Table Of ContentFunctional central limit theorems for
Markov-modulated infinite-server systems
J.Blom(cid:63),K.DeTurck ,M.Mandjes ,(cid:63)
† •
January13,2016
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Abstract
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In this paper we study the Markov-modulated M/M/∞ queue, with a focus on the correlation structure
n ofthenumberofjobsinthesystem. Themainresultsdescribethesystem’sasymptoticbehaviorundera
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particularscalingofthemodelparametersintermsofafunctionalcentrallimittheorem.Morespecifically,
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relyingonthemartingalecentrallimittheorem,thisresultisestablished,coveringthesituationinwhich
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thearrivalratesarespedupbyafactorN andthetransitionratesofthebackgroundprocessbyNα,for
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someα>0. Theresultsrevealaninterestingdichotomy,withcruciallydifferentbehaviorforα>1and
] α<1,respectively.ThelimitingGaussianprocess,whichisoftheOrnstein-Uhlenbecktype,isexplicitly
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identified,anditisshowntobeinaccordancewithexplicitresultsonthemean,variancesandcovariances
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ofthenumberofjobsinthesystem.
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t KEYWORDS. Queues(cid:63)infinite-serversystems(cid:63)Markovmodulation(cid:63)centrallimittheorems
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• Korteweg-deVriesInstituteforMathematics,UniversityofAmsterdam,SciencePark904,1098XH
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Amsterdam,theNetherlands.
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(cid:63) CWI,P.O.Box94079,1090GBAmsterdam,theNetherlands.
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1 † LaboratoireSignauxetSyste`mes(L2S,CNRSUMR8506),E´coleCentraleSupe´lec,Universite´Paris
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Saclay,3RueJoliotCurie,PlateaudeMoulon,91190Gif-sur-Yvette,France.
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2 M.MandjesisalsowithEURANDOM,EindhovenUniversityofTechnology,Eindhoven,theNetherlands,
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and IBIS, Faculty of Economics and Business, University of Amsterdam, Amsterdam, the Netherlands.
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1 M. Mandjes’ research is partly funded by the NWO Gravitation project NETWORKS, grant number
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024.002.003.
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1 Introduction
Thispaperstudiestheinfinite-serverqueuemodulatedbyafinite-stateirreduciblecontinuous-timeMarkov
chain J; when the so-called background process J is in state i, jobs arrive according to a Poisson process
withrateλ ,whilethedeparturerateisgivenbyµ . TheresultingMarkov-modulatedinfinite-serverqueue
i i
has attracted some attention during the past decades; see e.g. the early contributions [8, 12, 13]. In these
papersthemainresultswereintermsofsystemsof(partial)differentialequationscharacterizingprobability
generatingfunctionsrelatedtothesystem’stransientbehavior,andrecursionsenablingtheevaluationofthe
correspondingmoments.
Inaseriesofrecentpapers[1,2,3,4,5,6,7],substantialattentionhasbeenpaidtotheasymptoticbehavior
of Markov-modulated infinite-server queues in specific scaling regimes. In these parameter scalings the
arrivalratesaretypicallyinflatedbyafactorN,whilethetransitionratesofthebackgroundprocessaresped
upbyafactorNα forsomeα ≥ 0. Theobjectiveistoanalyzethetransientdistributionofthenumberof
jobsinthesystemattimet,tobedenotedbyM(N)(t),inthelimitingregimethatN growslarge.
The asymptotic results derived come in three flavors: (i) large deviations (LD) results, describing the tail
probabilities P(M(N)(t)/N ≥ a) for N large; (ii) central-limit-theorem (CLT) type of results, describing
theconvergenceofM(N)(t)(aftercenteringandnormalization)toaNormallydistributedrandomvariable;
and(iii)functionalcentrallimittheorems(FCLTs),describingtheconvergenceoftheprocessM(N)(·)toan
appropriateGaussianprocess.
Importantly,twomodelvariantscanbedistinguished,withtheirownspecificdepartureprocesses.
◦ Inthefirst,tobereferredtoasModel I,eachjobpresentisexperiencingadeparturerateµi whenJ
is in state i; as a consequence, this hazard rate may change during the job’s sojourn time (when the
backgroundprocessmakesatransition).
