M/D/1 queue

In queueing theory, a discipline within the mathematical theory of probability, an M/D/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times are fixed (deterministic). The model name is written in Kendall's notation.^[1] Agner Krarup Erlang first published on this model in 1909, starting the subject of queueing theory.^[2]^[3] An extension of this model with more than one server is the M/D/c queue.

Model definition

An M/D/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 system, including any currently in service.

Arrivals occur at rate λ according to a Poisson process and move the process from state i to i + 1.
Service times are deterministic time D (serving at rate μ = 1/D).
A single server serves customers one at a time from the front of the queue, according to a first-come, first-served discipline. When the service is complete the customer leaves the queue and the number of customers in the system reduces by one.
The buffer is of infinite size, so there is no limit on the number of customers it can contain.

The state space diagram for M/D/1 queue is as below:

Stationary distribution

The number of jobs in the queue can be written as an M/G/1 type Markov chain and the stationary distribution found for state i (written π_i) in the case D = 1 to be^[4]

{\begin{aligned}\pi _{0}&=1-\lambda \\\pi _{1}&=(1-\lambda )(e^{\lambda }-1)\\\pi _{n}&=(1-\lambda )\left(e^{n\lambda }+\sum _{k=1}^{n-1}e^{k\lambda }(-1)^{n-k}\left[{\frac {(k\lambda )^{n-k}}{(n-k)!}}+{\frac {(k\lambda )^{n-k-1}}{(n-k-1)!}}\right]\right)\quad (n\geq 2).\end{aligned}}

Delay

Define ρ = λ/μ as the utilization; then the mean delay in the system in an M/D/1 queue is^[5]

{\frac {1}{2\mu }}\cdot {\frac {2-\rho }{1-\rho }}.

and in the queue:

{\frac {1}{2\mu }}\cdot {\frac {\rho }{1-\rho }}.

Busy period

The busy period is the time period measured from the instant a first customer arrives at an empty queue to the time when the queue is again empty. This time period is equal to D times the number of customers served. If ρ < 1, then the number of customers served during a busy period of the queue has a Borel distribution with parameter ρ.^[6]^[7]

Finite capacity

Stationary distribution

A stationary distribution for the number of customers in the queue and mean queue length can be computed using probability generating functions.^[8]

Transient solution

The transient solution of an M/D/1 queue of finite capacity N, often written M/D/1/N, was published by Garcia et al in 2002.^[9]

References

^ Kendall, D. G. (1953). "Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain". The Annals of Mathematical Statistics. 24 (3): 338. doi:10.1214/aoms/1177728975. JSTOR 2236285.
^ Kingman, J. F. C. (2009). "The first Erlang century—and the next". Queueing Systems. 63: 3–4. doi:10.1007/s11134-009-9147-4.
^ Erlang, A. K. (1909). "The theory of probabilities and telephone conversations" (PDF). Nyt Tidsskrift for Matematik B. 20: 33–39. Archived from the original (PDF) on October 1, 2011. {{cite journal}}: Unknown parameter |deadurl= ignored (|url-status= suggested) (help)
^ Nakagawa, Kenji (2005). "On the Series Expansion for the Stationary Probabilities of an M/D/1 queue" (PDF). Journal of the Operations Research Society of Japan. 48 (2): 111–122.
^ Cahn, Robert S. (1998). Wide Area Network Design:Concepts and Tools for Optimization. Morgan Kaufmann. p. 319. ISBN 1558604588.
^ Tanner, J. C. (1961). "A derivation of the Borel distribution". Biometrika. 48: 222–224. doi:10.1093/biomet/48.1-2.222. JSTOR 2333154.
^ Haight, F. A.; Breuer, M. A. (1960). "The Borel-Tanner distribution". Biometrika. 47: 143. doi:10.1093/biomet/47.1-2.143. JSTOR 2332966.
^ Brun, Olivier; Garcia, Jean-Marie (2000). "Analytical Solution of Finite Capacity M/D/1 Queues". Journal of Applied Probability. 37 (4). Applied Probability Trust: 1092–1098. doi:10.1239/jap/1014843086. JSTOR 3215497.
^ Garcia, Jean-Marie; Brun, Olivier; Gauchard, David (2002). "Transient Analytical Solution of M/D/1/N Queues". Journal of Applied Probability. 39 (4). Applied Probability Trust: 853–864. JSTOR 3216008.

[1] Kendall, D. G. (1953). "Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain". The Annals of Mathematical Statistics. 24 (3): 338. doi:10.1214/aoms/1177728975. JSTOR 2236285.

[2] Kingman, J. F. C. (2009). "The first Erlang century—and the next". Queueing Systems. 63: 3–4. doi:10.1007/s11134-009-9147-4.

[3] Erlang, A. K. (1909). "The theory of probabilities and telephone conversations" (PDF). Nyt Tidsskrift for Matematik B. 20: 33–39. Archived from the original (PDF) on October 1, 2011. {{cite journal}}: Unknown parameter |deadurl= ignored (|url-status= suggested) (help)

[4] Nakagawa, Kenji (2005). "On the Series Expansion for the Stationary Probabilities of an M/D/1 queue" (PDF). Journal of the Operations Research Society of Japan. 48 (2): 111–122.

[5] Cahn, Robert S. (1998). Wide Area Network Design:Concepts and Tools for Optimization. Morgan Kaufmann. p. 319. ISBN 1558604588.

[6] Tanner, J. C. (1961). "A derivation of the Borel distribution". Biometrika. 48: 222–224. doi:10.1093/biomet/48.1-2.222. JSTOR 2333154.

[7] Haight, F. A.; Breuer, M. A. (1960). "The Borel-Tanner distribution". Biometrika. 47: 143. doi:10.1093/biomet/47.1-2.143. JSTOR 2332966.

[8] Brun, Olivier; