Sample-path analysis of queues with batch arrivals

• We consider a batch arrival single server queueing model.
• We use sample path analysis to give rigorous direct proofs under weak conditions.
• A general new result that relates mean workload and batch delays is given.
• We extend the P–K formula for a batch arrival single-server queueing model.
• New versions of conservation laws are given.

In this article we give a deterministic sample path general relationship that relates workload and batch delays, and use it to extend the Pollaczek–Khintchine formula for a batch arrival single-server queueing model. We also give a conservation law for the same system with multiple classes that leads to new versions of conservation laws for Poisson batch arrival models. Our results are completely rigorous and hold under weaker assumptions than those given in the literature. We do not make stochastic assumptions, so the results hold almost surely on every sample path of the stochastic process that describes the system evolution. The article is self contained in that it gives a brief review of necessary background material.