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A service system with two stages of waiting and feedback of customers. (English) Zbl 0544.60088

The authors consider the following variant of an M/G/1 queueing system. Customers arrive at event times of a Poisson process and upon arrival they are placed in a waiting room. Customers are served one at a time in a separate service room. Whenever the service room becomes empty it is replenished by transferring all customers in the waiting room and the addition of a random number \(N\geq 1\) of overhead customers. The epochs at which this occurs are called transfer times. A customer completing service is returned to the waiting room with probability p where \(0\leq p<1\) and he leaves the system with probability \(q=1-p\). All service times, inter-arrival times, feed back events and service room top ups are independent.
This set up generalises an earlier one of L. Takacs [RAIRO, Rech. Oper. 11, 345-354 (1977; Zbl 0375.60106)] in which \(N=1\) and the service times are exponentially distributed. Still earlier, D. G. Lampard [J. Appl. Probab. 5, 648-668 (1968; Zbl 0185.462)] mentioned the special case of Takacs’ model where \(p=0\), but with N a constant not necessarily unity.
The authors obtain the limiting distributions of some associated processes, viz., the number of customers in the service room immediately following transfers, the numbers in each room at any time and at service completions, and the virtual waiting time. The virtual waiting time process is discussed under various rules governing the priorities of transferred and overhead customers. Finally, the authors discuss algorithmic aspects of the case where N is constant and service times are exponentially distributed.
Reviewer’s comment. Theorem 1 is not correct. The given condition is sufficient for positive recurrence, but the imbedded Markov chain can be positive recurrent when \(\lambda \alpha =p\) and service times have infinite variance. In fact, when \(\lambda \alpha =p\), the necessary and sufficient condition for positive recurrence is \(\int^{1}_{0}(1- z)(P(z)-u)^{-1}dz<\infty\), see E. Seneta, ibid. 4, 489-495 (1967; Zbl 0178.196), section 4.
Reviewer: A.Pakes

MSC:

60K25 Queueing theory (aspects of probability theory)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
90B22 Queues and service in operations research
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