Queueing Systems of INAR(1) Processes with Compound Poisson Arrivals

Abstract

Integer valued autoregressive processes of order 1 (or INAR(1) processes) that may be interpreted as discrete time G/Geom/∞ 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.

Keywords:

• Burke's  Theorem
• Compound Poisson Distribution
• Discrete Time Queueing Model
• INAR(1) Processes
• Time Reversibility.

Mathematics Subject Classifications:

• 60J10
• 90B22
• 60K25.

Notes

It is worth noting that the notion of a queueing process without queueing seems a little odd. However, the infinite-server queue has established its importance in queueing theory.