# 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 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, fininshing 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.

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## Queue length

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

$\Pi(z) = \frac{(1-\rho)(1-z)G(z)}{G(z)-z}$

where $G(z)=B^*[\lambda(1-z)]$ and $B^*$ is the Laplace–Stieltjes transform of the service time distribution function. This can be found 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.[3][4]

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## 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.[5] The busy period probability density function $\phi(s)$ can be shown to obey the functional relationship[6]

$\phi(s) = B^*[s+\lambda - \lambda \phi(s)]$

where again $B^*$ is the Laplace–Stieltjes transform of the service time distribution function.

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## Arrival theorem

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

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## M/G/1 type Markov chains

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, ….[7] 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,[8] a term coined by M. F. Neuts.[9][10] The stationary distribution of an M/G/1 type Markov model can be computed using Ramaswami's formula.[11]

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## Multiple servers

Results for an extended version of this problem, an M/G/k queue with k > 1 servers remains an open problem.[12] 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."[13]

Various approximations for the average queue size,[14] average delay a job experiences,[13] stationary distribution[15][16] and approximation by a reflected Brownian motion[17][18] 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[19] or the Laplace transform of the distribution when the service time distribution is a mixture of exponential distributions.[20] The Laplace transform of queue length[21] and waiting time distributions[22] can be computed when the waiting time distribution has a rational Laplace transform.

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## References

1. ^ Gittins, John C. (1989). Multi-armed Bandit Allocation Indices. John Wiley & Sons. p. 77. ISBN 0471920592.
2. ^ a b Harrison, Peter; Patel, Naresh M. (1992). Performance Modelling of Communication Networks and Computer Architectures. Addison–Wesley.
3. ^ Pollaczek, F. (1930). "Über eine Aufgabe der Wahrscheinlichkeitstheorie". Mathematische Zeitschrift 32: 64–75. doi:10.1007/BF01194620. edit
4. ^ Khintchine, A. Y (1932). "Mathematical theory of a stationary queue". Matematicheskii Sbornik 39 (4): 73–84. Retrieved 2011-07-14.
5. ^ Ross, Sheldon M.; Seshadri, Sridhar (1999). "Hitting time in an M/G/1 queue". Journal of Applied Probability: 934–940. JSTOR 3215453.
6. ^ Mitrani, I. (1997). Probabilistic Modelling. doi:10.1017/CBO9781139173087. ISBN 9781139173087. edit page 91
7. ^ Stewart, William J. (2009). Probability, Markov chains, queues, and simulation. Princeton University Press. p. 510. ISBN 0-691-14062-6.
8. ^ 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.
9. ^ Neuts, M. F . (1989). Structured Stochastic Matrices of M/G/1 Type and Their Applications. New York: Marcel Dekk.
10. ^ Meini, B. (1998). "Solving m/g/l type markov chains: Recent advances and applications". Communications in Statistics. Stochastic Models 14: 479–496. doi:10.1080/15326349808807483. edit
11. ^ Bini, D. A.; Latouche, G.; Meini, B. (2005). Numerical Methods for Structured Markov Chains. doi:10.1093/acprof:oso/9780198527688.001.0001. ISBN 9780198527688. edit
12. ^ Kingman, J. F. C. (2009). "The first Erlang century—and the next". Queueing Systems 63: 3–4. doi:10.1007/s11134-009-9147-4. edit
13. ^ a b Tijms, H. C.; Van Hoorn, M. H.; Federgruen, A. (1981). "Approximations for the Steady-State Probabilities in the M/G/c Queue". Advances in Applied Probability 13 (1): 186–206. doi:10.2307/1426474. JSTOR 1426474. edit
14. ^ Ma, B. N. W.; Mark, J. W. (1995). "Approximation of the Mean Queue Length of an M/G/c Queueing System". Operations Research 43: 158. doi:10.1287/opre.43.1.158. JSTOR 171768. edit
15. ^ Breuer, L. (2008). "Continuity of the M/G/c queue". Queueing Systems 58 (4): 321–331. doi:10.1007/s11134-008-9073-x. edit
16. ^ Hokstad, Per (1978). "Approximations for the M/G/m Queue". Operations Research (INFORMS) 26 (3): 510–523. doi:10.1287/opre.26.3.510. JSTOR 169760. edit
17. ^ Kimura, T. (1983). "Diffusion Approximation for an M/G/m Queue". Operations Research 31 (2): 304–321. doi:10.1287/opre.31.2.304. JSTOR 170802. edit
18. ^ Yao, D. D. (1985). "Refining the Diffusion Approximation for the M/G/m Queue". Operations Research 33 (6): 1266–1277. doi:10.1287/opre.33.6.1266. JSTOR 170637. edit
19. ^ Knessl, C.; Matkowsky, B. J.; Schuss, Z.; Tier, C. (1990). "An Integral Equation Approach to the M/G/2 Queue". Operations Research 38 (3): 506. doi:10.1287/opre.38.3.506. JSTOR 171363. edit
20. ^ Cohen, J. W. (1982). "On the M/G/2 queueing model". Stochastic Processes and their Applications 12 (3): 231–248. doi:10.1016/0304-4149(82)90046-1. edit
21. ^ Hokstad, Per (1979). "On the Steady-State Solution of the M/G/2 Queue". Advances in Applied Probability (Applied Probability Trust) 11 (1): 240–255. JSTOR 1426776. edit
22. ^ Boxma, O. J.; Deng, Q.; Zwart, A. P. (2002). "Waiting-Time Asymptotics for the M/G/2 Queue with Heterogeneous Servers". Queueing Systems 40: 5. doi:10.1023/A:1017913826973. edit
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