The ∆(i)/GI/1 Queue: A New Model of Transitory Queueing

Harsha Honnappa∗, Rahul Jain∗,† and Amy R. Ward?

∗ EE Department, University of Southern California† ISE Department, University of Southern California

? IOM Department, Marshall School of Business, University of Southern California.

Abstract— We introduce the ∆(i)/GI/1 queue, a new modelof transitory queueing, where demand for service exists onlyin a fixed interval of time and a large, but finite, number ofcustomers enter the system. Customers independently arriveaccording to some given distribution F . Thus, the arrivaltimes are an ordered statistics, and the inter-arrival times aredifferences of consecutive ordered statistics. They are served bya single server which provides service according to a generaldistribution G, with independent service times. The exact modelis analytically intractable. Thus, we develop fluid and diffusionlimits for the various stochastic processes as the population sizeis increased. The fluid limit of the queue length is observed tobe a reflected process, while the diffusion limit is observed to bea function of a Brownian motion and a Brownian bridge, andgiven by a netput process and a directional derivative of theSkorokhod reflected fluid netput in the direction of a diffusionrefinement of the netput process.

I. INTRODUCTION

The traffic model assumed in a single-server queueingmodel can significantly affect the accuracy and robustnessof its predicted performance in practice. It is commonfor queueing theorists to assume that the traffic model isdescribed by i.i.d. inter-arrival times. In this case, the arrivalcounting process is a renewal process. A similar assumptionis made for the service counting process. This model ismathematically convenient as it offers full use of the toolsfrom renewal and ergodic theory. However, there are manysituations where users choose to enter the system at timesthat are independent of the others. Even if one assumes thatevery arriving user draws an arrival time from the samedistribution, this need not lead to a renewal arrival process.This situation is more likely in the case of ’transitory’queueing systems, where demand for service exists for afinite amount of time and/or there is only a finite numberof users who enter the service system. These scenarios donot seem to fit the standard, single-server models in queueingtheory such as the M/M/1 or M/G/1 etc. A goal of thepresent work is to propose queueing models, and developtheir theory, that are relevant for such transitory queueingsystems.

Models of transitory queueing behavior can arise whenservice starts at a certain time, and a finite number ofcustomers may choose to arrive early. For example, whencustomers go to a rock concert in a Greek theatre, they maychoose to arrive before the gates open, or arrive any timeafter until the gates close. Such a scenario was studied as theconcert arrival game in [11], [9]. Other scenarios where sucha model may be relevant include queueing outside stores for

black friday sales, outside Apple store before new productlaunches, DMV or postal offices, lunch cafeterias, etc. It iseasy to see that the queueing processes in such scenarios areinherently transient, and there really is no steady state forthe system. Unfortunately, transient performance measuresfor queueing systems can be very difficult to compute oftennecessitating the use of numerical schemes. This makes thestudy of transitory queueing systems very hard. A goal ofthe present work is to address this deficiency by introducinga model, theory and approximations for transitory queueingmodels.

We introduce a new queueing model that has a finitepopulation of customers, the arrival process is not a renewalprocess, and service occurs only during a finite time window;in particular, it is a transitory queueing model. Considern customers who arrive into a single-server queue. Eachcustomer’s time of arrival is sampled i.i.d. from a distributionF . Then, the times of arrival are an ordered statistics.Service times have a general distribution G and are i.i.d. Wecall this the ∆(i)/GI/1 queueing model in the notation ofKendall [13], where ∆(i) = (X(i)−X(i−1)) and Xi is the ithorder statistic from a sample of size n from the distributionF . Exact analysis of this model is impossible. Therefore, wedevelop fluid and diffusion limits for this queueing model.

Before describing the fluid and diffusion approximations,it is useful to note here that one could assume an i.i.d.inter-arrival time traffic model in the finite population case.However, it is easy to see that since the customers enter inorder of their arrival times (and hence are order statistics) inthe ∆(i)/GI/1 queue, the inter-arrival times are a weaklymixed exchangeable sequence, implying that our trafficmodel encompasses the i.i.d. model. On the other hand, it isinteresting to note that under the assumption of an absolutelycontinuous F , the arrival process is equal in distribution toa non-homogeneous Poisson process conditioned to have nevents by time T := inft ∈ R|F (t) = 1. We elaborate onthis interpretation in Section V.

To develop our fluid and diffusion limits for the∆(i)/GI/1 model, we scale up the population size andaccelerate the service process, and use the functional stronglaw of large numbers (FSLLN) and the Glivenko-Cantellitheorem to establish fluid limits for the service and arrivalprocesses. Then, using Skorokhod’s reflection mapping theo-rem [20], [8], we can obtain fluid limits for the queue lengthprocess and the workload (the time it would take to servecurrent jobs in the queue) process. The diffusion process

738

Fiftieth Annual Allerton ConferenceAllerton House, UIUC, Illinois, USAOctober 1 - 5, 2012

978-1-4673-4539-2/12/$31.00 ©2012 IEEE

limits of the service and arrival processes can be obtainedby use of the functional central limit theorem (FCLT) forrenewal processes and for empirical processes respectively.The queue length diffusion process limit involves a mapwhich can be interpreted as the directional derivative ofthe Skorokhod reflected fluid netput in the direction of adiffusion refinement of the netput process. We also note thatour diffusion process convergence results are in Skorokhod’sM1 topology on the space Dlim[0,∞), the space of functionsthat are right or left continuous at every point, and rightcontinuous at 0.

