DTIC ADA572201: A Fluid Queueing Model for Link Travel Time Moments

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A Fluid Queueing Model for Link Travel Time Moments†
Jeffrey P. Kharoufeh,1 Natarajan Gautam2
1Department of Operational Sciences, Air Force Institute of Technology, AFIT/ENS,
Building 640, 2950 Hobson Way, Wright Patterson AFB, Ohio 45433-7765
2Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, The
Pennsylvania State University, 310 Leonhard Building, University Park, Pennsylvania
Received 12 September 2002; revised 27 March 2003; accepted 28 August 2003
DOI 10.1002/nav.10114
Abstract: We analyze the moments of the random time required for a vehicle to traverse a
transportationnetworklinkofarbitrarylengthwhenitsspeedisgovernedbyarandomenvironment.
Theproblemismotivatedbystochastictransportationnetworkapplicationsinwhichtheestimation
of travel time moments is of great importance. We analyze this random time in a transient and
asymptoticsensebyemployingresultsfromthefieldoffluidqueues.Theresultsaredemonstratedon
twoexampleproblems.©2003WileyPeriodicals,Inc.*NavalResearchLogistics51:242–257,2004.
Keywords: linktraveltime;randomenvironment;stochasticmodeling
1. INTRODUCTION
Weconsiderthemovementofavehiclealongatransportationnetworklink(suchasafreeway
segment)whoselengthisx units.Thevehicle’sspeedvariesoverthecourseofitssojourndue
to the influence of an underlying environment process. Owing to the effects of this random
environment,thetimerequiredtotraversethislinkisarandomvariabledenotedbyT(x).Inthis
paper, our aim is to analyze the moments of the random travel time T(x), in the transient and
the asymptotic sense, with the ultimate objective of obtaining computationally expedient
measures that are extremely useful in a number of transportation contexts. To that end, we
demonstratethattheproblemcanbeviewedandanalyzedasafluidqueueingmodelfromwhich
such expedient measures may be derived. The approach allows us to consider the travel time
moments for an individual link of a transportation network subject to a randomly evolving
environment.Thisrandomenvironmentischaracterizedasacontinuous-timestochasticprocess
on a finite sample space.
†The views expressed in this paper are those of the authors and do not reflect the official policy or
positionoftheUnitedStatesAirForce,DepartmentofDefense,ortheU.S.Government.
Correspondenceto:J.P.Kharoufeh(e-mail:Jeffrey.Kharoufeh@afit.edu)
©2003WileyPeriodicals,Inc.*ThisarticleisaUSGovernmentworkand,assuch,isinthepublicdomain
intheUnitedStatesofAmerica.
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A Fluid Queueing Model for Link Travel Time Moments
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Naval Research Logistics, 51 (2), 242-257
14. ABSTRACT
We analyze the moments of the random time required for a vehicle to traverse a transportation network
link of arbitrary length when its speed is governed by a random environment. The problem is motivated by
stochastic transportation network applications in which the estimation of travel time moments is of great
importance. We analyze this random time in a transient and asymptotic sense by employing results from
the field of fluid queues. The results are demonstrated on two example problems.
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KharoufehandGautam:AFluidQueueingModelforLinkTravelTimeMoments 243
The dynamics of the model can be described as follows. If at time t (cid:1) 0 the underlying
finite-stateprocessdenotedby{Z(t):t (cid:1)0}isinstatei(cid:1)S(cid:1){1,2, ...,K},thenthespeed
of the vehicle is a strictly positive quantity V. The stochastic process {Z(t) :t (cid:1) 0} is the
i
random environment process. If D(t) denotes the cumulative distance travelled by the vehicle
up to time t (cid:2) 0, and {Z(t) :t (cid:1) 0} is a Markov process, then it can be shown that {(D(t),
Z(t)) : t (cid:1) 0} is also a Markov process. Moreover, the random variable D(t) is an additive
functional of Z(t). The process {(D(t), Z(t)) : t (cid:1) 0} is used to analyze the moments of the
random time T(x), which corresponds to a first passage time for the process {D(t) :t (cid:1) 0}.