◦ Inthesecond,ModelII,thejob’ssojourntimeissampleduponarrival: whenthebackgroundprocess
istheninstatei,ithasanexponentialdistributionwithmean1/µ ,andhencethecorrespondinghazard
i
rateisconstantoveritslifetime.
Fig.1summarizestheresultsthathavebeenestablishedsofar. IntheLDdomain,thepapers[3,6,7]cover,
forbothmodels,theregimeinwhichthebackgroundprocessisrelativelyslow(morespecifically,α=0)as
wellastheregimeinwhichitisessentiallyfasterthanthearrivalprocess(α>1).AlsointheCLTregimethe
pictureiscomplete,withresultsforModelsIandII,andwithbothslow(α<1)andfast(α>1)switching
of the background process. In terms of FCLTs, however, not all cases are covered: the only result derived
sofar[1]concernsthecasethatµi = µforalli, i.e., thecaseinwhichModels I and II actuallycoincide;
wemayrefertothismodelasto‘Model0’. Themaincontributionofthepresentpaperisthederivationof
FCLTs for Models I and II; this is done in Sections 5 and 6, respectively. These findings, with a limiting
Gaussian process of the Ornstein-Uhlenbeck type, turn out to be in accordance with explicit expressions
formeans, variances, andcovariancesinthesemodels, aswepresentinSections3and4. Weconcludein
Section7withsomenumericalexperiments.
2 Notation, preliminaries
Let J(t) denote an irreducible continuous-time Markov chain on the (finite) state space {1,...,d}, with
transition rate matrix Q = (q )d and (unique) invariant probability measure π. In addition, we let
ij i,j=1
p (t):=P(J(t)=j|J(0)=i).ItisassumedthatJ(0)isdistributedaccordingtoπππ.
ij
2
[3] [4,5]
ModelI
[3] [2,4,5]
‘Model0’:[1]
[7] [5]
ModelII
[6] [5]
LD CLT F-CLT
Figure1: GraphicalillustrationofliteratureonMarkov-modulatedinfinite-serverqueues. Upper-lefttrian-
gle: fast regime; lower-right triangle: slow regime. White area: regimes covered by earlier work; shaded
areas: regimesnotcoveredyet.
TheprocessJ(t)isreferredtoasthebackgroundprocess,andregulatesaninfinite-serverqueue. WhenJ(t)
is in state i, jobs arrive at the queueing resource according to a Poisson process of rate λ . Regarding the
i
wayinwhichthesejobsarehandled,twovariantsaredistinguished:
◦ InModelIthehazardrateofjobsleavingisµiwhenthebackgroundprocessisinstatei. Observethat
thishazardratemaychangeduringthelifetimeofthejob,whenthebackgroundprocessjumps.
◦ InModel II jobdurationsaresampleduponarrival: theyaredrawnfromanexponentialdistribution
withmean1/µ ifthebackgroundprocessisinstateiwhenthejobentersthesystem.
i
Throughoutthispaperwewriteλλλ := (λ ,...,λ )T andΛ := diag{λλλ}, andlikewiseµµµ := (µ ,...,µ )T
1 d 1 d
andM:=diag{µµµ}. Wealsodefineλ :=πππTλλλandµ :=πππTµµµ.
∞ ∞
InSections3and4weconsiderexplicitexpressionsforthemeans,variancesandcovariancesintheunscaled
system.TherewedenotebyM(t)thenumberofjobspresentattimet,fort≥0.Forsimplicity,itisassumed
thatthesystemstartsemptyattime0,i.e.,M(0)=0.
In Sections 3 and 4 we also analyze the obtained expressions for the mean, variance and covariance in a
specificparameterscaling,viz.wereplacethearrivalratesλλλbyNλλλ,andthegeneratormatrixQbyNαQ,
forsomeα > 0, andletN growlarge. Itisinthisasymptoticregimethatwealsoestablishour FCLTsin
Sections5and6. ForthesescaledmodelswewriteM(N)(t)forthenumberofjobsattimet,toemphasize
thedependenceonthescalingparameter.
In the sequel, we use the concept of deviation matrices. Define the (i,j)-th element of the exponentially
weighteddeviationmatrixD(γ),asafunctionofthevectorγγγ ∈Rd,by
+
(cid:90) ∞
D(γ) := e−γit(p (t)−π )dt.
ij ij j
0
ThematrixD := D(0) isthecanonicaldeviationmatrix. Inthesequel, alsothematrixΠ := 111πππT playsa
role, aswellasthefundamentalmatrixF := D+Π.Anumberofidentitieshold: QF = FQ = Π−I,
ΠF =FΠ=Π,andF111=Π111=111.
