a how hard are steady-state queueing simulations? · 2015-03-22 · how hard are steady-state...

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How Hard are Steady-State Queueing Simulations?

ERIC CAO NI and SHANE G. HENDERSON, Cornell University

Some queueing systems require tremendously long simulation runlengths to obtain accurate estimators ofcertain steady-state performance measures when the servers are heavily utilized. However, this is not uni-formly the case. We analyze a number of single-station Markovian queueing models, demonstrating thatseveral steady-state performance measures can be accurately estimated with modest runlengths. Our anal-ysis reinforces the meta result that if the queue is “well dimensioned,” then simulation runlengths will bemodest. Queueing systems can be well dimensioned because customers abandon if they are forced to wait inline too long, or because the queue is operated in the “quality and efficiency driven regime” where serversare heavily utilized but wait times are short. The results are based on computing or bounding the asymptoticvariance and bias for several standard single-station queueing models and performance measures.

Categories and Subject Descriptors: G.3 [Probability and Statistics]: Markov Processes, Queueing The-ory; I.6.6 [Simulation and Modeling]: Output Analysis

General Terms: Design, Performance, Theory

Additional Key Words and Phrases: Diffusion approximations, Markovian queues, asymptotic variance

ACM Reference Format:Eric C. Ni and Shane G. Henderson. 2013. How hard are steady-state queueing simulations? ACM Trans.Model. Comput. Simul. V, N, Article A (January YYYY), 20 pages.DOI:http://dx.doi.org/10.1145/0000000.0000000

1. INTRODUCTIONThere is a widely held perception that using simulation to estimate steady-state per-formance measures for queueing systems with heavily utilized servers is hard. By“heavily utilized” servers we mean that the fraction of time that the servers are busyis close to 1. By “hard” we mean that the runlengths needed to obtain narrow con-fidence intervals with the desired coverage level are very large. On the contrary, wewill argue that for “well-dimensioned” single-station queueing systems, the simulationrunlengths needed to obtain accurate estimates are often modest. Queueing systemscan be well dimensioned because customers abandon if they are forced to wait in linetoo long, or because the queue is operated in the “quality and efficiency driven regime”where servers are heavily utilized but wait times are short. Our argument is basedon extending existing results [Whitt 2006] that support this view to additional single-station queueing models with infinite waiting room and first-come-first served servicediscipline. See Srikant and Whitt [1996] for closely related results for loss-systems,which we do not explore.

To make this discussion more precise, let X = (X(t) : t ≥ 0) be a stochastic processrepresenting the number of customers or jobs in a queueing system as a function oftime, and suppose that X possesses a steady-state, i.e., there exists a random vari-

This work is partially supported by the National Science Foundation, under grant CMMI-1200315.Authors’ address: School of Operations Research and Information Engineering, Cornell UniversityPermission to make digital or hard copies of part or all of this work for personal or classroom use is grantedwithout fee provided that copies are not made or distributed for profit or commercial advantage and thatcopies show this notice on the first page or initial screen of a display along with the full citation. Copyrightsfor components of this work owned by others than ACM must be honored. Abstracting with credit is per-mitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any componentof this work in other works requires prior specific permission and/or a fee. Permissions may be requestedfrom Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212)869-0481, or [email protected]© YYYY ACM 1049-3301/YYYY/01-ARTA $15.00DOI:http://dx.doi.org/10.1145/0000000.0000000

ACM Transactions on Modeling and Computer Simulation, Vol. V, No. N, Article A, Publication date: January YYYY.

A:2 Ni and Henderson

able X(∞) say, for which X(t)⇒ X(∞) as t → ∞, where ⇒ denotes convergence indistribution. Furthermore, let f : {0, 1, 2, . . .} → R be a real-valued cost function andsuppose we wish to estimate the steady-state performance measure α = Ef(X(∞)).For example, if f(x) = x, then our goal is to estimate the mean steady-state number ofcustomers in the system.

A natural estimator of α is

α(t) =1

t

∫ t

0

f(X(s)) ds.

For a wide class of queueing systems and cost functions, it is known that as t→∞,√t(α(t)− α)⇒ σN(0, 1),

where N(0, 1) denotes a (standard) normal random variable with mean 0 and variance1, and σ2 is the asymptotic variance, which is also called the time-average variance.Accordingly, an asymptotic 100(1 − κ)% confidence interval for α is α(t) ± zκ/2σt−1/2,where zκ/2 is the 1−κ/2 quantile of a standard normal random variable. The confidenceinterval halfwidth is zκ/2σt−1/2, which is proportional to σ. Accordingly, the asymptoticvariance σ2, or the standard deviation σ, is an indicator of the absolute accuracy of theestimator α(t). Similarly, relative error, which is perhaps preferable to absolute error,is indicated through the ratio σ2/α2 or instead σ/α.

Whitt [1989] and Asmussen [1992] explore the magnitude of σ for a range of queue-ing systems and performance measures. The most important of their results showsthat for certain queueing systems with a fixed number of servers in which the serversare utilized for a large fraction ρ < 1 of the time, when estimating the mean steady-state number of customers in the system, σ is typically of order (1− ρ)−2 while α is oforder (1−ρ)−1. Accordingly, when ρ is close to 1, σ/α is of order (1−ρ)−1, and hence therunlengths required to obtain estimators of α with small relative error are very large.

This observation has been exploited within the simulation community in “stress test-ing” of output-analysis algorithms. Indeed, the heavily loaded M/M/1 queue is a stan-dard test problem for batching algorithms; see, e.g., Steiger et al. [2005].

It is now well understood that heavily loaded queueing systems as described aboverequire large simulation runlengths to obtain accurate estimators of steady-state per-formance measures, at least for steady-state moments of queue size and waiting time.But what of other performance measures, such as the steady-state probability of de-lay, i.e., that a customer will have to wait for service? Perhaps more importantly, suchheavily loaded queues do not necessarily reflect “real” queueing systems. In reality,customers will not queue forever; a common feature in queueing systems is customerabandonment, where customers leave without receiving service if they have to waittoo long. Furthermore, it is usually the case that the number of servers in a queueingsystem is chosen to ensure good customer service in the sense of short waiting times.

We call queueing systems in which customers may abandon, and/or where the num-ber of servers is chosen to deliver short waiting times “well dimensioned” queueingsystems. (The notion of “dimensioning” queueing systems is not ours, although our useof the term “well dimensioned” is specific to this paper; see Borst et al. 2004.) The keyquestion considered herein is how hard it is to accurately estimate various steady-stateperformance measures associated with well-dimensioned queueing systems.

To answer this question we compute the asymptotic variance in a range of Marko-vian queueing models, including the M/M/∞, M/M/c, and M/M/c + M models, andalso several diffusion models. We confine our attention to these tractable models, eventhough performance measures can be computed directly so that simulation is unneces-sary, specifically because they are tractable. This allows us to perform the needed cal-


How Hard are Steady-State Queueing Simulations? A:3

culations. We believe that similar conclusions will hold for many less-tractable queue-ing systems, partly because many of our results are obtained for diffusion models thatare known to accurately approximate more general queues.

