Delay-Optimal Scheduling for Queueing Systems with Switching Overhead

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by  Ping-Chun Hsieh

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Delay-Optimal Scheduling for Queueing Systems with
Switching Overhead
Ping-ChunHsieh · I-HongHou · XiLiu
7 Received:date/Accepted:date
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0
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Abstract Westudytheschedulingpolicesforasymptoticallyoptimaldelayinqueue-

n ing systems with switching overhead. Such systems consist of a single server that
a
serves multiple queues, and some capacity is lost whenever the server switches to
J
serveadifferentsetofqueues.Thecapacitylossduetothisswitchingoverheadcan
3
be significant in many emerging applications, and needs to be explicitly addressed
1
inthedesignofschedulingpolicies.Forexample,in60GHzwirelessnetworkswith

] directionalantennas,basestationsneedtotrainandreconfiguretheirbeampatterns
F
wheneverthey switch from one client to another. Considerable switching overhead
P
can also be observed in many other queueing systems such as transportation net-
.
s worksandmanufacturingsystems.WhilethecelebratedMax-Weightpolicyachieves
c
asymptoticallyoptimalaveragedelayforsystemswithoutswitchingoverhead,itfails
[
to preserve throughput-optimality,let alone delay-optimality,when switching over-
1
headistakenintoaccount.WeproposeaclassofBiasedMax-Weightschedulingpoli-
v
cies that explicitlytakes switching overheadinto account.The Biased Max-Weight
1
3 policycanuseeitherqueuelengthorhead-of-linewaitingtimeasanindicatorofthe
8 system status. We provethat our policies not only are throughput-optimal,butalso
3 canbemadearbitrarilyclosetotheasymptoticlowerboundonaveragedelay.Toval-
0
idatetheperformanceoftheproposedpolicies,weprovideextensivesimulationwith
.
1 varioussystem topologiesand differenttraffic patterns. We show that the proposed
0 policiesindeedachievemuchbetterdelayperformancethanthatofthestate-of-the-
7
artpolicy.
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:
v Keywords Delay-optimality· Scheduling· Switching overhead· Max-Weight·
i
X Throughput-optimality·Stability
r
a MathematicsSubjectClassification(2010) 60K25·68M20·90B22
Ping-ChunHsieh·I-HongHou·XiLiu
DepartmentofECE,TexasA&MUniversity,CollegeStation,Texas77843-3128,USA.
E-mail::{pingchun.hsieh,ihou,xiliu}@tamu.edu
2 Ping-ChunHsiehetal.
1 Introduction
Designofschedulingpoliciesisoneofthemostcriticalpartsinachievinggoodper-
formance for queueing systems. Based on the system information, such as queue
backlogorinstantaneousservicerate,theserverdynamicallyswitchesbetweendif-
ferent scheduling decisions. The delay required for each switch is traditionally as-
sumedtobeminimalcomparedtotheservicetimeofeachjob,andthereforecanbe
neglected.However,forapplicationsthatrequiredynamictuningorstrictsafetyguar-
antees,thisswitchingoverheadhastobeexplicitlyaddressedwithcaution.Wireless
networkingonthe 60GHz band[11,25,28], forexample,is inherentlyfeaturedby
theswitchingoverhead.Signalpropagationonthe60GHzbandsuffersfrommuch
more serious attenuation than on the widely-used 2.4 GHz or 5 GHz spectrum. To
cope with this increased attenuation, beamformingtechniques based on directional
antennashavebeenwidelyappliedtomitigatethisproblem[24].However,asstated
in[21],toreconfigurethebeamdirection,thebeamswitchingprocedurecantakeup
to severalhundredmicroseconds,whichis aboutthe transmissiontime of a packet.
Whilethisswitchingoverheadisusuallyoverlookedinmostwirelesscommunication
networks,thisbeam-switchinglatencyneedstobeexplicitlyaddressedindirectional-
antennaapplications.
Anotherexampleisthetrafficsignalcontrolforsignalizedintersections.During
each signal phase transition, the traffic signal controller has to undergo a yellow-
to-red period and an all-red period to guarantee safety. Without proper scheduling,
the switching overhead can greatly reduce the intersection capacity [12]. Existing
studies have shown that the conventional fixed time scheduling policy can result
in a significant amountof capacity loss. [2,33]. Moreover,this switching overhead
is expected to be even larger in mixed transportation networks with both human-
drivenand autonomousvehicles[17,19]. Recently,there has been growinginterest
indesigningschedulingalgorithmstoachievemaximumtrafficthroughputintrans-
portationnetworks[33,35].Meanwhile,moretraffic-responsiveschedulingpolicies
are now applicable due to the recent progress in both vehicle-to-infrastructureand
vehicle-to-vehiclecommunication.Indeed,toapproachmaximumthroughputinreal
trafficscenarios,theinfluenceofswitchingoverheadonnetworkcapacityhastobe
incorporatedandovercomeinschedulingdesign.
Theeffectofswitchingoverheadisalsocriticalinmanyotherapplications,such
asmulti-threadprocessors[8],passiveopticalnetworks[20],andmanufacturingsys-
tems[27].Asaclassicexample,pollingsystemisonewidelystudiedqueueingmodel
thatincorporatesswitchingoverhead.Insuchsystems,theserverservethequeuesin
cyclic order, and the server requires finite time to go from one queue to the next
queue.Thepollingsystemmodelhasbeenappliedinavarietyofapplications,such
ascommunicationnetworks[1] andmanufacturingsystems[27].Thesurveyof the
pollingmodelandsummaryofthetheoreticalresultscanbefoundin[18,29,30,34].
Despite the progressin the study of the polling model, there are still a vast variety
of applications that require multiple queues to be served simultaneously while the
pollingmodelassumesonlyonequeueisservedateachtime.
Forqueueingsystemsthatallowsimultaneousserviceofmultiplequeues,aclass
ofschedulingpoliciesnamedtheMax-Weightpolicies,introducedbyTassiulasand
Delay-OptimalSchedulingforQueueingSystemswithSwitchingOverhead 3
Ephremides in [31], has been shown to achieve optimal throughput for the multi-
hopqueueingsystems.Lateron,thereareseveralextendedworksoftheMax-Weight
policies,suchas[10,32].Inadditiontothroughput-optimality,theMax-Weightpol-
icy has also been shown to achieve good delay performance in different queue-
ing systems [9,13,15,16,22]. To obtain delay bound for the Max-Weightschedul-
ing, one common technique is to set the drift of a Lyapunov function to zero, as
in[9,13,15,16,22].Besides,EryilmazandSrikant[9]derivesanasymptoticallytight
upperboundonqueuelengthfortheMax-Weightpolicy.However,noneoftheabove
literature considersswitching overheadincurredby the transition betweendifferent
schedules.Infact,Max-Weightpolicyfailsto preservethroughput-optimalitywhen
the switching overhead is considered. The reason is that Max-Weight policy may
sufferfromexcessiveswitchingandthereforewasteasignificantportionoftimeon
switching.
Toremedytheinstability issue ofMax-Weightpolicy,ArmonyandBambos[3]
propose cone policies and batch policies and prove that the two families of poli-
cies are throughput-optimalwith non-zeroswitching overhead.Later on, Hungand
Chang[14]proposeanextendedversionofDynamicConepolicytoreducethecom-
plexity and improve delay performance of the conventional cone policies. Chan et
al. [6] proposes the Maximum-Weight-Matchingwith hysteresis (MWM-H) policy
toachieveoptimalthroughputwithdeterministicserviceratesaswellastoproperly
distributethequeuebacklogwhenthequeuesbecomeoverloaded.Besides,Celiket
al.[4]proposetheVariableFrame-BasedMax-Weight(VFMW)policywhichincor-
porateframestructuretoavoidfrequentswitching,andshowthatthepolicypreserves
throughput-optimality with nonzero switching overhead for interference networks.
However, all of the above studies focus only on throughput-optimality and do not
provide any theoretical bound on delay performance. In this paper, we regard the
VFMWpolicyin[4]asthestate-of-the-artpolicyandtheVFMWpolicyservesasa
referenceforthecomparisonofdelayperformance.Forthedelayanalysisofqueue-
ing systems with switching overhead, Perkins and Kumar [26] propose a class of
exhaustivepoliciestoachieveasymptoticallyoptimaldelayforsingle-serverqueue-
ingsystemswithdeterministicarrivalandserviceprocesses.However,itisnoteven
clear whether the exhaustive-type policies can also achieve optimal throughput in
systemswheremultiplequeuescanbeservedsimultaneously.