Our results are primarily motivated by problems in transportation and logistics where it is
important for decision-makers to know the travel time moments of individual vehicles in a
stochastictransportationnetwork.Themaincontributionoftheworkisthenovelapplicationof
fluidqueueingtechniquesforthe analysis ofanindividual vehiclewhosetime-variant speedis
modulated by a random environment. This is in contrast to an aggregated approach for all
vehicles usually found in the traffic flow theory (cf. Lighthill and Whitham [12] and Highway
Capacity Manual [7]). Our approach is intuitive in the sense that a plot of the vehicle’s speed
over time corresponds to a sample path of a continuous-time stochastic process, possibly on a
finitestatespace.Owingtoitsgenerality,themodelmaybeusedinanumberoftransportation
settingsdirectlyorbymakingsuitablealterations.Wedescribeindetailapplicationsinground
and maritime transportation.
● Ground transportation: Consider a vehicle that traverses a roadway segment with x
correspondingtothephysicallengthofthesegment.Severalfactorsinfluencethespeed
of the vehicle as it attempts to traverse the roadway segment. Some of those may be
physicalfactors(e.g.,roadwaygeometry,grades,visibility),trafficfactors(e.g.,density,
presence of heavy vehicles, merging traffic), or environmental factors (e.g., weather
conditions,speedlimits,etc.).Itisassumedthattheenvironmentprocess{Z(t) :t (cid:1) 0}
is known, and thus, may be used to obtain the moments of the random variable T(x).
These moments can then be applied to construct parametric distributions for stochastic
arc weights in transportation networks within the context of automatic route guidance
systemsdescribedin[6]orinleast-timestochastictransportationnetworkproblemssuch
as those described in [13] and [14].
● Maritime transportation: The environment process approach is similarly applicable in
maritime scenarios. In particular, consider a ship traversing one leg of its journey of
length x. Stochastic and dynamic weather conditions directly influence the speed with
which the ship may travel. In such case, the stochastic process {Z(t) :t (cid:1) 0} may be
used to model the set of meteorological variables which determines the ship’s speed at
agivenpointintime(andpossiblyspace).Byassigningacostforeachspeed(e.g.,fuel
consumption)andmakingappropriatealterations,ourmodelmaybeusedtocomputethe
expectedcostincurredfortraversingeachlegoftheship’ssojourn.Furthermore,ifeach
arcofanetworkisgovernedbyitsownenvironmentprocess,thenitmaybepossibleto
solve a stochastic and dynamic minimum cost problem such as that considered by
Psaraftis and Tsitsiklis [17].
Throughout the remainder of this paper, we concern ourselves with the general setting of a
vehicletraversingalinkoflengthx withtheunderstandingthatthismaypertaintoeitherofthe
above scenarios. Moreover, we may model a number of real-world contexts with appropriate
alterations to the problem parameters and their physical interpretations.
244 NavalResearchLogistics,Vol.51(2004)
The main contributions of this work can be summarized as follows. First, using the tools of
fluid queueing models, we present a transparent approach for implicitly incorporating the time
dependenceofspeedforavehicletraversingalinkoflengthx. Next,wegiveanexplicitmatrix
transformexpressionfortherthmomentofthelinktraveltimewhichgivesexactresultswhen
thetransformcanbealgebraicallyinverted,andveryaccurateapproximateresultswithnumer-
icalinversion.Third,wediscussasymptoticresultsforthefirstandsecondmomentsofthelink
traveltimewhichserveascomputationallyexpedientapproximationsorasparameterestimates
for surrogate link travel time distributions.