3
3 Mean, variance, and covariance for model I
Thefirstpartofthissectionpresentsexplicitformulaeforthemean,variance,andcovarianceinModelI. In
thesecondparttheseturnouttoallowforamoreexplicitcharacterizationinparticularasymptoticregimes.
3.1 Explicitformulae
OurgoalistodeviseamethodtocomputeCov(M(t),M(t+u)). Tothisend,theobjectthatwestudyfirst
is,foru≥0fixed,thebivariateprobabilitygeneratingfunction
(cid:16) (cid:17)
Ξ (z,w,t,u):=E zM(t)wM(t+u)1{J(t)=i,J(t+u)=j} ,
ij
which implicitly contains all information about the joint distribution of M(t) and M(t + u). In matrix
notation,weobtaininAppendixA,suppressingtheargumentsforeaseofnotation,
∂Ξ ∂Ξ ∂Ξ
=(z−1)ΛΞ+(w−1)ΞΛ−(z−1)M −(w−1) M+QTΞ+ΞQ. (1)
∂t ∂z ∂w
WenowpointouthowtocomputethecovariancebetweenM(t)andM(t+u)fromthissystemofpartial
differentialequations.
Tothisend,wefirstdefinethethreematrices
E(t,u)≡(E (t,u))d , whereE (t,u):=EM(t)1{J(t)=i,J(t+u)=j}
ij i,j=1 ij
G(t,u)≡(G (t,u))d , whereG (t,u):=EM(t+u)1{J(t)=i,J(t+u)=j}
ij i,j=1 ij
C(t,u)≡(C (t,u))d , whereC (t,u):=EM(t)M(t+u)1{J(t)=i,J(t+u)=j}
ij i,j=1 ij
(2)
Itfollowsfromthemoment-generatingpropertyofgeneratingfunctionsthat
∂Ξ ∂Ξ ∂2Ξ
E (t,u)= lim ij, G (t,u)= lim ij, C (t,u)= lim ij. (3)
ij z,w↑1 ∂z ij z,w↑1 ∂w ij z,w↑1∂z∂w
From the partial differential equation (1) that defines Ξ, we can find the following systems of ordinary
differentialequationsforthematricesE(t,u),G(t,u)andC(t,u). Wedemonstratehowthisisdoneforthe
equationinvolvingE(t,u).Differentiate(1)withrespecttoz,andtakethelimitofw,z ↑ 1. Recallingthat
J(0)isdistributedaccordingtoπππ,itisstraightforwardtoobtain,withK (u):=π p (u),
ij i ij
E(cid:48)(t,u)=ΛK(u)−ME(t,u)+QTE(t,u)+E(t,u)Q,
wherethederivativeintheleft-handsideisagainwithrespecttot. Wecanderivethe ODEsforG(t,u)in
thesamemanner,
G(cid:48)(t,u)=K(u)Λ−G(t,u)M+QTG(t,u)+G(t,u)Q.
Similarly,forC(t,u)wehave:
C(cid:48)(t,u)=ΛG(t,u)+E(t,u)Λ−MC(t,u)−C(t,u)M+QTC(t,u)+C(t,u)Q,
The above differential equations are matrix-valued systems of linear differential equations, which can be
converted into vector-valued systems of linear differential equations, relying on the concept of ‘vectoriza-
tion’. WeshowthisideaforthematrixE(t,u).WetakethecolumnsofE(t,u),andputthemintoavector
4
eee(t,u)ofdimensiond2,suchthatthefirstdentriesareE (t,u)uptoE (t,u),entriesd+1upto2dcorre-
11 d1
spondtoE (t,u)uptoE (t,u),etc.;wewriteeee(t,u):=vec(E(t,u)).Likewise,ggg(t,u):=vec(G(t,u)),
12 d2
ccc(t,u):=vec(C(t,u))andkkk(u):=vec(K(u)).