Inthispaper,weproposeaclassofschedulingpoliciescalledBiasedMax-Weight
(BMW) policies to achieve both throughput-optimalityand delay-optimality in the
presenceofswitchingoverhead.Theweightofeachschedulecanbedefinedbyeither
thequeuebacklogorthewaitingtimeofthehead-of-line(HOL)job.LiketheMax-
Weightpolicy,theBMWpolicystillmakesschedulingdecisionsbasedontheweight
ofeachschedule,butithasabiastowardsthecurrentschedule.Inotherwords,the
servermakesaswitchonlywhenthenewscheduleisbetterthanthepreviousschedule
by an enoughmargin.The BMW policiesshare similar design philosophywith the
VFMWpolicyandtheMWM-Hpolicytoavoidexcessiveswitchingaccordingtothe
current queue status. Different from those prior works, we are able to characterize
thequeuelengthstatistics andthereforetheaveragedelay.Toachievethisgoal,we
first prove that the BMW scheduling policy is throughput-optimalby showing that
the underlyingMarkovchain is positive recurrent.We furtherprovethat the BMW
4 Ping-ChunHsiehetal.
policyachievestheasymptoticallytightqueuelengthboundbychoosingparameter
usedbyBMWarbitrarilyclosetozero.Throughextensivesimulation,weshowthat
the BMW policystill providesgood delay performancewhen the server may serve
multiplequeuessimultaneously,subjectto someconflictingconstraintsimposedby
thesystem.
Inadditiontoaveragedelay,theper-queuedelayis also crucialformanyappli-
cations,suchastransportationnetworks.Wefurthercomparethequeue-length-based
BMW(Q-BMW)andthewaiting-time-basedBMW(W-BMW)intermsofper-queue
average delay. Under the Q-BMW policy, each queue has about the same average
queuelength.ByLittle’slaw,thequeuewithasmallermeanarrivalrateisexpected
to experience a larger average delay. On the other hand, the W-BMW policy fol-
lows the same switching scheme as the Q-BMW policy, but with the HOL waiting
timesasthestatevariables.SincetheHOLwaitingtimeservesasagoodindicatorof
per-queueaveragedelay,theW-BMWcanavoidholdingupthejobsfortoolongin
the queueswith lighter incomingtraffic. Simulation results show that the W-BMW
policy indeed achieves better fairness than the Q-BMW policy without sacrificing
system-wideaveragedelay.
The rest of the paper is organized as follows. Section 2 describes the system
modelaswellasthenotationsinuse.Section3providesmathematicalpreliminaries
ofqueuestabilityanddelay-optimalityandalsodescribesthefundamentaldilemma
in achieving delay-optimality.Section 4 describes the throughput-optimalityof the
Q-BMWpolicy.Then,Section5providesanasymptoticupperboundonaveragede-
layunderQ-BMWpolicy.FortheW-BMWpolicy,Section6showsthatitachieves
throughput-optimalityandthesamedelayupperboundastheQ-BMWpolicy.Sec-
tion7providesextensivesimulationresultsofqueueingnetworkswithdifferentcon-
straints.Finally,Section8concludesthepaper.
2 NotationandSystemModels
2.1 NetworkModel
We consider a time-slotted switched queueing system with one centralized server
andN ∈ N (N isashorthandforN∪{0})parallelqueues,whichareindexedby
0 0
N = {1,2,...,N}.Timeslotsareindexedbyt ∈N .Eachqueueisassociatedwith
0
anexogenoustrafficstream.Arrivingjobsarefirstbufferedinthequeueandleavethe
systemrightaftertheserviceiscompleted.Theservermaybeabletoservemultiple
queuessimultaneously.Asetofqueuesthatcanbeservedsimultaneouslyiscalleda
feasibleschedule.WerepresenteachfeasibleschedulebyanN-dimensionalbinary
vector I = (I ,I ,··· ,I ), where I is the indicator function of whether queue
1 2 N i
i is included in the schedule. Throughoutthe paper, we focus on maximal feasible
schedules:afeasiblescheduleismaximalifnoadditionalqueuecanbeaddedtothe
schedule.Withalittle abuseofnotation,weuse|I|todenotethenumberofqueues
includedinthescheduleI.SupposethereareJ maximalfeasibleschedulesdenoted
by I = {I(1),··· ,I(J)}. In each time slot t, the server selects a maximal feasible
scheduleI(t)∈I accordingtoitsschedulingpolicyη.
