Estimating mean sojourn time in the processor sharing M / G / 1 queue with inaccurate job size information (original) (raw)

Sojourn time asymptotics in Processor Sharing queues with varying service rate

Queueing Systems, 2007

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.

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

Queueing Systems - Theory and Applications, 2000

We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index -ν, ν non-integer, iff the sojourn time distribution is regularly varying of index -ν. 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

Sojourn times in a processor sharing queue with service interruptions

2000

We study the sojourn time of customers in an M/M/1 queue with processor sharing service discipline and service interruptions. The lengths of the service interruptions have a general distribution, whereas the periods of service availability are assumed to have an exponential distribution. A branching process approach is shown to lead to a decomposition of the sojourn time into independent contributions, that can be investigated separately. The Laplace-Stieltjes Transform of the distribution of the sojourn time is found through an integral equation. We derive the first two moments of the sojourn time conditioned on the amount of work brought into the system and on the number of customers present upon arrival. We show that the expected sojourn time of a customer that arrives at the system in steady state is not linear in the amount of work he brings with him. Finally, we show that the sojourn time conditioned on the amount of work, scaled by the traffic load, converges in heavy traffic to an exponential distribution. This study was motivated by a need for delay analysis of elastic traffic in modern communication networks. Specifically, the results are of interest for the performance analysis of the Available Bit Rate (ABR) service class in ATM networks, as well as for the best-effort services in IP networks.

Queues with Equally Heavy Sojourn Time and Service Requirement Distributions

2002

For the G G 1 queue with First-Come First-Served, it is well known that the tail of the sojourn time distribution is heavier than the tail of the service requirement distribution when the latter has a regularly varying tail. In contrast, for the M G 1 queue with Processor Sharing, Zwart and Boxma 26 showed that under the same assumptions on the service requirement distribution, the two tails are equally heavy". By means of a probabilistic analysis we provide a new insightful proof of this result, allowing for the slightly weaker assumption of service requirement distributions with a tail of intermediate regular variation. The new approach allows us to also establish the tail equivalence" for two other service disciplines: Foreground-Background Processor Sharing and Shortest Remaining Processing Time. The method can also be applied to more complicated models, for which no explicit formulas exist for transforms of the sojourn time distribution. One such model is the M G 1 Processor Sharing queue with service that is subject to random interruptions. The latter model is of particular interest for the performance analysis of communication networks.

The effect of higher moments of job size distribution on the performance of an M/G/s queueing system

ACM SIGMETRICS …, 2007

The M/G/K queueing system is the oldest model for multi-server systems, and has been the topic of performance papers for almost half a century. However, even now, only coarse approximations exist for its mean waiting time. All the closed-form (non-numerical) approximations in the literature are based on the first two moments of the job size distribution. In this paper we prove that no approximation based on only the first two moments can be accurate for all job size distributions, and we provide a lower bound on the inapproximability ratio. This is the first such result in the literature. The proof technique behind this result is novel as well and combines mean value analysis, sample path techniques, scheduling, regenerative arguments, and asymptotic estimates. Finally, our work provides insight into the effect of higher moments of the job size distribution on the mean waiting time.

Large deviations of sojourn times in processor sharing queues

Queueing Systems, 2006

This paper presents a large deviation analysis of the steady-state sojourn time distribution in the GI/G/1 PS queue. Logarithmic estimates are obtained under the assumption of the service time distribution having a light tail, thus supplementing recent results for the heavy-tailed setting. Our proof gives insight in the way a large sojourn time occurs, enabling the construction of an (asymptotically efficient) importance sampling algorithm. Finally our results for PS are compared to a number of other service disciplines, such as FCFS, LCFS, and SRPT.

Sojourn Times in the M/PH/1 Processor Sharing Queue

Queueing Systems, 2005

We give in this paper an algorithm to compute the sojourn time distribution in the processor sharing, single server queue with Poisson arrivals and phase type distributed service times. In a first step, we establish the differential system governing the conditional sojourn times probability distributions in this queue, given the number of customers in the different phases of the PH distribution at the arrival instant of a customer. This differential system is then solved by using a uniformization procedure and an exponential of matrix. The proposed algorithm precisely consists of computing this exponential with a controlled accuracy. This algorithm is then used in practical cases to investigate the impact of the variability of service times on sojourn times and the validity of the so-called reduced service rate (RSR) approximation, when service times in the different phases are highly dissymmetrical. For two-stage PH distributions, we give conjectures on the limiting behavior in terms of an M/M/1 PS queue and provide numerical illustrative examples.

Sojourn Time Approximations for a Discriminatory Processor Sharing Queue

ACM Transactions on Modeling and Performance Evaluation of Computing Systems, 2016

We study a multiclass time-sharing discipline with relative priorities known as discriminatory processor sharing (DPS), which provides a natural framework to model service differentiation in systems. The analysis of DPS is extremely challenging, and analytical results are scarce. We develop closed-form approximations for the mean conditional (on the service requirement) and unconditional sojourn times. The main benefits of the approximations lie in its simplicity, the fact that it applies for general service requirements with finite second moments, and that it provides insights into the dependency of the performance on the system parameters. We show that the approximation for the mean conditional and unconditional sojourn time of a customer is decreasing as its relative priority increases. We also show that the approximation is exact in various scenarios, and that it is uniformly bounded in the second moments of the service requirements. Finally, we numerically illustrate that the a...

Exponential Approximations for Tail Probabilities in Queues II: Sojourn Time and Workload

Operations Research, 1996

We continue to focus on simple exponential approximations for steady-state tail probabilities in queues based on asymptotics. For the G/GI/1 model with i.i.d. service times that are independent of an arbitrary stationary arrival process, we relate the asymptotics for the steady-state waiting time, sojourn time, and workload. We show that the three asymptotic decay rates coincide and that the three asymptotic constants are simply related. We evaluate the exponential approximations based on the exact asymptotic parameters and their approximations by making comparisons with exact numerical results for BMAP/G/1 queues, which have batch Markovian arrival processes. Numerical examples show that the exponential approximations for the tail probabilities are remarkably accurate at the 90th percentile and beyond. Thus, these exponential approximations appear very promising for applications.

Far-End-Tail Estimation of Queueing System Performance

Journal of Mathematical Sciences, 2020

In this work, we consider an approximation of the so-called far-end-tail distribution of processes describing quality of service (QoS) performance of queueing systems. This approximation is based on the asymptotic equivalence between the excess distribution over a high threshold and the generalized Pareto distribution, for a wide class of the governing distributions. The numerical results based on the regenerative estimation of a large deviation of the maximal workload over the regeneration cycle, in the queueing system M/G/1 with Pareto service time, are also presented, which verify theoretical results.