Dynamic Service Rate Control for a Single Server Queue with Markov Modulated Arrivals

TL;DR

This paper investigates optimal service rate control in a single server queue with Markov-modulated arrivals, revealing monotonicity properties and evaluating heuristics for practical implementation in non-stationary traffic scenarios.

Contribution

It establishes the monotonicity of the optimal service rate with respect to queue length and explores conditions under which heuristics approximate optimal control.

Findings

Optimal service rate increases with queue length.

Monotonicity depends on the stochastic properties of the arrival process.

Heuristics vary in performance across different traffic regimes.

Abstract

We consider the problem of service rate control of a single server queueing system with a finite-state Markov-modulated Poisson arrival process. We show that the optimal service rate is non-decreasing in the number of customers in the system; higher congestion rates warrant higher service rates. On the contrary, however, we show that the optimal service rate is not necessarily monotone in the current arrival rate. If the modulating process satisfies a stochastic monotonicity property the monotonicity is recovered. We examine several heuristics and show where heuristics are reasonable substitutes for the optimal control. None of the heuristics perform well in all the regimes. Secondly, we discuss when the Markov-modulated Poisson process with service rate control can act as a heuristic itself to approximate the control of a system with a periodic non-homogeneous Poisson arrival process. Not only is the current model of interest in the control of Internet or mobile networks with bursty traffic, but it is also useful in providing a tractable alternative for the control of service centers with non-stationary arrival rates.