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On \(\text{GI}^{X(u)}/\text{G}/ \infty\) queue. (English) Zbl 0841.60075

The authors study an infinite server queue with batch arrivals in which the size of each arriving batch depends on the time elapsed since the last arrival. Single server bulk queues with finite capacity are dependent of the size of an arriving batch of the time elapsed [the second author and R. Varughese, ibid. 3, No. 1, 131-147 (1992; Zbl 0753.60094)]. The interarrival times and service times are mutually independent. Arrival of customers in groups of variable size depends on the time elapsed since the last arrival. The number of busy servers or number of customers in the system in transient state can be modelled as a shot noise process. This method was used in queueing theory by L. Takacs (1958) and W. Smith [J. Appl. Probab. 10, 685-690 (1973; Zbl 0283.60100)]. In this case, when arriving batch sizes are independent and identically distributed random variables, the authors use the model considered by L. Liu, the third author and J. G. C. Templeton [ibid. 27, No. 3, 671-683 (1990; Zbl 0715.60114)]. The steady state results coincide for both models, as is also seen in the Bellman-Harris process and the Crump-Mode process (1968). Explicit expressions for the time-dependent behaviour of the system size at time \(t\) are presented in Section 3. Also from these results limiting behaviour of the system state probabilities and binomial moments are obtained (Theorems 3 and 4).

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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