Violation Probability in Processor-Sharing Queues

Processor-sharing queues are often used to model file transmission in networks. While sojourn time is a common performance metric in the queueing literature, average transmission rate is the more commonly discussed metric in the networking literature. Whereas much is known about sojourn times, there is little known about the average service rate experienced by jobs in processor-sharing queues. We focus here upon performance requirements in the form of an upper bound on the probability of failing to achieve a specified minimum transmission rate or a specified minimum average rate. For an M/G/l processor-sharing queue, we give a closed-form expression for this violation probability. We derive closed-form expressions for the marginal service rate with respect to the violation probability and to the minimum transmission rate, and characterize when each is binding. We then consider the effect of using connection access control by modeling an M/G/l/K processor-sharing queue, and discuss the relationship between queue service rate, queue limit, violation probability, and blocking probability. Finally, we consider a two-class discriminatory processor-sharing queue, and discuss what combinations of class weighting and service rate can be used to achieve specified minimum rate violation probabilities for both classes.