Sojourn time asymptotics in processor sharing queues

This paper addresses the sojourn time asymptotics for a GI/GI/• queue operating under the Processor Sharing (PS) discipline with stochastically varying service rate. Our focus is on the logarithmic estimates of the tail of sojourn-time distribution, under the assumption that the jobsize distribution has a light tail. Whereas upper bounds on the decay rate can be derived under fairly general conditions, the establishment of the corresponding lower bounds requires that the service process satisfies a samplepath large-deviation principle. We show that the class of allowed service processes includes the case where the service rate is modulated by a Markov process. Finally, we extend our results to a similar system operation under the Discriminatory Processor Sharing (DPS) discipline. Our analysis relies predominantly on large-deviations techniques

Tail asymptotics for processor sharing queues

The basic queueing system considered in this paper is the M/G/1 processor-sharing queue with or without impatience and with finite or infinite capacity. Under some mild assumptions, a criterion for the validity of the reduced-service-rate approximation is established when service times are heavy tailed. This result is applied to various models based on M/G/1 processor-sharing queues

A large-deviations analysis of the GI/GI/1 SRPT queue

We consider a GI/GI/1 queue with the shortest remaining processing time discipline (SRPT) and light-tailed service times. Our interest is focused on the tail behavior of the sojourn-time distribution. We obtain a general expression for its large-deviations decay rate. The value of this decay rate critically depends on whether there is mass in the endpoint of the service-time distribution or not. An auxiliary priority queue, for which we obtain some new results, plays an important role in our analysis. We apply our SRPT-results to compare SRPT with FIFO from a large-deviations point of view.Comment: 22 page

Bandwidth sharing with heterogeneous service requirements

We consider a system with two heterogeneous traffic classes. The users from both classes randomly generate service requests, one class having light-tailed properties, the other one exhibiting heavy-tailed characteristics. The heterogeneity in service requirements reflects the extreme variability in flow sizes observed in the Internet, with a vast majority of small transfers ('mice') and a limited number of exceptionally large flows ('elephants'). The active traffic flows share the available bandwidth in a Processor-Sharing (PS) fashion. The PS discipline has emerged as a natural paradigm for modeling the flow-level performance of bandwidth-sharing protocols like TCP. The number of simultaneously active traffic flows is limited by a threshold on the maximum system occupancy. We obtain the exact asymptotics of the transfer delays incurred by the users from the light-tailed class. The results show that the threshold mechanism significantly reduces the detrimen

Sojourn time asymptotics in the M/G/1 processor sharing queue

We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index $-nu$, $nu$ non-integer, iff the sojourn time distribution is regularly varying of index $-nu$. This result is derived from a new expression for the Laplace-Stieltjes transform of the sojourn time distribution. That expression also leads to other new properties for the sojourn time distribution. We show how the moments of the sojourn time can be calculated recursively and prove that the k-th moment of the sojourn time is finite iff the k-th moment of the service time is finite. In addition, we give a short proof of a heavy traffic theorem for the sojour

Scheduling for the tail: Robustness versus optimality

When scheduling to minimize the sojourn time tail, the goals of optimality and robustness are seemingly at odds. Over the last decade, results have emerged which show that scheduling disciplines that are near-optimal under light (exponential) tailed workload distributions do not perform well under heavy (power) tailed workload distributions, and vice-versa. Very recently, it has been shown that this conflict between optimality and robustness is fundamental, i.e., no policy that does not learn information about the workload can be optimal across both light-tailed and heavy-tailed workloads. In this paper we show that one can exploit very limited workload information (the system load) in order to design a scheduler that provides robust performance across heavy-tailed and light-tailed workloads