Delay-OptimalSchedulingforQueueingSystemswithSwitchingOverhead 5
Whentheserverswitchesfromonescheduletoservinganotherschedule,itneeds
to spend T ∈ N slots on preparing for the transition before working on the new
s
schedule.ThedelayT reflectstheswitchingoverheadwhichisusuallyoverlooked
s
in the queueingsystems but needs to be explicitly addressed in many applications,
such as directional-antenna systems as well as transportation systems described in
Section 1. Therefore,there are two operationmodesfor the server:ACTIVE mode
andSWITCH mode.We letM(t)betheindicatorfunctionofwhethertheserveris
in ACTIVE modeat time t. We use t to denotethe time when the server makesa
k
switchforthek-thtime,andsett =0.Thetimebetweentwoconsecutiveswitches
0
iscalledaninterval.LetT := t −t denotethelengthofthek-thinterval,for
k k+1 k
all k ∈ N . Note that T reflects how frequently the server is switching between
0 k
differentschedules.
Throughoutthepaper,weuse(N,I,T )todenoteaqueueingsystemdescribed
s
intheabove.
2.2 TrafficModel
Wemodelthearrivalprocess{A (t)} ofeachqueueibyasequenceofindependent
i t
and identically distributed (i.i.d.) non-negative random variables A (t) ∈ N with
i 0
E[A (t)]=λ forallt≥0.WefurtherassumethatA (t)isupperbounded,i.e.there
i i i
existsafiniteconstantA >0suchthatA (t)≤A foreveryt≥0.Similarly,
max i max
theserviceprocess{S (t)} ofeachqueueiismodeledbyasequenceofi.i.d.non-
i t
negative random variable S (t) ∈ N with E[S (t)] = µ and S (t) ≤ S , for
i 0 i i i max
all t ≥ 0. We also assume that the server does not collect any information about
theinstantaneousservicerates.Thereasonisthatwithnon-zeroswitchingoverhead
the server is not able to exploitthe time-varyingservice when the service rates are
independentacross time. Moreover,let ρ := λ /µ be the normalized traffic load
i i i
of queue i. For every queue i, each arriving job is labeled with a unique sequence
number.Foreachqueuei,thesequencenumbersstartfrom1andwouldindicatethe
arrivalorderofthejobs.Ateachtimet,weuseϕ (t)todenotethesequencenumber
i
of the latest completedjob of queue i beforetime t. Let {V (m)} be the inter-
i m≥0
arrival time process of queue i, where V (m) denote the inter-arrivaltime between
i
thetwojobswithsequencenumbersmandm+1inqueuei.
Tosimplifynotationsinlatersections,weuseboldfacelettersA,S,λ,µandρto
denotetheN-dimensionalvectorsofthearrivals,services,meanarrivalrates,mean
service rates, and normalized traffic loads, respectively. We also use λ , λ ,
max min
µ andµ astheshorthandsofthemaximummeanarrivalrate,minimummean
max min
arrivalrate,maximummeanservicerate,andminimummeanservicerateamongall
thequeues,respectively.
2.3 QueueDynamics
Let Q (t) ∈ N be the number of jobs buffered in queue i at time t. We assume
i 0
Q (0)=0foreveryqueuei.DefineSˆ(t):=min{Q (t),M(t)I (t)S (t)}tobethe
i i i i i
6 Ping-ChunHsiehetal.
amountofserviceactuallyusedbyeachqueueiattimet.Throughoutthispaper,we
considerthestore-and-forwardqueueingmodel,i.e.,foreachqueuei,
Q (t+1)=Q (t)−Sˆ(t)+A (t), ∀t≥0. (1)
i i i i
Forsimplicity,weuseQ(t) = (Q (t),...,Q (t)) ∈ NN todenotethequeuelength
1 N 0
status of the system at time t. Moreover, in our model the queue length process
{Q(t)} form a discrete-time Markov chain on a countablestate space NN. The
t≥0 0
total queue length of the system can be written as 1TQ(t), where 1T denotes the
N-dimensionalall-onesrowvector.
LetW (t)∈N bethewaitingtimeofthehead-of-line(HOL)jobofqueueiand
i 0
W(t) = (W (t),...,W (t)) ∈ NN bethecorrespondingHOLwaitingtimevector.
