Technical Note—Approximating Systems Fed by Poisson Processes with Rapidly Changing Arrival Rates

Zeyu Zheng (), Harsha Honnappa () and Peter W. Glynn ()
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Zeyu Zheng: Department of Industrial Engineering and Operations Research, University of California, Berkeley, Berkeley, California 94720
Harsha Honnappa: School of Industrial Engineering, Purdue University, West Lafayette, Indiana 47906
Peter W. Glynn: Department of Management Science and Engineering, Stanford University, Stanford, California 94305

Operations Research, 2021, vol. 69, issue 5, 1566-1574

Abstract: This paper introduces a new asymptotic regime for simplifying stochastic models having nonstationary effects, such as those that arise in the presence of time-of-day effects. This regime describes an operating environment within which the arrival process to a service system has an arrival intensity that is fluctuating rapidly. We show that such a service system is well approximated by the corresponding model in which the arrival process is Poisson with a constant arrival rate. In addition to the basic weak convergence theorem, we also establish a first order correction for the distribution of the cumulative number of arrivals over [ 0 , t ] , as well as the number-in-system process for an infinite-server queue fed by an arrival process having a rapidly changing arrival rate. This new asymptotic regime provides a second regime within which nonstationary stochastic models can be reasonably approximated by a process with stationary dynamics, thereby complementing the previously studied setting within which rates vary slowly in time.

Keywords: probability: stochastic model applications; Stochastic Modeling; counting processes; Poisson process; weak convergence; total variation convergence; compensator; intensity; infinite-server queue (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:inm:oropre:v:69:y:2021:i:5:p:1566-1574

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