Ford×dmatricesA,B,andC,andwithasusualA⊗B denotingtheKroneckerproductandA⊕B :=
A⊗I+I⊗BtheKroneckersumofthematricesAandB,recall
vec(AB)=(I⊗A)vec(B)=(BT⊗I)vec(A) and vec(ABC)=(CT⊗A)vec(B),
with I the d×d identity matrix. We thus obtain the following equations in terms of Kronecker sums and
products:
eee(cid:48)(t,u)=(I⊗Λ)kkk(u)−(I⊗M)eee(t,u)+(QT⊕QT)eee(t,u).
Anequationforggg(t,u)canbefoundanalogously:
ggg(cid:48)(t,u)=(Λ⊗I)kkk(u)−(M⊗I)ggg(t,u)+(QT⊕QT)ggg(t,u).
Alongthesamelinesweobtain
ccc(cid:48)(t,u)=(I⊗Λ)ggg(t,u)+(Λ⊗I)eee(t,u)−(M⊕M)ccc(t,u)+(QT⊕QT)ccc(t,u),
thederivativesintheleft-handsidesbeingagainwithrespecttot. ObservethatQ⊕Qisagainatransition
ratematrix,andM⊕Madiagonalmatrixwithnon-negativeentries.
Thesystemsdescribingeee(t,u)andggg(t,u)arestandardsystemsofnon-homogeneouslineardifferentialequa-
tions,whichcanbesolvedwithstandardtechniques. Thenthesolutioncanbepluggedintothedifferential
equation describingccc(t,u), which is then also a system of non-homogeneous linear differential equations.
Wesummarizetheresultsinthefollowingproposition.
Proposition1. Thematrix-valuedfunctionsE(t,u),G(t,u)andC(t,u)satisfythefollowingODEs:
E(cid:48)(t,u)=ΛK(u)−ME(t,u)+QTE(t,u)+E(t,u)Q,
G(cid:48)(t,u)=K(u)Λ−G(t,u)M+QTG(t,u)+G(t,u)Q.
C(cid:48)(t,u)=ΛG(t,u)+E(t,u)Λ−MC(t,u)−C(t,u)M+QTC(t,u)+C(t,u)Q,
Moreover, the vectorized versionseee(t,u),ggg(t,u) andccc(t,u), of the matrices E(t,u), G(t,u) and C(t,u)
satisfythefollowinglineardifferentialequations.
eee(cid:48)(t,u)=(I⊗Λ)kkk(u)−(I⊗M)eee(t,u)+(QT⊕QT)eee(t,u).
ggg(cid:48)(t,u)=(Λ⊗I)kkk(u)−(M⊗I)ggg(t,u)+(QT⊕QT)ggg(t,u).
ccc(cid:48)(t,u)=(I⊗Λ)ggg(t,u)+(Λ⊗I)eee(t,u)−(M⊕M)ccc(t,u)+(QT⊕QT)ccc(t,u).
Alloccurringderivativesarewithrespecttot.
WehavenowdevisedaproceduretocomputethecovarianceCov(M(t),M(t+u)).Tothisend,firstrealize
that,withe(t):=EM(t)and111denotingheread2-dimensionalall-onesvector,
e(t)=111Teee(t,u), e(t+u)=111Tggg(t,u).
Asaconsequence,
Cov(M(t),M(t+u))=111Tccc(t,u)−e(t)e(t+u).
5
3.2 Twospecificlimitingregimes
Inthissubsection,weconsidertwoparticularlimitingregimes,inwhichtheexpressionssimplifyconsider-
ably.
(cid:66) Letusfirstconsiderthebehaviorfort→∞. Itisreadilyverifiedthat
eee(∞,u):= lim eee(t,u)=(cid:0)(I⊗M)−(QT⊕QT)(cid:1)−1(I⊗Λ)kkk(u),
t→∞
andhence
e(∞)=111Teee(∞,u)=111T(cid:0)(I⊗M)−(QT⊕QT)(cid:1)−1(I⊗Λ)kkk(u)=111Tggg(∞,u).
Foru=0weobtainthesolutionfromO’CinneideandPurdue[13,Thm.3.1].
