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

We consider the problem of service rate control of a single server queueing system when the arrival process is governed by a finite-state Markov-modulated Poisson process. There are two main technical contributions. First, in keeping with intuition, we show that the optimal service rate is non-decreasing in the number of customers in the system. That is, 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 we show that the monotonicity is recovered. Together these results imply that we have reasonable conditions for the optimal control to follow a monotone switching curve in the current state of the system. Our numerical study also has two components. We examine several heuristics and show where those heuristics can be reasonable substitutes for the optimal control. None of the heuristics perform well in all the regimes we consider. Secondly, we discuss when the Markovmodulated Poisson process with service rate control can act as a heuristic itself. In particular, we show that it can approximate the optimal control of a system with a periodic non-homogeneous Poisson arrival process. Thus, 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.