As it turns out, the ∆(i)/GI/1 queue has a close connec-tion with single server queues that have a time-varying arrivalrate. Thus, it is pertinent to compare the ∆(i)/GI/1 modelwe introduce to the already studied Mt/Mt/1 queueingmodel. One of the earliest papers on this model is [16].Strong approximations for the model were later developed byMandelbaum and Massey [15]. However, all such models stillassume independent inter-arrival times (though not identi-cally distributed). Thus, the arrival process in the ∆(i)/GI/1model cannot be obtained simply by windowing the processfor the Mt/GI/1 queue since the first and last inter-arrivaltime in such a time-window need not be renewal periods.

Perhaps the work closest to the current paper is [14],where the author considers the same setup as we have, butdoes not allow early arrivals. The paper develops diffusionapproximations to the queue length in separate, distinctintervals and the maximum queue length process. However,without establishing a “process-level” convergence over alltime, such a result is rather incomplete. In fact, it is notdifficult to derive point-wise limits to the queue lengthprocess. Establishing “process-level” convergence for suchlimiting processes in an appropriate topology is the mainmathematical difficulty, as was also observed in [15] forthe Mt/Mt/1 model. Also, worth mentioning here is thework on queueing systems with periodic arrivals studied in[7] and [1]. These models are, of course, distinct from the∆(i)/GI/1 model that we introduce in this paper.

II. PRELIMINARIES

A. Queue Model

Consider a single server, infinite buffer queue that is non-preemptive and non-idling, and say, starts empty. Customersin the queue will be assumed served on a first-come-first-served (FCFS) basis, though this is not essential. Arrivingcustomers do not balk or renege once they commit toqueueing. We also allow early-bird arrivals so customerscan queue up before service starts . Most importantly, it isassumed there is a large but finite number of customers whoarrive for service over some finite time interval.

Let n be the customer population size. Arriving customersindependently sample an arrival time Ti, i = 1, . . . , n, froma fixed cumulative distribution function F that is assumedto have support [−T0, T ] ⊂ R, where −T0 ≤ 0 and T > 0.Thus, F (−T0) = 0 and F (T ) = 1. Customers enter thequeue in increasing order of the sampled arrival times. LetA(t) be the number of customers who have arrived by time

t, which is usually called the arrival process. This is definedto be

A(t) :=

n∑i=1

1Ti≤t. (1)

Let νi, i ≥ 1 be a sequence of independent and identicallydistributed (IID) random variables, independent of the arrivaltimes Ti, with mean Eνi = 1/µ and finite variance σ2,whose cumulative distribution function G has support [0,∞).Define S to be a renewal process in terms of νi as

S(t) := supm ≥ 1|V (m) ≤ t, ∀t ≥ 0, (2)

where V (m) :=∑m

i=1 νi. S(t) is interpreted as the serviceprocess of the queue. Here, νi is the service time for the ithpotential customer, and µ as the service rate offered by thesystem. Note that S is defined for all t ≥ 0, so servicestarts at time 0 in the ∆(i)/GI/1 model, and we defineS(t) = 0 for all t < 0. Thus, the service process S(t) can beinterpreted as the number of customers that could be servedif the server were busy all the time in the interval [0, t].

Now, V (m) can be interpreted as the amount of work (inunits of time) presented by m customers. Let Z representthe virtual waiting time process, defined using V as

Z(t) := V (A(t))−B(t)− t1t≤0, (3)

where B(t) is the busy time process defined as

B(t) :=

(∫ t

0

1Q(s)>0ds

)1t≥0, ∀t ∈ [−T0,∞). (4)

B(t) is the amount of time the server has been busy inthe interval [0, t], and clearly B(t) ≤ t. Thus, Z(t) is thedifference between the total amount of work presented bythe arrivals up to time t and the amount of work completedby the server by t. Z(t) can also be interpreted as the amountof time a virtual arrival at time t would have to wait till itenters service. Note that this definition varies slightly fromthe standard definition due to the fact that an arrival at timet < 0 before service starts has to wait an extra t units oftime for service to start, which accounts for the −t1t≤0term.

Finally, let Q represent the queue length process, or thenumber of customers in service and waiting in the buffer attime t. This is defined in terms of the arrival and serviceprocesses as

Q(t) := A(t)− S(B(t)), ∀t ∈ [−T0,∞). (5)

As noted before, S(t) is the number of service completionsif the server is always busy in [0, t]. Thus, S(B(t)) countsthe number of service completions by time t given that itis busy only for time B(t) until that time. This then is thenumber of departures from the queue in [0, t], and the queuelength is the difference between the number of arrivals andthe number of departures by time t.

We observe that this model is intractable to exact analysis.Therefore, we develop asymptotic approximations as thepopulation size increases.

