Dynamic Service Rate Control for a Single

Server Queue with Markov Modulated Arrivals

Ravi Kumar, Mark E. Lewis and Huseyin Topaloglu
Cornell University
School of Operations Research and Information Engineering
226 Rhodes Hall
Ithaca, New York 14853

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 \textbf{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 Markov-modulated 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.