M/G/1 queue

In queueing theory, an M/G/1 queue is a queue model where arrivals are Markovian (modulated by a Poisson process), service times have a General distribution and there is a single server.^[1] The model name is written in Kendall's notation, and is an extension of the M/M/1 queue, where service times must be exponentially distributed. The classic application of the M/G/1 queue is to model performance of a fixed head hard disk.^[2]

Model definition

A queue represented by a M/G/1 queue is a stochastic process whose state space is the set {0,1,2,3...}, where the value corresponds to the number of customers in the queue, including any being served. Transitions from state i to i + 1 represent the arrival of a new customer: the times between such arrivals have an exponential distribution with parameter λ. Transitions from state i to i − 1 represent a customer who has been being served, finishing being served and departing: the length of time required for serving an individual customer 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.

Queue length

Pollaczek–Khinchine method

The probability generating function of the stationary queue length distribution is given by the Pollaczek–Khinchine transform equation^[2]

\pi (z)={\frac {(1-z)(1-\rho )g(\lambda (1-z))}{g(\lambda (1-z))-z}}

where g(s) is the Laplace transform of the service time probability density function.^[3] In the case of an M/M/1 queue where service times are exponentially distributed with parameter μ, g(s) = μ/(μ + s).

This can be solved for individual state probabilities either using by direct computation or using the method of supplementary variables. The Pollaczek–Khinchine formula gives the mean queue length and mean waiting time in the system.^[4]^[5]

Matrix analytic method

Consider the embedded Markov chain of the M/G/1 queue, where the time points selected are immediately after the moment of departure. The customer being served (if there is one) has received zero seconds of service. Between departures, there can be 0, 1, 2, 3,… arrivals. So from state i the chain can move to state i – 1, i, i + 1, i + 2, ….^[6] The embedded Markov chain has transition matrix

P={\begin{pmatrix}a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&\cdots \\a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&\cdots \\0&a_{0}&a_{1}&a_{2}&a_{3}&\cdots \\0&0&a_{0}&a_{1}&a_{2}&\cdots \\0&0&0&a_{0}&a_{1}&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{pmatrix}}

where

a_{v}=\int _{0}^{\infty }e^{-\lambda u}{\frac {(\lambda u)^{v}}{v!}}{\text{d}}F(u)~{\text{ for }}v\geq 0

and F(u) is the service time distribution and λ the Poisson arrival rate of jobs to the queue.

Markov chains with generator matrices or block matrices of this form are called M/G/1 type Markov chains,^[7] a term coined by M. F. Neuts.^[8]^[9] The stationary distribution of an M/G/1 type Markov model can be computed using the matrix analytic method.^[10]

Busy period

The busy period is the time spent in states 1, 2, 3,… between visits to the state 0. The expected length of a busy period is 1/(μ−λ) where μ is the expected length of service time and λ the rate of the Poisson process governing arrivals.^[11] The busy period probability density function $\phi (s)$ can be shown to obey the functional relationship^[12]

\phi (s)=g[s+\lambda -\lambda \phi (s)]

where again g is the Laplace–Stieltjes transform of the service time distribution function.

Waiting/response time

Writing W^*(s) for the Laplace–Stieltjes transform transform of the waiting time distribution,^[13] is given by the Pollaczek–Khinchine transform as

W^{\ast }(s)={\frac {(1-\rho )sg(s)}{s-\lambda (1-g(s))}}

where g(s) is the Laplace–Stieltjes transform of serivice time probability density function.

Arrival theorem

As the arrivals are determined by a Poisson process, the arrival theorem holds.

Multiple servers

Many metrics for the M/G/k queue with k servers remain an open problem, though some approximations and bounds are known.

References

^ Gittins, John C. (1989). Multi-armed Bandit Allocation Indices. John Wiley & Sons. p. 77. ISBN 0471920592.
^ ^a ^b Harrison, Peter; Patel, Naresh M. (1992). Performance Modelling of Communication Networks and Computer Architectures. Addison–Wesley.
^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1088/0967-1846/3/1/003, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1088/0967-1846/3/1/003 instead.
^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/BF01194620, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1007/BF01194620 instead.
^ Khintchine, A. Y (1932). "Mathematical theory of a stationary queue". Matematicheskii Sbornik. 39 (4): 73–84. Retrieved 2011-07-14.
^ Stewart, William J. (2009). Probability, Markov chains, queues, and simulation. Princeton University Press. p. 510. ISBN 0-691-14062-6.
^ Neuts, Marcel F. (1981). Matrix-geometric solutions in stochastic models: an algorithmic approach (Johns Hopkins Studies in Mathematical Sciences). Johns Hopkins University Press. p. 2. ISBN 0-486-68342-7.
^ Neuts, M. F . (1989). "Structured Stochastic Matrices of M/G/1 Type and Their Applications". New York: Marcel Dekk. {{cite journal}}: Cite journal requires |journal= (help)
^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1080/15326349808807483, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1080/15326349808807483 instead.
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^ Ross, Sheldon M.; Seshadri, Sridhar (1999). "Hitting time in an M/G/1 queue" (PDF). Journal of Applied Probability: 934–940. JSTOR 3215453.
^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1017/CBO9781139173087, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1017/CBO9781139173087 instead. page 91
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[1] Gittins, John C. (1989). Multi-armed Bandit Allocation Indices. John Wiley & Sons. p. 77. ISBN 0471920592.

[harrison-2] Harrison, Peter; Patel, Naresh M. (1992). Performance Modelling of Communication Networks and Computer Architectures. Addison–Wesley.

[3] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1088/0967-1846/3/1/003, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1088/0967-1846/3/1/003 instead.

[4] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/BF01194620, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1007/BF01194620 instead.

[5] Khintchine, A. Y (1932). "Mathematical theory of a stationary queue". Matematicheskii Sbornik. 39 (4): 73–84. Retrieved 2011-07-14.

[6] Stewart, William J. (2009). Probability, Markov chains, queues, and simulation. Princeton University Press. p. 510. ISBN 0-691-14062-6.

[7] Neuts, Marcel F. (1981). Matrix-geometric solutions in stochastic models: an algorithmic approach (Johns Hopkins Studies in Mathematical Sciences). Johns Hopkins University Press. p. 2. ISBN 0-486-68342-7.

[8] Neuts, M. F . (1989). "Structured Stochastic Matrices of M/G/1 Type and Their Applications". New York: Marcel Dekk. {{cite journal}}: Cite journal requires |journal= (help)

[9] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1080/15326349808807483, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1080/15326349808807483 instead.

[10] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1093/acprof:oso/9780198527688.001.0001, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1093/acprof:oso/9780198527688.001.0001 instead.

[11] Ross, Sheldon M.; Seshadri, Sridhar (1999). "Hitting time in an M/G/1 queue" (PDF). Journal of Applied Probability: 934–940. JSTOR 3215453.

[12] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1017/CBO9781139173087, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1017/CBO9781139173087 instead. page 91

[diagle-13] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/0-387-22859-4_5, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1007/0-387-22859-4_5 instead.

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v t e Queueing theory
Single queueing nodes	D/M/1 queue M/D/1 queue M/D/c queue M/M/1 queue Burke's theorem M/M/c queue M/M/∞ queue M/G/1 queue Pollaczek–Khinchine formula Matrix analytic method M/G/k queue G/M/1 queue G/G/1 queue Kingman's formula Lindley equation Fork–join queue Bulk queue
Arrival processes	Poisson point process Markovian arrival process Rational arrival process
Queueing networks	Jackson network Traffic equations Gordon–Newell theorem Mean value analysis Buzen's algorithm Kelly network G-network BCMP network
Service policies	FIFO LIFO Processor sharing Round-robin Shortest job next Shortest remaining time
Key concepts	Continuous-time Markov chain Kendall's notation Little's law Product-form solution Balance equation Quasireversibility Flow-equivalent server method Arrival theorem Decomposition method Beneš method
Limit theorems	Fluid limit Mean-field theory Heavy traffic approximation Reflected Brownian motion
Extensions	Fluid queue Layered queueing network Polling system Adversarial queueing network Loss network Retrial queue
Information systems	Data buffer Erlang (unit) Erlang distribution Flow control (data) Message queue Network congestion Network scheduler Pipeline (software) Quality of service Scheduling (computing) Teletraffic engineering
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