Queue lengths and workloads in polling systems

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

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