G/G/1 queue

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In queueing theory, a discipline within the mathematical theory of probability, the G/G/1 queue represents the queue length in a system with a single server where interarrival times have a general (meaning arbitrary) distribution and service times have a (different) general distribution.[1] The evolution of the queue can be described by the Lindley equation.[2]

The system is described in Kendall's notation where the G denotes a general distribution for both interarrival times and service times and the 1 that the model has a single server.[3][4] Different interarrival and service times are considered to be independent, and sometimes the model is denoted GI/GI/1 to emphasise this.

Waiting time

Kingman's formula gives an approximation for the mean waiting time in a G/G/1 queue.[5] Lindley's integral equation is a relationship satisfied by the stationary waiting time distribution which can be solved using the Wiener–Hopf method.[6]

Multiple servers

Few results are known for the general G/G/k model as it generalises the M/G/k queue for which few metrics are known. Bounds can be computed using mean value analysis techniques, adapting results from the M/M/s queue model, using heavy traffic approximations, empirical results[7]: 189 [8] or approximating distributions by phase type distributions and then using matrix analytic methods to solve the approximate systems.[7]: 201 

In a G/G/2 queue with heavy-tailed job sizes, the tail of the delay time distribution us known to behave like the tail of an exponential distribution squared under low loads and like the tail of an exponential distribution for high loads.[9][10][11]

References

  1. ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/978-0-8176-4725-4_9, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1007/978-0-8176-4725-4_9 instead.
  2. ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1002/9780470400531.eorms0878, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1002/9780470400531.eorms0878 instead.
  3. ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1214/aoms/1177728975, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1214/aoms/1177728975 instead.
  4. ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1017/S0305004100028620, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1017/S0305004100028620 instead.
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  6. ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/978-3-642-80838-8_5, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1007/978-3-642-80838-8_5 instead.
  7. ^ a b Gautam, Natarajan (2012). Analysis of Queues: Methods and Applications. CRC Press. ISBN 9781439806586.
  8. ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1111/j.1937-5956.1993.tb00094.x, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1111/j.1937-5956.1993.tb00094.x instead.
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