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  • Queueing Syst (2015) 79:391426DOI 10.1007/s11134-014-9429-3

    Stochastic grey-box modeling of queueing systems:fitting birth-and-death processes to data

    James Dong Ward Whitt

    Received: 8 January 2014 / Revised: 17 November 2014 / Published online: 2 December 2014 Springer Science+Business Media New York 2014

    Abstract This paper explores grey-box modeling of queueing systems. A stationarybirth-and-death (BD) process model is fitted to a segment of the sample path of thenumber in the system in the usual way. The birth (death) rates in each state are estimatedby the observed number of arrivals (departures) in that state divided by the total timespent in that state. Under minor regularity conditions, if the queue length (number in thesystem) has a proper limiting steady-state distribution, then the fitted BD process hasthat same steady-state distribution asymptotically as the sample size increases, evenif the actual queue-length process is not nearly a BD process. However, the transientbehavior may be very different. We investigate what we can learn about the actualqueueing system from the fitted BD process. Here we consider the standard G I/G I/squeueing model with s servers, unlimited waiting room and general independent,non-exponential, interarrival-time and service-time distributions. For heavily loadeds-server models, we find that the long-term transient behavior of the original process,as partially characterized by mean first passage times, can be approximated by adeterministic time transformation of the fitted BD process, exploiting the heavy-trafficcharacterization of the variability.

    J. DongSchool of Operations Research and Information Engineering, Cornell University,287 Rhodes Hall, Ithaca, NY 14850, USAe-mail:

    W. Whitt (B)Department of Industrial Engineering and Operations Research,Columbia University, New York, NY 10027, USAe-mail:


  • 392 Queueing Syst (2015) 79:391426

    Keywords Birth-and-death processes Grey-box modeling Fitting stochasticmodels to data Transient behavior First passage times Heavy traffic

    Mathematics Subject Classification 60F17 60J25 60K25 62M09 90B25

    1 Introduction

    Queueing theory primarily involves white-box modeling, in which queueing modelsare defined from first principles and analyzed. To apply these models to analyze theperformance of queueing systems, we check that the assumptions of the models aresatisfied, for example, by performing statistical tests on system data, as in [7,28], andestimating model parameters, as in [4,5,36] and references therein.

    An alternative to white-box modeling of queueing systems is black-box modeling,in which models (for example, statistical time-series methods) are employed, withoutusing any structure of queueing models; see [42,59]. Black-box modeling is growing inpopularity as the amount of available data grows, as well as our ability to extract usefulinformation from it; see [22]. A modeling approach in between these two extremes,which might conceivably share the advantages of both, is grey-box modeling, whichexploits some model structure together with learning from data; see [6,29].

    We examine grey-box modeling applied to queueing processes. In this paper, thegrey-box model is a general stationary birth-and-death (BD) process, which exploitsthe common property of many queue-length (number in system) processes that theyalmost surely make only unit transitions, corresponding to separate arrivals and depar-tures. Indeed, BD queueing models such as the Markovian M/M/s model are wellstudied and have been extensively used. Moreover, BD models have been found to beuseful approximations for other more general but less tractable models, as illustratedby the BD approximation in [52] for the M/G I/s +G I model having customer aban-donment from the queue with non-exponential patience times (the +G I ) as well asnon-exponential service times.

    The main idea here is to approach an application where we do not know whatmodel is appropriate by fitting a stationary BD process to an observed segment of thesample path of a queue-length stochastic process, assuming only that it increases anddecreases in unit steps. Just as is commonly done for estimating rates in a BD process[5,26,57], we estimate the birth rate in state k from arrival data over an interval [0, t]by k k(t), the number of arrivals observed in that state, divided by the totaltime spent in that state, while we estimate the death rate in state k by k k(t),the number of departures observed in that state, divided by the total time spent inthat state. As usual, the steady-state distribution of that fitted BD model, denoted byek ek(t) (with superscript e indicating the estimated rates), is well defined (underregularity conditions, see [54]) and is characterized as the unique probability vectorsatisfying the local balance equations,

    ek k = ek+1k+1, k 0. (1.1)Throughout this paper, we assume that limiting values of the rates as t existand that our estimators are consistent, so we omit the t .


