Sojourn time asymptotics in Processor Sharing queues with varying service rate (original) (raw)
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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
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.
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.
Tail asymptotics for processor-sharing queues
Advances in Applied Probability, 2004
The basic queueing system considered in this paper is the M/G/1 processor sharing (PS) queue with or without impatience and with finite or infinite capacity. Under some mild assumptions, a criterion for the validity of the RSR (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.
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...
Sojourn Time Tails in the M/D/1 Processor Sharing Queue
Probability in the Engineering and Informational Sciences, 2006
We consider the sojourn time V in the M/D/1 processor sharing (PS) queue and show that P(V > x) is of the form Ce−γx as x becomes large. The proof involves a geometric random sum representation of V and a connection with Yule processes, which also enables us to simplify Ott's [21] derivation of the Laplace transform of V. Numerical experiments show that the approximation P(V > x) ≈ Ce−γx is excellent even for moderate values of x.
Law of Large Number Limits of Limited Processor-Sharing Queues
Mathematics of Operations Research, 2009
Motivated by applications in computer and communication systems, we consider a processor sharing queue where the number of jobs served is not larger than K. We propose a measure-valued fluid model for this limited processor sharing queue and show that there exists a unique associated fluid model solution. In addition, we show that this fluid model arises as the limit of a sequence of appropriately scaled processor sharing queues. studies of operating systems papers , as well as in more recent Web server design papers , and database implementation papers . So in the modeling of many computer and communication systems, a sharing limit is normally imposed, which results in an LPS model.
Exact and approximate analysis of sojourn times in finite discriminatory processor sharing queues
AEU - International Journal of Electronics and Communications, 2006
Exact analysis of discriminatory processor sharing (DPS) systems has proven to be extremely hard. We describe how the sojourn time distribution can be obtained in closedform for exponential service requirement distributions when there is admission control. The computational complexity suffers from the usual state-space explosion when the number of customer classes becomes large, or if the admission control allows for many concurrent customers. Through numerical experiments, we show that a time-scale decomposition approach provides an approximation that requires much less computational effort, while giving accurate results even when the classes do not have different time scales and are distinguished through the relative service shares only.
2017
We seek to approximate the mean sojourn time in the processor sharing M/G/1 queue with inaccurate job size information. Suppose we are given the arrival rate λ and random service time Ŝ = SX where X ∼ LN(0, σ) represents the inaccuracy. Denote the mean sojourn time in an M/G/1 queue with processor sharing with service time Ŝ by E(T̂) and with service time S by E(T). Finally, E(T̂) denotes the mean sojourn time of an M/G/1 queue with resampling service policy and service time distribution according to S. It can be shown that for exponential service time S, E(T) < E(T̂) < E(T̂) holds for any σ > 0.
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.