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Service with a queue and a random capacity cart: random processing batches and E-limited policies

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  • George C. Mytalas

    (New Jersey Institute of Technology)

  • Michael A. Zazanis

    (Athens University of Economics and Business)

Abstract

In this paper we examine a queueing model with Poisson arrivals, service phases of random length, and vacations, and its applications to the analysis of production systems in which material handling plays an important role. The length of a service phase can be interpreted as a “processing batch”, leading to a varying E-limited M/G/1 queue and the analysis is carried out separately for processing batch distributions with bounded and unbounded support. In the first case, standard techniques from the analysis of limited service systems are used, involving Rouché’s theorem, while in the second the analysis proceeds via Wiener–Hopf factorization techniques. Processing batches with size that is either geometrically distributed or distributed according to a combination of geometric factors lead to particularly simple solutions related to Bernoulli vacation models. In all cases, care is taken in the analysis in order to obtain the steady state distribution of the system under minimal assumptions, namely the finiteness of the first moment of the service and vacation distributions together with the stability condition. This is in contrast to most of the literature where usually the assumption that the service and vacation distribution is light-tailed is either explicitly stated or tacitly adopted.

Suggested Citation

  • George C. Mytalas & Michael A. Zazanis, 2022. "Service with a queue and a random capacity cart: random processing batches and E-limited policies," Annals of Operations Research, Springer, vol. 317(1), pages 147-178, October.
    • Handle: RePEc:spr:annopr:v:317:y:2022:i:1:d:10.1007_s10479-015-2077-0
    DOI: 10.1007/s10479-015-2077-0
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    References listed on IDEAS

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    1. S. W. Fuhrmann, 1984. "Technical Note—A Note on the M / G /1 Queue with Server Vacations," Operations Research, INFORMS, vol. 32(6), pages 1368-1373, December.
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    3. S. W. Fuhrmann & Robert B. Cooper, 1985. "Stochastic Decompositions in the M / G /1 Queue with Generalized Vacations," Operations Research, INFORMS, vol. 33(5), pages 1117-1129, October.
    4. Junmin Shi & Michael Katehakis & Benjamin Melamed, 2013. "Martingale methods for pricing inventory penalties under continuous replenishment and compound renewal demands," Annals of Operations Research, Springer, vol. 208(1), pages 593-612, September.
    5. Zazanis, Michael A, 1998. "A Palm calculus approach to functional versions of Little's law," Stochastic Processes and their Applications, Elsevier, vol. 74(2), pages 195-201, June.
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