  • Queueing Syst (2015) 79:391426 393

    The BD process is appealing as a grey-box model for queueing systems, because thefitted BD steady-state distribution {ek : k 0} in (1.1) closely matches the empiricalsteady-state distribution, {k : k 0}, where k k(t) is the proportion of totaltime spent in each state. Indeed, as has been known for some time (for example,see Chapter 4 of [19]), under regularity conditions, these two distributions coincideasymptotically as t (and thus the sample size) increases, even if the actual systemevolves in a very different way from the fitted BD process. For example, the actualprocess {Q(t) : t 0} might be periodic or non-Markovian; see [46].

    If we directly fit a BD process to data as just described, we should not concludewithout further testing that the underlying queue-length process actually is a BDprocess. In fact, the main point of [54] was to caution against drawing unwarrantedpositive conclusions from a close similarity in the steady-state distributions, becausethese two distributions are automatically closely related. Nevertheless, the fitted BDprocess might provide useful insight about the underlying system. The purpose of thepresent paper is to start investigating that idea.

    In fact, our earlier paper [54] was largely motivated by exploratory data analysiscarried out in Sect. 3 of [3]. They had fitted several candidate models to data on thenumber of patients in a hospital emergency department, and found that a BD processfit better than others. After writing our previous paper [54], cautioning against drawingunwarranted positive conclusions, we realized that fitting BD processes to data mightindeed be useful, and started the present investigation. See Sect. 3 of [3] for morediscussion about exploratory data analysis of an emergency department.

    There also is a much earlier precedent for exploiting fitted BD processes. This ideawas part of the program of operational analysis suggested by Buzen and Denning[8,9,15] in early performance analysis of computer systems. However, they expressedthe view that there was no need for an associated underlying stochastic model. Incontrast, as in Sects. 4.64.7 of [19], we think that an underlying stochastic modelshould play an important role. With that in mind, we want to see if the fitted BD modelcan provide useful insight into an unknown underlying stochastic model.

    Our goal in the present paper is to better understand this fitted BD model. Weprimarily want to answer two questions:

    (i) How are the fitted birth and death rates related to the key structural features ofan actual queueing model?

    (ii) How does the transient behavior of an actual queueing system differ from thetransient behavior of the fitted BD process?

    We start to address the first question by estimating the birth and death rates fromsimulation experiments for G I/G I/s models. When the interarrival-time (service-time) distribution is exponential, then the fitted birth (death) rate captures the truebehavior of the queue-length process, but otherwise it does not. For non-exponentialdistributions, we see that the fitted rates evidently have substantial structure, whichwe do our best to explain.

    We start to address the second question by looking at first passage times in theunderlying model and the fitted model. Perhaps our most interesting finding is a simpleconnection between the fitted BD process Qf {Qf(t) : t 0} and the queue-length (number in system) process Q {Q(t) : t 0} for a stationary G I/G I/s


  • 394 Queueing Syst (2015) 79:391426

    model, which holds approximately under two conditions: (i) if we focus on the long-term transient behavior, as measured by the expected first passage time from the pth

    percentile of the stationary distribution to the (1 p)th percentile and back again forp suitably small, for example, p = 0.1, and (ii) if we consider models with relativelyhigh traffic intensity, so that heavy-traffic approximations are appropriate. Under thesetwo conditions, we find that the two processes differ primarily in a one-dimensionalway, by the speed that they move through the state space. In other words, we suggestthe process approximation

    {Q(t) : t 0} {Qf(t/) : t 0}, (1.2)

    where is the speed ratio, which admits the conventional heavy-traffic approximation(traffic intensity near 1 with finitely many servers)

    c2a + c2s

    2, (1.3)

    with c2a and c2s being the squared coefficients of variation (scvs, variance divided by

    the square of the mean) of an interarrival time and a service time, respectively; seeTables 2, 3, 4, and 5. The approximating speed ratio in (1.3) can be recognized asthe heavy-traffic characterization of the variability in the G I/G I/s queuei


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