739

B. Basic results

Notation Unless noted otherwise, all intervals of time aresubsets of [−T0,∞), for a given −T0 ≤ 0. Let Dlim :=Dlim[−T0,∞) be the space of functions x : [−T0,∞)→ Rthat are right-continuous at −T0, and are either right or leftcontinuous at every point t > −T0. Note that this differsfrom the usual definition of the space D as the space offunctions that are right continuous with left limits (cadlagfunctions). We denote almost sure convergence by a.s−→ andweak convergence by ⇒. The topology of convergence isindicated by the tuple (S,m), where S is the metric spaceof interest and m is the metric topologizing S. Thus, Xn

a.s−→X in (Dlim, U) as n → ∞ indicates that Xn ∈ Dlim

converges to X ∈ Dlim uniformly on compact sets (u.o.c.)of [−T0,∞) almost surely. Similarly, Xn ⇒ X in (Dlim, U)as n → ∞ indicates that Xn ∈ Dlim converges weaklyto X ∈ Dlim uniformly on compact sets of [−T0,∞).(Dlim,M1) indicates that the topology of convergence isthe M1 topology. X indicates a fluid-scaled or fluid limitprocess. X and X are used to indicate diffusion-scaledand diffusion limit processes. We use is used to indicatecomposition of functions or processes. The indicator functionis denoted by 1· and the positive part operator by (·)+.

We now present known functional strong law of largenumbers (FSLLN) or fluid limits, and functional centrallimit theorem (FCLT) or diffusion limits, for the arrival andservice processes, as the population size n increases to ∞.Our convention is to superscript any process associated withthe model having population size n by n. These limits arepresented for the reader’s convenience, and will be useful inlater sections where fluid and diffusion limits for the queuelength and the virtual waiting time processes are derived.

We start with the arrival process. For the rest of thissection, let An := A be the arrival process indexed bythe population size n. Recall that customers sample a timeto arrive from a common distribution F that has compactsupport [−T0, T ] and is assumed to be absolutely continuous.Clearly, as n increases to∞, An(t) will increase to∞. Thus,to obtain a ‘fluid’ limit, we define the fluid-scaled arrivalprocess as

An :=An

n.

Now, for the service process, we use an accelerationtechnique wherein the service rate is accelerated by mul-tiplication with the population size so that µn := nµ.Correspondingly, the scaled service times are νni := νi/nfor i = 1, . . . , n. Using these, we define the acceleratedservice process as

Sn(t) := sup

m ≥ 1|

m∑i=1

νin≤ t

t ≥ 0,

and the fluid-scaled service process as

Sn :=1

nSn.

Finally, Recall that we have

V (m) :=

m∑i=1

νi.

This random variable can be interpreted as the amountof work (in units of time) presented to the server by mcustomers. We can define the offered work process (theamount of work offered by time t by a population of sizen), as

V n(t) :=

bntc∑i=1

νni ∀t ∈ [0,∞). (6)

Here, we have used the acceleration argument presented forthe service process.

The following lemma establishes the fluid limits for theseprocesses.

Proposition 1. As n→∞, in (Dlim, U)

(An(t), Sn(t)1t≥0, Vn(t)1t≥0)

a.s−→ (F (t), µt1t≥0,t

µ1t≥0).

Remarks 1. The proof of Proposition 1 follows easily fromstandard results: The fluid arrival process limit is given by theGlivenko-Cantelli Theorem (see [4]). The fluid limits of theservice process and the offered work process follow from thefunctional strong law of large numbers theorem for renewalprocesses (see [3]).

Next, using the fluid limits from Proposition 1, we presentfunctional central limit theorem or diffusion limits, to theappropriately standardized or diffusion-scaled processes. Thediffusion-scaled arrival process is defined as

An(t) :=√n

(An(t)− F (t)

)∀t ∈ [−T0,∞).

Similarly, the diffusion-scaled service and offered work pro-cesses are given by

Sn(t) :=√n

(Sn(t)− µt

)t ≥ 0

V n(t) :=√n

(V n(t)− 1

µt

)t ≥ 0.

The following proposition presents the diffusion limits forthese processes.

Proposition 2. As n→∞, in (Dlim, U),

(An, Sn, V n)⇒(W 0 F, σµ3/2W e,−σµ1/2W e

µ

)where W 0 is the standard Brownian Bridge process and W isthe standard Brownian motion process, both are independentprocesses. e : [0,∞)→ [0,∞) is the identity map.

Remarks 1. The proof of this proposition follows easilyfrom standard results: The FCLT limit for the diffusion-scaled arrival process, also called the empirical process,is a Brownian Bridge process by Donsker’s Theorem (seeSections 13 and 16 in [2]). Note that this limit also arisesin the study of the invariance principle associated with the

740

Kolmogorov-Smirnov statistic used to compare empiricaldistributions with candidate ones (see [23] for more detail).The limits for the diffusion-scaled service and offered workprocesses follow from the FCLT for renewal processes (seeSection 16 in [2] and Chapter 5 in [3]).2. Note that we have not placed any restriction on the arrivaldistribution F other than that of finite support. The proofsof the fluid limits to the performance metrics will hold forarbitrary distribution functions F . However, the diffusionlimits require F to be absolutely continuous, since the formof the limit process depends on this fact. Extending the resultto arbitrary F appears non-trivial, and left for future work.