The remainder of the paper is organized as follows. The next section reviews the pertinent
concepts from the theory of fluid queues and demonstrates the means by which the theory is
applied to the link travel time problem. Section 3 demonstrates how to compute the moments
of the link travel time for a link of arbitrary (but finite) length x. In Section 4, we use the
transform results of Section 3 to provide intuitive asymptotic expressions for the mean and
varianceofthelinktraveltime.Section5presentsnumericalresultsontwoexampleproblems
followed by our concluding remarks in Section 6.
2. MATHEMATICAL MODEL
2.1. Fluid Queueing Concepts
Inthissection,weprovideabriefoverviewofthenotationandrudimentaryconceptsoffluid
queueing models. A fluid queueing model can be described as one in which the input to a
stochasticsystemismodeledasacontinuousfluidthatentersabufferandthenleavesthebuffer
through an output channel (service mechanism) with constant output capacity c. One measure
ofimportanceforsuchsystemsistheamountoffluidcontainedinthebufferattimet denoted
bytherandomvariableX(t). Thestochasticprocess,{X(t) :t (cid:1) 0}, isoftenreferredtoasthe
buffercontentprocess.Thereexistsanexternalprocesscalledtherandomenvironmentprocess
that modulates the input of fluid to the buffer. That is, the state of the random environment
process dictates the rate at which fluid flows into the buffer.
Suppose{Z(t) :t (cid:1) 0} denotestherandomenvironmentprocessthatdrivesfluidgeneration.
Define R as entrance rate of fluid to the buffer at time t and let the drift function of this
Z(t)
process be
(cid:2) (cid:3)R (cid:4)c. (1)
Z(cid:3)t(cid:4) Z(cid:3)t(cid:4)
The overall storage capacity of the buffer is denoted by a fixed, deterministic value B. The
dynamics of the buffer-content process when B (cid:5) (cid:6)are given by
(cid:1)
dX(cid:3)t(cid:4) (cid:2) , X(cid:3)t(cid:4)(cid:5)0,
(cid:3) Z(cid:3)t(cid:4)
dt (cid:2)Z(cid:7)(cid:3)t(cid:4), X(cid:3)t(cid:4)(cid:3)0,
where w(cid:7) (cid:1) max{w, 0}. In case B (cid:8) (cid:6), the system is governed by
(cid:1)
(cid:2)(cid:7) , X(cid:3)t(cid:4)(cid:3)0,
dX(cid:3)t(cid:4) Z(cid:3)t(cid:4)
(cid:3) (cid:2) , 0(cid:6)X(cid:3)t(cid:4)(cid:6)B,
dt (cid:2)Z(cid:9)(cid:3)t(cid:4), X(cid:3)t(cid:4)(cid:3)B,
Z(cid:3)t(cid:4)
where w(cid:9) (cid:1) max{0, (cid:9)w}. The probability law of the buffer-content process, {X(t) :t (cid:1) 0}
isdictatedbytheformoftherandomenvironmentprocess,{Z(t) :t (cid:1) 0}, andtheassociated
KharoufehandGautam:AFluidQueueingModelforLinkTravelTimeMoments 245
function (cid:2). This function, referred to as the drift function, corresponds to the net input rate of
fluidtothebuffer(entrancerate(cid:9)exitrate).Thediagonalmatrix(cid:10) (cid:1)diag( (cid:2) , (cid:2) , ..., (cid:2) )
1 2 K
is called the drift matrix. The first time the buffer fluid crosses some fixed level x corresponds
directly to a first passage time for the stochastic process {X(t) :t (cid:1) 0}.
Many researchers in the field of telecommunications have recognized the utility of such
modelsforsolvingengineeringproblems.AfewimportantpapersinthisareaareduetoAnick,
Mitra, and Sondhi [2], Elwalid and Mitra [5], Kesidis, Walrand, and Cheng-Shang [8], and
KulkarniandGautam[11].Otherresearchershaveconsideredmoregeneralizedfluidqueueing
problemsandsomegoodexamplesarethepapersduetoAsmussen[3],Rogers[18],and,more
recently, Takada [19]. The approach of our paper is to apply a fluid queueing model for the
mathematical characterization of link travel time moments as simple expressions that may be
computedinacomputationallyexpedientmanner.Wenextdemonstratethemeansbywhichthis
may be accomplished.