Assuming a Poisson arrival process is often appropriate, as justified by the Palm-Khintchine theorem; see Karlin and Taylor [1975, p. 221], Cinlar [1972], and Nelson[2013, p. 107]. Exponential service times are sometimes reasonable, but usually someother distribution is more appropriate. Finally, assuming exponential customer pa-tience times, (the “+M” in the M/M/c+M queue) is not ideal, although the results ofZeltyn and Mandelbaum [2005] suggest that in many queueing systems the value ofthe density of the patience time distribution at 0 is the key quantity, in which case thefull distribution is unimportant and assuming an exponential distribution with rateequal to this density value is an accurate approximation. (See the excellent surveysDai and He [2011; 2012] for much more on queueing systems with abandonment.) Inany case, our goal is to obtain the right “order of magnitude” of the asymptotic vari-ance, so as long as our results are interpreted as applying to queueing systems thatare robust in this sense, we believe that confining attention to Markovian systemsis reasonable. So, for example, one should not attempt to extend our conclusions toqueueing systems with heavy-tailed interarrival and/or service time distributions, norto systems in which the sequences of these quantities exhibit long-range dependence.(See Whitt 2002 for much more on such queues.)

In addition to considering the asymptotic variance of estimators, we also considertheir asymptotic bias. It turns out that either variance or bias can be more problem-atic in terms of delivering narrow confidence intervals that have the desired coverage,depending on the performance measure and queueing system. In fact, bias is often themore important property, at least in certain asymptotic regimes, as previously noted inSrikant and Whitt [1996] for a variety of estimators of loss probabilities in loss models.

Our overall approach and philosophy is mostly adopted from Whitt [2006], who ana-lyzed Markovian single-server queues and infinite server queues in some detail, alongwith some results for multiserver queues. Indeed, on p. 411, Whitt stated that “Otherimportant classes of stochastic models should be analyzed in the same way.” We actu-ally work with the same stochastic processes that Whitt did, except that we emphasizethe phenomenon of customer abandonment, we work with a greater variety of perfor-mance measures, we consider what happens in queues when the number of servers ischosen so as to ensure that large backups do not arise, and we use slightly differenttechnical tools, especially for diffusion models. The paper Srikant and Whitt [1996]mentioned above is also relevant. In that paper, asymptotic approximations are de-rived for the asymptotic variance and bias for four loss-probability estimators in losssystems. Similar calculations to those we employ for diffusion models are used in Wangand Glynn [2014], where the properties of a certain bias reduction scheme are studied.

Our primary contribution is to reinforce the meta result that for well-dimensionedqueueing systems, estimating steady-state performance measures using simulation isnot hard.

This meta-result, supported by analysis in Srikant and Whitt [1996] and Whitt[2006] and reinforced here, shows that not only does abandonment or appropriate siz-ing of server pools relieve congestion (as is well understood), but the benefits extendto simulation models in the sense that the runlengths required to obtain high-qualityconfidence intervals for a number of steady-state performance measures are modest.

The remainder of this paper is structured as follows. In Section 2 we explain themathematical tools used to obtain the asymptotic variance and bias, and the inter-pretation of those quantities in choosing runlengths that deliver high-quality confi-dence intervals. Then, in Section 3 we review the so-called “efficiency-driven” regime,which is the source of the common view that simulating heavily loaded queues is hard.



We then present some results in Section 4 that show that the presence of customerabandonment changes the situation dramatically. In Section 5 we turn to the so-called“quality and efficiency driven” regime associated with queueing systems with manyheavily-loaded servers, but where customer wait times are also modest. Finally, inSection 6 we discuss and compare our results.

2. PRELIMINARIESThe primary queueing models we consider in this paper are the M/M/c model witharrival rate λ, service rate µ and c servers with λ < cµ, and the M/M/c + M modelwhere a patience time is associated with each customer, and each customer is willingto wait in queue only up to its patience time, at which point it abandons, i.e., leaveswithout receiving service. In these systems, the sequences of customer interarrivaltimes, service times and patience times are mutually independent iid sequences ofexponential random variables. If X(t) gives the number of customers in the system(for either queueing model) at time t ≥ 0, then X := {X(t) : t ≥ 0} is an irreducible,positive-recurrent continuous-time Markov chain on the state space S = {0, 1, 2, . . .}.

Let π be the unique stationary distribution, and with an abuse of notation, let π(k) =πk = π({k}), k ≥ 0. Let f : S 7→ R+ be a cost function and let α :=

∑∞k=0 f(k)π(k) be the

desired performance measure, namely the expected steady-state cost. We approximateα by

α(t) := t−1∫ t

0

f(X(s)) ds, (1)

the average cost over [0, t]. The regenerative strong law of large numbers ensures thatα(t) → α as t → ∞ almost surely. See, e.g., Resnick [1992, p. 123, p. 396], and Craneand Iglehart [1974a; 1974b; 1975] for an introduction to the regenerative method forsteady-state simulation output analysis.

Let A be the rate matrix for X, and define the function V : S → [1,∞) by V (k) = aebk

for k = 0, 1, 2, . . ., where we leave a, b > 0 unspecified. It is straightforward to show, foreach of our queueing models, that there exist strictly positive constants a, b, β, δ suchthat

AV (k) ≤ −βV (k) + δ, (2)for all k ∈ S, which is known as a Lyapunov drift criterion. (In this expression we takeV to be a column vector with kth component V (k), k ≥ 0, so that AV is a matrix-vectorproduct.) This condition implies that the chainX is “V-uniformly ergodic,” which allowsus to make a number of conclusions below; see Meyn and Tweedie [1993b, Theorem 7.1]and also Down et al. [1995]. It turns out that one can also apply this same theory tothe diffusion models we consider in this paper to ensure that the same results apply tothose models, with only modest modifications, e.g., the rate matrix A in (2) is replacedby the so-called generator of the diffusion process.

The Lyapunov drift criterion (2) implies [Glynn and Meyn 1996, Theorem 4.3] thecentral limit theorem (CLT)

√t(α(t)− α)⇒ N(0, σ2), (3)

as t→∞, where⇒ denotes convergence in distribution, provided that for some γ > 0,f2(k) ≤ γV (k) for all k. (An expression for the asymptotic variance constant σ2 is givenbelow.) The functions f we consider grow at most linearly, so this condition is assuredand the CLT indeed holds. The CLT establishes that an asymptotic confidence intervalfor α is given by

α(t)± zσ√t, (4)



where z is an appropriate quantile of the standard normal distribution. (In practice wemust replace σ with an estimator thereof, but that is not important for our present pur-pose.) If we want the half-width of this confidence interval to be smaller than some ab-solute error tolerance ε > 0, then we require that the simulation runlength t ≥ z2σ2/ε2.Hence, the (asymptotic) variance constant σ2 provides information on the accuracy ofthe estimator α(t) in terms of the amount of simulated time that is required to obtain anarrow confidence interval. This remains true if we want to assure that the half-widthof the confidence interval, relative to the true performance measure, is smaller thansome relative error tolerance ε > 0. In that case we require

t ≥ z2σ2/(ε2α2). (5)

(While relative error is typically the more relevant quantity, there is no additionalwork to obtaining absolute error as well, so we discuss both measures.)

