Queue lengths and workloads in polling systems

We consider a polling system: a queueing system of $N \geq 1$ queues with Poisson arrivals $Q_1, \ldots, Q_N$ visited in a cyclic order (with or without switchover times) by a single server. For this system we derive the probability generating function $D(.)$ 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 $D$ and $W$ in the probability generating functions of the joint queue length distribution at visit beginnings, $V_{b_i}(.)$, and visit completions, $V_{c_i}(.)$, at $Q_i, i=1, \ldots, N$. It is well known that $V_{b_i}$ and $V_{c_i}$ can be computed in a broad variety of cases. Furthermore, we establish a workload decomposition result.