2.2. Fluid Model for Vehicle Displacement
Now consider a vehicle that must traverse a link of length x whose speed is governed by a
randomenvironmentonafinitestatespaceS(cid:5){1,2, ...,K},whereK(cid:1)N,thesetofnatural
numbers.Definetherandomenvironmentprocessby{Z(t):t (cid:1)0}sothatattimet,thevehicle
assumesa(strictlypositive)speedV . Sincetheenvironmentprocesshasafinitestatespace,
Z(t)
the vehicle may assume speeds in the finite set {V : i (cid:5) 1, 2, ..., K}, where V (cid:2) 0 for all
i i
i. Moreover, define the matrix V (cid:5) diag(V , V , ..., V ). Hence, the time dependence of
1 2 K
vehiclespeediscapturedimplicitlythroughtheenvironmentprocess,{Z(t):t (cid:1)0}.Theinitial
conditionsexperiencedbythevehiclearecapturedbytheinitialstateoftheenvironmentZ(0),
andwedenotetheinitialdistributionoftheenvironmentprocess,arowvector,z (cid:5)[P{Z(0)(cid:5)
0
i}]. Define by D(t), the total displacement of the vehicle up to time t and the associated
stochasticprocess{D(t) :t (cid:1) 0}. Withthesedefinitions,wearenowpreparedtoformalizethe
analogy to a fluid queueing model. Table 1 summarizes the relationship between the concepts
of a fluid queueing model and those of the link travel time model.
Thoughmanyanalogousconceptsexistbetweenthetwomodels,therearesomenoteworthy
distinctions.Inourmodel,welimitthemovementofavehicletothepositivedirectiononlyand
weallowonlypositivevelocities.Oncethevehiclebeginsitssojourn,itdoesnotstop,nordoes
itmoveinthenegativedirection.Theanalogoussituationinthefluidqueueingmodelisthatthe
output capacity of the system is zero (c (cid:5) 0) and the buffer accumulates fluid until it first
reaches the threshold value x. Hence, our link travel time model is a special case of a general
fluid queueing model in which the drift rates ({(cid:2)}) are all positive, and the D process,
i
correspondingtothebuffercontentprocessX,possessesmonotonicallyincreasingsamplepaths.
The cumulative distance travelled by the vehicle up to time t (cid:2) 0 is defined by
Table1. Analogousfluidqueueingandtransportationconcepts.
Fluidqueueingconcept Transportationanalogy
EnvironmentprocessZ EnvironmentprocessZ
Drift function (cid:2), i (cid:1) S Velocity function V, i (cid:1) S
i i
Driftmatrix(cid:10) VelocitymatrixV
BuffercontentprocessX VehicledisplacementprocessD
FluidlevelcrossingtimeT(x) VehicledistancecrossingtimeT(x)
246 NavalResearchLogistics,Vol.51(2004)
(cid:2)
t
D(cid:3)t(cid:4)(cid:3) V du. (2)
Z(cid:3)u(cid:4)
0
Equation (2) indicates that D is a Markov additive functional of Z and that the random travel
time T(x) is given by the first passage time
T(cid:3)x(cid:4)(cid:3)inf(cid:11)t:D(cid:3)t(cid:4)(cid:5)x(cid:12). (3)
When the process {Z(t) :t (cid:1) 0} is an irreducible, continuous-time Markov chain (CTMC)
on S (cid:5) {1, 2, ..., K} with infinitesimal generator matrix Q (cid:5) [q ], the dynamics of the
ij
buffer-contentprocessarewellunderstood.Forexample,Rogers[18]solvedforthestationary
probability law of the buffer-content process when the buffer is finite or infinite using the
Wiener-HopffactorizationforthegeneratormatrixofthegoverningMarkovprocess.Asmussen
[3] derives the stationary distribution in two cases, with and without an additional Brownian
component. Recently, Takada [19] presented a fluid queueing model that generalizes previous
results for the MAP/G/1 queue by allowing the buffer content process to additionally have
jumps. These works provide mathematically elegant approaches for computing the invariant
probability distribution of the buffer-content process. By contrast, the link travel time problem
requiresthetransientprobabilitylawasindicatedbyEquation(3).Forthepurposesofthiswork,
weconsideratransformapproachfortheprobabilitydistributionofD(t)fromwhichweareable
to obtain both transient and asymptotic measures for the link travel time moments. The
following brief review of this distribution follows from [9].
