Waiting Times in an Exponential Queue with Hysteretic Service Rate Control: A Numerical Investigation

Abstract

Increasing the service rate when a queue is long is a common practice in many service systems. The control mechanism can be uni-level, where the service rate is increased whenever a preset threshold is exceeded and returns to normal when the number of customers drops below the same threshold. In a hysteretic control, the service rate remains at high level until the number in the queue is brought down to a value much lower than the threshold for increasing the service rate. When the server can operate at k levels, a k-level hysteretic control can be used. While explicit results are obtained for the equilibrium distribution of the number of customers in the system, there is no study dealing with the waiting time characteristics of the customers entering the system. The difficulty arises from the fact that service rates, and hence the waiting times, are affected by arrivals during a customer’s sojourn through the system. In this paper algorithms are developed to compute the equilibrium probability distributions of customer waiting times and the times spent by the server at each of the service levels during each visit to that state. Numerical results are presented to describe the behavior of the system under different traffic and control structures.