M/G/k queue: Difference between revisions

Revision as of 18:35, 12 June 2013

Revision as of 18:39, 12 June 2013

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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]]. Most performance metrics for this queueing system are not known and remain an [[open problem]].<ref>{{cite doi|10.1007/s11134-009-9147-4}}</ref>

==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''&nbsp;+&nbsp;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''&nbsp;&minus;&nbsp;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]].

==Steady state distribution==

==Results==

Results for an extended version of this problem, an M/G/''k'' queue with ''k''&nbsp;>&nbsp;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>

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.

==Delay/waiting time distribution==

===M/G/2 queue===

Approximations exist for the average delay a job experiences.<ref name="tijms" /> It is known that no approximation using only the first two moments can be accurate in all cases.<ref>{{cite doi|10.1007/s11134-009-9133-x}}</ref>

A [[Markov–Krein]] characterisation has been shown to produce tight bounds.<ref>{{cite doi|10.1007/s11134-011-9248-8}}</ref>

==Two servers==

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.

==References==