III. FLUID APPROXIMATIONS

We now derive the fluid limit to the queue length processfor an arbitrary continuous arrival distribution and constantservice rate. Recall the queue length process from (5). Thecorresponding fluid-scaled queue length process is

Qn(t)

n=

1

nAn(t)− 1

nSn(Bn(t)), (7)

where Bn(t) is the fluid-scaled version of the busy timesteprocess (4) defined as

Bn(t) :=

(∫ t

0

1Qn(s)>0ds

)1t≥0. (8)

Now, we can write (9) by adding and subtracting thefunctions F and µBn to obtain Qn(t)

n =(An(t)

n− F (t)

)−

(Sn(Bn(t))

n−

µBn(t)

)+

(F (t)− µBn(t)

).

Similarly, adding and subtracting the function µt1t≥0, weget Qn(t)

n := µIn(t)+(An(t)

n−F (t)

)−(Sn(Bn(t)

n−µBn(t)

)+

(F (t)−µt1t≥0

),

where In(t) = t1t≥0 − Bn(t) is the fluid-scaled idletime process. Now, we rewrite the fluid-scaled queue lengthprocess (9) as follows:

Qn(t)

n= Xn(t) + µIn(t), ∀t ∈ [−T0,∞), (9)

where Xn(t) is defined to be Xn(t) :=(An(t)

n−F (t)

)−(Sn(Bn(t))

n−µBn(t)

)+(F (t)−µt1t≥0).

(10)Let Qn(t) := Qn(t)/n denote the fluid-scaled queue lengthprocess. Theorem 1 below proves two things about Qn. First,that it satisfies the Skorokhod reflection mapping theorem(see Chapter 6 of [3]), and can be expressed uniquely interms of Xn. Second, using this unique representation andthe continuous mapping theorem, we obtain the the fluid limitfor the queue length process.

Q(t)

t-T0 !1 !2 !30

F(t)

µt

Fig. 1. An example of a ∆(i)/GI/1 queue that will undergo multiple“regime changes”. The fluid queue length process is positive on [−T0, τ1)and [τ2, τ3), and 0 on [τ1, τ2) and [τ,∞)

Recall that the Skorokhod reflection map is a continuousfunctional (Φ,Ψ) : Dlim → Dlim ×Dlim defined as

x 7→ Ψ(x) := sup−T0≤s≤t

(−x(s))+,

andx 7→ Φ(x) := x+ Ψ(x), ∀x ∈ Dlim.

Theorem 1 (Fluid Limit). The pair (Qn, µIn) has aunique representation (Φ(Xn),Ψ(Xn)) in terms of Xn.Furthermore, as n→∞,

(Qn, µIn)a.s−→ (Φ(X),Ψ(X)) in (Dlim, U),

where X(t) = (F (t)− µt1t≥0).

The proof is in the [10].Remarks 1. Note that X is the difference between thefluid limits of the arrival and service processes, and is oftenreferred to as the fluid limit of the netput process. In effect,it is the amount of net potential fluid inflow to the system.2. Theorem 1 shows that the fluid limit of the queue lengthprocess for t ∈ [−T0,∞) is Q(t)

= (F (t)− µt1t≥0) + sup−T0≤s≤t

(−(F (s)− µs1s≥0))+.

Q can be interpreted as the sum of the fluid netput processand the potential amount of fluid lost from the system.Suppose that (F (t)−µt1t≥0) < 0 so that the fluid serviceprocess has “caught up” and exceeded the cumulativeamount of fluid arrived in the system by time t (forsimplicity assume t > 0). Further, suppose f(t) − µ < 0,implying that the netput process is decreasing at t. In thiscase, sup−T0≤s≤t(−(F (s)− µs1s≥0))+ = −(F (t)− µt).This is the amount of extra fluid that could have beenserved, but is now lost.

From Theorem 1, we can see that the fluid limit of thefluid-scaled busy time process is

B(t) := t1t≥0 −1

µΨ(X(t)), ∀t ∈ [−T0,∞) (11)

Note that B(t) = 0 for all t ≤ 0, as Ψ(X)(t) = 0 on thatinterval.

741

The limit for the busy time process will prove useful inestablishing a fluid limit for the virtual waiting time process(3). The fluid-scaled virtual waiting time process is

Zn(t) = V n

(n

(An(t)

n

))− Bn(t)− t1t≤0.(12)

The following result establishes a fluid limit for the virtualwaiting time process.

Proposition 3 (Fluid Transient Little’s Law). As n→∞,

Zn a.s−→ Z in (Dlim, U), (13)

where Z(t) := Q(t)/µ− t1t≤0.

Remarks 1. The fluid limit in Corollary 3 can be interpretedas a fluid ’transient’ Little’s Law in the fluid limit for thismodel since it relates the virtual waiting time fluid queuelength but there is a transient term, t1t≤0 which accountsfor the fact that an arrival at time t < 0 would have to have−t time units for service to start.

