On Queues with Markov Modulated Service Rates

In this paper, we consider a queue whose service speed changes according to an external environment that is governed by a Markov process. It is possible that the server changes its service speed many times while serving a customer. We derive first and second moments of the service time of customers in system using first step analysis to obtain an insight on the service process. In fact, we obtain an intriguing result in that the moments of service time actually depend on the arrival process! We also show that the mean service rate is not the reciprocal of the mean service time. Further, since it is not possible to obtain a closed form expression for the queue length distribution, we use matrix geometric methods to compute performance measures such as average queue length and waiting time. We apply the method of large deviations to obtain tail distributions of the workload in the queue using the concept of effective bandwidth. We present two applications in computer systems: 1) Web server with multi-class requests and 2) CPU with multiple processes. We illustrate the analysis and various methods discussed with the help of numerical examples for the above two applications.