Alongthesamelines,
ccc(∞,u):= lim ccc(t,u)=(cid:0)(M⊕M)−(QT⊕QT)(cid:1)−1((I⊗Λ)ggg(∞,u)+(Λ⊗I)eee(∞,u)).
t→∞
WehavethusderivedanexpressionforthelimitofCov(M(t),M(t+u))ast→∞:
lim Cov(M(t),M(t+u))=111Tccc(∞,u)−(e(∞))2.
t→∞
(cid:66) Next,weconsiderthefollowingscaling:wereplaceλ(cid:55)→NλandQ(cid:55)→NαQ,forα>0. Inthisregime,
thepacewithwhichthearrivalprocessisspedup,differsfromthatcorrespondingtothebackgroundprocess.
As we will see below, the situation α > 1 crucially differs from α < 1; this was already observed earlier
ine.g.[1,5]. Asmentionedbefore,tostressthedependenceonN,wewriteM(N)(t)ratherthanM(t).It
is this scaling that is imposed in Section 5, and under which an FCLT is established. We now identify the
associatedmeanand(co-)variance,relyingonelementarytechniques.
Letmmm(N)(t) ≡ mmm(t) the d-dimensional row-vector, with EM(N)(t)1{J(t)=i} on the i-th position. Ac-
cordingto[13,Thm.3.2],mmm(t)satisfiesthefollowingnon-homogeneouslineardifferentialequation:
πππTNΛ−mmm(t)(M−NαQ)=mmm(cid:48)(t). (4)
In[5]weprovedthat,with(cid:37)(I) :=λ /µ ,
∞ ∞
EM(N)(t)=N(cid:37)(I)(1−e−µ∞t)+o(N). (5)
Nowdefine(cid:37)(I)(t):=(cid:37)(I)(1−e−µ∞t)and
(cid:90) t (cid:16) (cid:17) (cid:16) (cid:17)
ς(I)(t):=2 e−2µ∞(t−s)πππT Λ−M(cid:37)(I)(s) D Λ−M(cid:37)(I)(s) 111ds. (6)
0
InAppendixBitisshownthat
Cov(M(N)(t),M(N)(t+u)) (cid:16) (cid:17)
lim =v(I)(t,u):=e−µ∞u ς(I)(t)1 +(cid:37)(I)(t)1 . (7)
N→∞ Nmax{1,2−α} {α≤1} {α≥1}
We conclude that under this parameter scaling the covariance exhibits the same dichotomy as the one ob-
servedin[5]forthevariance,i.e.,behavingcruciallydifferentforα<1andα>1. Inthelatterregime,the
systemessentiallybehavesasa(non-modulated)M/M/∞queue,witharrivalrateλ andservicerateµ ,
∞ ∞
whereasforα<1thefulltransitionratematrixQplaysarole(asς(I)(t)involvesthedeviationmatrixD).
6
4 Mean, variance, and covariance for model II
As we saw above, for Model I the mean, variance and covariance can be determined by solving specific
non-homogeneous linear differential equations; for Model II, however, the analysis is simpler, and can be
performed by relying on the law of total (co-)variance, as shown in Section 4.1. Focusing on the same
limitingregimesaswehavestudiedforModelI,theexpressionsbecomemoreexplicit;seeSection4.2.
4.1 Explicitformulae
ThemeanofM(t)forModel II wasalreadydeterminedin[2]; recallingfrome.g.[8]theobservationthat
M(t)obeysaPoissondistributionwiththerandomparameterE(M(t)|J),weconcludethat
EM(t)=E(E(M(t)|J))=E(cid:18)(cid:90) tλJ(s)e−µJ(s)(t−s)ds(cid:19)=(cid:88)d πiµλi (cid:0)1−e−µit(cid:1)=:(cid:37)(II)(t),
0 i=1 i
withJ ≡(J(s))t .
s=0
NowconcentrateontheevaluationofthecovariancebetweenM(t)andM(t+u);assume,withoutlossof
generality,thatu≥0. The‘lawoftotalcovariance’entailsthat
Cov(M(t),M(t+u))=E(Cov(M(t),M(t+u)|J))+Cov(E(M(t)|J),E(M(t+u)|J)). (8)
InAppendixD,weevaluatebothterms,soastoobtain
d
Cov(M(t),M(t+u))=(cid:88)π λi (cid:0)1−e−µit(cid:1)e−µiu+λλλTK (t,u)λλλ+λλλTL(t,u)λλλ; (9)
iµ
i
i=1
thepreciseformofthematricesK (t,u)andL(t,u)isgiveninAppendixDaswell.