IV. DIFFUSION APPROXIMATIONS

A. Queue Length Process

Define the diffusion-scaled queue length process as

Qn(t)√n

:=An(t)√

n− Sn(Bn(t))√

n∀t ∈ [−T0,∞) (14)

Rewriting it after introducing the term√nµt1t≥0, we have

Qn(t)√n

=

(An(t)√

n−√nF (t)

)−(Sn(Bn(t))√

n−√nµBn(t)

)+√n(F (t)− µt1t≥0) +

√nµ(t1t≥0 −Bn(t)).

Using the definition of the idle time process√nIn(t) =

√n(t1t≥0 −Bn(t)),

and we can express Qn/√n as

Qn

√n

= Xn +√nX +

√nµIn, (15)

where Xn(t) :=(An(t)√

n−√nF (t)

)−(Sn(Bn(t))√

n−√nµBn(t)

)= An(t)− Sn(Bn(t)), ∀t ∈ [−T0,∞). (16)

Recall from Theorem 1 that X(t) = (F (t) − µt1t≥0)is the fluid netput process. We can think of Xn as adiffusion refinement of the netput process. Now, Lemma 1gives a diffusion limit of Xn(t) as a direct consequence ofProposition 2.

Lemma 1. As n→∞,

Xn ⇒ X := W 0 F − σµ3/2W B in (Dlim, U) (17)

where B is defined in (13), and W 0 and W are independentstandard Brownian Bridge and standard Brownian motionprocesses.

The proof is straightforward and can be found in [10].

Remarks 1. Note that using a classical time change (see[12]) it is possible to see that the Brownian Bridge processis equal in distribution to a time changed Brownian motionprocess, and X is equal in distribution to a stochastic integral

X(t)d=

∫ t

−T0

√g′(s)dWs, ∀t ∈ [−T0, T ]

−σµ3/2W (B(T )), ∀t > T.(18)

where g(t) = F (t)(1 − F (t)) + σ2µ3B(t) and W is aBrownian motion independent of W 0 and W . Thus, the pro-cess X can also be interpreted as a time-changed Brownianmotion, on the interval [−T0, T ], and then its sample path isa constant on (T,∞).

In the rest of this section, we will use Skorokhod’salmost sure representation theorem [19], [22], and replacethe random processes above that converge in distribution bythose defined on a new probability space, that have the samedistribution as the original processes and converge almostsurely. Remarkably, the requirements for the almost surerepresentation are quite mild - the underlying topologicalspace needs to be Polish (a seperable and complete metricspace). We note without proof that the space Dlim, as definedin this paper, is Polish when endowed with the M1 topology.This conclusion follows from Theorem 2.6 of [21] and thefact that the proof there extends easily to the case of the M1

topology. The authors in [15] also point out that [17] has amore general proof of this fact.

Thus, we replace the weak convergence in (8) by

(An, Sn, V n)a.s−→

(W 0 F, σµ3/2W,− σµ1/2W h

µ

)in (Dlim, U), where abusing notation we denote the new

limit random processes by the same letters as the old ones.This implies that in Lemma 1, as n→∞, we actually have

Xn a.s−→ X in (Dlim, U).

Now, our goal is to establish a functional central limittheorem for the centered queue length process

Qn(t) :=√n

(Qn(t)

n− Q(t)

). (19)

We achieve this by using the Skorokhod reflection mappingtheorem [19], [3], [23] and express (Qn(t)/

√n,√nµIn(t))

uniquely in terms of Xn and X . Using this representation,we redefine Qn in terms of Xn and X , and then establishthe necessary limit as n → ∞. Note that we will establishthe limit in the weaker topology M1, as opposed to the morecommon U (uniform) or J1 topologies. This is because a di-rectional derivative reflection mapping lemma (Lemma 2) wewill use is only available for (Dlim,M1). For convenience,we define the function

Y n(t) :=√nµIn(t)−

√nΨ(X(t)). (20)

Recall that (Φ,Ψ) is the Skorokhod reflection map. Wedenote the directional derivative of the Skorokhod reflectionmap by

∇Xt = −T0 ≤ s ≤ t|X(s) = Ψ(X)(t), (21)

742

which is a set correspondence of points upto time t wherethe fluid netput process achieves an infimum. The diffusionlimit result is a direct consequence of the following lemma.

Lemma 2. Let x and y be real-valued continuous functionson [0,∞), and Ψ(z)(t) = sup0≤s≤t(−z(s)), for any processz ∈ Dlim. Let yn ⊂ Dlim be a sequence of functions suchthat yn

a.s→ y as n → ∞. Then, with respect to Skorokhod’sM1 topology, yn := Ψ(

√nx + yn) −

√nΨ(x) −→ y :=

sups∈∇xt(−y(s)) as n → ∞, where ∇x

t = 0 ≤ s ≤t|x(s) = x(t).

Now, the following theorem proves the diffusion limit forthe queue length process.

Theorem 2 (Diffusion Limit). The pair (Qn, Y n) has aunique representation in terms of Xn and

√nX given by(

Φ(Xn +√nX)−

√nQ,Ψ(Xn +

√nX)−

√nΨ(X)

),

where Q = X + Ψ(X) is the fluid limit of the queue lengthprocess. Furthermore, as n→∞

(Qn, Y n)⇒ (X + Y , Y ) in (Dlim,M1),

where X(t) = W 0(F (t))− σµ3/2W (B(t)), and Y (t) =maxs∈∇X

t(−X(s)) ∀t ∈ [−T0,∞).