The estimator α(t) is almost always biased, owing to the fact that X(0) cannot usu-ally be generated from the stationary distribution π. This bias can deteriorate thecoverage probability of the confidence interval (4). Suppose that we initiate the chainX in some fixed state x, and let Ex and Px denote the corresponding expectation andprobability. As in, e.g., Proposition 2.1 of Awad and Glynn [2007], the bias is then

α(t)− α =1

tEx

∫ t

0

[f(X(s))− α] ds

=1

t

∫ t

0

[Exf(X(s))− α] ds (6)

=1

t

∫ ∞0

[Exf(X(s))− α] ds− 1

t

∫ ∞t

[Exf(X(s))− α] ds

=g(x)

t− o(t−1), (7)

where the bias constant

g(x) =

∫ ∞0

[Exf(X(s))− α] ds,

provided that the interchange (6) is valid, and that∫∞0|Exf(X(s))− α|ds <∞. These

conditions are satisfied for our examples as assured by (2); see Down et al. [1995].If we consider the (standard) regime where the desired confidence interval halfwidth

ε→ 0, then the required runlength according to the relative error criterion (5) is of theorder ε−2. For such a runlength, the asymptotic bias is, according to (7), of the order ε2,which is asymptotically negligible compared to the confidence interval halfwidth.

We instead consider a different asymptotic regime, where ε is held fixed and the lim-iting behavior of σ and g(x) are considered as a function of some other quantity, suchas the arrival rate of customers and/or the number of servers, in order to understandhow desired runlengths scale with these quantities.

We adopt the philosophy that we want to ensure that confidence intervals are of adesired width and the coverage of the confidence interval is not unduly affected bybias. From this perspective, it is important that bias is small relative to the confidenceinterval width. The confidence interval width is of the order σt−1/2 while the bias isof the order g(x)/t. In order to achieve a narrow confidence interval, we must chooset so that t1/2 is large relative to σ, i.e., t is large relative to σ2. Likewise, to ensurethat the bias g(x)/t is small, we must take t large relative to g(x). Relative to thesimulation runlength t then, the appropriate comparison is between variance σ2 andbias g(x). This may seem strange if one is used to measuring the quality of an estimator



through its mean-squared error, where variance and squared bias are often balanced.The difference arises from our goal of having the bias be negligible relative to theconfidence interval width.

If we instead consider relative error, then the relative confidence interval width (rel-ative to the performance measure α) is proportional to (σ/α)t−1/2 and the relative biasis (g(x)/α)/t, so in terms of desired runlengths we then compare σ2/α2 with g(x)/α.

In the limiting regimes we consider, the confidence interval width criterion can dom-inate the bias criterion or vice versa, and there are also situations where neither cri-teria dominates. When the bias dominates the variance, or is of the same order asthe variance, then confidence interval coverage will be affected, and one might turnto bias mitigation schemes such as initial transient deletion or careful choice of theinitial conditions.

But how can we compute the variance σ2 and bias constant g(x) for a particularmodel and choice of parameters?

It is known (see Meyn and Tweedie 1993a, Section 17.4 for the result for discrete-time chains, and Steckley and Henderson 2006, Section 6 for a direct proof for thecontinuous-time chains corresponding to our queueing models) that

−Ag = f := f − α (8)

where A is the rate matrix of the chain and f is the “centered” cost function in thesense that π>f = π>f − α = 0. (Here > denotes the usual matrix transpose.) In fact, gis the π-integrable solution of these equations that satisfies π>g = 0.

We can therefore compute g, and hence the bias constant g(x) for any initial condi-tion X(0) = x, by identifying the π-integrable solution to Poisson’s equation (8) thatsatisfies π>g = 0. It turns out that this also allows us to compute σ, because [Glynnand Meyn 1996, Theorem 4.3]

σ2 = 2

∞∑k=0

f(k)g(k)π(k). (9)

Thus, in the sections to come, we will compute the stationary distribution π of theappropriate Markov process, use this to compute α = π>f and hence f = f−α where frepresents the performance measure in question, solve (8) for the π-integrable solutiong satisfying π>g = 0, and hence obtain the bias constant g(x) for any fixed initial condi-tion (X(0) = x), and compute the variance constant using (9). The magnitude of thesequantities then tells us how “hard” it is to estimate certain steady-state performancemeasures of Markovian queues using simulation. Whitt [2006] uses a very similarapproach for continuous time Markov chains, with the key differences being that weemphasize the phenomenon of customer abandonment, we work with a greater vari-ety of performance measures, we consider what happens in queues when the numberof servers is chosen so as to ensure that large backups do not arise, and we use aslightly different version of Poisson’s equation. For birth-death processes the method-ology above is the same as that of Whitt [1992], except that we use what Whitt callsthe “alternate form of Poisson’s equation.” For diffusions we work with the infinitesi-mal generator of the process, as employed in Glynn and Meyn [1996].

A similar agenda could be followed to analyse estimators other than those consideredhere, provided that they can be represented as a time-average for a suitably definedcost function f(·) as in (1).

3. THE EFFICIENCY-DRIVEN REGIMEConsider as performance measure the steady-state expected number of customers insystem (the expected occupancy), so we take f(k) = k. In this section we analyze this



performance measure within what is known as the efficiency-driven regime, first look-ing at the M/M/c special case and then at general GI/GI/c queues. The results inthis section are known, but our derivations are included in an appendix because themethod of derivation is instructive of our general approach and is new in some casesthat we clarify there.

One might be tempted to apply a similar analysis to the steady-state delay proba-bility, i.e., the steady-state probability that a customer will have to wait. In doing so,one might exploit the “Poisson arrivals see time averages” property, e.g., Wolff [1989,Section 5.16], taking f(k) = I(k ≥ c), i.e., f(k) equals 1 if k ≥ c and 0 otherwise.Indeed, we were so tempted, but as pointed out by a referee, in the efficiency-drivenregime, this delay probability converges to 1, so there is (asymptotically) no value inusing simulation if the error precision ε remains fixed. Moreover, the neglected term inthe bias approximation (7) can in fact be non-negligible in the regime we consider, sowe do not attempt to analyze this performance measure in this section. More refinedtools are needed.

3.1. The M/M/c QueueSuppose we initiate a simulation of the M/M/c queue with X(0) = 0, and consider theefficiency-driven regime where we keep c and µ fixed while λ → cµ from below, i.e.,ρ→ 1 from below.

From (7) the bias in the estimator t−1∫ t0X(s) ds is asymptotically g(0)/t, which cal-

culations in the appendix show is asymptotically −c−1(1− ρ)−3t−1 (taking µ = 1). Theasymptotic variance is of the order 4c−1(1− ρ)−4/t as ρ→ 1. (These values agree withthe M/M/1 special case in Whitt [2006].) Recall from the discussion in Section 2 thatto obtain a desired absolute error (confidence interval halfwidth) of ±ε, the requiredsimulation runlength t is z2σ2/ε2. For a 95% confidence interval, z ≈ 2, so if µ = 1,then the desired runlength is 4σ2ε−2 ∼ 16c−1(1 − ρ)−4ε−2 as ρ → 1. To ensure theasymptotic bias, g(0)/t, is smaller than ε, we require a runlength that is of the orderc−1(1 − ρ)−3ε−1. Consequently, as ρ → 1, the variance is the dominant criterion. Con-sidering relative error rather than absolute error, the simulation runlength needed isasymptotically t = z2σ2/(ε2α2) which is of the order 16c−1(1− ρ)−2ε−2. Also, to ensurethat the bias relative to α, g(0)/(tα), is smaller than ε requires a runlength of orderc−1(1−ρ)−2ε−1, which is of the same order (in terms of ρ) as that required from the per-spective of the confidence interval width. Nevertheless the constant multipliers ensurethat variance is the primary driver of runlengths. These conclusions reinforce similarconclusions given in Whitt [2006] for the M/M/1 queue.