Define the joint distribution function
H(cid:3)x, t(cid:4)(cid:3)P(cid:11)D(cid:3)t(cid:4)(cid:7)x, Z(cid:3)t(cid:4)(cid:3)i(cid:12), i(cid:1)S. (4)
i
Moreover, define the cumulative distribution function for T(x) by
(cid:4)
G(cid:3)x, t(cid:4)(cid:3)P(cid:11)T(cid:3)x(cid:4)(cid:7)t(cid:12)(cid:3)1(cid:4)P(cid:11)D(cid:3)t(cid:4)(cid:7)x(cid:12)(cid:3)1(cid:4) H(cid:3)x, t(cid:4). (5)
i
i(cid:1)S
Let H(x, t) (cid:5) [H(x, t)] denote a 1 (cid:13) K vector. It can be shown that the joint probability
i i(cid:1)S
distribution H(x, t) satisfies the partial differential equation (PDE),
(cid:8)H(cid:3)x, t(cid:4) (cid:8)H(cid:3)x, t(cid:4)
(cid:9) V(cid:3)H(cid:3)x, t(cid:4)Q, (6)
(cid:8)t (cid:8)x
withinitialconditionH(x, 0)(cid:5) z . DenotebyH*(x, s ), theLaplacetransform(LT)ofH(x,
0 2
t) with respect to t given by
(cid:2)
(cid:6)
H*(cid:3)x, s (cid:4)(cid:3) e(cid:9)s2tH(cid:3)x, t(cid:4) dt, (7)
2
0
and the Laplace-Stieltjes transform (LST) of H*(x, s ) with respect to x as
2
KharoufehandGautam:AFluidQueueingModelforLinkTravelTimeMoments 247
(cid:2)
(cid:6)
H˜*(cid:3)s , s (cid:4)(cid:3) e(cid:9)s1xdH*(cid:3)x, s (cid:4). (8)
1 2 2
0
so that H˜*(s , s ) (cid:5) [H˜*(s , s )] is a 1 (cid:13) K row vector of transform expressions. The
1 2 i 1 2 i(cid:1)S
following result of [9] can be used in conjunction with Eq. (5) to obtain the probability
distribution of the random link travel time T(x).
THEOREM 1: The solution to the differential Eq. (6) in the transform space is given by
H˜*(cid:3)s , s (cid:4)(cid:3)z (cid:3)s V(cid:9)s I(cid:4)Q(cid:4)(cid:9)1, (9)
1 2 0 1 2
whereH˜*(s , s ) isdefinedbyEq.(8),z istheinitialenvironmentdistribution,ands ands
1 2 0 1 2
are complex transform variables with Re(s ) (cid:2) 0 and Re(s ) (cid:2) 0.
1 2
Finally, it is easy to see that the cumulative distribution function for T(x) is given by
1
G˜*(cid:3)s , s (cid:4)(cid:3) (cid:4)z H˜*(cid:3)s , s (cid:4)e, (10)
1 2 s 0 1 2
2
wheree denotesaK-dimensionalcolumnvectorofones.ItshouldbenotedthattheLSTofthe
link travel time distribution can be obtained as a 1-dimensional transform with respect to the
temporalvariabletinaslightlymoremathematicallyelegantmanner.Inparticular,wenotethat
ourproblemisaspecialcaseoftheproblemsconsideredbybothRogers[18]andAsmussen[3]
with the exception that Eq. (9) corresponds to a transient distribution.
