Throughput-Optimal Scheduling for Multi-Hop Networked Transportation Systems With Switch-Over Delay

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by  Ping-Chun Hsieh

Throughput-Optimal Scheduling for Multi-Hop Networked
Transportation Systems With Switch-Over Delay
Ping-Chun Hsieh, Xi Liu, Jian Jiao, I-Hong Hou, Yunlong Zhang, P. R. Kumar
TexasA&MUniversity
{pingchun.hsieh,xiliu,jiaojian,ihou,yz61,prk}@tamu.edu
ABSTRACT munication. Withconnected-vehicletechnologies,infrastruc-
7 The emerging connected-vehicle technology provides a new turescanobtainaccurateandreal-timeinformationaboutthe
1 dimensionindevelopingmoreintelligenttrafficcontrolalgo- numberofvehicleswaitingineachlane[24]. Thescheduling
0 problem in networked transportation systems then becomes
rithms for signalized intersections in networked transporta-
2 very similar to that in computer networks. In this analogy,
tionsystems.Animportantchallengefortheschedulingprob-
each intersection corresponds to a router, each lane corre-
n lem in networked transportation systems is the switch-over
spondstoaqueue,andeachvehiclecorrespondstoapacket.
a delaycausedbytheguardtimebeforeanytrafficsignalchange.
J Theswitch-overdelaycanresultinsignificantlossofsystem Indeed,therehavebeensomeeffortstoapplythewell-known
Max-Pressurepolicyincomputernetworks[23]tonetworked
9 capacityandhenceneedstobeaccommodatedintheschedul-
transportationsystems[26].
1 ingdesign. Totacklethischallenge,weproposeadistributed
Currently,mostschedulingalgorithmsmanagetrafficflows
online scheduling policy that extends the well-known Max-
at intersections via traffic signals, whose color switches pe-
] Pressurepolicytoaddressswitch-overdelaybyintroducinga
I riodically between red and green. When the color is green,
N bias factor toward the current schedule. We prove that the
thetrafficflowalongthecorrespondingdirectionobtainsthe
proposed policy is throughput-optimal with switch-over de-
. access to the intersection. The access will be revoked when
s lay. Furthermore,theproposedpolicyremainsoptimalwhen
c therearebothconnectedsignalizedintersectionsandconven- the color switches to red. Transition from green phase to
[ red phase is not instantaneous, but requires sufficient guard
tionalfixed-timeonesinthesystem. Withconnected-vehicle

2 technology, the proposed policy can be easily incorporated timeforsafety,whichusuallylastsfor3-8seconds[15]. The
throughputduringthistransitionphaseisnearlyzero. Inad-
v intothecurrenttransportationsystemswithoutadditionalin-
dition,thereisalsothroughputlosswhenanewgreenphase
1 frastructure. Through extensive simulation in VISSIM, we
starts or ends because of acceleration or deceleration of ve-
9 show that our policy indeed outperforms the existing popu-
hicles. We capture such capacity loss by introducing switch-
9 larpolicies.
over delay in this paper. The switch-over delay needs to be
3
explicitly addressed in designing scheduling policies for in-
0
1. INTRODUCTION tersections. Unfortunately, most of the existing literature on
.
1 Traffic congestion in urban area has been an increasingly schedulingofintersectionsviatrafficsignalsignoretheeffect
0 severeprobleminallcitiesofdifferentsizes. Accordingtoa of switch-over delay. In fact, Ghavami et al. [10] demon-
7 recentstudy[22],everydrivingcommuterintheU.S.spends stratethat,whilethedynamicsignalcontrolpolicieslikethe
1
on average 30 to 60 hours of extra time on the road each Max-Pressure policy outperform the conventional fixed-time
:
v year. Furthermore,abouttwothirdsoftheextratimecomes policyingeneral,theperformanceofthedynamicsignalcon-
i from road congestion. For an urban transportation network trolpoliciescanbeseriouslyaffectedwhencapacitylossdue
X
which consists of intersections as nodes and roads between toswitch-overdelayisconsidered.
r intersections as edges, intersections are often the source of Furthermore,duringthetransitionbetweentraditionaltrans-
a
roadcongestionaswellastheaccident-pronearea[17]. portation system and a fully connected system, only part of
Recently, considerable works are exploring novel schedul- theintersectionsareequippedwithsensorsandV2I/V2Vcom-
ing strategies for intersections from the perspective of net- munication [16], while other intersections need to rely on
workedtransportationsystems,whichincorporatetheemerg- conventional fixed-time control policies. In such partially-
ingconnected-vehicletechnologiessuchasvehicle-to-vehicle connectedsystems,thenewlyproposedpoliciesarerequired
(V2V)communicationandvehicle-to-infrastructure(V2I)com- tocoexistwellwithconventionalones.
This paper aims to address all the above challenges. We
proposeadistributedschedulingpolicyfornetworkedtrans-
portation systems and formally prove that the proposed pol-
icyisthroughput-optimalundertheexistenceofswitch-over
Permissiontomakedigitalorhardcopiesofallorpartofthisworkfor delay. The proposed policy accommodates the switch-over
personalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesare delay by adding a bias factor toward to the current sched-
notmadeordistributedforprofitorcommercialadvantageandthatcopies ule. Moreover, we introduce a superframe structure which
bearthisnoticeandthefullcitationonthefirstpage.Tocopyotherwise,to
achievessynchronizationamongconnectedintersectionsand
republish,topostonserversortoredistributetolists,requirespriorspecific
serves as a natural structure for stability analysis. Our main
permissionand/orafee.
Copyright20XXACMX-XXXXX-XX-X/XX/XX...$15.00.
contributioncanbesummarizedasfollows: andproposeafamilyofdynamicconepoliciesandbatchpoli-
• Switch-overdelayisconsideredandtackledinthepro- ciestoachieveoptimalthroughput. Subsequently,Hungand
posedpolicy,whichisprovedtobethroughput-optimal Chang [14] present a generalized version of dynamic cone
undertheexistenceofswitch-overdelay. policy to reduce the complexity of the original cone policy.
Chan [7] also presents a Max-Weight type policy with hys-
• Theproposedpolicyisdistributedwithlowimplemen-
teresisandprovethatitisthroughput-optimalforasystemof
tationcomplexity,andthereforeitscaleswellwithnet-
parallelqueueswithdeterministicserviceprocesses. Besides,
worksize.
Celik et al. [5] propose a family of generalized Max-Weight
• Throughput-optimalityoftheproposedpolicydoesnot policiesandprovethatanypolicysatisfyingtheproposedcri-
dependonanyknowledgeoftrafficdemands. teria is throughput-optimal. As an exemplar policy in [5],
• Throughput-optimality of the proposed policy is pre- the Variable Frame-Based Max-Weight (VFMW) policy intro-
servedwhentherearebothconnectedintersectionsand duces a frame structure to avoid excessive capacity loss due
fixed-time intersections in the system. Therefore, the to switch-over delay. However, all of the above policies are
proposedpolicycanstillperformwellinpartially-connected designedspecificallyforsingle-hopsystemsandhencetheop-
transportationsystems. timalityresultsmaynotbeguaranteedinmulti-hopsystems.