One way to potentially reduce bias is to choose the initial state to be “repre-sentative of steady-state conditions,” which one might interpret as meaning takingX(0) = (1 − ρ)−1, the approximate steady-state mean. In the appendix we computethe exact solution to Poisson’s equation and then obtain its order as ρ → 1. This en-ables us to conclude that, when estimating the mean occupancy, the bias constant isg((1−ρ)−1) ∼ −(2cµ)−1(1−ρ)−3, which is of the same order as g(0) so, at least in order,bias is not reduced.

3.2. The GI/GI/c QueueThe results above shed light on what happens in heavily loaded Markovian queues.The assumption that the arrival process is Poisson is often easily justified, owing tothe Palm-Khintchine theorem; see, e.g., Karlin and Taylor [1975, p. 221], Cinlar [1972],and Nelson [2013, p. 107]. However, service times are often not well modeled as expo-nential random variables, with, e.g., the lognormal distribution often fitting empiricaldata. We now review the GI/GI/c queue where the sequences of interarrival and ser-



vice times are independent and each consists of i.i.d. random variables. Such queuesdefy exact analysis in general. We rely on a reflected Brownian motion approximationfor the queue-size process due to Iglehart and Whitt [1970a; 1970b]. See Whitt [2002,Theorems 10.2.1 and 10.2.3] for a recent review. We develop similar results to those ofWhitt [1989] and Whitt [2006] using the tools sketched in Section 2. The derivationsare given in the appendix.

Let Xρ = (Xρ(s) : s ≥ 0) be the stochastic process giving the number of customersin the system over time as a function of ρ, the utilization of the servers. Iglehart andWhitt [1970a; 1970b] established thatXρ can be approximated by a reflected Brownianmotion (RBM) on [0,∞) with drift −η and variance δ2, where η = cµ(1 − ρ) and δ2 =cµ((cµσU )2 + (µσV )2). (The exact sense in which this approximation is appropriate isdescribed in the appendix.) We take this approximation as exact in the sense that wecompute results (bias and variance constants) for the approximating RBM rather thanthe original intractable queueing model, and use those to develop our conclusions.

Consider the steady-state mean occupancy. The bias constant when the simulationis initiated at 0 is of the order −(1 − ρ)−3 as seen in our M/M/c results. The varianceσ2 is of order (1− ρ)−4. Thus, exactly as with the M/M/c results, from the perspectiveof absolute error the variance dominates, while from the perspective of relative error,both variance and bias are of the same order, so that bias mitigation schemes shouldbe considered.

Accordingly, we come to the same conclusions for general GI/GI/c queues that wedid for the M/M/c queue in that the bias becomes important to consider as ρ→ 1. Wemight try to mitigate bias by initializing the simulation in the (deterministic) statecorresponding to the steady-state mean of the approximating RBM. In doing so, theinitial bias when estimating the mean occupancy remains of order (1− ρ)−3. Unfortu-nately, our tools are too crude to quantify the benefits from initiating a simulation ofthe queue from the steady-state distribution of the diffusion (or an analog thereof inthe original queueing model), since we are confining our analysis to diffusion modelsand for the diffusion the initial bias is then exactly 0.

3.3. The M/M/c+M QueueZeltyn and Mandelbaum [2005] defined an ED regime for queues with abandonmentin an asymptotic setting where the number of servers and the arrival rate both in-crease, while the patience time and service time parameters remain constant. Theyassumed that c = c(λ) = (1 − γ)λ/µ where γ ∈ (0, 1) is fixed. Thus the queue has in-sufficient servers to meet demand. As a result, some fraction of customers must aban-don to ensure stability, and this fraction approaches γ as λ → ∞. We do not analyzethis queueing system in this paper, because we believe that the quality and efficiencydriven regime that we analyze later is almost always more relevant in practice; seeDai and He [2011; 2012] for more discussion about this regime.

4. THE IMPACT OF ABANDONMENTIn the models we considered in the previous section, customers are willing to wait in-definitely, and this leads to very large queue sizes and persistent periods of congestionwith the associated very large asymptotic variance constants. However, in almost alltrue queueing systems, customers will not wait indefinitely, and this can lead to dra-matic differences in performance. Consider the M/M/c + M (or Erlang-A) queue inwhich customers are only willing to wait for an exponentially distributed patience timewith mean θ−1 ∈ (0,∞). Patience times of successive customers are iid and indepen-dent of the sequences of interarrival and service times.



−6 −5 −4 −3 −2 −1 0 1

14166425610240

2

4

log10

(θ)

ρ = 0.95

c

log 10

(σ2 )

−6 −5 −4 −3 −2 −1 0 1

14166425610240

5

10

log10

(θ)

ρ = 1

c

log 10

(σ2 )

−3 −2 −1 0 1

141664256102402468

log10

(θ)

ρ = 1.02

c

log 10

(σ2 )

Fig. 1. Asymptotic variance σ2 for the average number of jobs in the system under µ = 1

4.1. The M/M/∞ QueueSuppose that θ = µ so that the mean patience time and mean service times are thesame. In this case, the queue-size stochastic process X = (X(t) : t ≥ 0) coincides withthat of the M/M/∞ queue. Even if θ 6= µ, the stochastic process X is stochasticallydominated by the queue size in an infinite-server queue with service rate min{µ, θ}.Therefore, the M/M/∞ queue is an interesting first model to consider.

Let ρ = λ/µ. (We use this notation even though ρ no longer represents the serverutilization, which is 0.) Whitt [2006] showed that when estimating the mean steady-state number of customers in the system, the bias is −ρ/µ and the asymptotic varianceconstant is 2ρ/µ. We conclude that in terms of absolute error, the asymptotic varianceand bias are both of the same order in the regime where λ→∞ with µ held constant.Consequently, to ensure satisfactory confidence interval coverage, bias reduction mustbe explicitly considered. Interestingly, Whitt [2006] shows that when one considersrelative error in this same regime, then the bias becomes the dominant criterion. Thishappens because the runlength required to achieve a given confidence interval widthrelative to the true performance measure ρ is proportional to 1/ρ, while the bias rela-tive to ρ remains constant.

4.2. The M/M/c+M QueueIn general, when 0 < θ 6= µ, the solution to Poisson’s equation can be computed but iscomplicated, and we turn to numerical experimentation to illustrate the effect of aban-donment. We report computational results for the asymptotic variance σ2 and asymp-totic bias under different levels of λ, c and θ, with µ = 1 held fixed, for the expectedsteady-state number of customers in the system. Additional numerical results for theperformance measures steady-state probability of delay and steady-state probabilityof abandonment are reported in Section 5.

Figure 1 shows that σ2 decreases significantly in the presence of abandonment rel-ative to the no-abandonment, ED-regime case. Inspecting the plots, we see that for0 � θ < µ, we have, approximately, that σ2 ∝ θ−2 which is similar to the M/M/∞case where σ2 ∝ µ−2, except that the abandonment rate θ replaces the service rateµ. Recalling that σ2 ∝ (1 − ρ)−4 in the ED regime for the M/M/c queue, this resultsuggests that the reduction in asymptotic variance is of order θ2(1 − ρ)−4, even whenθ � µ, i.e., the abandonment rate is a small fraction of the service rate. Furthermore,the “plateau” we see in the plot of variance when ρ = 0.95 suggests that when ρ < 1, thevariance constant σ2 is upper bounded by the M/M/c variance as θ → 0. Also, whenρ ≥ 1, the queue without abandonment would be overloaded, but with abandonmentthe results are very much like those for an M/M/∞ queue with service rate θ.