4.2 Twospecificlimitingregimes
Inthissubsection,weconsiderthetwoparticularlimitingregimesthatwestudiedearlier,inSection3.2,for
ModelI. Asitturnsout,intheseregimestheexpressionssimplifyconsiderably.
(cid:66) Inthefirstregime,weconsiderCov(M(t),M(t+u))fort→∞.Goingthroughthecalculations,relying
ontheexplicitexpressionsforK (t,u)andL(t,u)asgiveninAppendixD,weobtain
d d d
lim Cov(M(t),M(t+u)) = (cid:88)π λie−µiu+(cid:88)(cid:88)π λiλj e−µjuD(µ)
t→∞ iµi iµi+µj ij
i=1 i=1j=1
+ (cid:88)d (cid:88)d π λiλj (cid:90) ue−µju+µiw(p (w)−π )dw
jµ +µ ji i
i=1j=1 i j 0
+ (cid:88)d (cid:88)d π λiλj (cid:90) ∞eµiu−µjw(p (w)−π )dw,
jµ +µ ji i
i=1j=1 i j u
alsoentailingthat
d d d
lim VarM(t)=(cid:88)π λi +2(cid:88)(cid:88)π λiλj D(µ).
t→∞ iµi iµi+µj ij
i=1 i=1j=1
7
(cid:66) In the second limit, we replace λ (cid:55)→ Nλ and Q (cid:55)→ NαQ, for α > 0. The FCLT under this scaling is
proveninSection6;weherefindthecorrespondingmeanand(co-)variance. Itturnsoutthat,forN large,
d d
Cov(cid:16)M(N)(t),M(N)(t+u)(cid:17)∼N(cid:88)e−µiu(cid:37)(II)(t)+N2−α(cid:88)e−µiuς(II)(t),
i i
i=1 i=1
with(cid:37)i(II) :=πiλi/µiand(cid:37)i(II)(t):=(cid:37)(iII)·(1−e−µit)and
d
ς(II)(t):=(cid:88) λiλj (cid:16)1−e−(µi+µj)t(cid:17)(π D +π D ).
i µ +µ j ji i ij
i j
j=1
Weconcludethat
lim Cov(M(N)(t),M(N)(t+u)) =v(II)(t,u):=(cid:88)d e−µiu(cid:16)ς(II)(t)1 +(cid:37)(II)(t)1 (cid:17). (10)
N→∞ Nmax{1,2−α} i {α≤1} i {α≥1}
i=1
WeobservethatthesamedichotomyappliesastheonewehaveobservedforModelI:forα>1thenumber
ofjobsinthesystembehaves‘Poissonian’,withmeanandvariancescalingessentiallylinearlywithN,both
withproportionalityconstant(cid:37)(II)(t). Forα<1,asseenearlierine.g.[5],thevariancegrowssuperlinearly
withN,withaproportionalityconstantthatinvolvesthedeviationmatrixD.
5 Functional central limit theorem for Model I
In Section 3.2 we considered the covariance of the number of jobs in the system under a specific scaling:
λ (cid:55)→ Nλ and Q (cid:55)→ NαQ, for α > 0. In this section, we prove that for a given t the random variable
M(N)(t) obeys a central limit theorem; moreover, we prove the stronger property that after centering and
normalizingtheprocessM(N)(t),thereisweakconvergencetoaspecificGaussianprocess. Weessentially
adopt the methodology used in [10]; some steps that are fully analogous to those in [10] are described
concisely. Inthesequel,weletZ(N)(t)betheindicatorfunctionoftheevent{J(N)(t)=i},whereJ(N)(t)
i
isaMarkovchainwithtransitionratematrixNαQ.
Firstobservethat,withP (·)andP (·)twoindependentunit-ratePoissonprocesses,itisstraightforwardto
1 2
seethatM(N)(t)canbewrittenas
(cid:32) (cid:90) t d (cid:33) (cid:32)(cid:90) t d (cid:33)
M(N)(t)=P N (cid:88)λ Z(N)(s)ds −P (cid:88)µ M(N)(s)Z(N)(s)ds . (11)
1 i i 2 i i
0 i=1 0 i=1
Nowimposethefollowingcenteringandnormalization,withβ :=max{1,2−α}/2,
(cid:16) (cid:17)
M˜(N)(t):=N−β M(N)(t)−N(cid:37)(I)(t) ,
where(cid:37)(I)(t):=(cid:37)(I)(1−e−µ∞t);theobjectiveofthissectionistoestablishtheconvergenceofM˜(N)(·)to
aspecificGaussianprocess,essentiallyrelyingonthemartingalecentrallimittheorem;seeforbackground
onthemartingalecentrallimittheoreme.g.[11,14].
