185 research outputs found
Asymptotic Expansions for the Conditional Sojourn Time Distribution in the -PS Queue
We consider the queue with processor sharing. We study the
conditional sojourn time distribution, conditioned on the customer's service
requirement, in various asymptotic limits. These include large time and/or
large service request, and heavy traffic, where the arrival rate is only
slightly less than the service rate. The asymptotic formulas relate to, and
extend, some results of Morrison \cite{MO} and Flatto \cite{FL}.Comment: 30 pages, 3 figures and 1 tabl
Heavy-traffic limits for Discriminatory Processor Sharing models with joint batch arrivals
We study the performance of Discriminatory Processor Sharing (DPS) systems, with exponential service times and in which batches of customers of different types may arrive simultaneously according to a Poisson process. We show that the stationary joint queue-length distribution exhibits state-space collapse in heavy traffic: as the load Ï tends to 1, the scaled joint queue-length vector (1âÏ)Q converges in distribution to the product of a determin
Asymptotic Expansions for the Sojourn Time Distribution in the -PS Queue
We consider the queue with a processor sharing server. We study the
conditional sojourn time distribution, conditioned on the customer's service
requirement, as well as the unconditional distribution, in various asymptotic
limits. These include large time and/or large service request, and heavy
traffic, where the arrival rate is only slightly less than the service rate.
Our results demonstrate the possible tail behaviors of the unconditional
distribution, which was previously known in the cases and (where it
is purely exponential). We assume that the service density decays at least
exponentially fast. We use various methods for the asymptotic expansion of
integrals, such as the Laplace and saddle point methods.Comment: 45 page
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
Sojourn time approximations for a discriminatory-processor-sharing queue
International audienceWe study a multi-class 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 approximation for exponential, hyperexponential and Pareto service requirements is accurate across a broad range of parameters
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 , non-integer, iff the sojourn time distribution is regularly varying of index . 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
Interpolation approximations for the steady-state distribution in multi-class resource-sharing systems
International audienceWe consider a single-server multi-class queue that implements relative priorities among customers of the various classes. The discipline might serve one customer at a time in a non-preemptive way, or serve all customers simultaneously. The analysis of the steady-state distribution of the queue-length and the waiting time in such systems is complex and closed-form results are available only in particular cases. We therefore set out to develop approximations for the steady-state distribution of these performance metrics. We first analyze the performance in light traffic. Using known results in the heavy-traffic regime, we then show how to develop an interpolation-based approximation that is valid for any load in the system. An advantage of the approach taken is that it is not model dependent and hence could potentially be applied to other complex queueing models. We numerically assess the accuracy of the interpolation approximation through the first and second moments
Sojourn time asymptotics in processor sharing queues
This paper addresses the sojourn time asymptotics for a GI/GI/âą 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
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