Recall that in the M/M/∞ queue, the bias constant differs from σ2 by a multiplica-tive constant -2. We observe a similar scaling relationship in the M/M/c + M case inthe plots of Figure 2, which are approximately proportional to the plots in Figure 1.



−6 −5 −4 −3 −2 −1 0 1

1416642561024

0123

log10

(θ)

ρ = 0.95

c

log 10

(|bi

as|)

−6 −5 −4 −3 −2 −1 0 1

14166425610240

5

log10

(θ)

ρ = 1

c

log 10

(|bi

as|)

−3 −2 −1 0 1

14166425610240

2

4

6

log10

(θ)

ρ = 1.02

c

log 10

(|bi

as|)

Fig. 2. Absolute asymptotic bias for the average number of jobs in the system under µ = 1

5. THE QUALITY AND EFFICIENCY DRIVEN REGIMEIn this section, we consider Markovian queues operating in the Halfin-Whitt regimenamed in honor of Halfin and Whitt [1981], which is now also known as the “qualityand efficiency driven” regime, a name coined by Avi Mandelbaum, because not onlyare customers served quickly, but the servers are also heavily utilized. This regime ismost relevant for systems with moderate to large numbers of servers, so we will beinterested in asymptotics where both the arrival rate λ and the number of servers cincrease, with the service rate µ held fixed. More precisely, we require that for somefinite constant β,

(1− ρ)√c→ β

as c → ∞, where ρ = λ/(cµ). Hence, for a given value of c, the arrival rate is λ = cµ −βµ√c. When there is no abandonment (θ = 0) we must have β > 0 so that the system

is stable, but this restriction is not necessary when the abandonment rate θ > 0, sinceabandonment stabilizes the system.

We continue to think of “hardness” in terms of the simulation runlength t neededto obtain high-quality confidence intervals, although this is imperfect in the followingsense. The computational effort required to simulate to simulated-time t is propor-tional to the number of random variates generated over the interval [0, t], which isproportional to λt. In the asymptotic regime considered here both c and λ increasewithout bound. So the computational effort required to simulate to time t is betterrepresented by λt, than by t alone. In previous sections where λ was bounded, thesequantities are equivalent in order, but now that λ → ∞, they are not. Nevertheless,we continue to estimate and report the asymptotic variance and bias constants, whichimply a desirable t, and which can in turn be scaled by λ (or cµ, since cµ and λ are ofthe same order in the asymptotic regime we consider) to estimate the computationaleffort required.

Exact calculations for the continuous-time Markov chain models can be performed,but it appears to be difficult to extract insight from the results. Accordingly, we employa combination of analytical results from diffusion models and numerical results forcontinuous-time Markov chain models.

5.1. The M/M/c QueueConsider a sequence of M/M/c queueing systems, indexed by c = 1, 2, . . . All systemshave a fixed service rate µ and are assumed to start out empty. The arrival rate in thecth system is chosen to ensure that

√c(1 − ρ) is constant and equal to β > 0, where

ρ = λ/(cµ). Let Xc = (Xc(t) : t ≥ 0) be the stochastic process giving the number ofcustomers in the system over time in the cth system. Halfin and Whitt [1981] provedthat

Xc(·)− c√c

⇒ Y (·) (10)



as c→∞, where Y is a diffusion on (−∞,∞) with drift function

µ(y) =

{−βµ y > 0

−µ(β + y) y ≤ 0,

and infinitesimal variance 2µ, with Y (0) = 0. (In fact, Halfin and Whitt 1981 proved aversion of this result for a sequence of GI/M/c queues, but we restrict attention to aPoisson arrival process.)

The convergence result (10) suggests the process approximationXc(·) ≈ c+

√cY (·). (11)

We take this approximation as an equality, which then allows us to obtain a number ofinsights that agree with our numerical results for exact calculation for M/M/c models.In other words, we redefine Xc to be the right-hand side of (11), which is a diffusion,and compute the order of magnitude of the variance and bias for our performancemeasures for these diffusions that are indexed by c. The scaling relationship makesthis calculation quite tractable, but it is certainly not trivial, because the asymptoticbias depends on growth rates in the solution to Poisson’s equation for the process Y (·),which we therefore need to obtain.

To begin, consider the cost function f(x) = x, so that we wish to estimate the ex-pected steady-state number of customers in the system, with estimator t−1

∫ t0Xc(s) ds.

We can compute the asymptotic variance and bias of this estimator as in Section 3.2,but to emphasize the role of the scaling we relate these quantities to similar ones as-sociated with the process Y . Let gc be the desired solution to Poisson’s equation for Xc

and f(x) = x, and let gY be the solution to Poisson’s equation for Y and f(y) = y. Letαc = Ef(Xc(∞)) be the expected steady-state cost for Xc, and define αY similarly, sothat αc = c+

√cαY . The functions gc and gY are related, since

gc(x) =

∫ ∞0

E[Xc(t)− αX |Xc(0) = x] dt

=

∫ ∞0

E[c+√cY (t)− (c+

√cαY )|(Xc(0)− c)/

√c = (x− c)/

√c] dt

=√c

∫ ∞0

E[Y (t)− αY |Y (0) = (x− c)/√c] dt

=√cgY ((x− c)/

√c).

Thus, the asymptotic bias constant gc(0) =√cgY (−

√c), and so we will need to compute

gY . Before doing so, consider the calculation of the asymptotic variance. Let σ2c be the

asymptotic variance of Xc and σ2Y be the asymptotic variance of Y (for the cost function

f(x) = x). Let πc and πY be the stationary densities of Xc and Y respectively, and notethat πc(x) = c−1/2πY (c−1/2(x− c)). Hence,

σ2c = 2

∫ ∞−∞

(x− αX)gc(x)πc(x) dx

= 2

∫ ∞−∞

√c

(x− c√

c− αY

)√cgY (c−1/2(x− c)) 1√

cπY (c−1/2(x− c)) dx

= c2

∫ ∞−∞

(y − αY )gY (y)πY (y) dx

= cσ2Y ,

so that σ2c grows linearly in c, with multiplicative constant σ2

Y .



So now we return to obtaining the asymptotic bias√cgY (−

√c), for which we need to

compute gY , the πY -integrable solution to Poisson’s equation, with πY integral 0, thatsatisfies the differential equation

µg′′Y (y) + µ(y)g′Y (y) = −y + αY .

The solution for y ≤ 0 is

gY (y) = A1 +y

µ− β + αY

µ

∫ 0

y

Φ(y + β)

φ(y + β)dy,

where the constant A1 does not depend on c and is not important for our purposes.Now we use the fact that

limy→−∞

yΦ(y + β)

φ(y + β)= 1

so that

gY (−√c) ∼ A1 −

√c

µ− β + αY

2µln c ∼ −

√c

µ(12)

for large c. We conclude that the asymptotic bias is of order −c/µ as c→∞.A cautionary note is necessary at this point. The diffusion approximation (11) is most

appropriate for measuring fluctuations in the process of order√c around the “central”

value c. In considering the bias starting from initial state 0, we are considering a largerfluctuation that is of order c �

√c, so we are extrapolating past the usual range over

which we can expect the diffusion approximation to accurately match the dynamics ofthe continuous-time Markov chain it approximates. If we instead take as initial statec − a

√c for some a ≥ 0, then the diffusion approximation gives the asymptotic bias

constant as√cgY (−a), which is of order

√c. So we might expect that the bias starting

from initial state 0 is at least of order√c, and furthermore that bias is reduced to order√

c by choosing the initial state as c rather than 0. Our numerical experiments belowsupport the view that the bias starting from 0 is of order c. Furthermore, as pointedout by a referee, the fluid model also suggests that the asymptotic bias starting fromthat state is of order −c/µ; see Section 6.

