Abstract
A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: The distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, with probability one occur individually, i.e., not in batches. We show that it is easy to write down somewhat similar balance equations for multidimensional queue-length processes for a quite general network of multiclass multiserver queues. We formally derive those balance equations under a general framework. They are called distributional relationships and are obtained for any external arrival process and state-dependent routing as long as certain stationarity conditions are satisfied and external arrivals and service completions do not simultaneously occur. We demonstrate the use of these balance equations, in combination with PASTA, by (1) providing very simple derivations of some known results for polling systems and (2) obtaining new results for some queueing systems with priorities. We also extend the distributional relationships for a nonstationary framework.
Highlights
We extend the distributional relationships for a nonstationary framework
A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: The distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch
We shall argue that it is easy to write down a more global balance equation for multidimensional queue length processes for a large class of queues and queueing networks— when service times are not exponentially distributed, and even when arrivals may occur in batches
Summary
A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: The distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. We shall explore that fact to obtain a simple relation between the steady-state joint queue-length distribution at arrival epochs (which under various circumstances is equal to the time average distribution) and at service completion epochs. The research for the present paper was initially motivated by the desire to provide an intuitive explanation of a result in [4] regarding the steady-state joint queue-length distribution in a large class of polling models. Literature review Hébuterne [11] provides a generalization of the above-mentioned classical result of Burke in two directions: He allows (i) batch arrivals, with batches of random size, and (ii) batch services, with batches of fixed size He points out that emptying the queue up to N customers is beyond the scope of the analysis, because the batch sizes are not independent of the system state.
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