Series Jackson networks and non-crossing probabilities

TL;DR

This paper links the queue length process in series Jackson networks to non-crossing probabilities, enabling new insights into queueing dynamics and confirming a conjecture about relaxation times in such networks.

Contribution

It expresses Markov transition probabilities as sums of non-crossing probabilities, connecting queueing theory with non-crossing probability analysis and proving a conjecture on relaxation times.

Findings

Relaxation time of a series Jackson network equals that of an equivalent M/M/1 queue.

Transition probabilities can be written as finite sums of non-crossing probabilities.

Established a link between queueing behavior and non-crossing probability theory.

Abstract

This paper studies the queue length process in series Jackson networks with external input to the first station. We show that its Markov transition probabilities can be written as a finite sum of non-crossing probabilities, so that questions on time-dependent queueing behavior are translated to questions on non-crossing probabilities. This makes previous work on non-crossing probabilities relevant to queueing systems and allows new queueing results to be established. To illustrate the latter, we prove that the relaxation time (i.e., the reciprocal of the `spectral gap') of a positive recurrent system equals the relaxation time of an M/M/1 queue with the same arrival and service rates as the network's bottleneck station. This resolves a conjecture of Blanc, which he proved for two queues in series.