In this paper, we regard VFMW as the reference policy for
• Weevaluatetheproposedpolicyviarealisticmicroscopic
comparison in simulation. In Section 7, we will show that
simulation on a standard simulator for transportation
theVFMWpolicy,whichisthroughput-optimalforsingle-hop
research.
systems,canactuallyperformpoorlyinmulti-hopsystems.
Whilethispaperfocusesonnetworkedtransportationsys-
tems,ourtheoreticalresultsarealsoapplicabletomanyother
applicationswithswitch-overdelay,suchasopticalnetworks 3. SYSTEMMODEL
[19], wireless networks with directional antennas [21], and
Wemodelamulti-hoptransportationsystembyadirected
multi-thread operating systems [9]. The rest of the paper is
graph (V,L), where V denotes the set of intersections and
organizedasfollows. Section3describesthemodelofinter-
L is the set of directional links connecting the intersections.
sectionsandmulti-hoptransportationsystems. Theproposed
Eachlinkhasastartnodeandanendnode. Inthispaper,we
schedulingpolicyisillustratedandtheproofofoptimalityis
usethetermsnodeandintersectioninterchangeably. Forcon-
provided in Section 5. Section 7 presents the simulation re-
venience, we also include one common virtual source node
sults. Section8concludesthepaper.
v as well as one common virtual destination node v in the
s d
directedgraph. Weassumetimeisslotted. Thelinkscanbe
2. RELATEDWORK
furtherdividedintothreecategories:internallinksL ,entry
int
In current transportation systems, traffic signals are often linksLentry,andexitlinksLexit. Eachentrylinkhasthesame
adaptivelycontrolledbyproprietarytrafficcontrolsuites,such start node vs and an end node v ∈ V where v (cid:54)= vd. Simi-
asSCATS[18]andSCOOT[2]. Followingthefixed-timecon- larly,eachexitlinkhasthesameendnodev andastartnode
d
trol paradigm, these software suites require real-time traffic v ∈ V where v (cid:54)= vs. Therefore, entry links and exit links
statisticstooptimizecyclesplitsandoffsetsinthetimingplan together characterize the boundary of a system. This model
forsomegivenobjectivefunctions. However,sincetrafficde- canalsotakegaragesintoaccountbymodellingeachgarage
mandscanchangerapidlywithtime,itmightbedifficultand asanentrylinkplusanexitlink.
costlytocollectthestatisticsinatimelymanner. Giventwolinksi,j ∈ Lincidenttothesameintersection,
Different from the fixed-time approach, scheduling design linkiiscalledadownstreamlinkofj (orequivalently,iisan
basedonreal-timequeuelengthinformationisattractingmore upstream link of j) if the end node of link i is the same as
andmoreattentionduetotherecentprogressinconnected- thestartnodeoflinkj. WeuseD(i)andU(i)todenotethe
vehicle technology. For example, adaptive control based on setofallthedownstreamlinksandthesetofalltheupstream
queuelengthisproposedin[24],wherequeuelengthisesti- links of each link i, respectively. Without loss of generality,
mated via probe vehicles with V2I and V2V communication. we suppose that each link has at most Umax upstream links.
On the other hand, inspired by the results in computer net- Moreover, the link pair (i,j) forms a movement of vehicles.
works[23],Varaiya[26]andWongpiromsarnetal.[27]pro- WedenoteMv tobethesetofmovementsofeachintersec-
poseindividuallyaMax-Pressurepolicyforsignalcontroland tionv ∈ V anddefineM := ∪v∈VMv. Besides,acollection
formally prove that the Max-Pressure policy is throughput- ofnon-conflictingmovementsiscalledanadmissiblephaseof
optimal when the queue capacity is infinite and the routing an intersection. For example, Figure 1 shows a standard in-
rates are known. To relax the assumption of infinite queue tersectionofeightmovementsandfouradmissiblephases.
capacity,Xiaoetal.[28]presentavariationofMax-Pressure
policy that is throughput-optimal within a reduced capacity
8 7 8 7 8 7 8 7
region when the queue capacity is finite but large enough.
Torelaxtheassumptiononroutingrates,Gregoireetal.[11]
1 6 1 6 1 6 1 6
alsoproposeaback-pressure-basedsignalcontrolpolicyand
2 5 2 5 2 5 2 5
prove that it is throughput-optimal with unknown routing
rates.Despitetheaboveprogress,noneofthesepoliciestakes 3 4 3 4 3 4 3 4
theswitch-overdelayintoaccount. (a) (b) (c) (d)
Amongtheexistingliteratureontheschedulingdesignfor
systemswithswitch-overdelay,[4,14,7,5]arethemostrel-
evant to the scope of this paper. First, Armony and Bambos Figure1: Atypicalintersectionwitheightmovementsand
[4]studyasystemofparallelqueueswithswitch-overdelay 4admissiblephases.
Inthistypicalintersection,eachlinkhastwoupstreamlinks eachintersection,thetimebetweentwoswitch-overeventsis
and two downstream links. For ease of explanation, we as- calledaframe.
sumethatvehiclescanonlygostraightorturnleft,butcannot In this paper, each intersection is either a fixed-time inter-
turnright,inthisexample. Eachmovement(i,j)hasanasso- section or a connected intersection. For a fixed-time intersec-
ciatedqueueQ holdingincomingvehicles. Inotherwords, tion, it simply follows the weighted round-robin policy with
i,j
we assume that there exists a separate queue for each left- the weights determined a priori according to long-term av-
turnandthroughmovement. Weassumethateachqueuehas erage traffic demands. In contrast, a connected intersection
infinitesizesuchthatthereisnooverfloworblockageateach dynamically makes scheduling decisions based on real-time
intersection. Throughout this paper, we use the three-tuple informationobtainedviaconnected-vehicletechnology, such
G =(V,L,M)todenoteatransportationsystem. asqueuelength.WeuseV andV todenotethesetoffixed-
F C
Externalvehiclesenterthesystemonlyviatheentrylinks. timeintersectionsandconnectedintersections,respectively.
