Adaptive Route Choice Models in Stochastic Time-Dependent Networks

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Stohasti Time-Dependent Networks

S. Gao E. Frejinger M. Ben-Akiva

July 7, 2007

Report TRANSP-OR 070730

Transportand Mobility Laboratory

Shool ofArhiteture, Civil andEnvironmentalEngineering

Eole Polytehnique Federale de Lausanne

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Abstract

We study adaptive route hoie models that expliitly apture

travelers' route hoie adjustments aording to information on

real-ized network onditions instohasti time-dependent networks. Two

typesof adaptive route hoie modelsareexplored: anadaptive path

model where a sequene of path hoie models are applied at

inter-mediate deision nodes; and a routingpoliy hoie model wherethe

alternativesorrespondtoroutingpoliiesratherthanpathsatthe

ori-gin. Aroutingpoliyinthispaperisadeisionrulethatmapsfromall

possible(node,time)pairstonextlinksoutofthenode. Apoliy-size

Logit model is proposed forthe routing poliy hoie, where

poliy-sizeisageneralization ofpath-sizeinpathhoie models totakeinto

aount the overlapping of routing poliies. The speiations of

es-timating the two adaptive route hoie models are established and

the feasibility of estimation from path observations is demonstrated

on an illustrative network. Predition results from three models

-non-adaptive path model, adaptive path model, and routing poliy

model - are ompared. The routing poliy model is shown to better

apture the option value of diversion than the adaptive path model.

Thedierenebetweenthetwoadaptivemodelsandthenon-adaptive

modelislargerintermsofexpetedtraveltime,ifthenetworkismore

stohasti, indiating that the benet of being adaptive is more

sig-niant ina morestohastinetwork.

1

Introduction

Transportation systems are inherently unertain due to disturbanes suh

as inidents,vehile breakdowns,workzones,bad weather onditions,

spe-ialeventsandsoforth. Ontheotherhand,real-timeinformationinvarious

formatsisavailable,frompersonalobservations, websites, variablemessage

signs (VMS), radio broadasts, ell phones to personal in-vehile systems.

Real-time information an redue the unertainty of the traÆ network,

and therefore potentially help traveler make better route hoie deisions.

Travelers usually obtain updated information at various deision points

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essofaseriesofroutehoieswiththepreseneofreal-timeinformationin

astohastinetwork,isofgreatinterest,sineitisritialtotheevaluation

of any real-time informationsystem. We refer tomodels that apture this

proessas adaptiveroutehoiemodels, inthatthe hoies areadaptedto

the realized network onditionsrevealed byreal-time information.

Mostdisretehoiemodelsforroutehoieanalysisarebasedon

deter-ministinetworks. ExamplesofsuhmodelsarePathSizeLogit(Ben-Akiva

and Ramming, 1998; Ben-Akiva and Bierlaire, 1999), C-Logit (Casetta

etal.,1996),Cross-NestedLogit(VovshaandBekhor,1998),andLogit

Mix-ture (Ramming, 2001; Bekhor et al., 2002; Frejinger and Bierlaire, 2007).

In this paper we refer to these models as non-adaptive path hoie

mod-els beause travelers are assumed to make their omplete path hoie at

the origin. The fat that travelers adjust their route hoies en-route in

response to revealed traÆ onditionsis therefore ignored.

A seemingly natural way to build adaptive route hoie models is to

have a sequene of path hoie models at deision nodes, where the

at-tributesof alternativepathstothedestinationreetupdatedinformation.

Anyof the abovementionedroute hoie modelswith adequate

inorpora-tion of real-time information ould in priniple be applied suessively in

a stohasti network to model adaptive route hoie behavior. DynaMIT

(Ben-Akiva et al., 2002) and DYNASMART (Mahmassani, 2001) are

ex-amples of dynami traÆ assignment models that apply an adaptive path

hoiemodel. Calibrationof DynaMIT'sroute hoie modelbasedon eld

data is reported in Balakrishna(2006) and Balakrishna et al. (2007).

There have been a large number of studies of path hoie models with

real-time information, both pre-trip and en-route, and a reent literature

reviewanbefoundinAbdel-AtyandAbdalla(2006). Somemodelspredit

the deision to swith from a previous hosen or experiened route (e.g.

Polydoropoulou et al., 1996; Abdel-Aty and Abdalla, 2004; Mahmassani

and Liu, 1999; Srinivasan and Mahmassani, 2003); others are route hoie

models with expliit paths as alternatives (e.g. Bogers et al., 2005; Peeta

and Yu, 2005; Abdel-Atyand Abdalla, 2006).

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spe-formation is available or not (e.g. Abdel-Aty and Abdalla, 2004;

Poly-doropoulou et al.,1996; SrinivasanandMahmassani, 2003; Abdel-Atyand

Abdalla,2006), proxies suh asqueuelength (Bogerset al.,2005)or travel

time (Mahmassani and Liu, 1999; Srinivasan and Mahmassani, 2003) or

fuzzyvariableswithontinousmembershipfuntions(PeetaandYu,2005).

Most of these models were estimated with interative simulation data or

syntheti data(with theexeption of Polydoropoulouet al.(1996)),whih

suggests the diÆulty of obtaining eld data. Among the models,

Srini-vasan and Mahmassani (2003) and Abdel-Aty and Abdalla (2006) use

ob-servations from all deision nodes during a trip to estimate the models,

whih ould be treatedas paneldata,whileothers use ross-setional data

whihdonotonsiderthesuessiveroute hoieadjustmentduringatrip.