1 N 0
Then,theHOLwaitingtimecanbeupdatedasfollows:
Sˆi(t)
W (t+1)=max 0, W (t)− V (ϕ(t)+j) . (2)
i i i
 
 (cid:18) Xj=1 (cid:19)
Similartothequeuelengthprocess,theHOLwaitingtimeprocess{W(t)} also
t≥0
formadiscrete-timeMarkovchainonacountablestatespaceNN.
0
3 Preliminaries
3.1 CapacityRegionandaLower-BoundforDelay
Inpreparationforthediscussionofdelayperformance,wefirstintroducethefunda-
mentalconceptsofqueuestability,throughput-optimality,anddelay-optimality.First,
weformallyintroduceonecommonlyuseddefinitionofqueuestability.
Definition1 (Strongstability)Thequeueingsystemisstronglystableunderaschedul-
ingpolicyif
t−1 E 1TQ(τ)
limsup τ=0 <∞. ⊓⊔ (3)
t
t→∞ P (cid:2) (cid:3)
Basedontheabovedefinitions,wecanclassifythearrivalratevectorsbyqueue
stabilityanddefinethecapacityregion.
Definition2 (Admissible arrival rates) An arrival rate vector λ = (λ ,...,λ ) is
1 N
said to be admissible if there exists a scheduling policy under which the queueing
systemisstronglystable.⊓⊔
Definition3 (Capacityregion)ThecapacityregionΛ⊂RN ofthesystemisdefined
+
astheclosureofthesetthatconsistsofalltheadmissiblearrivalratevectors.⊓⊔
The following lemma shows that the capacity region can be fully characterized
bythenormalizedtrafficloadvectorandthemaximalfeasibleschedules.
Delay-OptimalSchedulingforQueueingSystemswithSwitchingOverhead 7
Lemma1 (Characterizationofcapacityregion)Foranyqueueingsystemdescribed
inSection2,giventhemeanserviceratevectorµ,thecapacityregioncanbechar-
acterizedas
J J
Λ= λ ∃β ≥0with β ≤1suchthatρ≤ β I(j) . ⊓⊔ (4)
j j
 (cid:12) 
 (cid:12)(cid:12) Xj=1 Xj=1 
(cid:12)
Proof Thisisadir(cid:12)(cid:12)ectresultofTheorem1in[5].⊓⊔ 
Here,weintroducethenotionofthroughput-optimality:
Definition4 (Throughput-optimality)Aschedulingpolicyηissaidtobethroughput-
optimalifforanyinteriorpointλofΛ,thesystemisstronglystableunderη.⊓⊔
Givenλ,µ,andI,wecanfurtherdescribethetrafficloadofthewholesystem.
Definition5 (Utilizationfactor)GiventhequeueingsystemdescribedinSection2,
wedefinetheutilizationfactoras
β∗ := min 1Tβ. (5)
β:PJj=1βjIj≥ρ
Forconvenience,wealsodefineǫ∗ :=1−β∗,whichreflectsthe”distance”fromthe
boundaryofthecapacityregion.⊓⊔
Fromthestudyin[9],theaveragedelayofaqueueingsystemiscloselyrelatedto
ǫ∗.Ausefullowerboundfortheaveragedelayisprovidedhereasaneasyreference.
Lemma2 (Lowerboundonqueuelengthwithoutswitchingoverhead)Givenasta-
blequeueingsystemQ=(N,I,T )describedinSection2withqueuelengthprocess
s
{Q(t)} andT =0,underanyschedulingpolicyη,theexpectedtotalqueuelength
t s
insteadystatescalesas
E[1TQ(t)]=Ω(1/ǫ∗).⊓⊔ (6)
Proof ThisisadirectresultofLemma6in[9].⊓⊔
Remark1 By Little’s law, (6) implies that the total average delay also scales as
Ω(1/ǫ∗)forsystemswithoutswitchingoverhead.
Remark2 In [9], the lower bound (6) is obtained by constructing a hypothetical
single-queuesystemfromtheoriginalmulti-queuesystem,whosetotalqueuelength
islargerinstochasticorderingthanthatoftheconstructedsingle-queuesystem.Fol-
lowing thisargument,we can also derivea similar lower boundon expectedqueue
lengthforthequeueingsystemswithswitchingoverhead.