Hence, the bias and variance are both of the same order, being asymptotically linearin c, and the bias can be reduced to order

√c by starting from initial state c.

Next, consider the cost function f(x) = I(x ≥ c), so that we wish to estimatethe steady-state probability that an arriving customer must wait, with estimatort−1

∫ t0I(Xc(s) ≥ c) ds. Let us redefine gc to be the desired solution to Poisson’s equation

for Xc and f(x) = I(x ≥ c), and let gY be the solution to Poisson’s equation for Y andf(y) = I(y ≥ 0). Redefine αc = P (Xc(∞) ≥ c) and αY = P (Y (∞) ≥ 0) similarly, so thatαc = αY .

Using the same arguments used for the cost function f(x) = x, we find that gc(x) =gY (c−1/2(x− c)), σ2

c = σ2Y is constant, and

gY (y) = A2 −αYµ

∫ 0

y

Φ(y + β)

φ(y + β)dy, (13)

for y < 0. Hence the asymptotic bias constant when initiating in State 0 is

gc(0) ∼ −αY2µ

ln c,

while the bias is reduced to constant order if we initiate in State c.



(The same cautionary note above about the range of applicability of the diffusionapproximation also applies here.)

We conclude that when estimating the steady-state probability of delay, the bias isof order ln c, while the variance is constant in c, suggesting that at least for large c,the bias is the dominant criterion. However, given that ln c grows extremely slowly, itis likely that both quantities are important to consider, and this remains true even ifwe reduce bias by initiating in State c.

5.2. The M/M/c+M QueueNow consider the case where θ > 0, so that customers abandon if their waiting timesare too long. Again consider a sequence of M/M/c + M queueing systems, indexed byc = 1, 2, . . . All systems have a fixed service rate µ and are assumed to start out empty.The arrival rate in the cth system is chosen to ensure that

√c(1 − ρ) is constant and

equal to β, where ρ = λ/(cµ). Hence, we use exactly the same asymptotic regime as inthe previous section where customers did not abandon, except that we explicitly allowβ ≤ 0, since abandonment ensures that the systems are stable. Let Xc = (Xc(t) : t ≥ 0)be the stochastic process giving the number of customers in the system over time inthe cth system. Garnett et al. [2002] proved that

Xc(·)− c√c

⇒ Y (·) (14)

as c→∞, where Y is a diffusion on (−∞,∞) with drift function

µ(y) =

{−(βµ+ θy) y > 0

−µ(β + y) y ≤ 0,

and infinitesimal variance 2µ, with Y (0) = 0. We see that abandonment modifies thedrift function for y > 0, but otherwise the diffusion is unchanged.

We again take the process approximation implied by (14) as exact, so that we rede-fine

Xc(·) = c+√cY (·). (15)

Consider the cost function f(x) = x, so that we wish to estimate the expected steady-state number of customers in the system, with estimator t−1

∫ t0Xc(s) ds. We can com-

pute the asymptotic variance and bias of this estimator exactly as in the M/M/c caseabove. Redefining all the quantities of interest for the case in point, we find thatαc = c +

√cαY , gc(x) =

√cgY ((x − c)/

√c), σ2

c = cσ2Y , gY (y) is given, for y ≤ 0 by

(12) although with a different additive constant, and gc(0) ∼ −c/µ as c → ∞. Hence,our conclusions for the M/M/c queue continue to hold in the case of abandonment,although with a different variance constant σ2

Y . This is perhaps to be expected, sincethe Halfin-Whitt regime corresponds to a situation where a nontrivial fraction of cus-tomers have to wait, but they have to wait for a vanishingly small amount of time as cincreases, and so abandonment has very little effect asymptotically.

The analysis for the cost function f(x) = I(x ≥ c) follows similar, albeit nontrivial,lines, and we omit the details. The asymptotic bias is of order ln c while the asymptoticvariance does not depend on c.

There is an additional cost function we should consider for this model. Some man-agers use the steady-state probability of abandonment as a performance measure fordesign, so it is worth understanding how this measure might be estimated, along withthe asymptotic bias and variance of the estimator. The discrete-time process consistingof the indicators of whether successive customers abandon or not is not very tractable.Fortunately, there is an alternative based on the system-size process [Garnett et al.



−1.5−1 −0.50 0.5 1 1.5 23264

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2

2.5

3

Number of jobs in the system

βc

log 10

(σ2 )

−1.5−1 −0.50 0.5 1 1.5 23264

128256

5121024

0

0.2

Probability of having to wait

βc

σ2

−1.5−1 −0.500.5 1 1.5 2

3264

128256

5121024

−4

−2

β

Probability of abandonment

c

log 10

(σ2 )

Fig. 3. Asymptotic variance σ2 for various performance measures for the M/M/c + M queue under theQED regime with µ = θ = 1

2002]. When there are x customers in the system, the abandonment rate is [x − c]+θ.On the other hand, the long run abandonment rate is λαX , where αX is the steady-state probability that an arriving customer will abandon. Thus

αX =θ

λE[Xc(∞)− c]+,

which can be estimated viaθ

λ

1

t

∫ t

0

[Xc(s)− c]+ dt. (16)

First consider the cost function f(x) = [x − c]+. Following our now familiar argu-ment, we redefine αX = E[Xc(∞) − c]+ =

√cE[Y (∞)]+ =

√cαY . Again, gc(x) =√

cgY (c−1/2(x − c)), and gY for this model is of the same form as (13) with differentconstants, so the asymptotic bias is of the order c1/2 ln c and the asymptotic variance iscσ2Y . The bias can be reduced to order c1/2 by initiating the simulation in State x = c.Of course, we are more interested in the cost function θ

λ [x − c]+, and since λ ∼ cµ

as c → ∞, the asymptotic bias is of the order c−1/2 ln c and the asymptotic variance isθ2µ−2σ2

Y /c. Hence, when estimating the probability of abandonment using (16), boththe bias and the variance decay as c grows, with the bias being asymptotically of largerorder.

5.3. Numerical ExamplesWe derived the results above assuming that the diffusion approximation was exact.We now confirm the predictions of that approximation by numerically computing theasymptotic variance σ2 and bias for the M/M/c+M queue under the QED regime. Wepresent the results in Figures 3 and 4. In these plots, we fix µ and θ, and for each valueof β and c considered we choose λ so that (1− ρ)

√c = β. We then choose the scaling of

the c axis and vertical axis according to the predictions made by the diffusion models,and find that both (scaled) σ2 and bias on the vertical axis appear to be linear withrespect to (scaled) c. This suggests that the diffusion model estimates the true ordersof the variance and bias accurately.

