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Queueing systems of INAR(1) processes with compound Poisson arrivals. (English) Zbl 1329.60323

Summary: Integer-valued autoregressive processes of order 1 (or INAR(1) processes) that may be interpreted as discrete time \(\mathrm{G}/\mathrm{Geom}/\infty\) queue length processes are considered. The arrivals are assumed to be compound Poisson distributed. It is shown that then the stationary distribution of the queue length process as well as the distribution of the departures from the system are again members of the class of compound Poisson distributions. This reveals remarkable invariance properties of the model. The derived explicit expressions allow for the calculation of important performance measures. It is further shown that time-reversibility of the queue length process as well as an analogue of Burke’s theorem hold only if the arrival process is Poisson.

MSC:

60K25 Queueing theory (aspects of probability theory)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
90B22 Queues and service in operations research
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