Sojourn time asymptotics in Processor Sharing queues with varying service rate

Regina Egorova 0 1 Michel Mandjes 0 1 Bert Zwart 0 1 0 B. Zwart Stewart School of Industrial and Systems Engineering, Georgia University of Technology , 765 Ferst Drive, Atlanta, GA 30332, USA 1 M. Mandjes Korteweg-de Vries Institute for Mathematics, University of Amsterdam , Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands 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 sample-path largedeviation 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. - Based on the traffic characteristics and Quality-of-Service requirements, traffic flows in communication networks can be roughly divided into two categories: streaming flows (voice, video, etc.) and elastic flows (data files, Web pages, etc.), see e.g. [20]. Streaming flows require strict delay guarantees for the duration of its connection time, whereas elastic traffic is less demanding. One way of handling both types of traffic is to meet these Quality-of-Service requirements by prioritizing streaming traffic. The bandwidth remaining from the transmission of streaming traffic is made available to elastic traffic. It is widely agreed upon that the protocols for handling elastic traffic are such that each elastic flow obtains roughly an equal share. Motivated by the above application, one could consider the following model. Let elastic flows (we use the word jobs throughout) arrive at a queueing resource, according to a renewal process, and let these jobs be independent samples from some common distribution. The jobs are served in a Processor Sharing (PS) manner, but the capacity available (to be interpreted as the service rate left over by the streaming flows) fluctuates in time. The streaming flows do not see the elastic flows, so their performance can be evaluated by using traditional models. The performance experienced by the elastic flows, however, can be regarded as a GI/GI/ queue with a service rate that varies in time (according to some stochastic process), operating under PS, and is more involved. In this paper we study the asymptotic properties of the sojourn-time distribution of the elastic flows. It is worth noting that in the case of processor-sharing queues with constant service capacity, the sojourn time has been studied in many different settings, and this has already proven to be a rather challenging task. The (conditional) sojourn time distribution in the M/G/1-PS queue was analyzed in terms of LaplaceStieltjes transforms (LST) by e.g. Yashkov [23], Schassberger [21], Ott [18], NezQueija [16], and Zwart and Boxma [24]. Unfortunately, analytic inversion of these LST s has appeared to be hard, and only partial results are available. Another classical subject of research is the derivation of the asymptotic behavior of sojourn times in PS-queues. Notably, one of the major insights is that there is a fundamental difference between sojourn-time asymptotics under heavy-tailed and light-tailed jobs. A so-called reducedload approximation for queues with heavy-tailed distributed job sizes was proven in different settings by, e.g., Zwart and Boxma [24], Nez-Queija [16] and Jelenkovic and Momcilovic [13]; importantly, long sojourn times are essentially due to the tagged job itself being large. For PS queues with a light-tailed job-size distribution long sojourn times are predominantly caused by the jobs that arrive during the sojourn time of the tagged job. For the light-tailed case, exact asymptotics are known in a few special cases, see [4, 10, 11]; for a survey, see [6]. Recent work by Mandjes and Zwart [15] addressed the logarithmic asymptotics of the sojourn time in the GI/GI/1PS queue, under technical assumptions which guarantee that the tail distribution of the service time is not too light and not too heavy. More precisely, they proved under specific conditions that the sojourn time V obeys where (s) is the so-called (asymptotic) cumulant function of total amount of work fed to the queue, i.e., ( ) = xlim x1 log E[eA(0,x)], with A(0, x) the amount of traffic offered to the system in (0, x]. The goal of the present paper is to generalize the result (1.1) of [15] to a setting in which the available service capacity varies according to some stochastic process. Again the job sizes should be from a light-tailed distribution (but not too light). We extend (1.1) by constructing asymptotic lower and upper bounds, which coincide as x becomes large. The upper bounds can be established under rather general conditions, whereas the lower bound requires that the service process obeys a sample-path large-deviation principle. More specifically, the main result of our work is that we can express the exponential decay rate of P(V > x) through log P(V > x) = inf (( ) + c( )), 0 where () and c() are the cumulant functions of the arrival and service processes, respectively. The exact statement of the result is given in Theorem 3.4. As a special case, we study service processes that have a so-called Markov-fluid structure. Under the extra assumption of the interarrival times of the jobs being exponential (rather than renewal), we derive for these service processes an explicit upper bound on the tail probability, rather than just an upper bound on the exponential decay rate. Our proofs predominantly rely on large-deviation tools, such as the classical Chernoff bound, as well as the application of sample-path large deviations principles. An important role, however, is also played by the insight that, for PS systems with load larger than 1, the queue length increases roughly at a linear rate. As a by-product, the proofs also show that the sojourn-time asymptotics resemble busyperiod asymptotics (in the sense that their exponential decay rates coincide). Although our results are an extension of the results in [15], we have succeeded to simplify the proofs; in particular, we have eliminated the need to use detailed fluidlimit results for overloaded PS queues, as used in [15]. Finally, our methods allow us to obtain an extension of the result to the system operating under the Discriminatory Processor Sharing (DPS) discipline. As for the single-class case, we allow the service process to be random, but note that this result is also new for the standard DPS queue with a fixed service rate. More spe (...truncated)