Queue lengths and workloads in polling systems

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