5.4. The M/M/c+GI QueueWe conclude this section with a brief comment about M/M/c + GI queues. In thesequeues, the patience times are still iid, but may not have an exponential distribution.Zeltyn and Mandelbaum [2005] proved that (14) still holds for such queues, with theproviso that the term θ in the drift function of the limiting diffusion Y is redefined to bethe value of the density of the patience time distribution at zero. To understand why,note that in the QED regime, customer wait times become very small, being of orderc−1/2 as c → ∞. Hence, while a nontrivial fraction of customers have to wait, theirwaiting times are almost all very small. Consequently, customers have very little time



−1.5−1 −0.50 0.5 1 1.5 23264

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2

3

Number of jobs in the system

βc

log 10

(|bi

as|)

−1.5−1 −0.50 0.5 1 1.5 23264

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5121024

1

2

3

Probability of having to wait

βc

|bia

s|

−1.5−1 −0.50 0.5 1 1.5 2

0.2

0.4

β

Probability of abandonment

c−1/2ln(c)

|bia

s|

Fig. 4. Absolute asymptotic bias for various performance measures for the M/M/c +M queue under theQED regime with µ = θ = 1

to abandon, and the patience time distribution is relevant only in terms of its behaviornear 0. Assuming the patience distribution has a positive continuous density at 0, ourconclusions about the order of the variance and bias for the performance measures weanalyzed for M/M/c+M queues remain valid for M/M/c+GI queues, assuming thatour approximation (15) does not introduce significant error.

6. DISCUSSION AND COMPARISONSTable I summarizes our results. The values given represent the highest-order termin the property (variance or bias accordingly) and do not include any multiplicativeconstants. For example, when estimating the steady-state probability of delay in theM/M/c queue operated in the QED regime, one can expect the asymptotic bias whenstarting the simulation in State 0 to be O(ln c), while the asymptotic bias is O(1) whenstarting the simulation in State c, which is more representative of steady-state condi-tions. These values are also proportional to the order of magnitude of the simulationrunlength t required to give a confidence interval of a fixed width in the case of vari-ance, or to obtain a fixed bias respectively. In interpreting these results, recall that inthe QED regime, the arrival rate is of the same order as c, so that the computationaleffort needed is of the order ct.

Table I. A summary of our results. Values represent the order of magnitude of thevariance or bias, ignoring multiplicative constants, for the stated steady-state perfor-mance measure and model in the stated regime. The three performance measures arethe mean number of customers in the system, the probability of delay and the prob-ability of abandonment. The columns labelled Bias0 and Biasα respectively give theorder of the bias constant when initiating simulations with an empty system or wheninitiating from an approximation to the steady-state mean α obtained from the diffusionapproximation.

Performance Regime Model Variance |Bias0| |Biasα|MeasureEX ED M/M/c (1− ρ)−4 (1− ρ)−3 (1− ρ)−3

QED M/M/c c c c1/2

QED M/M/c+M c c c1/2

P (X ≥ c) QED M/M/c 1 ln c 1QED M/M/c+M 1 ln c 1

P (Ab) QED M/M/c+M c−1 c−1/2 ln c c−1/2

The values in Table I are appropriate when errors are measured in absolute terms. Ifwe instead measure errors relative to the true values of the performance measure, thenas discussed in Section 2 we must divide the variance by the square of the performancemeasure, and the bias by the performance measure. Doing so yields Table II below.

The values in Table II are striking in the sense that the bias when initiating with anempty system is of larger order than the variance in all cases, except for the M/M/c



Table II. Values are interpreted as in Table I above, except that variance is relativeto the square of the performance measure, while bias is relative to the performancemeasure.

Performance Regime Model Variance |Bias0| |Biasα|MeasureEX ED M/M/c (1− ρ)−2 (1− ρ)−2 (1− ρ)−2

QED M/M/c c−1 1 c−1/2

QED M/M/c+M c−1 1 c−1/2

P (X ≥ c) QED M/M/c 1 ln c 1QED M/M/c+M 1 ln c 1

P (Ab) QED M/M/c+M 1 ln c 1

queue in the ED regime, where the two properties are equal in magnitude. This sug-gests that bias should receive careful consideration in simulations of heavily-loadedqueues, in agreement with results for loss models in Srikant and Whitt [1996], andresults for the M/M/∞ queue in Whitt [2006]. To mitigate this bias, Whitt [2006]suggested simulating starting from an initial state where all servers are busy, withresidual service times sampled from the equilibrium residual-life distribution, insteadof starting with an empty system. Our results suggest that this would substantiallyreduce bias in the QED regime, as seen in the final columns of the tables above. Forexample, in estimating the expected steady-state number of customers in the M/M/cqueue in the QED regime, the absolute bias would then be of the order

√c rather than

c, and in estimating the probability of delay the bias would be of order 1 rather thanln c. However, as seen in the ED results for EX, the order of the bias reduction maydepend on the performance measure; an order of magnitude in bias reduction is notguaranteed.

Even more substantial bias reduction might result if the initial state of the sim-ulation is randomly chosen with a distribution that is related to the stationary dis-tribution of the heavy-traffic approximation. While we expect bias to be reduced, ourmethods cannot shed light on the effect, because we analyze the bias reduction fromthe perspective of the heavy-traffic approximation itself. Thus our prediction of theresulting bias would be 0, and a deeper analysis is needed.

It is interesting that in Table II the asymptotic variances relative to the square ofthe mean in estimating EX in the QED regime are of order c−1, showing that thesimulation runlength needed to obtain confidence interval widths with given relativeerror shrinks as c→∞. It is worth keeping in mind that in the QED regime the arrivalrate is approximately proportional to c, so that the total number of customer arrivalssimulated is constant. This phenomenon was noted in Srikant and Whitt [1996] andin Whitt [2006] for related performance measures and systems. This is a striking ob-servation, especially when one compares it with the situation in the ED regime inthe absence of abandonment, where the number of customer arrivals that need to besimulated is of the order (1− ρ)−2, which grows extremely rapidly as ρ→ 1.

Although the relative bias is of equal or larger order than the relative variance in allcases, it is important to keep in mind the discussion from Section 2 that in the usualasymptotic setting where the desired confidence interval width ε → 0, the confidenceinterval width will eventually dominate the bias. The comments above apply to thesetting where ε is fixed and ρ→ 1 (in the case of ED) or c→∞ (in the case of QED).

A referee suggested that fluid models underlie and explain the large difference inbias results for the multi-server (c remaining bounded) and many-server (c → ∞)regimes that we obtained through tractable diffusion models. This suggests that ourresults, and others, might instead be obtained by studying the even-more tractablefluid models associated with these processes. For example, for the M/M/c queue in theQED regime of Section 5, the fluid model initiated in State 0 is x′(t) = λ − µmin(c, x)



with x(0) = 0. The solution when λ < cµ is

x(t) =λ

µ(1− e−µt) t ≥ 0.

The corresponding approximation for g(0) is

g(0) ≈∫ ∞0

(x(t)− λ

µ) dt = − λ

µ2.

This is asymptotically of order −c/µ since λ ∼ cµ in the QED regime, matching ourorder of the bias computed using the diffusion model.

A. APPENDIXHere we provide further details on the calculations in Section 3.

A.1. The M/M/c QueueThe stationary distribution π is given by

π0 =

[c−1∑k=0

(cρ)k

k!+

(cρ)c

c!