The distributions of this section can be accurately computed using a two-dimensional
numerical inversion algorithm such as the one due to Moorthy [15]. However, it is possible to
generateapproximatedistributionswithfarlesscomputationaleffortbycomputingthemoments
of the vehicle link travel time and using them in surrogate, parametric distributions. This
approachmaybeespeciallyusefulintheanalysisofstochastictransportationnetworkswherein
the entire cumulative distribution function is needed to compute stochastically shortest paths.
However, in lieu of the link travel time distribution, the moments of this random time can be
computed in a simple fashion. Moreover, asymptotic approximations of the link travel time
momentscanbeobtainedasclosed-formanalyticalexpressions.InSection3,weshowhowto
compute the moments of the random link travel time using Eq. (9) when the length of the link
(x) is finite.
3. TRANSIENT LINK TRAVEL TIME MOMENTS
In this section, we derive an expression using the fluid queueing approach for the moments
of the link travel time whenever the link length is finite. By Eq. (5),
(cid:4)
G(cid:3)x, t(cid:4)(cid:3)1(cid:4) H(cid:3)x, t(cid:4), (11)
i
i(cid:1)S
where G is the CDF of the random link travel time, H(x, t) (cid:5) P{D(t) (cid:7) x, Z(t) (cid:5) i}, and
i
S is the finite state space of the random environment process {Z(t) :t (cid:1) 0} that modulates
248 NavalResearchLogistics,Vol.51(2004)
vehicle speed. It is well-known that the rth moment of the random variable T(x) may be
obtainedbyevaluatingat0,therth-orderderivativeoftheLaplace-Stieltjestransform(LST)of
G which is given by
(cid:2)
(cid:4)
(cid:6)
G˜(cid:3)x, s (cid:4)(cid:3) e(cid:9)s2t dG(cid:3)x, t(cid:4)(cid:3)1(cid:4) H*(cid:3)x, s (cid:4)s , (12)
2 i 2 2
0 i(cid:1)S
whereH*(x, s ) isgivenbyEq.(7).Therthmomentofthelinktraveltime,denotedbym (x),
i 2 r
is
(cid:8)r
m(cid:3)x(cid:4)(cid:3)E(cid:14)(cid:3)T(cid:3)x(cid:4)(cid:4)r(cid:15)(cid:3)(cid:3)(cid:9)1(cid:4)r G˜(cid:3)x,s (cid:4)(cid:5) . (13)
r (cid:8)sr 2 s2(cid:5)0
2
Next define
(cid:6) (cid:7)
(cid:4) (cid:4)
(cid:8)rG˜(cid:3)x,s (cid:4) (cid:8)rH*(cid:3)x,s (cid:4) (cid:8)r(cid:9)1H*(cid:3)x,s (cid:4)
Kr(cid:3)x(cid:4)(cid:3)(cid:3)(cid:9)1(cid:4)r 2 (cid:3)(cid:3)(cid:9)1(cid:4)r(cid:7)1 s i 2 (cid:9)r i 2 . (14)
s2 (cid:8)sr 2 (cid:8)sr (cid:8)sr(cid:9)1
2 i(cid:1)S 2 i(cid:1)S 2
Equation (14), which is derived from Eq. (12), implies that
(cid:4) (cid:8)
(cid:8)r(cid:9)1H*(cid:3)x,s (cid:4)
m(cid:3)x(cid:4)(cid:3)Kr(cid:3)x(cid:4)(cid:3)(cid:3)(cid:9)1(cid:4)r(cid:7)1r i 2 . (15)
r 0 (cid:8)sr(cid:9)1
i(cid:1)S 2 s2(cid:5)0
Inordertosolvethedifferentialequation(15),transformmethodsareagainemployed.TheLST
of m (x) with respect to x is
r
(cid:2)
(cid:6)
m˜ (cid:3)s (cid:4)(cid:3) e(cid:9)s1x dm(cid:3)x(cid:4).