For any entry link i and its downstream link j ∈ D(i), let For simplicity of notation, we use boldface fonts for vec-
{A (t)} beani.i.d. sequenceofexternalarrivalsatQ tors and matrices throughout the paper. For example, λ =
i,j t≥0 i,j
withaverageexternalarrivalrateλ >0andA (t)≤A (λ ) denotes the per-link external arrival rate vector and
i,j i,j max i i∈L
atanytimet. Foranynon-entrylinkiandj ∈D(i),wesim- Q(t) = (Q (t)) denotes the queue length vector of
i,j (i,j)∈M
ply let A (t) = 0 for all t and hence λ = 0. For ease of allthequeuesinthesystem.
i,j i,j
(cid:80)
later discussion, we also define λ := λ to be the
i j∈D(i) i,j
total external arrival rate through each link i. Similarly, let 4. CAPACITYREGION
{S (t)} beani.i.d. sequenceofpotentialserviceratesof
i,j t≥0 Tostudythroughput-optimality,wefirstneedtocharacter-
the movement (i,j) with average service rate µ , for each
i,j izethecapacityregionofamulti-hoptransportationsystem.
movement (i,j) ∈ M. We also assume that S (t) ≤ S ,
i,j max
foranymovement(i,j)andanytimet. S (t)capturesthe
i,j
variation in the passage time required by different vehicles. DEFINITION 1. Amulti-hoptransportationsystemisstrongly
Since S (t) depends on the instantaneous conditions such stableunderaschedulingpolicyπif
i,j
asvehiclespeedanddriverbehavior,itisdifficultforthetraf-
T−1
fic scheduler to obtain the information about potential ser- limsup 1(cid:88) (cid:88) E(cid:2)Q (τ)(cid:3)<∞. (1)
vice rates. Therefore, we presume that the traffic scheduler T→∞ T i,j
τ=0(i,j)∈M
only has the information of average service rate, which is of-
ten called saturation flow in the transportation community. Meanwhile,wesaythatthepolicyπstabilizesthesystem.
Besides, the average service rate of a movement is roughly Next, we introduce the definition of feasible external arrival
proportionaltothenumberoflanesofthatmovement[25]. ratevectors:
Inourmulti-hopmodel,vehiclesareroutedinaprobabilis-
tic manner. When a vehicle enters a link i, it will choose to DEFINITION 2. Given a multi-hop transportation system G,
join a downstream link j ∈ D(i) independently with proba- an external arrival rate vector λ = (λi)i∈L is feasible if there
bilityr with(cid:80) r =1. Hereweassumetherouting exists a scheduling policy under which the system is strongly
i,j j∈D(i) i,j stablewithλ.
probabilityr >0,forallmovements(i,j)∈M. LetR (t)
i,j i,j
denotetheportionofvehiclesthatjoinQi,j amongthevehi- From the definition of feasibility, we can define the capacity
cles entering link i at time t, where 0 ≤ Ri,j(t) ≤ 1. Since regionasfollows:
each vehicle chooses its route independently, then we know
thatE[R (t)]=r foranytimetbythebasicpropertiesof DEFINITION 3. Thecapacityregionisdefinedastheclosure
i,j i,j
ofthesetofallthefeasibleexternalarrivalratevectorλ.
multinomialrandomvariables. Notethattheabovemodelof
arrivals, service, and routing is similar to that of the classic To explicitly characterize the capacity region, we first ob-
openJacksonnetwork. taintheeffectivearrivalrate,whichincludebothexternalar-
For each intersection, at each time slot exactly one of the rivalsandarrivalsfromupstreamlinks,ofeachlinkandthen
admissible phases is chosen to have the right of way based providethenecessaryandsufficientconditionofthecapacity
on its scheduling policy. Let Ii,j(t) be the indicator function region. Letλ∗i betheeffectivearrivalrateoflinki. According
of whether Qi,j is scheduled at the corresponding intersec- to our model, we have λ∗i = λi for all i ∈ Lentry. For any
tionattimet. Therefore,foreachintersectionv ∈V,wecan linkj ∈ L\L , theeffectivearrivalrateisdeterminedby
entry
usea|Mv|-dimensionalbinaryvectortorepresentthesched- λ∗j = (cid:80)i:j∈D(i)λ∗iri,j. Let λ∗ = (λ∗i)i∈L be the effective ar-
uledphaseoftheintersection. LetIv bethecollectionofthe rivalratevectorandR=(r ) betheroutingprobability
i,j i,j∈L
schedule vectorsof all theadmissible phasesat the intersec-
matrix. Then,wecanwritethesystemoftrafficequationsin
tion v. Then, under a scheduling policy, each intersection v
matrixform:
determinesI (t)∈I ateachtimet.
Moreover,vin orderv to guarantee absolute safety, it takes λ∗ =λ+R(cid:124)λ∗, (2)
non-zero time for an intersection to switch the right of way where R(cid:124) is the transpose of the routing probability matrix.
from the current schedule to the next. Such loss of service Notethat(2)issimilartothesystemoftrafficequationsofan
time during traffic signal change is modelled as switch-over openJacksonnetwork.Let1bean|L|×|L|identitymatrix.It
delay,duringwhichallthemovementsattheintersectionare iseasytoverifythattheequationin(2)hasauniquesolution
prohibited and hence the throughput is zero. For simplicity, asλ∗ =(1−R(cid:124))−1λ,where(1−R(cid:124))isinvertible(Section
weassumethattheswitch-overdelayisTS slot(s)forallthe 2.1in[8]).
intersections. Besides,anintersectionissaidtobeactiveifit Foreachfixed-timeintersectionv,letξ ∈(0,1)betheav-
v
isnotinswitch-over. LetXi,j(t)betheindicatorfunctionof eragefractionoftimeinwhichtheintersectionvisinswitch-
theeventthatthemovement(i,j)isactiveattimeslott. For over. Sincethepolicyofeachfixed-timeintersectionisgiven
apriori,thenξ isalsofixed. LetΛbethesetofalltheexter- 5.1 AThroughput-OptimalSchedulingPolicy
v
nalarrivalratevectorsλwithwhichthefollowingconditions Tobeginwith,wedefinepressureasfollows:
hold: (i)Foreachfixed-timeintersectionv,thereexists(cid:15)>0
and a vector Σ = (Σ ) in the convex hull of I DEFINITION 5. For any time t, the pressure of a movement
v i,j (i,j)∈Mv v (i,j)∈Misdefinedasthedifferencebetweenthequeuelength
suchthattheeffectivearrivalratessatisfythat
of(i,j)andtheweightedaverageofthequeuelengthsof(j,k)
ξvµi,jΣi,j >λ∗iri,j+(cid:15), ∀(i,j)∈Mv. (3) foreveryk∈D(j),i.e.
In other words, a fixed-time intersection v needs to have at W (t):=Q (t)− (cid:88) r Q (t). (5)
i,j i,j j,k j,k
leastasmallservicemarginforeverymovementatv.
k:k∈D(j)
(ii) For each connected intersection v ∈ V there exists
C
(cid:15)>0andavectorΣ =(Σ ) intheconvexhullof Inaddition,foranyintersectionv,thepressureofanyadmissi-
v i,j (i,j)∈Mv (cid:80)
Iv suchthat blephaseIv =(Ii,j)∈Ivisdefinedas i,j∈Mvµi,jIi,jWi,j(t).
