M/G/k queue: Difference between revisions
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In [[queueing theory]], an '''M/G/k queue''' is a queue model where arrivals are '''M'''arkovian (modulated by a [[Poisson process]]), service times have a '''G'''eneral [[probability distribution|distribution]] and there are ''k'' servers. The model name is written in [[Kendall's notation]], and is an extension of the [[M/M/c queue]], where service times must be [[exponential distribution|exponentially distributed]]. |
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#REDIRECT[[M/G/1 queue]] |
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==Model definition== |
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A queue represented by a M/G/''k'' queue is a stochastic process whose [[state space]] is the set {0,1,2,3...}, where the value corresponds to the number of members in the queue, including any being served. Transitions from state ''i'' to ''i'' + 1 represent the arrival of a new queue member: the times between such arrivals have an [[exponential distribution]] with parameter λ. Transitions from state ''i'' to ''i'' − 1 represent a member who has been being served, finishing being served and departing: the length of time required for serving an individual queue member has a general distribution function. The lengths of times between arrivals and of service periods are [[random variable]]s which are assumed to be [[statistically independent]]. |
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==Results== |
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Results for an extended version of this problem, an M/G/''k'' queue with ''k'' > 1 servers remains an open problem.<ref>{{cite doi|10.1007/s11134-009-9147-4}}</ref> In this model, arrivals are determined by a [[Poisson process]] and jobs can be served by any one of ''k'' servers. Tijms ''et al.'' believe it is "not likely that computationally tractable methods can be developed to compute the exact numerical values of the steady-state probability in the M/G/''k'' queue."<ref name="tijms">{{cite jstor|1426474}}</ref> |
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Various approximations for the average queue size,<ref>{{cite doi|10.1287/opre.43.1.158}}</ref> average delay a job experiences<ref name="tijms" /> (where it is known no approximation using only the first two moments can be accurate in all cases<ref>{{cite doi|10.1007/s11134-009-9133-x}}</ref> and a [[Markov–Krein]] characterisation has been shown to produce tight bounds<ref>{{cite doi|10.1007/s11134-011-9248-8}}</ref>), stationary distribution<ref>{{cite doi|10.1007/s11134-008-9073-x}}</ref><ref>{{cite jstor|169760}}</ref> and approximation by a [[reflected Brownian motion]]<ref>{{cite doi|10.1287/opre.31.2.304}}</ref><ref>{{cite doi|10.1287/opre.33.6.1266}}</ref> have been offered by different authors. See the [[M/M/c queue]] article for the case where service times are exponentially distributed. |
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===M/G/2 queue=== |
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For an M/G/2 queue, the model with two servers, the problem of determining marginal probabilities can be reduced to solving a pair of [[integral equation]]s<ref>{{cite doi|10.1287/opre.38.3.506}}</ref> or the Laplace transform of the distribution when the service time distribution is a mixture of exponential distributions.<ref>{{cite doi|10.1016/0304-4149(82)90046-1}}</ref> The Laplace transform of queue length<ref>{{cite jstor|1426776}}</ref> and waiting time distributions<ref>{{cite doi|10.1023/A:1017913826973}}</ref> can be computed when the waiting time distribution has a rational Laplace transform. |
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==References== |
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{{Reflist}} |
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{{Queueing theory}} |
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{{Stochastic processes}} |
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{{DEFAULTSORT:M G k queue}} |
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[[Category:Single queueing nodes]] |
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[[Category:Stochastic processes]] |
Revision as of 18:35, 12 June 2013
In queueing theory, an M/G/k queue is a queue model where arrivals are Markovian (modulated by a Poisson process), service times have a General distribution and there are k servers. The model name is written in Kendall's notation, and is an extension of the M/M/c queue, where service times must be exponentially distributed.
Model definition
A queue represented by a M/G/k queue is a stochastic process whose state space is the set {0,1,2,3...}, where the value corresponds to the number of members in the queue, including any being served. Transitions from state i to i + 1 represent the arrival of a new queue member: the times between such arrivals have an exponential distribution with parameter λ. Transitions from state i to i − 1 represent a member who has been being served, finishing being served and departing: the length of time required for serving an individual queue member has a general distribution function. The lengths of times between arrivals and of service periods are random variables which are assumed to be statistically independent.
Results
Results for an extended version of this problem, an M/G/k queue with k > 1 servers remains an open problem.[1] In this model, arrivals are determined by a Poisson process and jobs can be served by any one of k servers. Tijms et al. believe it is "not likely that computationally tractable methods can be developed to compute the exact numerical values of the steady-state probability in the M/G/k queue."[2]
Various approximations for the average queue size,[3] average delay a job experiences[2] (where it is known no approximation using only the first two moments can be accurate in all cases[4] and a Markov–Krein characterisation has been shown to produce tight bounds[5]), stationary distribution[6][7] and approximation by a reflected Brownian motion[8][9] have been offered by different authors. See the M/M/c queue article for the case where service times are exponentially distributed.
M/G/2 queue
For an M/G/2 queue, the model with two servers, the problem of determining marginal probabilities can be reduced to solving a pair of integral equations[10] or the Laplace transform of the distribution when the service time distribution is a mixture of exponential distributions.[11] The Laplace transform of queue length[12] and waiting time distributions[13] can be computed when the waiting time distribution has a rational Laplace transform.
References
- ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/s11134-009-9147-4, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with
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