The proof is available in [10].Remarks 1. Observe that the diffusion limit to the queuelength process is a function of a Brownian Bridge and aBrownian motion process. This is significantly different fromthe usual limits obtained in a heavy-traffic or large populationapproximation to a single server queue. For instance, in theG/G/1 queue one would expect a reflected Brownian motionin the heavy-traffic setting. In [15] it was shown that thediffusion limit process to the Mt/Mt/1 queue is a timechanged Brownian motion reflected through the directionalderivative reflection map we used in Lemma 2. There arevery few examples of heavy-traffic limits involving a diffu-sion that is a function of a Brownian Bridge and a Brownianmotion process. However, there have been some results innon-conventional queueing models where a Brownian bridgearises in the limit. In [18], for instance, a Brownian Bridgelimit arises in the study of a many-server queue in the Halfin-Whitt regime.2. We noted in the remarks after Theorem 1 that the fluidlimit can change between being positive and zero in thearrival interval for a completely general F . One can thenexpect the diffusion limit to change as well, and switchbetween being a ‘free’ diffusion, a reflected diffusion anda zero process. This is indeed the case. Figure 2 illustratesthis for the example in Figure 1. Note that ∀t ∈ [−T0, τ1)Ψ(X)(t) = −X(−T0), implying that the set ∇X

t is asingleton. On the other hand, at τ1 ∇X

t = −T0, τ1.For t ∈ (τ1, τ2], Ψ(X)(t) = 0 = X(t), implying that∇X

t = (τ1, t]. On (τ2, τ3), Ψ(X)(t) = 0, but X(t) > 0, sothat ∇X

t = (τ1, τ2]. Finally, the fluid queue length becomeszero when the fluid service process exceeds the fluid arrival

Fig. 2. An example of a ∆(i)/GI/1 queue that will undergo multiple“regime changes”. The diffusion limit switches between a free Brownianmotion (BM), a reflected Brownian motion (RBM), and the zero process

process in [τ3,∞), implying that Ψ(X)(t) = −(F (t)−µt) >0. It can be seen that ∇X

t = t in this case.1) Uniform Arrival Distribution: Note that ∇X

t is a setcorrespondence that maps each time t to the set of points(upto t) at which the fluid netput process is equal to itsinfimum at t. Theorem 2 shows that the diffusion limit to thequeue length process is in fact piecewise continuous since Yis. We now specialize the diffusion limit results to the caseof a uniform F with early-bird arrivals. This illustrates withgreater clarity the discontinuous nature of the limit processes.

Corollary 1. Let F be the uniform distribution on [−T0, T ],where −T0 < 0. Then,

Q(t) =

W 0(F (t))− σµ 3

2W (t), ∀t ∈ [−T0, τ)

(W 0(F (τ))− σµ 32W (τ))

+(−(W 0(F (τ))− σµ 32W (τ)))+, t = τ

0, ∀t ∈ (τ,∞),

where τ = −T0 ≤ t <∞|F (t) = µt.

The proof is available in [10].The time τ can be interpreted as the first time that the fluid

service process catches up with the fluid arrival process. Fora uniform F there is at most one such point, but in generalthere can be many such points.

Interestingly, the nature of the discontinuity at Q(τ)depends on the the sign of X(τ). The following corollaryclarifies this statement. Recall that t is a point of right-discontinuity for a function x ∈ Dlim if x is left-continuousat t, and x(t−) > x(t+). On the other hand, t is a point ofleft-discontinuity if x is right-continuous at t, and x(t+) >x(t−).

743

Corollary 2. Let F be the uniform distribution function over[−T0, T ], where T0 > 0, and τ = −T0 ≤ t < ∞|F (t) =µt1t≥0. Then, for the process Q in Corollary 1, we have

(i) [−T0, τ) ∪ (τ,∞) are points of continuity.(ii) τ is a point of right-discontinuity, when X(τ) ≥ 0.

(iii) τ is point of left-discontinuity, when X(τ) < 0.

The proof is available in [10].Remarks: 1. In Corollary 1, Q is piecewise continuouson [−T0,∞), with a single point of discontinuity at τ .Interestingly, Q(τ) is determined by the value of the processat τ−. If Q(τ−) is non-positive, then the value of Q(τ)is 0. On the other hand, if Q(τ−) = X(τ) > 0, thenQ(τ) = X(τ) = Q(τ−). However, at τ+ the queue lengthimmediately falls to 0 and remains there forever after, as thereflection regulator map becomes positive for all time thatfollows τ .2. A useful way to interpret the discontinuity at τ inCorollary 1 is to consider the process on the two sub-intervals separately and try to “patch” them together. IfQ(τ−) = X(τ) = Q(τ) > 0 we should expect a freediffusion path on the interval [−T0, τ ], and a reflected processsuch that the path is 0 on (τ,∞). Further, Q(τ) becomesthe “starting state” for the process on the interval (τ,∞),and the reflection operator is applied an instant after τ . Onthe other hand, if Q(τ−) = X(τ−) ≤ 0 we have a freediffusion on [−T0, τ) and the zero process on [τ,∞); i.e.,the process drops to zero at τ . Thus, Q(τ−) provides thestarting conditions for the new “regime” of the diffusion, asthe process transitions from [−T0, τ) to (τ,∞).3. We note that in [14], a diffusion approximation to thequeue length process is derived independently for differentoperating regimes. However, these limit results have not been“patched” together to obtain a “process-level” convergenceresult, which is precisely where the mathematical challengeslie. We also note that diffusion limits in [14] do not involvedirectional derivative maps since the processes are contin-uous are continuous over the intervals on which they arestudied.4. It is also pertinent to mention that the limit results in [14]are obtained in the uniform topology at what are the conti-nuity points of the limit process between regime changes.However, as we noted above, there are discontinuities inthe limit process at points such as τ where regimes change,precluding the possibility of establishing a “process-level”limit in the uniform topology and necessitating the need toestablish the limit in a weaker topology.