µi,jΣi,j >λ∗iri,j+(cid:15), ∀(i,j)∈Mv. (4) Wealsointroduceausefuldefinition:
Besides, letΛbetheclosureofΛ. ThefollowingTheorem1 DEFINITION 6. Aschedulingpolicyπissaidtobemax-pressure-
providesasufficientconditionforcapacityregion. at-switch-over if π always schedules the phase with the maxi-
mumpressureateachswitch-overevent.
Now, we formally present the Biased Max-Pressure (B-MP)
THEOREM 1. Foramulti-hoptransportationsystemwith
schedulingpolicyinAlgorithm1. InB-MP,timeisdividedinto
switch-over delay, an external arrival rate vector λ =
consecutive superframes. At the beginning of a superframe,
(λ ) isfeasibleifλ∈Λ.
i i∈L
the duration of a superframe is calculated by (6). When-
everaconnectedintersectionswitches, italwaysswitchesto
PROOF. Thiscanbeprovedbyfindingaproperfixed-time the phase with the maximum pressure, and therefore B-MP
policyforeachconnectedintersection. ByTheorem1in[26], ismax-pressure-at-switch-over. Aconnectedintersectionwill
wedirectlyknowthatgivenanyλ ∈ Λ, thereexistsafixed- onlyswitchundertwoconditions:(i)atthebeginningofeach
timepolicyforeachconnectedintersectionsuchthatthewhole superframe,or(ii)whenconditionspecifiedby(7)and(8)is
systemisstronglystable. Hence,λmustbefeasibleifλ∈Λ. satisfied. Fromcondition(7)-(8),wecanseethatB-MPonly
makeaswitchwhenthemaximumpressureislargerthanthe
pressureofthecurrentphasebyacertainportion. Condition
Next,weprovideanecessaryconditionforcapacityregionin (7)canbeinterpretedasaddingabiasfactortowardthepres-
Theorem2. sureofthecurrentphase,andhencethenameB-MP.Thisbias
towardthecurrentphaseistopreventthetrafficsignalfrom
significantcapacitylossduetofrequentswitch-overs.
THEOREM 2. Foramulti-hoptransportationsystemwith Moreover, within one superframe, each connected inter-
switch-overdelayandwithanexternalarrivalratevector sectionunderB-MPcanmakeschedulingdecisionsindepen-
λ, if λ ∈/ Λ, then there exists no policy under which the dentlybasedononlythelocalqueuelengthinformation.There-
systemisstronglystable. fore,B-MPisfullydistributedwithineachsuperframeandthe
coordination among the connected intersections is minimal.
Weuset todenotethebeginningofthek-thsuperframeand
k
PROOF. ThisisadirectresultofTheorem1in[26]. set t = 0. Let T := t −t be the length of the k-th
0 k k+1 k
superframe. Besides, let Mv be the number of switch-over
k
Hence,byTheorem1andTheorem2,thecapacityregioncan eventsinthek-thsuperframeforeachconnectedintersection
becharacterizedasfollows: v. Since each superframe may contain different number of
framesatdifferentconnectedintersections,weusetv tode-
k,l
note the time of the l-th switch-over at intersection v in the
THEOREM 3. Givenamulti-hoptransportationsystem k-thsuperframeandsettvk,0 =tk.
G withswitch-overdelay,thecapacityregionofG isΛ.
Algorithm1BiasedMax-PressurePolicy(B-MP)
Given the knowledge of the capacity region, the concept of
1: Attimet=t ,obtainthelengthofthek-thsuperframe:
k
throughput-optimalityisdefinedasfollows:
(cid:16) (cid:88) (cid:17)β
T = Q (t ) , β ∈(0,1). (6)
DEFINITION 4. Given a multi-hop transportation system G, k i,j k
a scheduling policy π is said to be throughput-optimal if the (i,j)∈M
systemisstronglystableunderπwithanyexternalarrivalrate Calculatethebeginningofthenextsuperframeast =
k+1
vectorλ∈Λ. t +T .
k k
2: Findthephasewiththelargestpressureatcurrenttimet,
5. SCHEDULINGFORTHROUGHPUTOP- i.e.
TIMALITY I∗(t)∈argmax (cid:88) µ I W (t).
v I∈Iv i,j i,j i,j
Inthissection,weintroduceourschedulingpolicyforcon- (i,j)∈Mv
nected intersections and prove that it is throughput-optimal
Tiesarebrokenarbitrarily.
withswitch-overdelay.
3: If I∗(t) (cid:54)= I∗(t−1), initiate switch-over for the next T whereQ(t)(cid:124) isthetransposeofthequeuelengthvector. De-
v v S
slotsandthenapplythenewscheduleI∗(t)foroneslot. finetheLyapunovdriftoverthek-thsuperframeas∆L(t ):=
v k
Else,directlyapplyI∗(t)foroneslot. L(Q(t ))−L(Q(t )). Then,wehave
v k+1 k
4: Foranyt∈[tv ,tv )intherestofthek-thsuperframe, (cid:124) (cid:124)
k,l k,l+1 ∆L(t )=2Q(t ) ∆Q(t )+∆Q ∆Q(t ), (15)
find the phase I∗(t) that has the largest pressure. If the k k k k
v
intersectionisnotinswitch-overattimet,theintersection where∆Q(t ):=Q(t )−Q(t ). GivenQ(t ),thesizeof
k k+1 k k
willmakeaswitchifthefollowingconditionissatisfied: the k-th superframe is known and therefore the conditional
driftoverthek-thsuperframeiswell-defined. Notethatitis
(cid:16)1+B (tv )(cid:17)(cid:18) (cid:88) µ I∗ (t−1)W (t)(cid:19)+ (7) actuallynotstraightforwardtocalculatetheconditionaldrift
v k,l i,j i,j i,j
overonesuperframe:
(i,j)∈Mv
(cid:18) (cid:19)+ • Foranyintersection,therecouldbemultipleframesand
< (cid:88) µi,jIi∗,j(t)Wi,j(t) , (8) hencemultiplephasesscheduledinastochasticsequence
(i,j)∈Mv inonesuperframe.
where x+ is a shorthand for max{x,0} and B (·) is the • Differentintersectionscouldpossiblyhavetotallydiffer-
v entframesizesinthesamesuperframe.
biasfunctiondefinedas
• Giventhequeuelengthinformationatthebeginningof
(cid:40) (cid:18)(cid:20) (cid:21)+(cid:19)−α(cid:41)
(cid:88) asuperframe, itisstillnotclearwhenswitch-overwill
B (t)=ζT min 1, W (t) (9)
v S i,j betriggeredandwhichphasewillbescheduledateach
(i,j)∈Mv intersection since the arrival and service processes are
withα∈(0,1)andζ >0. Otherwise,stayatthecurrent stochastic.
phase. Despite the above challenges, the conditional drift over one
5: RepeatStep3andStep4untiltheendofthek-thsuper- superframecanstillbecharacterizedforthemax-pressure-at-
frame. switch-overpolicies. Wefirstprovideanupperboundonthe
conditionaldriftinthefollowinglemma.