In this paper, we propose a novel adaptive route hoie model where

the alternativesare not paths, but routing poliies. Generally speaking,a

routing poliy is a deision rule that maps all possible network states in a

stohasti networkto deisions,whileapath is axedset of links without

inoporatinginformationorstohastiity. Thedenitionofaroutingpoliy

depends on the underlying stohasti network and the information aess

(Gao and Chabini, 2006). Some researhers refer to it as strategy,

hyper-path oronlinepath withreourse. Theliterature inludesanumbers of

al-gorithmistudies ofoptimalroutingpoliyproblems(e.g.Hall,1986;

Poly-hronopoulos and Tsitsiklis, 1996; Marotte and Nguyen, 1998; Pretolani,

2000; Miller-HooksandMahmassani,2000; Miller-Hooks,2001;Wallerand

Ziliaskopoulos, 2002; Gao, 2005; Gao and Chabini, 2006), however

eono-metri models of routing poliy hoie is a new area. Ukkusuri and Patil

(2006) applied sequential logit loading of hyperpath ows in an

equilib-rium traÆ assignment, where travelers were assumed to learn realized

travel times on outgoing links. However the estimation problem was not

addressed. This paper therefore is the rst researh eort to develop an

estimator of arouting poliy hoie model anddemonstrate the feasibility

of estimatingsuh a model.

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the dierenebetween thetwo adaptivemodels. InSetion 4weformulate

the estimationproblemsof adaptivepath hoie androuting poliy hoie

in astohasti time-dependent network, where the observations are

man-ifested paths,and the hoie of routing poliyis latent. Setion 5 ontains

the numerialexperiment setup, estimationresults of three models -

non-adaptive path,adaptive pathand routingpoliy- fromsyntheti dataand

disussions of predition results to gain insights into the adaptive route

hoie models. Conlusionsand future researh diretionsare provided in

Setion 6.

2

Background

Westudyadaptiveroutehoiemodelsinastohastitime-dependent

net-work, where the travel time on eah link

ℓ

= (

v, w

)

, with soure node

v

and sink node

w

, for an arrival time

t

at

v

is a random variable

e

T

ℓ,t

with nite number of disrete, positive and integral support points. A support

point is dened as a distintive value (vetor of values) a disrete random

variable (vetor) an take, and thus the probability mass funtion (PMF)

of arandom variable (vetor)is theombinationof supportpointsandthe

assoiated probabilities. In this paper, a symbol with a

∼

over it is a ran-dom variable,whilethe samesymbol withoutthe

∼

is one speivalue of the random variable, whihsometimes mightbe supersripted with an

in-dexfor supportpoint. Sine linktravel timeis random,a travelerentering

a link at a given time might exit the link at dierent times, whih might

result in dierent travel time PMFs on the next link. A traveler with a

hosen path will take a xed link out of an intermediate node regardless

of the possible dierentarrivaltimes at thenode. In anotherword, apath

is purely topologial. On the other hand, if atraveler has a priori

knowl-edgeofthePMFsoftime-dependentlinktraveltimesaswellastherealized

arrivaltimesatnodes,he/sheanmakeadaptiveroutehoiesaordingly.

Two types of adaptive route hoie models are studied in this paper:

adaptive path model and routing poliy model. Travelers are assumed to

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model, at eah intermediate deision node and for eah possible arrival

time,thetravelerseletsamongasetof pathstothedestination,andtakes

the rstlink ofthe hosen path. Onethetravelerarrives atthesink node

of the link(with randomarrivaltime),he/shemakes anotherhoieout of

a new set of paths fromthat node to the destination,whose attributesare

updated based on the atual arrival time. He/She then again follows the

rst linkof thehosen path,whihisnotneessarilytheseondlinkonthe

hosen path from the previous deision node. An adaptive path appears

to besuperior toa non-adaptive pathwhih ignoresinformation onatual

arrival time at intermediate nodes, yet the hoie is stillshort-sighted. At

eah deision node, the next link is hosen based on a path, and thus the

fat thathe/sheanbeadaptiveatsubsequent deisionpoints isnottaken

into aount.

The routing poliy model, on the other hand, fully onsiders future

adaptive hoies. Generally speaking, a routing poliy is a mapping from

network states to hoies of next link, where the set of network states

dependsontheassumptionsonstohastinetworksandinformationaess.

In this paper, a routing poliy is a mapping

(

v, t

)

→

ℓ

from node

v

at arrivaltime

t

tonext link

ℓ

∈

O

(

v

)

where

O

(

v

)

is the set of outgoing links of node

v

. For example, denote

e

(

v, t

)

as the minimum expeted travel time of a routing poliy from node

v

at time

t

to a given destination. A travelerwhominimizesexpetedtravel timewouldhoosealink

ℓ

= (

v, w

)

suh that

E

(e

T

ℓ,t

+

e

(

w, t

+

T

e

ℓ,t

))

is the minimum among all the outgoing links, where

E

(

X

e

)

standsfor the expetedvalueof random variable

X

e

. The seond addend

e

(

w, t

+

e

T

ℓ,t

)

is theexpetedtraveltime of a routing poliy from the sink node

w

to the destination,and thus future adaptivehoies are taken into aount. For eah support point of the random network, a

routing poliy will manifest as a path, but the manifested path hanges

over support points. In this sense, a routing poliy an be viewed as a

olletionofpaths,eahwithaertainprobability. Thereadersarereferred

toGaoandChabini(2006)andGao(2005)foradetailedaountofoptimal

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3

Illustrative Example

We use an example to larify the onepts related to the two adaptive

hoies. Figure 1 gives the topology of the stohasti and time-dependent

network and the PMFs of relevant link travel times,where

T

i

denotes the travel time on link

i

. Travelers are going from nodes A to D at departure time 0. The possible (node, time) pairs a traveler ould enounter during

the trip are:

(

A, 0

)

,

(

B, x

1

)

,

(

B, y

1

)

,

(

C, x

2

)

,

(

C, y

2

)

and the sets of outgoing linksfor all deision nodes are:

O

(

A

) =

{

0, 1

}

, O

(

B

) =

{

2, 3

}

, O

(

C

) =

{

4, 5

}

Theoretially the number of routing poliies are

2

5

, sine there are 5

possible (node, time) pairs and eah pair an be mapped to two possible

next links. However, one a traveler is at node

B

, the mapping at node

C

does not aet his/her remaining trip and therefore do not need to be speied. The same argumentan be made atnode

C

where the mapping at node

B

is not needed. Therefore there are 8 routing poliies as shown in Figure 1. Note that a path is a speial routing poliy, suh that the

mapping from a (node, time) pair is the same regardless of the arrival

time. Disussions of alulating attributes of the routing poliies an be

found in Setion 5.

We use general symbols for the PMFs, but for illustrative purpose, we

make the example simple by assuming

a

=

f, P

1

=

P

2

=

0.5, x

0

=

x

1

=

x, y

0

=

y

1

=

y

.

Travel times on links 0 and 1 at departure time 0 are random. It is

assumed that these two random variables are independent of eah other.

There are no restritions on the values of

x

and

y

, but for illustrative purpose, we assume

x < y

and denote the situation where link 0 or 1 has a travel time of

x

as the normal ase, and that where link 0 or 1 has a travel time of

y

as the inident ase. Travel times on links 2 and 4 are deterministi, but are dependent on the arrival times at soure nodes of

the links, whih ould be either

x

or

y

. A later arrival time at node

B

(alternatively

C

) leads to a longer travel time on link 2 (alternatively 4)

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A D B C

e

T

0

=

x

0

, w.p. 1

−

P

0

=

y

0

T

3

=

c

∀

t

e

T

1

=

x

1

, w.p. 1

−

P

1

y

1

, w.p. P

1

, t

=

0

T

4

=

f, t

=

x

1

d, t

=

y

1

T

5

=

e

∀

t

A D B C Poliy 8(Path 4) A D B C if

y

1

if

x

1

Poliy 7 A D B C Poliy 5(Path 3) A D B C if

x

1

if

y

1

y

0

if

x

0

x

0

if

y

0

Poliy 2 Figure1: Network

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(

b > a, d > f

=

a

). This ouldbedue tothefat travelers whoarriveslate (

y

) are aught in peak traÆ, whilethose with an earlier arrival (

x

) ould have avoided it. Travel times on links 3 and 5 are both deterministi and

time-independent.

The relationships among link travel time variables are:

a

=

f <

(

a

+

b

)

/2 < e < c < b < d

. The peak traÆ onditionon link4 is more severe than that on link 2, suh that the travel time on link 4 at time

y

(

d

) is higher thanthatof link2 (

a

). However, both linkshave diversions. Link2 has link 3 as the diversion link with a travel time of

c

, and link4 has link 5 as the diversion link with a travel time of

e

. Link5 is abetter diversion than link 3, sine

e < c

.

A traveler has a priori knowledge on the time-dependent link travel

time PMFsof all linksin thenetworkbefore atrip starts. During thetrip,

thetravelerobtainsadditionalonlineinformationontheatualarrivaltime

at the seond node (

x

or

y

). Depending on the arrival time, the traveler hooses the next link to take to minimize expeted travel time.

Consider rst the route hoie proess in an adaptive path model. At

node

A

, four paths are available: path 1 with an expeted travel time

(

x

+

a

+

y

+

b

)

/2

, path 2 with an expeted travel time

(

x

+

y

)

/2

+

c

, path 3 with anexpeted traveltime

(

x

+

f

+

y

+

d

)

/2

, and path4 with an expetedtraveltime

(

x

+

y

)

/2

+

e

. Path1hastheminimumexpetedtravel time, and thus the traveler takes link 0 whih is the rst link along that

path. The traveler then arrives at node

B

at either time

x

or

y

, eah with probability0.5. Ifthe arrival time is

x

(o peak), the traveler takes link 2 withatraveltimeof

a

;andifthearrivaltimeis

y

(peak),thetravelertakes a detour whih is link 3 with a travel time of

c

. Therefore the expeted travel time from node

A

to node

D

by making suessive path hoies is

(

x

+

a

+

y

+

c

)

/2

.

