Queue lengths and workloads in polling systems

We consider a polling system: a queueing system of N ≥ 1 queues with Poisson arrivals Q1, . . . , QN visited in a cyclic order (with or without switchover times) by a single server. For this system we derive the probability generating function Q(·) of the joint queue length distribution at an arbitrary epoch in a stationary cycle, under no assumptions on service disciplines. We also derive the Laplace-Stieltjes transform W (·) of the joint workload distribution at an arbitrary epoch. We express Q and W in the probability generating functions of the joint queue length distribution at visit beginnings, Vbi(·), and visit completions, Vci(·), at Qi, i = 1, . . . , N . It is well known that Vbi and Vci can be computed in a broad variety of cases. Furthermore, we establish a workload decomposition result.