6: Att = t ,gobacktostep1andrepeattheabovepro-
k+1
cedureforthenextsuperframe.
LEMMA 1. Givenanyλ∈Λ,underanymax-pressure-
at-switch-over policy with superframe structure, the con-
5.2 ProofofThroughput-Optimality
ditionaldriftoveronesuperframeisupperboundedas
daTteoostvuedryonsyestseumpesrtfarabmiliety.,Dweeficnoen∆sidQert(hte)q:u=euQele(ntgth)up−- E(cid:2)∆L(tk)(cid:12)(cid:12)Q(tk)(cid:3)≤−2(cid:15)Tk (cid:88) Wi,j(tk)+ (16)
i,j k i,j k+1
Q (t ).Foranymovement(i,j)withlinki∈L ,wehave (i,j)∈M
i,j k entry
(cid:18) (cid:19)
∆Qi,j(tk) (10) +C1 (cid:88) Mkv (cid:88) Wi,j(tk)+ (17)
=−tk(cid:88)+1−1(cid:16)Si,j(t)Ii,j(t)Xi,j(t)∧Qi,j(t)(cid:17)+tk(cid:88)+1−1Ai,j(t), +C2v(cid:88)∈VC (cid:88)(i,j)W∈Mi,jv(tk)++C3Tk2+C4Tk(18)
t=tk t=tk v∈VF(i,j)∈Mv
(11)
whereC ,C ,C andC arefinitepositiveconstantsand
1 2 3 4
where(x∧y)isashorthandformin{x,y}. Notethatthefirst x+isashorthandformax{x,0}.
term of (11) represents the number of vehicles that actually
leaves Q during the k-th superframe and the second term
isthetotia,ljexternalarrivalsatQ inthek-thsuperframe. PROOF. Withthemax-pressure-at-switch-overproperty,we
i,j are able to quantify the pressure of the scheduled phases at
Ontheotherhand,foranymovement(i,j)∈Mwithlink
any t ∈ [t ,t ) even if the scheduling decision of each
i∈/ L ,wehave k k+1
entry frame is not known. The complete proof is provided in the
tk(cid:88)+1−1(cid:16) (cid:17) AppendixA.
∆Q (t )=− S (t)I (t)X (t)∧Q (t) (12)
i,j k i,j i,j i,j i,j
t=tk REMARK 1. Note that (16) represents the negative drift re-
quired for system stability. Besides, (17) and the first term of
tk(cid:88)+1−1 (cid:88) (cid:16) (cid:17) (18)representthelossofserviceduetoswitch-overatconnected
+ S (t)I (t)X (t)∧Q (t) R (t).
m,i m,i m,i m,i i,j
intersections and the fixed-time intersections, respectively. The
t=tk m:(m,i)∈M secondandthethirdtermof(18)standfortheservicelossdue
(13)
topossibleemptinessofthescheduledqueues.
Notethat(13)representsthetotalnumberofvehiclescom-
ingfromtheupstreamlinksofiduringthek-thsuperframe. REMARK 2. Notethatin(17)theservicelossduetoswitch-
To study the throughput performance of such policies, we over is basically a direct sum of the service loss contributed by
apply Lyapunov drift analysis and study the Lyapunov drift each connected intersection. In other words, the performance
acrossonesuperframe. DefineaLyapunovfunctionas ofconnectedintersectionsarecompletelydecoupled. Duetothis
feature, Lemma1stillholdsifdifferentconnectedintersections
L(Q(t)):=Q(t)(cid:124)Q(t)= (cid:88) Q (t)2, (14) followdifferentmax-pressure-at-switch-overpolicieswithsuper-
i,j
framestructure.
(i,j)∈M
To show that B-MP is throughput-optimal, we introduce a
sufficientconditionforstrongstabilityinthefollowinglemma.
LEMMA 3. For any queue length vector Q = (Qi,j)
anditscorrespondingpressurevectorW = (W ),there
i,j
mustexistaconstantδ>0suchthat
(cid:18) (cid:19)
LEMMA 2. Foranymax-pressure-at-switch-overschedul- (cid:88) W + ≥δ (cid:88) Q . (26)
i,j i,j
ingpolicywithsuperframedeterminedby(6),ifthereex-
(i,j)∈M (i,j)∈M
ists some constant B > 0, (cid:15) > 0 and the conditional
0 0
driftsatisfiesthat
(cid:18) (cid:19)1+β PROOF. Weprovideasketchoftheproof:Wefirstconstruct
E(cid:2)∆L(tk)(cid:12)(cid:12)Q(tk)(cid:3)≤B0−(cid:15)0 (cid:88) Qi,j(tk) , a new system by adding several dummy links and dummy
movementstotheoriginalsystemandshowthatthenewsys-
(i,j)∈M
(19) temisstronglyconnectedandthecorrespondingroutingma-
thenwehave trixisinvertible. ByapplyingthePerron-FrobeniusTheorem
totheroutingmatrix,weobtainastrictlypositiveeigenvector
T−1
limsup 1(cid:88) (cid:88) E(cid:2)Q (t)(cid:3)<∞. (20) withapositiveeigenvalue. Basedontheeigenvectorproper-
T i,j ties,weshowthattheremustexistaconstantδ>0suchthat
T→∞
t=0(i,j)∈M theinequality(26)holds. Thecompleteproofisprovidedin
AppendixC.
PROOF. DefineH(tk):=(cid:80)Tt=k0−1(cid:80)(i,j)∈MQi,j(tk+t).Then, NotethatB-MPisamax-pressure-at-switch-overpolicyand
wehave
therefore Lemma 1 holds under the B-MP policy. To charac-
terize the number of switch-over events in one superframe
H(t )≤T(cid:88)k−1 (cid:88) (cid:16)Q (t )+T(cid:88)k−1A (t +s)(cid:17) under the B-MP policy, we provide an upper bound on the
k i,j k i,j k
sizeofeachframeasfollows.
t=0 i∈Lentry,j∈D(i) s=0
T(cid:88)k−1 (cid:88)
+ Q (t ).
i,j k
t=0 i∈Lint,j∈D(i) LEMMA 4. Under the B-MP policy, there exists a con-
stantC > 0suchthatthelengthofeachframeislower
5
AftertakingconditionalexpectationofH(t ),wehave boundedas
k
E(cid:2)H(tk)(cid:12)(cid:12)Q(tk)(cid:3) (21) Tkv,l ≥C5Bv(tvk,l)(cid:16) (cid:88) Wi,j(tvk,l)+(cid:17). (27)
(cid:18) (cid:19) (cid:18) (cid:19) (i,j)∈Mv
≤T2 (cid:88) λ∗r +T (cid:88) Q (t ) (22)
k i i,j k i,j k
i∈Lentry,j∈D(i) (i,j)∈M PROOF. TheproofisprovidedinAppendixD.
(cid:18) (cid:19)1+β
(cid:88)
≤B Q (t ) (23)
1 i,j k WithLemma4,wearereadytoprovidealowerboundon
(i,j)∈M the number of switch-over events in one superframe under
where B = 1+(cid:80) λ∗r is a positive constant inde- B-MP.