Considernextthe hoieproessin aroutingpoliymodel. Atnode

A

, the traveler is atuallyomparing the attrativeness of links 0 and 1. The

traveler knows that one arriving at the next node, he/she would make a

hoie based on realized arrive time, therefore it is better to onsider all

the possible diversions. The optimalrouting poliyfrom node

B

is totake the faster of links 2 and 3: if arrival time is

x

, take link 2 with a travel

(10)

time

a

; if arrival time is

y

, take link 3 with a travel time

c

. Similarly, the optimal routing poliy at node

C

is to take the faster of links 4 and 5: if arrival time is

x

f

; if arrival time is

y

e

. With this alulation in hand, the travelerevaluatesatnode

A

anddeidesthattakinglink1isoptimal,sine

(

x

+

a

+

y

+

e

)

/2 <

(

x

+

f

+

y

+

c

)

/2

(note that

a

=

f

). Reallingthat the expeted travel timeof makingsuessivepathhoies is

(

x

+

a

+

y

+

c

)

/2

, the optimalrouting poliy is thus more eÆient as a result of onsidering

future adaptive possibilities.

4

Model Specifications

In this setion we present disrete hoie model formulations for the

pre-viously disussed adaptive path and routing poliy hoies. Note that in

the datafor model estimation,only themanifestedpath is observed. Eah

pathobservation

i

of individual

n

isanordered setof hosen links

I

i

. Also known are the departure time and the arrival time

t

atthe soure node

v

of eah link

ℓ

∈ I

i

. Suh information are is available, for example, from Global Positioning System (GPS), see Bierlaire and Frejinger (2007) for a

disussion on route hoie data.

4.1

Adaptive Path Choice Model

This model assumes that a traveler hooses at the soure node

v

of eah observed link

ℓ

∈ I

i

a path

p

from

v

to the destination. We therefore dene anindividualandtime spei hoie set

C

vtn

of pathsfrom

v

tothe destination. Hene, for eah observation there are as many hoie sets as

there are links in the observed path.

Theprobabilityofan observationis denedas theprodutof the

prob-ability of hoosing eah link

ℓ

in the observed path, onditional on arrival time

t

atthe soure node:

P

n

(

i

) =

Y

ℓ

∈I

i

P

n

(

ℓ

|

t, v

) =

Y

ℓ

∈I

i

X

p

∈C

vtn

P

(

ℓ

|

p

)

P

(

p

|

C

vtn

;

β

)

(1)

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P

n

(

ℓ

|

t, v

)

is dened by the sum of the probabilitiesfor eah path that be-gins with

ℓ

. The path hoie model

P

(

p

|

C

vtn

;

β

)

(

β

denotes the vetor of parameters to be estimated) is therefore multiplied with a binary variable

P

(

ℓ

|

p

)

thatequalsoneiftherstlinkinpath

p

is

ℓ

andzerootherwise. Note

thatthe pathhoiean be modeled withanyof theexistingnon-adaptive

models.

4.2

Routing Policy Choice Model

Consider the model for the hoie of routing poliy among a hoie set

G

of routing poliies at the origin. Note that the adaptive behavior is alreadyapturedinthedenitionof aroutingpoliy. Thehoieofrouting

poliy is latent and only the manifested path is observable. A support

point is fully dened by the realized travel times on all random links. We

assumethattherealizedsupportpointforeahobservationisknowntothe

modeler through, for example, adequately dispersed GPS observations or

probevehiles thatover allrandom links. Thetravelerdoesnot knowthe

realized supportpointat theorigin;his/her informationn aessis dened

in the routing poliy, and in this paper it is the arrival times at deision

nodes. We model the the probabilityof a path observation onditionalon

support point

r

and hoie set of routing poliies

G

as

P

n

(

i

|

r

) =

X

γ

∈G

P

(

i

|

γ, r

)

P

(

γ

|

G

)

(2)

where

γ

is a routing poliy. As desribedin Setion 3, for agivensupport point a routing poliy is manifested as a path. However, several dierent

routing poliies an be manifested as the same path. We therefore sum

over allrouting poliiesin

G

and multiplythe routing poliyhoiemodel

P

(

γ

|

G

)

with a binary variable

P

(

i

|

γ, r

)

that equals one if

i

orresponds to

γ

for support point

r

and zero otherwise.

Gao (2005) propose the poliy size logit to model

P

(

γ

|

G

)

whih is the routing poliy version of the path size logit model (Ben-Akiva and

Ram-ming, 1998; Ben-Akiva and Bierlaire, 1999). It adds a term, poliy size

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P

(

γ

) =

e

lnPoS

γ

+

V

γ

P

k

∈G

e

lnPoS

k

+

V

k

(3)

where

V

γ

is the deterministiutilityof

γ

andthe formulationof PoS

PoS

γ

=

R

X

r

=

1





X

ℓ

∈I

r

γ

T

r

ℓ

T

r

γ

1

M

r

ℓ





P

(

r

)

(4)

maybe viewed as \expeted pathsize" with notations

R

: numberof supportpoints of link travel time distribution;

I

r

γ

: set of links of the realized path of routing poliy

γ

for support point

r

;

T

r

ℓ

: travel time of link

ℓ

for support point

r

;

T

r

γ

: realizedtravel time of routing poliy

γ

for support point

r

;

M

r

ℓ

: numberof routing poliies using link

ℓ

for support point

r

;

P

(

r

)

: probabilityof support point

r

.