Itisfirstrealizedthat,asadirectimplicationof(11),forsomemartingaleκ(N)(·),
dM(N)(t)=NλTZ(N)(t)dt−µTZ(N)(t)M(N)(t)dt+dκ(N)(t).
ThenwerewritethisequationintermsofoneforM˜(N)(t):
(cid:16) (cid:17)(cid:48)
dM˜(N)(t) = N1−βλTZ(N)(t)dt−N−βµTZ(N)(t)M(N)(t)dt+N−βdκ(N)(t)−N1−β (cid:37)(I) (t)dt
= N1−βλTZ(N)(t)dt−µTZ(N)(t)M˜(N)(t)dt−N1−βµTZ(N)(t)(cid:37)(I)(t)dt
(cid:16) (cid:17)(cid:48)
+N−βdκ(N)(t)−N1−β (cid:37)(I) (t)dt.
8
Followingtheideasof[10],wenowintroduce
(cid:16) (cid:17) (cid:90) t
Y(N)(t):=exp µTζ(N)(t) M˜(N)(t), where ζ(N)(t):= Z(N)(s)ds.
0
Itthusfollowsthat,usingstandardstochasticdifferentiationrules,
(cid:16) (cid:17)(cid:18) (cid:16) (cid:17)T (cid:16) (cid:17)(cid:48) (cid:19)
dY(N)(t)=exp µTζ(N)(t) N1−β λ−µ(cid:37)(I)(t) Z(N)(t)dt+N−βdκ(N)(t)−N1−β (cid:37)(I) (t)dt .
Nowobservethat,fromthedefinitionofthefunction(cid:37)(I)(t),wefindthat
(cid:16) (cid:17)T (cid:16) (cid:17)(cid:48)
λ−µ(cid:37)(I)(t) π =λ e−µ∞t = (cid:37)(I) (t),
∞
andhenceitisobtainedthat
(cid:16) (cid:17)(cid:18) (cid:16) (cid:17)T(cid:16) (cid:17) (cid:19)
dY(N)(t)=exp µTζ(N)(t) N1−β λ−µ(cid:37)(I)(t) Z(N)(t)−π dt+N−βdκ(N)(t) .
Wenowanalyzethetwotermsinthepreviousdisplayseparately.
◦ We first concentrate on the first term. In [10], relying on the methodology developed in [11], it was
shownthatthefollowingweakconvergenceholds:
(cid:90) · (cid:16) (cid:17)(cid:16) (cid:17)T(cid:16) (cid:17) (cid:90) ·
Nα/2exp µTζ(N)(s) λ−µ(cid:37)(I)(s) Z(N)(s)−π ds→ eµ∞sdG(s),
0 0
wherethestochasticprocessG(·)issuchthat
(cid:104)G(cid:105) =V(t) := (cid:90) t(cid:16)λ−µ(cid:37)(I)(s)(cid:17)T(cid:0)diag{π}D+DTdiag{π}(cid:1)(cid:16)λ−µ(cid:37)(I)(s)(cid:17)ds
t
0
(cid:90) t (cid:16) (cid:17) (cid:16) (cid:17)
= 2 πT Λ−M(cid:37)(I)(s) D Λ−M(cid:37)(I)(s) 1ds;
0
cf. Eqn. (6). (It is noted that in [10] the background process was sped up by a factor N rather than
√
Nα;thisexplainsthattherethegrowthrate N wasfound,whileinoursetupwehaveNα/2.)
Importantly,fromtheaboveweconcludethatthefullfirsttermindY(N)(t)behavesessentiallypro-
portionaltoN1−β−α/2,whichconvergestoaconstantifα≤1,andvanishesotherwise.
◦ Wenowconsiderthesecondterm.Wenotethat,recallingthefactthatP (·)andP (·)areindependent
1 2
unit-ratePoissonprocessesincombinationwithstandardpropertiesforpurejumpprocesses,
d
(cid:104)κ(N)(cid:105) =NλTZ(N)(t)+µTZ(N)(t)M(N)(t),
dt t
andconsequently
1 (cid:90) t (cid:90) t M(N)(s)
(cid:104)κ(N)(cid:105) = λTZ(N)(s)ds+ µTZ(N)(s) ds.