1 i∈Lentry i i,j
pendentofQ(t ). Then,by(19),
k
E(cid:2)∆L(tk)(cid:12)(cid:12)Q(tk)(cid:3)≤B0− B(cid:15)0 E(cid:2)H(tk)(cid:12)(cid:12)Q(tk)(cid:3). (24) icyLwEiMthMbAia5s.fuFnocrtiaonnydienfitenresdecbtyio(n9v),uwnedheravthee∀Bk-≥MP0,pol-
1
Bysumming(107)overallthesuperframes,wehave Mv(cid:16) (cid:88) W (t )+(cid:17)=o(cid:18)(cid:16) (cid:88) Q (t )(cid:17)1+β(cid:19).
k i,j k i,j k
(cid:88)E(cid:2)∆L(tk)(cid:12)(cid:12)Q(tk)(cid:3)≤(cid:88)(cid:16)B0− B(cid:15)0 E(cid:2)H(tk)(cid:12)(cid:12)Q(tk)(cid:3)(cid:17). (i,j)∈Mv (i,j)∈M (28)
1
k≥0 k≥0
(25)
PROOF. TheproofisprovidedinAppendixE.
GivenafiniteinitialconditionQ(0),wehaveL(0) < ∞and
(cid:80)k≥0E(cid:2)∆L(tk)(cid:12)(cid:12)Q(tk)(cid:3)≥−L(0). Hence,weconcludethat We are ready to show that B-MP is throughput-optimal.
limsup(cid:80)Tt=−01E(cid:2)(cid:80)(i,j)∈MQi,j(t)(cid:3) ≤ B1(cid:0)B0+L(0)(cid:1) <∞.
T (cid:15)
T→∞ 0 THEOREM 4. The B-MP policy is throughput-optimal
foranyα∈(0,1),β ∈(0,1).
Next,sinceLemma1involvesbothqueuelengthandpres- PROOF. Since B-MP is a max-pressure-at-switch-over pol-
sure,weprovideausefulinequalitybetweentotalqueuelength icy with superframe structure, then Lemma 1 holds under
andtotalpressureasfollows. B-MP. Therefore, by Lemma 3 and the fact that W (t)+ ≤
i,j
Q (t)foranymovement(i,j)andanytimet,wehave Queueoverflowoftenoccurswhenthesystemoperatesunder
i,j
oversaturatedtraffic(evenmerelyforashortperiodoftime).
E(cid:2)∆L(tk)(cid:12)(cid:12)Q(tk)(cid:3)≤−2(cid:15)δ0Tk (cid:88) Qi,j(tk) (29) Theoverfloweffectcanleadtosignificantservicelossaswell
(i,j)∈M asseveredelay. Giventheinformationaboutqueuecapacity,
+C (cid:88) Mv(cid:16) (cid:88) W (t )+(cid:17) (30) we can properly choose qi,j for each movement (i,j) to re-
1 k i,j k duce the chance of queue overflow. For example, choosing
v∈VC (i,j)∈Mv q tobeinverselyproportionaltothequeuecapacityofQ
+C (cid:88) (cid:88) Q (t )+C T2+C T . (31) isi,jsuggested in [12]. In Section 7, we show an exampleio,jf
2 i,j k 3 k 4 k
v∈VF(i,j)∈Mv applyingweightedqueuelengthinsimulation.
(cid:80)
ByLemma5andthechoiceofT ,weknowT Q (t )
k k (i,j)∈M i,j k
isthedominatingtermamong(29)-(31).Therefore,thereex- 6.2 Estimated Queue Length With Bounded
istsaconstantB >0suchthat Error
(cid:18) (cid:19)1+β
E(cid:2)∆L(tk)(cid:12)(cid:12)Q(tk)(cid:3)≤B−(cid:15)δ0 (cid:88) Qi,j(tk) . (32) orIenxpneentwsiovreketodotrbatanisnpocrotmatpiolenteslyystaecmcus,raittemqiugehutebleendgiftfihcuinlt-
(i,j)∈M formation due to the latency in communication or random
ByLemma2,weknowthatthesystemisstronglystableunder errorinsensordetection. LetQ† (t)andW† (t)betheesti-
i,j i,j
the B-MP policy for any external arrival rate λ ∈ Λ. Hence, matedqueuelengthandthecorrespondingpressure,respec-
theB-MPpolicyisthroughput-optimal. tively. If the estimation error of queue length is always up-
perbounded, thentheB-MPisstillthroughput-optimalwith
REMARK 3. By Theorem 4, B-MP can achieve throughput- the estimated queue length. We still consider the Lyapunov
optimalitywithanyαbetween0and1. Meanwhile,thechoice functionL(Q(t))=(cid:80) Q (t)2andthecorresponding
(i,j)∈M i,j
ofαcanindeedaffecttheaveragedelayperformance. Theissue drift conditioned on Q† (t ). Then, we have the following
onchoosingαforachievingoptimaldelayinmulti-hopsystems i,j k
upperboundontheconditionaldrift:
withswitch-overdelaywillbeourfutureworkandisbeyondthe
scopeofthispaper.
REMARK 4. Theparameterβdeterminesthesuperframesize LEMMA 6. Given any λ ∈ Λ, under the B-MP policy
forcoordinationamoingtheintersections. Tominimizetheco- using estimated queue length (Q†i,j(t)), if there exists a
(cid:12) (cid:12)
ordinationoverhead,βisrecommendedtobecloseto1. constantB >0suchthat(cid:12)Q (t)−Q† (t)(cid:12)≤B forall
(cid:12) i,j i,j (cid:12)
(i,j)andallt, theconditionaldriftoveronesuperframe
6. EXTENSIONSOFBIASEDMAX-PRESSURE
isupperboundedas:
POLICY
E(cid:2)∆L(tk)(cid:12)(cid:12)Q†(tk)(cid:3)≤−2(cid:15)Tk (cid:88) Wi†,j(tk)+ (34)
6.1 WeightedQueueLength (i,j)∈M
(cid:18) (cid:19)
Theconceptofpressurecanbefurthergeneralizedbyusing +C† (cid:88) Mv (cid:88) W† (t )+ (35)
weightedqueuelegnth: 1 k i,j k
v∈VC (i,j)∈Mv
DEFINITION 7. Let qi,j > 0 be the pre-determined weight +C2† (cid:88) (cid:88) Wi†,j(tk)++C3†Tk2+C4†Tk (36)
factorofmovement(i,j). Foreachmovement(i,j), wedefine
the weighted queue length as Qˆ (t) := q Q (t), for all t. v∈VF(i,j)∈Mv
i,j i,j i,j
Then,thegeneralizedpressureisdefinedas whereC1†,C2†,C3†andC4†arefinitepositiveconstants.