Notethat ifthe support pointis unknown due to dataunavailability,a

path annot unambiguously be mathed with a given routing poliy. The

model presented in Equation(2) an then be generalizedto

P

n

(

i

) =

X

γ

∈G

P

(

i

|

γ

)

P

(

γ

|

G

) =

X

γ

∈G

R

X

r

=

1

P

(

i

|

γ, r

)

P

(

γ

|

G

)

(5)

5

Numerical Results

Wearry outnumerialtests of theproposedadaptiveroute hoiemodels

on a hypothetial network. The objetives of the tests are to: 1)

demon-strate the feasibility of estimating the two adaptive route hoie models;

and 2) gain insights into the adaptive route hoie models by omparing

preditionresults.

5.1

Test Settings

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travel time of the routing poliy6 (the one disussed in Setion 3) an be

either

x

1

+

f

or

y

1

+

d

, with probability

1

−

P

1

and

P

1

respetively. The expeted traveltime andstandarddeviationof theroutingpoliythenan

be alulated.

The alulation of poliy size is more involved. As shown in

Equa-tion (4), poliy size is the expeted value of path sizes over all support

points of the random network. As there are two random links in the

net-workeahwithtwopossiblerealizationsoftraveltimes,therearealtogether

four support points. Let

(

T

, x

1

)

as an example to illustrate how poliy size is alulated.

Considerrstthemappingfromroutingpoliiestopaths. Forexample,

in support point

(

x

0

, x

1

)

, routing poliy2 takes link 2at node

B

at arrival time

x

0

, and therefore is manifested as path 1. Let

γ

be a routing poliy and

p

apath, we obtain manifestationof all routing poliiesas follows:

γ

1

→

p

1

, γ

2

→

p

1

, γ

3

→

p

2

, γ

4

→

p

2

, γ

5

→

p

3

, γ

6

→

p

3

, γ

7

→

p

4

, γ

8

→

p

4

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Onethemanifestedpathisknown,weanountthenumberofrouting

poliiesthat use a given linkin support point

(

x

0

, x

1

)

asfollows.

M

ℓ

0

=

4

(

γ

1

, γ

2

, γ

3

, γ

4

)

M

ℓ

1

=

4

(

γ

5

, γ

6

, γ

7

, γ

8

)

M

ℓ

2

=

2

(

γ

1

, γ

2

)

M

ℓ

3

=

2

(

γ

3

, γ

4

)

M

ℓ

4

=

2

(

γ

5

, γ

6

)

M

ℓ

5

=

2

(

γ

7

, γ

8

)

(14)

The routing poliy sizes in support point

(

x

0

, x

1

)

are then PoS

1

=

x

0

x

0

+

a

1

4

+

a

x

0

+

a

1

2

PoS

2

=

x

0

x

0

+

a

1

4

+

a

x

0

+

a

1

2

PoS

3

=

x

0

x

0

+

c

1

4

+

c

x

0

+

c

1

2

PoS

4

=

x

0

x

0

+

c

1

4

+

c

x

0

+

c

1

2

PoS

5

=

x

1

x

1

+

f

1

4

+

f

x

1

+

f

1

2

PoS

6

=

x

1

x

1

+

f

1

4

+

f

x

1

+

f

1

2

PoS

7

=

x

1

x

1

+

e

1

4

+

e

x

1

+

e

1

2

PoS

8

=

x

1

x

1

+

e

1

4

+

e

x

1

+

e

1

2

Similarly we an alulate the poliy sizes in all other support points.

Take the expetation over the foursupport points and we obtain the nal

poliysizes.

We also need to alulate path sizes at the origin node

A

to be used in both the non-adaptive path model and the adaptive path model. Path

sizes arebased on expeted traveltimes.

PS

1

=

x

0

(

1

−

P

0

+

a

(

1

−

P

0

) +

bP

0

(

x

0

+

a

)(

1

−

) +

y

0

c

1

2

+

c

x

0

(

1

−

P

0

) +

y

0

P

0

+

c

PS

3

=

x

1

(

1

−

P

1

) +

y

1

P

1

(

x

1

+

f

)(

1

−

P

1

) + (

y

1

+

d

)

P

0

1

2

+

f

(

1

−

P

1

) +

dP

1

(

x

1

+

f

)(

1

−

P

1

) + (

y

1

+

d

)

P

1

PS

4

=

x

1

(

1

−

P

1

) +

y

1

P

1

x

1

(

1

−

P

1

) +

y

1

P

1

+

e

1

2

+

e

x

1

(

1

−

P

1

) +

y

1

P

1

+

e

5.2

Observation Generation

(15)

V

γ

=

1.0

lnPoS

γ

−

0.4

ExpetedTime

γ

−

0.1

TimeSTD

γ

,

∀

γ

∈ G

A routing poliy is not observable, and only the manifestedpath for a

givensupportpointisobserved. Wespeifytherangeof linktraveltimeto

be[10, 40℄ (min). To generateonepath observation fromthe poliy hoie

model, we followthe steps:

1. Sample anumber froma uniform distributionbetween10 and 40 for

eah link travel time variable:

a, b, c, d, e, f

;

2. Sample a number from a uniform distribution between 0 and 1 for

eah link travel time probabilityvariable:

x

0

, y

0

, x

1

, y

1

;

3. Calulate

P

(

γ

k

)

, the hoie probability of routing poliy

k,

∀

k

using the poliy-sizelogit model (Equation (3));

4. Calulate

Q

k

,theumulativeprobabilityofhoosingpoliies

γ

1

, γ

2

, . . . , γ

k

,

k

=

0, 1, . . . , 8

,where

Q

0

=

0

and

Q

8

=

1

;

5. Sampleanumber

r

fromauniformdistributionbetween0and1,and

γ

k

is hosen if

Q

k

−

1

< r < Q

k

;

6. Calulate

S

k

,theumulativeprobabilityofsupportpoints

r

1

, r

2

, . . . , r

k

,

k

=

0, 1, . . . , 4

,where

S

0

=

0

and

S

4

=

1

; 7. Sample a number

r

′

from a uniform distribution between 0 and 1;

and support point

r

k

is realized if

S

k

−

1

< r

′

< S

k

;

8. The hosen routing poliy is manifested as a path depending on the

supportpoint,usingasimilarlogiasinthemappingofEquation(6).