B. Virtual Waiting Time Process

Now, consider the centered virtual waiting time processgiven by

Zn(t) =√n(Zn(t)− Z(t)) ∀t ∈ [−T0,∞), (22)

where Z(t) is defined in (15) and Zn(t) is defined in (14).Corollary 4 proves the diffusion limit to this process.

Proposition 4 (Diff. transient Little’s Law). As n→∞,

Zn ⇒ Z :=1

µQ+σµ1/2WB−σµ1/2WF in (Dlim,M1).

(23)

The proof is omitted due to a lack of space, and is availablein [10].

V. TRAFFIC MODEL

Notice that the inter-arrival time random variables in the∆(i) arrival process are “weakly” dependent due to thefact that the arrival times are ordered statistics. In general,working with independent inter-arrival time random variablesis mathematically easier. A natural problem that suggestsitself is whether there is an independent increments processthat closely approximates the ∆(i) process in some sense.While we do not have a definite answer to this questionyet, some classical results suggest a particular independentincrements process that could approximate the ∆(i) process.

Let (Ω,F , P ) be a probability space with respect to whichN(·) is a unit rate homogeneous Poisson process. Let f bea locally integrable function, so that for any 0 ≤ s < t∫ t

sf(u)du < ∞. Let F (t) =

∫ t

0f(u)du, then N F is a

non-homogeneous Poisson process with rate function f , andT := inft ≥ 0|F (t) = 1. The following property of pointprocesses provides an interesting relationship between pointprocesses and order statistics of random variables.

Definition 1. A point process is said to have the OS propertyif, when conditioned on the event that the process has n ∈ Npoint events by a fixed time τ ∈ [0,∞), it has the samedistribution as the order statistics of n i.i.d. random samplesdrawn from a distribution G(t) = m(t)/m(τ), where m(·)is the cumulative mean function of the point process andt ∈ [0, τ ] is fixed.

A well known classical result is that the non-homogeneousPoisson process possesses the OS property (see [6]). Itfollows that for a fixed time t ∈ [0, T ] (T is as definedabove), the distribution of the ∆(i) process with n eventsshould equal that of a non-homogeneous Poisson processconditioned to have n events by time T , and whose cumu-lative mean function matches the arrival distribution in thesupport of the distribution.

We would like to show that the conditioned Poissonprocess, when scaled appropriately, converges to a BrownianBridge process just as the ∆(i) process does. In effect, weneed to show that the measure induced on the space D[0, T ]by conditioning on the event N F (T ) = n converges tothe Brownian bridge measure on C[0, T ]. For brevity, let Nrepresent the non-homogeneous Poisson process.

It is straightforward to see that ∀t ∈ [0, T ]

P (N(t) = k|N(T ) = n)

=P (N(t) = k)P (N(T )−N(t) = n− k)

P (N(T ) = n)

,

where the right hand side follows easily by the fact thatN is a Levy process. Substituting for the probabilities and

744

simplifying the resulting expression we have

P (N(t) = k|N(T ) = n) =

(n

k

)(F (t))k(1− F (t))n−k.

For brevity let Pn(N(t) = k) := P (N(t) = k|N(T ) = n).Let N :=

√n(N(t)/n − F (t)), then it follows from the

De Moivre-Laplace Theorem (see Chapter 3 of [4]) that asn→∞ we have

Pn(a ≤ N(t) ≤ b)→∫ b

ae−x

2/(2F (t)(1−F (t)))dx√2πF (t)(1− F (t))

,

for any a < b. Thus, for a fixed t N(t) ⇒ Z(t), whereZ(t) is zero mean Gaussian random variable with varianceF (t)(1−F (t)). It is easy to see from the multinomial centrallimit theorem that for any 0 < t1 < t2 < . . . < tl ≤ T(N(t1), N(t2), . . . , N(tl)⇒ (Z(t1), Z(t2), . . . , Z(tl).

If the tightness of the measures Pn could be shown, thenthe next claim would establish a “process level” convergenceof N to a Brownian bridge process. However, at this pointthis is still a conjecture as we do not have a proof yet. Wedo have reason to believe that it is true, as a (scaled) Poissonprocess can be strongly approximated by a Brownian motionprocess (see Chapter 7 of [5]). Proving this conjecture is partof our ongoing research.

