae to: 1) compae computationally the optimal expected tavel times of each of the thee patial online infomation vaiants and no online infomation (NOI) with pefect online infomation (POI); 2) compae computationally the optimal expected tavel times of patial online infomation vaiants with the same type of infomation but diffeent system paametes; 3) show the value of infomation and veify the theoetical esult deived in Section 2.5 that moe infomation is always bette (o at least not wose) in a flow-independent netwok. 4.2 The Test Netwok Figue 4-1 The test netwok The test netwok is shown in Figue 4-1 with 6 nodes and 8 diected links. Thee ae divesion possibilities at nodes 0, 1 and 2. The study peiod is fom 6:30am to 8:00am. The time esolution is 1 minute fo depatues and aivals at intemediate nodes, and thee ae 90 time peiods in total. The tavel time is in seconds. 32

43 The link tavel time distibution is geneated though an exogenous simulation with the mesoscopic supply simulato of DynaMIT fom Ben-Akiva, et al. (2001). The demand between OD (0, 5) is low between 6:30am to 7:00am and highe late on. Thee ae andom incidents in the netwok defined as follows: 1) Thee is at most one incident fo any given day; 2) The incident has a positive pobability of occuence on link 0, 2, 4 and 6, but zeo on link 1, 3, 5 and 7; and 3) If an incident occus on a link, the stat time can be evey 10 minutes with equal pobability. The 4 possible locations and 9 possible stat times esult in (no incident) =37 suppot points. Details of the netwok can be found in Gao (2005). 4.3 Test Results Algoithm DOT-PART is un fo the thee patial online infomation schemes, no online infomation (NOI) and pefect online infomation (POI) to deive the minimum expected tavel times fom each of the vaiants fom node 0 to node 5 fo all depatue times and all event collections. The esults ae aggegated by depatue time, by taking an expectation ove all event collections at a given time. Figues 3.a and 3.b show the esult fo the LAG (global infomation with time lag ) vaiant: LAG5 indicates thee is a 5 minutes infomation time lag, and LAG10 and LAG15 espectively a 10 minutes and 15 minutes lag. It can be seen that the following elationship holds: POI LAG5 LAG10 LAG15 NOI. Figue 3.c shows the esults fo the PRE (global pe-tip infomation) vaiant. It can be seen that the following elationship holds: POI PRE NOI. 33