5.3

Estimation

Three models are estimated basedon the generated path observations:

(16)

3. Non-adaptivepath hoie model with path size logit.

Thedeterministiutilityfuntionshavealinear-in-parametersspeiation

of the same attributes as the true model, namely, expeted travel time,

travel time standarddeviation and path (poliy) size.

In the adaptive path hoie model, the hoie probabilityof a path is

the produt of hoosing all links along the path. The log likelihood of the

pathis thenthesum ofthe loglikelihoodof allthelinks. Therefore wean

treat eah hosen linkas an observation, and the linkhoie probabilityis

the sum of hoie probabilitiesof paths beginning with the the link. Note

that the path attributes are dependent on the realized arrival time at the

upstream node of the link. For example,assume

γ

2

is hosen and a travel time of

x

0

is realized on link 0, and thus the manifested path is path 1. Path 1 is omposed of two links: 0 and 2. Choie probability of link 0 is

the sum of those of paths 1 and 2. At time 0 and node

A

, travel time on link 0 is still random and therefore the expeted travel time of path 1 is

(

x

0

+

a

)(

1

−

P

0

)+(

y

0

+

b

0

. Now onsider the hoie of link 2. At time

x

0

and node

B

, travel time on link 2 (oinidingwith one of thepath alternatives out of node

B

) is xed as

a

, and suh value should be used in the observation data.

All models are estimated with BIOGEME (Bierlaire, 2003; Bierlaire,

2005). The estimation results are shown in Table 1. The oeÆient

esti-matesofroutingpoliyhoiemodelarenotsigniantlydierentfromthe

postulatedvalues whihshowsthatapoliyhoiemodelanbeestimated

based on path observations using Equation (2). The oeÆient estimates

of the other two models have their appropriate signs and are signiantly

dierent from zero. As expeted the model t of the routing poliymodel

is better than the adaptive path model and the non-adaptive path model

has the worst modelt.

5.4

Prediction

(17)

path path

b

β

PS 1.03 1.23 2.75 Std error 0.452 0.437 0.344 T-test 2.28 2.80 8.00

b

β

exptime -0.402 -0.28 -0.265 Std error 0.00805 0.00467 0.0049 T-test -49.97 -60.00 -54.02

b

β

stdtime -0.108 -0.071 -0.0451 Std error 0.00857 0.00923 0.00643 T-test -12.60 -7.69 -7.02 Finallog-likelihood -3257.097 -3536.324 -3932.998 Adj. rho-square 0.608 0.574 0.527

Number of observed paths: 6000

Nulllog-likelihood: -8317.766

BIOGEME (Bierlaire,2003; Bierlaire, 2005) hasbeen

used for all model estimations

(18)

a

=

f

=

10, b

=

30, c

=

26, d

=

38, e

=

22

x

0

=

x

1

=

14, y

0

=

y

1

=

18

P

0

=

P

1

=

P

The value of

P

is a parameter of the preditiontests and varies from 0 to 1, with an inrement of 0.1. The values of the link travel time variables

satisfy the ondition in Setion 3, i.e.

a

=

f <

(

a

+

b

)

/2 < e < c < d

. Therefore the analysis in Setion 3 applies here: path 1 is the minimum

expeted travel time path; links 2 and 4 have the same travel time under

normal ondition(when

x

0

or

x

1

isrealized); link4 ismore ongestedthan link 2 under inident ondition (when

y

0

or

y

1

is realized); links 3 and 5 are diversions for links2 and4 in inidentondition,and link5is a better

diversion thanlink 3 (

e < c

).

Sine the network is stohasti with all the support points known, we

obtain distributions of variables suh as path shares, path travel times,

origin-destination travel time and so forth. We take expetations of these

variabls over the four support points, where the probability of eah

sup-port is a funtion of

P

. We present the summary statistis (mean and/or standard deviation) togain a high-levelunderstanding of the results.

Figure 2 shows the expeted shares of all four paths. Eah subgraph

is for a path, and results from all three models are plotted as funtions of

inident probability

P

. Reall that paths 1 and 3 ontain the links that an beaeted bytheinidentsdue tothe time-dependenyof their travel

times, while paths 2 and 4 ontain the respetive diversion links that are

not aeted by theinidents. Therefore it is intuitivelyorret thatfor all

three models, the shares of paths 1 and 3 are dereasing funtions of

P

, whileshares of paths 2 and 4 areinreasing funtions of

P

.