1

1− ρ

]−1, and (17)

πk =

{π0

(cρ)k

k! , 0 < k < c,

π0ρkcc

c! , k ≥ c.(18)

With this stationary distribution and the cost function f(i) = i, the long-run averagecost π>f is cρ+ Cρ/(1− ρ), where the constant C is the delay probability

C =

∞∑k=c

πk =(cρ)c/[(1− ρ)c!]∑c−1k=0

(cρ)k

k! + (cρ)c

(1−ρ)c!

. (19)

Poisson’s equation (8) can be solved directly to yieldg(0) = K1 −K2,

g(j) =

{K1 −K2 + 1

µ

∑ji=1

∑i−1k=0

α−k(cρ)i−k

(i−1)!k! , 0 < j < c,

aj2 + bj −K2, j ≥ c,

where

a =1

2cµ(1− ρ),

b = a

[1 + 2

(ρ

1− ρ(1− C)− cρ

)],

and the constant K1 is determined by

K1 = ac2 + bc− 1

µ

c∑i=1

i−1∑k=0

α− k(cρ)i−k

(i− 1)!

k!(20)

We then select K2 so that π>g = 0, which gives

K2 = (1− C)K1 +1

µ

c−1∑j=1

πj

j∑i=1

i−1∑k=0

α− k(cρ)i−k

(i− 1)!

k!+ π0

cc

c!

∞∑j=c

(aj2 + bj)ρj . (21)



The asymptotic bias when initiating with an empty system is g(0) = K1 −K2. Thuswe need to understand the asymptotics of this quantity as ρ → 1, while we hold c andµ fixed. First consider K1 as in (20), which, in turn, depends on C, a and b, all of whichdepend on ρ. First a is of order (1 − ρ)−1. Second, 1 − C can be seen to be of the order(1−ρ) as ρ→ 1, and thus b is asymptotically (1−ρ)−1(3−2c)/(2cµ). Thus, the first twoterms in (20) are of order (1− ρ)−1. As to the last term, as ρ→ 1,

α = α(ρ) ∼∞∑j=c

jπj = π0cc

c!

∞∑j=c

jρj .

One can verify that∑∞j=c j

rρj ∼ (1− ρ)−(r+1)r! for r = 1, 2, . . ., and that π0cc/c! ∼ 1− ρ.Thus, α ∼ (1 − ρ)−1 as ρ → 1. Hence α − k in (20) is of order (1 − ρ)−1, which, whentaken out as a common factor, leaves a quantity that is bounded in ρ as ρ → 1. Weconclude that K1 is at most of order (1− ρ)−1 as ρ→ 1.

Using similar reasoning, we see that in (21), the first two terms are asymptoticallyof order 1 and the final term is asymptotically (1 − ρ)−3/(cµ). Hence g(0)/t is asymp-totically −(1− ρ)−3/(cµt) as ρ→ 1, agreeing with the M/M/1 special case discussed inWhitt [2006].

Turning to the asymptotic variance, σ2, substituting the expressions for π and g into(9) gives

σ2 = 2K1

c−1∑j=0

πj(j − α) +2

µ

c−1∑j=0

πj(j − α)

j∑i=1

i−1∑k=0

α− k(cρ)i−k

(i− 1)!

k!

+ π0cc

c!

∞∑j=c

(j − α)(aj2 + bj)ρj .

Using the same asymptotic-order calculations, the dominant term in this expres-sion is the last one, and it is asymptotically 4(cµ)−1(1 − ρ)−4, again agreeing with theM/M/1 special case in Whitt [2006].

This particular form of derivation of the asymptotic constants where we directlycompute the solution to Poisson’s equation and then estimate the asymptotic order ofthe expressions is, to the best of our knowledge, new for the M/M/c queue, althoughthe order of the constants has been known for some time.

A.2. The GI/GI/c QueueConsider a family of queueing systems all of which have c servers serving jobs in first-in-first-out order, indexed by ρ ∈ (0, 1), constructed as follows. Let U = (Ui : i ≥ 1)denote an iid sequence of unscaled interarrival times, and let V = (Vi : i ≥ 1) denotean iid sequence of service times. We assume that EV1 = µ−1, EU1 = (cµ)−1, and thatboth U1 and V1 have finite variances σ2

U and σ2V respectively. In the ρth system, the

service time sequence is V , and the interarrival time sequence is ρ−1U , so that the ithinterarrival time is Ui/ρ. All systems are initially empty at time 0, so that the firstcustomer arrives at time U1/ρ. Let Xρ(s) be the number of customers in the system attime s in the ρth system and let Xρ = (Xρ(s) : s ≥ 0) be the corresponding stochasticprocess.

For constants a, b > 0, let aXρ(·/b) be the stochastic process taking the value aXρ(t/b)at time t. Iglehart and Whitt [1970a; 1970b] proved that

(1− ρ)Xρ

(·

(1− ρ)2

)⇒ R(·;−cµ, cµ((cµσU )2 + (µσV )2), 0), (22)



where ⇒ denotes convergence in distribution of stochastic processes as in Billingsley[1968], and R(·; r0, r1, r2) is a reflected Brownian motion (RBM) on [0,∞) with drift r0,variance r1, and initial state r2. Using scaling properties of RBM as in Whitt [2006],(22) suggests the process approximation

Xρ(·) ≈ R(·;−η, δ2, 0), (23)

where η = cµ(1− ρ) and δ2 = cµ((cµσU )2 + (µσV )2). The stationary distribution of thisRBM is known [Harrison 1990, p. 94] to be exponential with mean δ2/(2η).

Consider the steady-state mean occupancy. We approximate all quantities for thisperformance measure (bias, variance etc) by the corresponding values for the approxi-mating RBM (23). Accordingly,

α =δ2

2η,

which simplifies to (1 − ρ)−1 for M/M/c queues where δ2 = 2cµ, agreeing (in orderas ρ → 1) with the exact result. Poisson’s equation for the RBM takes the form of adifferential equation (see Mandl 1968, p. 39 and Karlin and Taylor 1981, p. 305), andis

δ2

2g′′(x)− ηg′(x) = −x+ α g′(0) = 0.

The solution we seek (with zero steady-state mean) is

g(x) =x2

2η− δ4

4η3.

Accordingly, the bias constant when the simulation is initiated at 0 is g(0) or −δ4/(4η3)which is of order −(1 − ρ)−3. In the special M/M/c case where δ2 = 2cµ, the biasconstant is −[cµ(1− ρ)3]−1. In either case, the order of the bias constant is (1− ρ)−3 asreflected in our earlier results.

The variance is

σ2 = 2

∫ ∞0

f(x)g(x)π(dx) = δ6/(2η4),

which is of order (1 − ρ)−4 in general as pointed out in Whitt [1989], and equal to4(cµ(1− ρ)4)−1 in the M/M/c case where δ2 = 2cµ.

If we initialize the RBM in the state α = δ2/(2η) instead of 0, then the initial biaswhen estimating the mean occupancy is −δ4/(8η3) which remains of order (1− ρ)−3.

ACKNOWLEDGMENTS

It is a privilege to contribute to this issue honoring Don Iglehart and his distinguished career. As a graduatestudent, I (Henderson) took classes, including an independent reading course, from Don. I could not haveasked for a better role model. Don is simultaneously a scholar of the highest quality, a superb mentor, andone of the kindest people you could hope to meet. I will strive to emulate his humble excellence to the bestof my ability.

We are grateful to the editorial team for highly insightful comments that greatly improved the paper.

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Received July 2013; revised July 2013; accepted July 2013


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