Simple and explicit bounds for multi-server queues with $1/(1-\rho)$ scaling

TL;DR

This paper derives simple, explicit bounds for the steady-state queue length and delay in multi-server queues that scale as 1/(1-ρ), applicable under finite second moment conditions and extending to large server numbers and heavy traffic regimes.

Contribution

It provides the first explicit bounds for multi-server queues that scale as 1/(1-ρ) under minimal moment assumptions, generalizing Kingman's bound to multiple servers.

Findings

Bounds for tail of steady-state queue length and delay.

Bounds that scale well in heavy traffic and many-server regimes.

Explicit tail decay rates depending on service distribution moments.

Abstract

We consider the FCFS $GI/GI/n$ queue, and prove the first simple and explicit bounds that scale as $\frac{1}{1-\rho}$ under only the assumption that inter-arrival times have finite second moment, and service times have finite $2+\epsilon$ moment for some $\epsilon > 0$. Here $\rho$ denotes the corresponding traffic intensity. Conceptually, our results can be viewed as a multi-server analogue of Kingman's bound. Our main results are bounds for the tail of the steady-state queue length and the steady-state probability of delay. The strength of our bounds (e.g. in the form of tail decay rate) is a function of how many moments of the service distribution are assumed finite. Our bounds scale gracefully even when the number of servers grows large and the traffic intensity converges to unity simultaneously, as in the Halfin-Whitt scaling regime. Some of our bounds scale better than $\frac{1}{1-\rho}$ in certain asymptotic regimes. In these same asymptotic regimes we also prove bounds for the tail of the steady-state number in service. Our main proofs proceed by explicitly analyzing the bounding process which arises in the stochastic comparison bounds of Gamarnik and Goldberg for multi-server queues. Along the way we derive several novel results for suprema of random walks and pooled renewal processes which may be of independent interest. We also prove several additional bounds using drift arguments (which have much smaller pre-factors), and point out a conjecture which would imply further related bounds and generalizations. We also show that when all moments of the service distribution are finite and satisfy a mild growth rate assumption, our bounds can be strengthened to yield explicit tail estimates decaying as $O\big(\exp(-x^{\alpha})\big)$, with $\alpha \in (0,1)$ depending on the growth rate of these moments.