In order to better appreiate the dierenes among the three models,

weaggregatetheresultsfromFigure2 toyielddataforFigure3,wherethe

expetedsharesofgoingleftandrightattheoriginnodeareplotted. Reall

that the rightbranh hasa better diversion (link 5). In the routing poliy

model,as

P

inreases,theimportaneofdiversionbeomesmoresigniant

(19)

when

P

=

0

, both paths 1 and 3 (belongingto the left and right branhes respetively)havethesameminimumtraveltime(

a

=

f

)andzerostandard deviation. While as

P

inreases, path 3 has higher disutility(

d > b

), and therefore path 1 is more dominant and gains more share. However, as

P

inreases to a ertain value, path 4 beomes the path with minimum disutility,and thus the right share starts to inrease.

If we inspet Figures 2 and 3 together, we nd that although the

left-right shares of non-adaptive and adaptive path models are roughly the

same, the distributionof the ows at the seond nodes are dierent. This

is beausetheadaptive pathmodelredistributesowsatthe seond nodes

depending on the atual arrival times. On the other hand, boththe

adap-tivepathmodelandnon-adaptivepathmodelpreditmoreowtakingthe

left branh than the routing poliy model does. This is beause future

di-version possibilityisnotonsideredineitherof themodels,andthebranh

with lessexpeted pathtravel time(path1)is favored,althoughlink3 isa

worse diversion. In another word, therouting poliymodel better apture

the option value of diversion thanthe adaptive path model.

Figure 4 shows the expeted value and standard deviation of average

path traveltime wheretheaverageistakenoverallfourpaths weightedby

pathshares. As

P

isapproahing0or1,links0and1aremorelikelytobein the sameonditionandthenetworkisless stohastithanwhen

P

is inthe middle. The gure shows that as

P

approahes the middle point between 0 and 1, the two adaptive models and non-adaptive models are farther

away from eah other in terms of expeted average travel time. This is

in aordanewith the intuitionthat being adaptiveis more advantageous

when the networkis more unertain.

Figure4 alsoshows thatadaptive pathand routingpoliymodels have

similarexpetedaveragetraveltime,buttheirstandarddeviationsarequite

dierent. This is beause the eet of a diversion is two-fold. A better

diversion provides shorter travel time under inident ondition. On the

other hand, under normal ondition it also results in more ow moving

fromthefaster link,beauseitsdisutilityis notasfar awayfromthefaster

(20)

predits more ow to the left branh (worse diversion) than the routing

poliymodel,ithaslongertraveltimeunderinidentondition,butshorter

travel time under normal ondition. Hene both models predit roughly

the same expeted average travel time, but the adaptive path model has

larger standard deviations.

6

Conclusions and Future Directions

This paper develops the rst eonometri estimator for the routing poliy

hoie model and demonstrates the feasibilityof estimating the model. A

routing poliyingeneral is adeision rulethat mapsfromall possible

net-work states tonext links out of deision nodes. It ollapses to a path in a

deterministi networks. The onept of routing poliy expliitly aptures

travelers' route hoie adjustments aording to information on realized

network onditions in stohasti time-dependent networks. Sine the

in-formationomponentis embeddedin the dention of a routingpoliy,the

estimator of the routing poliy model is a general one and an be applied

to a large variety of information situations. This paper demonstrates one

of the information situations: the realized arrival times at deision nodes.

Other information situationswillbe the subjets of futureresearh.

Therouting poliy hoie model is alsoompared to the adaptivepath

model, whih is anaturalapproahto anadaptive routehoiemodel. An

adaptive path model is atuallya sequene of path hoie models applied

at intermediate deision nodes; while a routing poliy hoie model is one

model at the originwhere the alternatives are routing poliies. Predition

resultsshowthattheroutingpoliymodelapturesbettertheoptionvalue

of diversion than the adaptive path model beause of the foresight of a

routingpoliy. Whenthenetworkismorestohasti,thedierenebetween

the two adaptive models and the non-adaptive model is larger in terms of

expeted traveltime. As expeted, this result indiates thatthe benet of

being adaptive is more signiantin a more unertain network.

(21)

pol-poliies.

References

Abdel-Aty, M. A. and Abdalla, M. F. (2006). Examination of multiple

mode/route-hoieparadigms underatis, IEEE Transations on

In-telligent Transportation Systems

7

(3): 332{348.

Abdel-Aty, M. and Abdalla, M. F. (2004). Modeling drivers' diversions

from normalroutes under atis using generalized estimatingequations

and binomialprobit link funtion, Transportation

31

: 327{348.

Balakrishna, R. (2006). O-line alibration of dynami traÆ

assign-ment models, PhD thesis, Massahusetts Institute of Tehnology,

Cambridge, MA,USA.

Balakrishna, R., Ben-Akiva, M. and Koutsopoulos, H. N. (2007).

O-line alibration of dynami traÆ assignment: simultaneous

demand-supply estimation,Transportation Researh Reord

2003

.

Bekhor, S., Ben-Akiva, M. and Ramming,M. (2002). Adaptation of logit

kernel to route hoie situation, Transportation Researh Reord

1805

: 78{85.

Ben-Akiva,M. andBierlaire,M. (1999). Disretehoiemethodsandtheir

appliationstoshort-termtraveldeisions,inR.Hall(ed.),Handbook

of Transportation Siene, Kluwer, pp. 5{34.

Ben-Akiva, M., Bierlaire, M., Koutsopoulos, H. N. and Mishalani, R.

(2002). Real-time simulation of traÆ demand-supply interations

withindynamit, inM. Gendreau and P.Marotte (eds), T

ransporta-tionandNetworkAnalysis: CurrentTrends.MiselleneainHonor

(22)

modelsoftravelerbehaviorinnetworks.PreparedforAdvaned

Meth-odsforPlanningandManagementofTransportationNetworks.Capri,

Italy.

