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

• Published in: Annals of operations research 2022-10, Vol.317 (1), p.147-178
• Main Authors: Mytalas, George C., Zazanis, Michael A.
• Format: Article
• Language: English
• Subjects: Binomial distribution; Business and Management; Combinatorics; Customer service; Materials handling; Operations research; Operations Research/Decision Theory; Queueing; Queues; Queuing theory; Theory of Computation; Vacations
• ISSN: 0254-5330; 1572-9338
• DOI: 10.1007/s10479-015-2077-0

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.