Flow coupling and stochastic ordering of throughputs in linear networks

TL;DR

This paper introduces a novel method for comparing the throughput of linear queueing networks by coupling flow processes, providing more practical insights than traditional population-based stochastic orderings.

Contribution

It develops a new approach to order network flows via augmented Markov processes, relaxing strong monotonicity assumptions of classical methods.

Findings

Flow coupling provides tighter throughput bounds

Augmented state-flow Markov couplings enable practical comparisons

Method applies to complex queueing networks

Abstract

Robust estimates for the performance of complicated queueing networks can be obtained by showing that the number of jobs in the network is stochastically comparable to a simpler, analytically tractable reference network. Classical coupling results on stochastic ordering of network populations require strong monotonicity assumptions which are often violated in practice. However, in most real-world applications we care more about what goes through a network than what sits inside it. This paper describes a new approach for ordering flows instead of populations by augmenting network states with their associated flow counting processes and deriving Markov couplings of the augmented state-flow processes.