Wˆ (t):=Qˆ (t)− (cid:88) r Qˆ (t). (33)
i,j i,j j,k j,k
k:k∈D(j) PROOF. The proof is similar to that of Lemma 1, and the
maindifferencesare:(i)Sincethedriftisnowconditionedon
BysubstitutingWˆ (t)forW (t),theB-MPpolicyremains
i,j i,j Q†(t ) instead of Q(t ), the estimation error introduces an
throughput-optimal: k (cid:104) k (cid:105)
extraterminE Q(t )(cid:124)∆Q(t )|Q†(t ) . Duetothebound-
k k k
edness of estimation error, this extra term is at most of the
THEOREM 5. The B-MP policy using the generalized same order as Tk. (ii) For connected intersections, B-MP
pressureinDefinition7isstillthroughput-optimalforany using Q†(tk) makes scheduling decisions based on W†(tk).
α∈(0,1),anyβ ∈(0,1). Therefore, B-MP is max-pressure-at-switch-over in terms of
W†(t )insteadofW(t ). Besides, sinceQ (t)−Q† (t) ∈
k k i,j i,j
[−B,B],wealsohaveW (t)−W† (t) ∈ [−2B,2B],forall
PROOF. This can be proved by considering the drift of a (i,j)andallt. Asaresulti,,jthebounid,jederrorinpressureonly
Lyapunov function: Lˆ(Q(t)) = (cid:80) q Q (t)2. The
(i,j)∈M i,j i,j affects the coefficients of the existing terms in the original
rest of the proof is similar to that of Theorem 4 and hence driftexpression. ThecompleteproofisinAppendixF.
omittedduetospacelimitation.
One important application of weighted queue length is to
designacapacity-awareversionoftheB-MPpolicyinorderto Now,wearereadytoprovethatB-MPisthroughput-optimal
mitigatethequeueoverfloweffectduetofinitequeuecapacity. withestimatedqueuelengths.
WeevaluatetheproposedpolicyinVISSIM[1],whichisa
standardmicroscopictrafficsimulatorfortransportationsys-
THEOREM 6. IfthereexistsaconstantB >0suchthat
(cid:12) (cid:12) tems.Inadditiontothebuilt-infeaturesforconventionaltraf-
(cid:12)Q (t)−Q† (t)(cid:12) ≤ B for all (i,j) and all t, then B-
(cid:12) i,j i,j (cid:12) ficsignalcontrol,VISSIMalsoprovidesprogrammingintegra-
MPisstillthroughput-optimalusingtheestimatedqueue tionwithMATLABtosupportuser-customizabletrafficcontrol
length(Q† (t)). algorithms.
i,j
Weconsiderasystemofsixsignalizedintersectionsasshown
inFigure7. Intotal,thereare10entrylinks(4majorentries
PROOF. First, we have Qi,j(t) − Q†i,j(t) ∈ [−B,B] and from the East and the West along with 6 minor entries from
W (t)−W† (t)∈[−2B,2B],forall(i,j)andallt,Besides, theNorthandtheSouth)and10exitlinks. Besides,thenum-
i,j i,j
Lemma3holdsregardlessoftheschedulingpolicy.Therefore, beroflanesofeachthrough-trafficlinkandleft-turnlinkare
wecanrewritetheupperboundinLemma6as 3and1,respectively.
E(cid:2)∆L(tk)(cid:12)(cid:12)Q†(tk)(cid:3)≤−2(cid:15)δ0Tk (cid:88) Q†i,j(tk) (37)
(i,j)∈M
+C‡ (cid:88) Mv(cid:16) (cid:88) W† (t )+(cid:17) (38)
1 k i,j k
v∈VC (i,j)∈Mv
+C‡ (cid:88) (cid:88) W† (t )+C‡T2+C‡T , (39)
2 i,j k 3 k 4 k
v∈VF(i,j)∈Mv
whereC‡, C‡, C‡, C‡ arefinitepositiveconstants. Further-
1 2 3 4
more, with a slight modification of the proof we know that
Lemma 4 and Lemma 5 still hold when W(t ) is replaced
k
byW†(t )underB-MP.Bythesameargumentasthatinthe
k
proof of Theorem 4, we know that −2(cid:15)T (cid:80) Q† (t )
k (i,j)∈M i,j k
is the dominating term in (37)-(39). Therefore, there must
existaconstantB† >0suchthat Figure2: SystemtopologyinVISSIM.
(cid:18) (cid:19)1+β
E(cid:2)∆L(tk)(cid:12)(cid:12)Q†(tk)(cid:3)≤B†−(cid:15)δ0 (cid:88) Q†i,j(tk) . Accordingtotheofficialstatistics[25],thesaturationflow
ofeachlinkissettobe1900vehiclesperhourperlane. Ve-
(i,j)∈M
hicles enter the system from the entry links and are routed
(40)
towards an exit link in a probabilistic manner. We set the
BythesimilarprocedureasinLemma2,weknowthat(40)is routingprobabilitytobe0.2and0.8forleft-turnmovement
alsoasufficientconditionforstrongstability. Hence,wecon- andthroughmovement,respectively. Weuseλ ,λ ,λ ,and
E W N
cludethatB-MPremainsthroughput-optimalwhentheerror λ todenotethearrivalratesoftheentrylinkscomingfrom
S
inqueuelengthisbounded. theEast,theWest,theNorth,andtheSouth,respectively. We
use the default driver behavior and lane-change model pro-
FromTheorem4,weknowthatB-MPisalsorobusttoesti- vided in VISSIM. The speed limit of each vehicle is 40 miles
mationerrorinqueuelengthinformation. perhour. Eachintersectionhasfouradmissiblephasesasde-
scribed in Figure 1. Throughout the simulation, we choose
6.3 LimitationsonGreenPeriod
the slot time to be 1 second which is sufficient for updating
Conventionally, the timing plan of traffic signals includes theschedulingdecisions. Theswitch-overdelayissettobe5
a minimum green time to accommodate the vehicle startup seconds,whichincludesanamberperiodof3secondsandan
delay. Under the B-MP policy, the minimum green time can all-redperiodof2seconds. AnimportantfeatureofourVIS-
beeasilyincorporatedbyintroducingaminimumframesize SIM simulation is that we consider the effect of finite buffer
T >T . Accordingly,(27)inLemma4wouldbe size. When a link is fully occupied by vehicles, VISSIM will
G,min S
prohibitnewvehicles,whichcanbeeitherfromtheexternal
(cid:40) (cid:41)
Tv ≥max T ,C B (tv )(cid:16) (cid:88) W (tv )+(cid:17) . orfromupstreamlinks,fromjoiningthelinkandhencelower
k,l G,min 5 v k,l i,j k,l thethroughput.