Bierlaire,M. (2003). Biogeme: afree pakagefor theestimationof disrete

hoie models, Proeedings of the 3rd Swiss Transport Researh

Conferene, Asona, Switzerland.

Bierlaire, M. (2005). An introdution to Biogeme version 1.4.

http://biogeme.ep.h.

Bierlaire,M.andFrejinger,E.(2007).Routehoiemodelingwith

network-free data, Transportation Researh Part C . Aepted for

publia-tion.

Bogers, E. A. I., Viti, F. and Hoogendoorn, S. P. (2005). Joint modeling

of advaned travel information servie, habit, and learning impats

on routehoiebylaboratorysimulatorexperiments, Transportation

Researh Reord

1926

: 189{197.

Casetta, E., Nuzzolo, A., Russo, F. and Vitetta, A. (1996). A modied

logitroutehoiemodeloveromingpathoverlappingproblems.

Spe-iationandsomealibration resultsfor interurbannetworks,inJ. B.

Lesort (ed.), Proeedings of the 13th International Symposium on

Transportation and TraÆ Theory, Lyon, Frane.

Frejinger, E. and Bierlaire, M. (2007). Capturing orrelation with

sub-networks in route hoie models, Transportation Researh Part B

41

(3): 363{378.

Gao, S. (2005). Optimal Adaptive Routing and TraÆ Assignment in

Stohasit Time-Dependent Networks, PhD thesis, Massahusetts

Institute of Tehnology.

Gao, S. and Chabini, I. (2006). Optimal routing poliy problems in

(23)

time-dependenttravel times,Transportation Siene

20

(3): 91{101.

Mahmassani, H. (2001). Dynami network traÆ assignment and

simu-lation methodology for advaned system management appliations,

Network and Spatial Eonomis

1

: 267{292.

Mahmassani, H. and Liu,Y. (1999). Dynamisof ommuting deision

be-havior under advaned traveler information systems, Transportation

Researh Part C

7

(2/3):91{107.

Marotte, P. and Nguyen, S. (1998). Hyperpath formulations of traÆ

as-signmentproblems,inP. MarotteandS. Nguyen (eds),Equilibrium

and Advaned Transportation Modeling, KluwerAademi

Publish-ers, pp. 175{199.

Miller-Hooks,E. (2001). Adaptiveleast-expeted time pathsin stohasti,

time-varyingtransportation and datanetworks, Networks

37

(1): 35{

52.

Miller-Hooks, E. D. and Mahmassani, H. S. (2000). Least expeted time

paths in stohasti, time-varying transportation networks, T

rans-portation Siene

34

(2): 198{215.

Peeta, S. and Yu, J. W. (2005). A hybrid model for driver route hoie

inorporating en-route attributes and real-time information eets,

Networks and Spatial Enonomis

5

: 21{40.

Polyhronopoulos,G. andTsitsiklis,J.N. (1996). Stohasti shortestpath

problemswith reourse, Networks

27

: 133{143.

Polydoropoulou, A., Ben-Akiva, M., Khattak, A. and Lauprete, G.

(1996). Modling revealed and stated en-route travel response to

adavaned traveler information systems, Transportation Researh

Reord

1537

: 38{45.

Pretolani, D. (2000). A direted hyperpath model for random time

de-pendent shortest paths, European Journal of Operational Researh

(24)

Massahusetts Institute of Tehnology, Cambridge,MA, USA.

Srinivasan,K.K.andMahmassani,H.(2003).Analyzingheterogeneityand

unobserved strutural eets in route-swithing behavior under atis:

adynamikernallogitformulation,Transportation ResearhPart B

37

: 793{814.

Ukkusuri, S. V. and Patil, G. R. (2006). Exploring user behavior in

on-line network equilibrium problems, 11th International Conferene

on Travel Behavior Researh Compendium of Papers CD-ROM,

Kyoto, Japan.

Vovsha,P. and Bekhor,S. (1998). Link-nested logit model of route hoie,

Overoming route overlapping problem, Transportation Researh

Reord

1645

: 133{142.

Waller,S.T.and Ziliaskopoulos,A.K.(2002). Onthe onlineshortestpath

(25)

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Path 1

Incident Probability

Expected Path Share

Routing Policy

Adaptive path

Non−adaptive path

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Path 2

Incident Probability

Expected Path Share

Routing Policy

Adaptive path

Non−adaptive path

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Path 3

Expected Path Share

Routing Policy

Adaptive path

Non−adaptive path

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Path 4

Expected Path Share

Routing Policy

Adaptive path

Non−adaptive path

(26)

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Paths 1 and 2 (Left)

Expected Path Shares

Routing Policy

Adaptive Path

Non−Adaptive Path

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Paths 3 and 4 (Right)

Incident Probability

Expected Path Shares

Routing Policy

Adaptive Path

Non−Adaptive Path

Figure3: Expeted Shares for Leftand Right at Origin

0.0

0.2

0.4

0.6

0.8

1.0

20

25

30

35

40

45

Expected Average Time

Routing Policy

Adaptive path

Non−adaptive path

0.0

0.2

0.4

0.6

0.8

1.0

0

2

4

6

8

Standard Deviation of Average Time

Routing Policy

Adaptive path

Non−adaptive path