Corollary1 (Lowerboundonqueuelengthwithswitchingoverhead)Givenastable
queueingsystem Q = (N,I,T ) describedin Section2 with queuelengthprocess
s
{Q(t)} andT >0,underanyschedulingpolicyη,theexpectedtotalqueuelength
t s
insteadystateis
E[1TQ(t)]=Ω(1/ǫ∗). ⊓⊔ (7)
8 Ping-ChunHsiehetal.
Proof Given a queueing system Q, by following the same procedureas in Lemma
′
6 of [9], we can construct a single-queue system Q with switching overhead T .
s
ThenweknowthequeuelengthprocessofQislargerinstochasticorderingthanthe
′ ′′
queuelengthprocessofQ .Next,wecanconstructanothersingle-queuesystemQ
′ ′
from Q but with T = 0. Then we know the queue length process of Q is larger
s
′′
in stochastic orderingthan the queue lengthprocess of Q . By Lemma 2, we have
E[1TQ(t)]=Ω(1/ǫ∗).⊓⊔
BasedonCorollary1,wecandefineasymptoticdelay-optimalityasfollows.
Definition6 (Delay-optimality) A scheduling policy is delay-optimal if in steady
state,thetotalaveragequeuelengthsatisfiesthat
E[1TQ(t)]=O(1/ǫ∗). (8)
Inotherwords,O(1/ǫ∗)isanasymptoticallytightdelayupperbound.⊓⊔
3.2 TheVariable-FrameMax-WeightSchedulingPolicy
AsdiscussedinSection1,despitetheprogressinthroughput-optimalschedulingfor
systems with switching overhead, it is still not clear how to achieve optimal delay
performanceforsuchqueueingsystemswithstochasticarrivalandserviceprocesses.
Inthepriorwork[4],theVariable-FrameMax-Weight(VFMW)policyhasbeenpro-
posedto achievethroughputoptimalityqueueingsystems with switching overhead.
UndertheVFMWpolicy,weneedtodetermineasublinearfunctionforcalculating
framesizesuchthattheserverstayswiththesamescheduletilltheendoftheframe.
While the frame size function has no effecton throughput-optimality(as long as it
issublinear),itcanindeedgreatlyaffectthedelayperformance.Intheexampledis-
α
cussed in[4], the framefunctionis chosentobe Q (t) , whereα is between
i i
0 and 1. Besides, [4] also suggests that α should be chosen as close to 1 as possi-
(cid:0)P (cid:1)
ble based on their simulation results. However,the α value with the smallest delay
can actually differ in different scenarios. This phenomenoncan be easily observed
throughsimulationasfollows.
Forexample,weconsiderasingle-seversystemof4queueswithBernoulliarrival
andserviceprocesses.TheswitchingoverheadT =1.Weprovidesimulationresults
s
of two scenarios with differentmean arrivalrates and mean service rates in Figure
1(a)and1(b).
– ScenarioI:λ=(0.119,0.119,0.119,0.119),µ=(0.5,0.5,0.5,0.5)
– ScenarioII:λ=(0.08,0.25,0.09,0.01),µ=(0.8,0.5,0.3,0.2)
Note thatthe utilizationfactor of bothscenariosis 0.95.InScenario I, the smallest
totaldelayisachievedwhenαiscloseto1.However,theoptimalαisabout0.6in
Scenario II. This exampledemonstratesthatthere doesnotexist a fixed value of α
thatcanachievetheoptimaldelayperformancefortheVFMWpolicy.Westudythe
trace files of our simulations and find that the VFMW policy can suffer from poor
delayperformancefromtwoconflictingfactors:
Delay-OptimalSchedulingforQueueingSystemswithSwitchingOverhead 9
– If α is large, say close to 1, then the frame size growsfast. The VFMW policy
maysticktoaninefficientfeasibleschedulefortoolongandtherebysuffersfrom
largedelay.
– If α is small, say close to 0, then the frame size is very small for most of the
time andbecomesveryinsensitive to the changein queuestatus. Consequently,
theVFMWpolicymayswitchtoofrequentlyandgetsseverelyimpactedbythe
switchingoverhead.
