# Queue-length balance equations in multiclass multiserver queues and their generalizations

## TL;DR

This paper generalizes classical queue-length balance equations to complex multiclass multiserver networks, enabling simpler derivations of known results and discovery of new insights, including priority systems and non-stationary cases.

## Contribution

It introduces a general framework for distributional balance equations in multidimensional queueing networks, extending classical results to more complex, non-singleton batch scenarios.

## Key findings

- Derived balance equations for general multiclass multiserver networks.
- Simplified derivations of existing results in polling systems.
- New results for priority queueing systems and non-stationary frameworks.

## 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 {\em 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 (i) providing very simple derivations of some known results for polling systems, and (ii) obtaining new results for some queueing systems with priorities. We also extend the distributional relationships for a non-stationary framework.