Conjecture 1. N ⇒W 0 F in (C, U) as n→∞.

Here W 0 is the standard Brownian bridge process.It follows that a conditioned non-homogeneous Poisson

process (or Mt process) can be used to study the ∆(i) arrivalprocess. However, it is important to note that this does notimply that the ∆(i)/GI/1 queue is equivalent to studying aMt/GI/1 queue in [0, T ]. Notice that while the Mt/GI/1queue can only be studied in the transient setting, it is not atransitory queue as there are arrivals to the system over theinterval [0,∞). The performance metrics of the Mt/GI/1queue are completely different over the interval [0, T ] fromthe ∆(i)/GI/1, and the precise relationship between themis still an open problem that requires investigation.

VI. CONCLUSIONS

In this paper, we introduced a novel single server queueingmodel which we call the ∆(i)/GI/1 queue. Customers froma finite population independently choose (sample) their timeof arrival at the queue from a common distribution function.The arrival times are order statistics, and the inter-arrivaltimes are differences of consecutive order statistics, andnot necessarily renewal intervals. Service times are i.i.d.with some general distribution G, and the service rate isfixed. Studying the finite model is analytically intractable.Furthermore, the queue itself is transitory, and thus can onlybe studied in the transient setting. In fact the ∆(i)/GI/1queue does not have a stationary state.

Thus, we developed pathwise asymptotic approximationsto the system performance metrics as the population sizeincreases. We first derive a fluid limit to the queue lengthprocess, and show that the fluid queue length process isthe difference between the arrival distribution and the fluid

service process, with a reflecting barrier at 0. We then derivethe diffusion limit to the queue length process in terms of thenetput process and a directional derivative of the Skorokhodreflected fluid netput in the direction of a diffusion refinementof the netput process. This weak convergence result is shownto hold in the M1 topology on the space Dlim.

Finally, we have also shown an explicit connection be-tween the traffic models in the ∆(i)/GI/1 and Mt/GI/1queues. However, the precise relationship between the per-formance metrics of the two models requires further inves-tigation. We leave this for future work.

REFERENCES

[1] N. Bambos and J. Walrand. On queues with periodic inputs. J. ofApplied Probability, pages 381–389, 1989.

[2] P. Billingsley. Convergence of Probability Measures. Wiley & Sons,1968.

[3] H. Chen and D. Yao. Fundamentals of Queueing Networks: Perfor-mance, asymptotics, and optimization. Springer, 2001.

[4] R. Durrett. Probability: Theory and Examples, 4th Ed. CambridgeUniversity Press, 2010.

[5] S. Ethier and T. Kurtz. Markov processes: Characterization andconvergence. John Wiley and Sons, 1986.

[6] R. Gallager. Discrete stochastic processes, volume 321. Springer,1996.

[7] B. Hajek. A Queue with Periodic Arrivals and Constant Service Rate.Probability, Statistics and Optimization - A Tribute To Peter Whittle,pages 147–157, 1994.

[8] J. Harrison. Brownian motion and stochastic flow systems. J. Wiley,1985.

[9] H. Honnappa and R. Jain. Strategic Arrivals into Queueing Networks:The Network Concert Queueing Game. Submitted to OperationsResearch, 2011.

[10] H. Honnappa, R. Jain, and A. Ward. ∆(i)/GI/1:A New QueueingModel For Transitory Queueing Systems. submitted, 2012.

[11] R. Jain, S. Juneja, and N. Shimkin. The Concert Queueing Game: ToWait or To be Late. Discrete Event Dynamic Systems, 21(1):103–134,2011.

[12] I. Karatzas and S. E. Shreve. Brownian Motion and StochasticCalculus. Springer, 1991.

[13] D. G. Kendall. Stochastic Processes Occurring in the Theory ofQueues and their Analysis by the Method of Imbedded Markov Chain.Annals of Mathematical Statistics, 24(3):338–354, 1953.

[14] G. Louchard. Large finite population queueing systems. The single-server model. Stochastic Processes and their Applications, 53(1):117– 145, 1994.

[15] A. Mandelbaum and W. Massey. Strong Approximations For Time-dependent Queues. Math. of Operations Research, 20(1), 1995.

[16] W. Massey. Asymptotic analysis of the time dependent M/M/1 queue.Math. of operations research, pages 305–327, 1985.

[17] J. L. Pomarede. A Unified Approach via Graphs to Skorohod’sTopologies on the Function Space D. Ph.D. Thesis, Yale University,1976.

[18] A. Puhalskii and J. Reed. On many-server queues in heavy traffic.Annals of Applied Probability, 20(1):129–195, 2010.

[19] A. Skorokhod. Limit Theorems For Stochastic Processes. Theory ofProbability And Its Applications, 1(3), 1956.

[20] A. Skorokhod. Stochastic equations for diffusion processes in abounded region. Theory of Probability and its Applications, 6:264,1961.

[21] W. Whitt. Some Useful Functions for Functional Limit Theorems.Math. of Operations Research, 5(1):67–85, 1980.

[22] W. Whitt. Internet Supplement To Stochastic Process Limits. 2001.[23] W. Whitt. Stochastic Process Limits. Springer, 2001.

745