GI/G/1 processor sharing queue in heavy traffic

1994 ◽  
Vol 26 (02) ◽  
pp. 539-555 ◽  
Author(s):  
Sergei Grishechkin
Keyword(s):  
Queue Length ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Processor Sharing ◽  
Traffic Intensity ◽  
Random Measures ◽  
Processing Times ◽  
Processor Sharing Queues ◽  
Limiting Behaviour ◽  
Residual Processing

Consider GI/G/1 processor sharing queues with traffic intensity tending to 1. Using the theory of random measures and the theory of branching processes we investigate the limiting behaviour of the queue length, sojourn time and random measures describing attained and residual processing times of customers present.

GI/G/1 processor sharing queue in heavy traffic

1994 ◽  
Vol 26 (2) ◽  
pp. 539-555 ◽  
Author(s):  
Sergei Grishechkin
Keyword(s):  
Queue Length ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Processor Sharing ◽  
Traffic Intensity ◽  
Random Measures ◽  
Processing Times ◽  
Processor Sharing Queues ◽  
Limiting Behaviour ◽  
Residual Processing

Consider GI/G/1 processor sharing queues with traffic intensity tending to 1. Using the theory of random measures and the theory of branching processes we investigate the limiting behaviour of the queue length, sojourn time and random measures describing attained and residual processing times of customers present.

On a relationship between processor-sharing queues and Crump–Mode–Jagers branching processes

1992 ◽  
Vol 24 (3) ◽  
pp. 653-698 ◽  
Author(s):  
Sergei Grishechkin
Keyword(s):  
Processing Time ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Processor Sharing ◽  
Batch Arrivals ◽  
Processor Sharing Queues ◽  
Residual Processing

The M/G/1 queue with batch arrivals and a queueing discipline which is a generalization of processor sharing is studied by means of Crump–Mode–Jagers branching processes. A number of theorems are proved, including investigation of heavy traffic and overloaded queues. Most of the results obtained are also new for the M/G/1 queue with processor sharing. By use of a limiting procedure we also derive new results concerning M/G/1 queues with shortest residual processing time discipline.

On a relationship between processor-sharing queues and Crump–Mode–Jagers branching processes

1992 ◽  
Vol 24 (03) ◽  
pp. 653-698
Author(s):  
Sergei Grishechkin
Keyword(s):  
Processing Time ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Processor Sharing ◽  
Batch Arrivals ◽  
Processor Sharing Queues ◽  
Residual Processing

The M/G/1 queue with batch arrivals and a queueing discipline which is a generalization of processor sharing is studied by means of Crump–Mode–Jagers branching processes. A number of theorems are proved, including investigation of heavy traffic and overloaded queues. Most of the results obtained are also new for the M/G/1 queue with processor sharing. By use of a limiting procedure we also derive new results concerning M/G/1 queues with shortest residual processing time discipline.

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.

On berry–esseen rate for queue length of the GI/G/K system in heavy traffic

1988 ◽  
Vol 25 (03) ◽  
pp. 596-611
Author(s):  
Xing Jin
Keyword(s):  
Limit Theorem ◽  
Waiting Time ◽  
Queue Length ◽  
Queueing System ◽  
Heavy Traffic ◽  
Traffic Intensity ◽  
Main Method ◽  
Number Of Customers

This paper provides Berry–Esseen rate of limit theorem concerning the number of customers in a GI/G/K queueing system observed at arrival epochs for traffic intensity ρ > 1. The main method employed involves establishing several equalities about waiting time and queue length.

scholarly journals Stability of a Processor-Sharing Queue with Varying Throughput

2008 ◽  
Vol 45 (04) ◽  
pp. 953-962 ◽  
Author(s):  
Pascal Moyal
Keyword(s):  
Stability Criterion ◽  
Processor Sharing ◽  
Processing Times ◽  
Processor Sharing Queues ◽  
Point Measure ◽  
Number Of Customers

In this paper we present a stability criterion for processor-sharing queues, in which the throughput may depend on the number of customers in the system (such as in the case of interferences between users). Such a system is represented by a point measure-valued stochastic recursion keeping track of the remaining processing times of the customers.

On berry–esseen rate for queue length of the GI/G/K system in heavy traffic

1988 ◽  
Vol 25 (3) ◽  
pp. 596-611 ◽  
Author(s):  
Xing Jin
Keyword(s):  
Limit Theorem ◽  
Waiting Time ◽  
Queue Length ◽  
Queueing System ◽  
Heavy Traffic ◽  
Traffic Intensity ◽  
Main Method ◽  
Number Of Customers

This paper provides Berry–Esseen rate of limit theorem concerning the number of customers in a GI/G/K queueing system observed at arrival epochs for traffic intensity ρ > 1. The main method employed involves establishing several equalities about waiting time and queue length.

PERFORMANCE ESTIMATION OFM/DK/1 QUEUE UNDER FAIR SOJOURN PROTOCOL IN HEAVY TRAFFIC

2008 ◽  
Vol 23 (1) ◽  
pp. 61-74
Author(s):  
Yingdong Lu
Keyword(s):  
Markov Process ◽  
Waiting Time ◽  
Service Time ◽  
Infinitesimal Generator ◽  
Heavy Traffic ◽  
Queuing System ◽  
Performance Estimation ◽  
Processor Sharing ◽  
Average Waiting Time ◽  
Processor Sharing Queues

We study the performance of aM/DK/1 queue under Fair Sojourn Protocol (FSP). We use a Markov process with mixed real- and measure-valued states to characterize the queuing process of system and its related processor sharing queue. The infinitesimal generator of the Markov process is derived. Classifying customers according to their service time, using techniques in multiclass queuing system, and borrowing recently developed heavy traffic results for processor-sharing queues, we are able to derive approximations for average waiting time for the jobs.

scholarly journals Stability of a Processor-Sharing Queue with Varying Throughput

2008 ◽  
Vol 45 (4) ◽  
pp. 953-962 ◽  
Author(s):  
Pascal Moyal
Keyword(s):  
Stability Criterion ◽  
Processor Sharing ◽  
Processing Times ◽  
Processor Sharing Queues ◽  
Point Measure ◽  
Number Of Customers

In this paper we present a stability criterion for processor-sharing queues, in which the throughput may depend on the number of customers in the system (such as in the case of interferences between users). Such a system is represented by a point measure-valued stochastic recursion keeping track of the remaining processing times of the customers.

Note on the strong limiting behaviour of busy periods in GI/G/1 queues under heavy traffic

1993 ◽  
Vol 30 (2) ◽  
pp. 483-488
Author(s):  
Josef Steinebach ◽  
Hanqin Zhang
Keyword(s):  
Heavy Traffic ◽  
Traffic Intensity ◽  
Limiting Behaviour

In this note, the strong limiting behaviour of busy periods in GI/G/1 queues is studied under the condition that the traffic intensity equals unity.

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