(i,j)∈Mv WecomparetheB-MPpolicyagainsttheconventionalfixed-
(41)
timepolicy,Max-Pressure(MP)policy,andtheVariableFrame-
WithaslightmodificationoftheproofofLemma5,theB-MP
Based Max-Weight (VFMW) policy. For the fixed-time pol-
policy with a minimum frame size still remains throughput-
icy, the timing plan is calculated by Synchro [3], which is a
optimal. Ontheotherhand,amaximumgreentimeissome-
widely-usedoptimizationtoolfortimingplandesignintrans-
times applied in the actuated version of fixed-time policy to
portation research. Throughout the simulation, we assume
avoidexcessivedelayoftheminorroads. Whilethiscanalso
that the fixed-time policy has perfect information about the
be included in B-MP by introducing a maximum frame size
average traffic statistics of each link and therefore is able to
T , setting a maximum frame size can result in loss of
G,max optimizethetimingplanaccordingly. ForVFMW,wechoose
soyvsetremwotuhlrdouaglwhpayustbsiencgeretahteerfrtahcatnionoroefqtuimaletsopeTnGTt,mSoanx.switch- gtheestefrdamine[s6i]z.e tFoorbethTeSB+-M(cid:0)P(cid:80)p(oi,ljic)∈y,MwveQci,hj(otoks)e(cid:1)0α.9 =as 0su.0g1-
and β = 0.99 as discussed in Section 5.2. Besides, to miti-
7. SIMULATION
gatepossiblequeueoverflowduetofinitequeuecapacity,we
use weighted queue length with q = 3 for through-traffic Next, we measure the performance with λ¯ between 1200
i,j
queuesandq = 1forleft-turnqueuesasdiscussedinSec- and 2800. Figure 4(a) and Figure 4(b) show the system
i,j
tion 6.1. First, we consider the arrival traffic pattern as fol- throughputandaveragedelaywithdifferentarrivalrates.Note
lows: that the average delay here is defined as the difference be-
Scenario1: λ =λ =λ¯andλ =λ =0.5·λ¯(veh/hr). tween the actual traveling time and the traveling time with-
E W N S
outanystopattheintersections. InFigure4(a),weseethat
Under this traffic pattern, the maximum achievable λ¯ is
under B-MP the throughput grows linearly with the arrival
about 2600 veh/hr according to the traffic equations given rate for λ¯ up to 2600. With λ¯ = 2800, the throughput un-
by (2). The total simulation time is 1800 seconds. Figure 3
derB-MPgetssaturatedsimplybecausethearrivalrateisal-
shows the total number of vehicles in the system with λ¯ =
ready beyond the capacity region. For the fixed-time policy,
2400 under the four policies. We observe that B-MP indeed itcansupportλ¯ onlyupto2200duetothecapacitylossre-
achievesthesmallesttotalqueuelengthwhilethetotalqueue
sulting from the switch-over delay. For MP and VFMW, they
lengthkeepsincreasingunderanyoftheotherthreepolicies.
bothsufferfromseverecapacitylossduetofrequentswitch-
ing of traffic signals. In Figure 4(b), the B-MP still achieves
3000 thesmallestdelayforeveryλ¯. Fortheheavytrafficcondition
VFMW with λ¯ = 2600, compared to the fixed-time policy with per-
gth2500 MFTP fectknowledgeoftrafficstatistics,B-MPreducestheaverage
en B-MP delaybymorethan40%withoutanyarrivalrateinformation.
e L2000 ForVFMW,weonlyshowtheaveragedelayforλ¯below1800
eu simplebecauseitperformsmuchmorepoorlythantheother
Qu1500 threepoliciesforλ¯above2000.
al Next,wefurtherconsidertime-varyingarrivalrates:
Tot1000 Scenario2:
500 • 0sto1200s:(λW,λE,λN,λS)=(2000,2000,1000,1000).
200 600 1000 1400 1800
• 1201sto2400s:(λ ,λ ,λ ,λ )=(2500,1500,1500,500).
Time (s) W E N S
• 2401sto3600s:(λ ,λ ,λ ,λ )=(1500,2500,500,1500).
W E N S
Figure3: Totalqueuelengthofthesystemunderthefour
policieswithλ¯=2400. Notethatthetotalarrivalrateofthewholesystemremains
thesameundertheabovetrafficpattern. Figure5showsthat
totalqueuelengthunderthethreepolicies. Hereweomitthe
VFMW policy simply because it has much larger total queue
8000
B-MP length. Again, B-MP still achieves the smallest total queue
eh) FT length at any time. It is notable that the total queue length
v7000
put ( MVFPMW underB-MPdoesnotchangemuchwiththetime-varyingpat-
gh6000 tern. In contrast, the fixed-time policy suffers from much
ou morecongestionduringtime1200sto3600s. Thisisbecause
Thr5000 the fixed-time policy optimizes its timing plan based on the
m
e average arrival rates and thus fails to accommodate traffic
yst4000 dynamics. SimilartoFigure3,MPstillperformsquitepoorly
S
duetotheservicelossincurredbytheswitch-overdelay.
3000
1200 1600 2000 2400 2800 Lastly,weconsiderapartially-connectedsystemwherethree
Arrival Rate of Major Entries (veh/hr) of the intersections are connected under a user-customized
(a) Systemthroughput policy (B-MP, MP, or VFMW) and the rest are fixed-time in-
tersectionsasusual.Figure6(a)andFigure6(b)showtheav-
eragedelayandsystemthroughputofthepartially-connected
400 system with different arrival rates. Compared to the pure
VFMW
FT fixed-time system, partial inclusion of the B-MP policy still
s)300 MP providesimprovementinboththroughputandaveragedelay.
y ( B-MP Besides, B-MP still outperforms the other two policies by a
a
e Del200 laabrogveemsiamrguilnatiinont,hweepadretmiaollny-sctorantneetchtaetdBs-yMstPemin.deTehdropurgohvidthees
g
a
er significantimprovementovertheotherthreepopularpolicies.
Av100
8. CONCLUSION
0 In this paper, we study the scheduling problem for net-
1200 1600 2000 2400 2800
worked transportation systems with switch-over delay. We
Arrival Rate of Major Entries (veh/hr)
propose a distributed scheduling policy that is throughput-
(b) Averagedelay
optimalwithswitch-overdelaywithouttheknowledgeoftraf-
ficdemands. Moreover,theproposedpolicystillremainsop-
timal when there are both fixed-time intersections and con-
Figure 4: Delay and throughput performance under the nectedintersections. Hence,theproposedpolicycanstillper-
fourpolicieswithdifferentarrivalrates. form well in partially-connected systems. Simulation results
showthattheproposedpolicyindeedoutperformstheother [5] CELIK,G.,BORST,S.C.,WHITING,P.A.,AND
existingpolicies. MODIANO,E.DynamicSchedulingwith
ReconfigurationDelays.QueueingSyst.TheoryAppl.83,
1600 1-2(Jun2016),87–129.
MP
gth1400 FT [6] CELIK,G.D.,ANDMODIANO,E.Schedulingin
n B-MP networkswithtime-varyingchannelsand
e
L1200 reconfigurationdelay.IEEE/ACMTransactionson
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