r 1 r
0
By taking the LST of Eq. (15) on both sides,
(cid:4) (cid:8) (cid:8)
(cid:8)r(cid:9)1H˜*(cid:3)s ,s (cid:4) (cid:8)r(cid:9)1H˜*(cid:3)s ,s (cid:4)
m˜ (cid:3)s (cid:4)(cid:3)(cid:3)(cid:9)1(cid:4)r(cid:7)1r i 1 2 (cid:3)(cid:3)(cid:9)1(cid:4)r(cid:7)1r 1 2 e, (16)
r 1 (cid:8)sr(cid:9)1 (cid:8)sr(cid:9)1
i(cid:1)S 2 s2(cid:5)0 2 s2(cid:5)0
where H˜*(s , s ) is the matrix transform of Eq. (9). Assuming the existence of all derivatives
1 2
of H(x, s ) at s (cid:5) 0, inversion of Eq. (16) yields the rth moment of the random link travel
i 2 2
time. The following lemma will be needed to derive a matrix expression for Eq. (16).
LEMMA 1: The kth order partial derivative of the vector H˜*(s , s ) with respect to s is
1 2 2
(cid:8)kH˜*(cid:3)s , s (cid:4)
1 2 (cid:3)(cid:3)(cid:9)1(cid:4)kk!z (cid:3)s V(cid:9)s I(cid:4)Q(cid:4)(cid:9)k(cid:9)1, k(cid:1)0.
(cid:8)sk 0 1 2
2
KharoufehandGautam:AFluidQueueingModelforLinkTravelTimeMoments 249
Lemma 1 can be easily proved by mathematical induction and is next used to derive a general
expression for the rth moment of the link travel time.
THEOREM 2: The Laplace-Stieltjes transform of m (x) is given by
r
m˜ (cid:3)s (cid:4)(cid:3)r!z (cid:3)s V(cid:4)Q(cid:4)(cid:9)re. (17)
r 1 0 1
PROOF: Applying Lemma 1 to Eq. (16) directly shows that
(cid:8)r(cid:9)1
m˜ (cid:3)s (cid:4)(cid:3)(cid:3)(cid:9)1(cid:4)r(cid:7)1r H˜*(cid:3)s ,s (cid:4)(cid:5) e
r 1 (cid:8)sr(cid:9)1 1 2 s2(cid:5)0
2
(cid:3)(cid:3)(cid:9)1(cid:4)r(cid:7)1r(cid:3)(cid:9)1(cid:4)r(cid:9)1(cid:3)r(cid:4)1(cid:4)!z (cid:14)(cid:3)s V(cid:4)Q(cid:4)(cid:9)1(cid:15)re(cid:3)r!z (cid:3)s V(cid:4)Q(cid:4)(cid:9)re,
0 1 0 1
and the proof is complete. (cid:1)
Equation(17)givesanexactanalyticalexpressionfortheLSToftherthmomentoftherandom
linktraveltime,providedthatallderivativesexistats (cid:5) 0. Insomecases,thetransformmay
2
be inverted algebraically for an exact solution. However, very close approximations may be
obtainedvianumericalinversioninonlyonedimensionbyusinganumberofwidelyavailable
inversion algorithms such as the one due to Abate and Whitt [1].
4. ASYMPTOTIC LINK TRAVEL TIME MOMENTS
In this section, the asymptotic behavior of the moments of the random link travel time is
considered.Insubsections4.1and4.2,weinvestigatethefirstandsecondmomentsofthelink
traveltimeintheasymptoticregion(asx3(cid:6))whenthespeedismodulatedbyaCTMC,{Z(t)
: t (cid:1) 0}. Insubsection4.3,theasymptoticvarianceisconsidered.Asymptoticapproximations
often yield computationally expedient expressions that can drastically reduce computational
effortbyeliminatingthe needfor iterativealgorithms.Furthermore,theseapproximationsmay
be used to construct surrogate, parametric distributions for the link travel time.
