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Using a birth‐and‐death process to estimate the steady‐state distribution of a periodic queue
Author(s) -
Dong James,
Whitt Ward
Publication year - 2015
Publication title -
naval research logistics (nrl)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.665
H-Index - 68
eISSN - 1520-6750
pISSN - 0894-069X
DOI - 10.1002/nav.21672
Subject(s) - steady state (chemistry) , queue , queueing theory , poisson distribution , burke's theorem , mathematics , distribution (mathematics) , parametric statistics , function (biology) , state (computer science) , bulk queue , birth–death process , computer science , mathematical optimization , statistics , mathematical analysis , algorithm , fork–join queue , queue management system , population , chemistry , demography , sociology , evolutionary biology , biology , programming language
Abstract If the number of customers in a queueing system as a function of time has a proper limiting steady‐state distribution, then that steady‐state distribution can be estimated from system data by fitting a general stationary birth‐and‐death (BD) process model to the data and solving for its steady‐state distribution using the familiar local‐balance steady‐state equation for BD processes, even if the actual process is not a BD process. We show that this indirect way to estimate the steady‐state distribution can be effective for periodic queues, because the fitted birth and death rates often have special structure allowing them to be estimated efficiently by fitting parametric functions with only a few parameters, for example, 2. We focus on the multiserver M t / GI / s queue with a nonhomogeneous Poisson arrival process having a periodic time‐varying rate function. We establish properties of its steady‐state distribution and fitted BD rates. We also show that the fitted BD rates can be a useful diagnostic tool to see if an M t / GI / s model is appropriate for a complex queueing system. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 664–685, 2015