N t N
0 0
Usingtheergodictheorem,thefirstintegralintheright-handsideofthepreviousdisplayconvergesto
λTπ·t=λ t. Likewise,thesecondintegralconvergesto
∞
1 (cid:90) t(cid:88)d (cid:16) (cid:17)
lim µ E M(N)(s)1{J(s)=i} ds,
N→∞N 0 i=1 i
9
which,duetoargumentssimilartothoseunderlying(5),turnsouttoequal
(cid:90) t d
(cid:88)
µ π (cid:37)(I)(1−e−µ∞s)ds.
i i
0 i=1
HenceN−1(cid:104)κ(N)(cid:105) converges,asN →∞,to
t
(cid:90) t
W(t):=λ t+ µ (cid:37)(I)(1−e−µ∞s)ds.
∞ ∞
0
√
Weconcludefromtheabovethatκ(N)(·)/ N convergestoanappropriatelyscaledBrownianmotion.
Inaddition,thissecondtermindY(N)(t)isessentiallyproportionaltoN1/2−β,i.e.,convergingtoa
constantifα≥1,andvanishesotherwise.
Summarizing,wehavethatY(N)(t)convergesweaklytoaprocessY(t)whichisthesolutiontothefollow-
ingstochasticdifferentialequation:
(cid:113)
dY(t)= V(cid:48)(t)1 +W(cid:48)(t)1 dB(t),
{α≤1} {α≥1}
whereweusedthepropertythatforastandardBrownianmotionB andadifferentiablefunctionf,wehave
thatB(f(t))isequalindistributionto(cid:112)f(cid:48)(t)Bˆ(t),whereBˆdenotesanotherBrownianmotion,butwiththe
samedistribution. Also,duetotheergodictheoremwehavethatexp(cid:0)µTζ(N)(t)(cid:1)convergestoexp(µ t).
∞
From the definition of Y(N)(t), we thus conclude the following weak convergence: M˜(N)(·) → M˜(·),
whereM˜(·)solvesthestochasticdifferentialequation
(cid:113)
dM˜(t)=−µ M˜(t)dt+ V(cid:48)(t)1 +W(cid:48)(t)1 dB(t),
∞ {α≤1} {α≥1}
for a standard Brownian motion B(·). Its solution is that the limiting process M˜(·) is a centered Gaussian
processoftheOrnstein-Uhlenbecktype,characterizedbyitscovariancev(I)(t,u),asgivenin(7).
Theorem5.1. ConsiderModelI.AsN →∞,theprocessM˜(N)(·)convergesweaklytoacenteredGaussian
process,withcovariancestructurev(I)(·,·)givenin(7).
6 Functional central limit theorem for Model II
WenowshiftourattentionfromModel I toModel II. Essentiallythesameapproachcanbefollowed,with
animportantdifferencebeingthatnowonehastokeeptrackofthenumberofjobspresentofeachtype,tobe
denotedbyM(N)(t)fortypei,where‘type’referstothestatethebackgroundprocesswasinuponarrival
i
ofthejob. Weuseanapproachsimilartotheoneusedintheprevioussection, butitisnotedthataviable
alternativeistoadapttheapproachfollowedin[1]forthecasethatthedepartureratesarestate-independent,
tothatofModelII.
Asintheprevioussection, westartbywritingtheM(N)(t), fori = 1,...,dintermsofunit-ratePoisson
i
processes;inself-evidentnotation,wenowhave
(cid:18) (cid:90) t (cid:19) (cid:18)(cid:90) t (cid:19)
M(N)(t)=P N λ Z(N)(s)ds −P µ M(N)(s)ds .
i 1,i i i 2,i i i
0 0
Asbefore,weapplycenteringandnormalization,inthatwewillstudy,recallingthat(cid:37)(II) := π λ /µ and
i i i i
(cid:37)(II)(t):=(cid:37)(II)·(1−e−µit),
i i
(cid:16) (cid:17)
M˜(N)(t):=N−β M(N)(t)−N(cid:37)(II)(t) ,
i i i
10