4.1. Asymptotic First Moment of T(x)
In order to prove our result for the asymptotic mean of the random link travel time, we first
need the following lemma.
LEMMA2:LetQ betheinfinitesimalgeneratormatrixfortheenvironmentprocess,{Z(t) :
t (cid:1) 0}, having stationary distribution p (cid:5) [p] . Then the matrix Qˆ (cid:5) V(cid:9)1Q is an
j j(cid:1)S
infinitesimal generator for a CTMC, {Zˆ(t) :t (cid:1) 0}, with limiting distribution pˆ (cid:5) [pˆ ]
j j(cid:1)S
given by
pV
pˆ (cid:3) j j, j(cid:1)S, (18)
j pv
which satisfies pˆQˆ (cid:5) 0 and pˆe (cid:5) 1, where v (cid:5) Ve and pv (cid:5) ¥ pV.
j j j
250 NavalResearchLogistics,Vol.51(2004)
PROOF: The proof is immediate since V(cid:9)1Q is positive recurrent and clearly possesses the
stated unique, stationary distribution. (cid:1)
Theorem 3 provides an intuitive result for the mean link travel time, namely, that the long-run
averagelinktraveltimedividedbyitsdisplacementconvergestothereciprocalofthelong-run
average speed of the vehicle.
THEOREM 3: As x 3 (cid:6),
m (cid:3)x(cid:4) 1
1 3 . (19)
x pv
PROOF: By the asymptotic properties of the LST, it is well known (Kulkarni [10], p. 583)
that
lim s m˜ (cid:3)s (cid:4)(cid:3)limm (cid:3)x(cid:4)/x.
1 1 1 1
s130 x3(cid:6)
Thus, it will be shown that s m˜ (s ) 3 (pv)(cid:9)1 as s 3 0. Applying Theorem 2,
1 1 1 1
s m˜ (cid:3)s (cid:4)(cid:3)s z (cid:3)s V(cid:4)Q(cid:4)(cid:9)1e(cid:3)z s (cid:3)s I(cid:4)V(cid:9)1Q(cid:4)(cid:9)1V(cid:9)1e. (20)
1 1 1 1 0 1 0 1 1
ByLemma2,Qˆ (cid:5) V(cid:9)1Q isageneratormatrixfortheCTMC,{Zˆ(t) :t (cid:1) 0}, withprobability
transition matrix Pˆ(t) satisfying the forward equation,
dPˆ(cid:3)t(cid:4)
(cid:3)Pˆ(cid:3)t(cid:4)Qˆ. (21)
dt
Transform methods are employed to solve Eq. (21). After simplification, the LST of Pˆ(t),
denoted by, (cid:16)(s ), is
1
(cid:16)(cid:3)s (cid:4)(cid:3)s (cid:3)s I(cid:4)Qˆ(cid:4)(cid:9)1. (22)
1 1 1
By the limiting properties of the LST (Kulkarni [10]),
lim (cid:16)(cid:3)s (cid:4)(cid:3)limPˆ(cid:3)t(cid:4)(cid:3)Pˆ(cid:3)(cid:6)(cid:4), (23)
1
s130 t3(cid:6)
wherethejthcolumnofPˆ((cid:6)) haspˆ ofEq.(18)foreachrow,providedtheCTMCisergodic.
j
Now, substituting Eqs. (22) and (17) into Eq. (20) gives
lim s m˜ (cid:3)s (cid:4)(cid:3)lim z s (cid:3)s I(cid:4)V(cid:9)1Q(cid:4)(cid:9)1V(cid:9)1e(cid:3)z Pˆ(cid:3)(cid:6)(cid:4)V(cid:9)1e(cid:3)(cid:3)pv(cid:4)(cid:9)1,
1 1 1 0 1 1 0
s130 s130
and the result is obtained. (cid:1)