The above arguments highlight the fundamental difficulty in achieving delay-
optimality for queueing systems with switching overhead:If a policy switches too
frequently,itsuffersfromtoomuchcapacitylossduetoswitchingoverhead.Onthe
other hand, if a policy does not switch often enough, it may stay with a schedule
that is no longer efficient for too long. In the next section, we propose our online
schedulingpolicythatsolvessuchdilemma.
800 1400
VFMW VFMW
y y1200
a a
el600 el
D D1000
e e
g g
era400 era 800
v v
al A200 al A 600
Tot Tot 400
0 200
0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8
α value α value
(a) ScenarioI. (b) ScenarioII.
Fig. 1 Average total delay versus different α value under VFMW policy with frame function
α
(cid:0)PiQi(t)(cid:1) .
4 Throughput-OptimalSchedulingforSystemsWithSwitchingOverhead
4.1 SchedulingPolicyandIntuitiveDescription
WeproposetwotypesofBiasedMax-Weightschedulingpolicytoachieveboththroughput-
optimality and the asymptotically tight delay bound. We first introduce the queue-
length-basedBiasedMax-Weight(Q-BMW)policyasfollows.
Q-BMW policy: Let F(·) : RN → [1,∞) be a function chosen by the server. At
+
eachtimetinthek-thinterval,ifthesystemsatisfiesthat
T
1+ s I(t )TQ(t) ≤ max (I(j))TQ(t) , (9)
(cid:18) F(Q(tk))(cid:19)(cid:16) k (cid:17) (cid:18)j:1≤j≤J (cid:19)
10 Ping-ChunHsiehetal.
then the server makesa switch to servethe schedulewith the largestsum of queue
lengthsattimet(tiesarebrokenarbitrarily).Otherwise,theserverstayswiththecur-
rentschedule.⊓⊔
Note that the Q-BMW policy does not rely on the frame structure adopted by
the VFMW policy. Instead, the Q-BMW policy overcomesthe switching overhead
by givingan intentionalbiasto the currentschedule.Moreover,the Q-BMW keeps
checkingtheconditionof(9)ineachtimeslot.Intuitively,theQ-BMWpolicyavoids
thedilemmahighlightedintheprevioussectionbecause:
– SincetheQ-BMWpolicychecks(9)ineachtimeslot,itcannotsticktoaninef-
ficientschedulefortoolong,regardlessofthechoiceofthefunctionF(·).
– Since the Q-BMW favors the current schedule, it can avoid switching too fre-
quentlyaslongasthebiastothecurrentscheduleisnottoosmall.Thissuggests
thatoneshouldchooseafunctionF(·)thatincreasesveryslowlywithQ(t).
4.2 Throughput-OptimalityoftheQ-BMWScheduling
ToshowthattheQ-BMWisthroughput-optimal,wefirstintroducealowerboundon
T inthefollowinglemma.
k
Lemma3 Supposetheserver canserve atmostK queuesatatime. Underthe Q-
BMWpolicy,foranyk ≥ 0andforeverysamplepath,inthek-thintervalwehave
T ≥C (1TQ(t ))/F(Q(t )), (10)
k 0 k k
whereC =T / NK A +(1+T )S .⊓⊔
0 s max s max
(cid:0) (cid:0) (cid:1)(cid:1)
Proof Suppose at time t = t + τ, τ > 0 in the k-th interval, the server enters
k
SWITCHmodeandstartsswitching.Then,thereexistssomescheduleI(m) 6= I(t )
k
suchthat
T
1+ s I(t )TQ(t +τ) ≤ (I(m))TQ(t +τ) . (11)
F(Q(t )) k k k
(cid:18) k (cid:19)(cid:16) (cid:17) (cid:16) (cid:17)
Moreover,bytheboundednessofthearrivalprocesses,weknow
I(m) T Q(t )+τA 1 ≥ I(m) TQ(t +τ). (12)
k max k
(cid:0) (cid:1) (cid:16) (cid:17) (cid:0) (cid:1)
From(11)and(12),wehave
I(m) T Q(t )+τA 1 ≥ I(t )TQ(t +τ) 1+ Ts (13)
k max k k F(Q(t ))
(cid:0) (cid:1) (cid:16) (cid:17) (cid:16) (cid:17)(cid:18) k (cid:19)T
≥ I(t )T Q(t )−τS 1 1+ s
k k max F(Q(t ))
(cid:16) (cid:0) (cid:1)(cid:17)(cid:18) k ((cid:19)14)