scholarly journals The heavy traffic limit of an unbalanced generalized processor sharing model

2008 ◽  
Vol 18 (1) ◽  
pp. 22-58 ◽  
Author(s):  
Kavita Ramanan ◽  
Martin I. Reiman
Keyword(s):  
Heavy Traffic ◽  
Processor Sharing ◽  
Generalized Processor Sharing ◽  
Heavy Traffic Limit

scholarly journals A diffusion model for two parallel queues with processor sharing: transient behavior and asymptotics

1999 ◽  
Vol 12 (4) ◽  
pp. 311-338 ◽  
Author(s):  
Charles Knessl
Keyword(s):  
Diffusion Approximation ◽  
Queue Length ◽  
Heavy Traffic ◽  
Transient Behavior ◽  
Processor Sharing ◽  
Parallel Queues ◽  
Poisson Arrival ◽  
Heavy Traffic Limit ◽  
Service Rates ◽  
Dependent Probability

We consider two identical, parallel M/M/1 queues. Both queues are fed by a Poisson arrival stream of rate λ and have service rates equal to μ. When both queues are non-empty, the two systems behave independently of each other. However, when one of the queues becomes empty, the corresponding server helps in the other queue. This is called head-of-the-line processor sharing. We study this model in the heavy traffic limit, where ρ=λ/μ→1. We formulate the heavy traffic diffusion approximation and explicitly compute the time-dependent probability of the diffusion approximation to the joint queue length process. We then evaluate the solution asymptotically for large values of space and/or time. This leads to simple expressions that show how the process achieves its stead state and other transient aspects.

scholarly journals Heavy-Traffic Comparison of a Discrete-Time Generalized Processor Sharing Queue and a Pure Randomly Alternating Service Queue

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2723
Author(s):  
Arnaud Devos ◽  
Joris Walraevens ◽  
Dieter Fiems ◽  
Herwig Bruneel
Keyword(s):  
Functional Equation ◽  
Discrete Time ◽  
Heavy Traffic ◽  
The Other ◽  
Queueing Models ◽  
Single Server ◽  
Processor Sharing ◽  
Two Dimensional ◽  
Heavy Traffic Limit ◽  
Heavy Traffic Approximations

This paper compares two discrete-time single-server queueing models with two queues. In both models, the server is available to a queue with probability 1/2 at each service opportunity. Since obtaining easy-to-evaluate expressions for the joint moments is not feasible, we rely on a heavy-traffic limit approach. The correlation coefficient of the queue-contents is computed via the solution of a two-dimensional functional equation obtained by reducing it to a boundary value problem on a hyperbola. In most server-sharing models, it is assumed that the system is work-conserving in the sense that if one of the queues is empty, a customer of the other queue is served with probability 1. In our second model, we omit this work-conserving rule such that the server can be idle in case of a non-empty queue. Contrary to what we would expect, the resulting heavy-traffic approximations reveal that both models remain different for critically loaded queues.

AnRG-Factorization Approach for a BMAP/M/1 Generalized Processor-Sharing Queue

2005 ◽  
Vol 21 (2-3) ◽  
pp. 507-530 ◽  
Author(s):  
Quan-Lin Li ◽  
Zhaotong Lian ◽  
Liming Liu
Keyword(s):  
Processor Sharing ◽  
Factorization Approach ◽  
Generalized Processor Sharing
Sign in / Sign up

Export Citation Format

Share Document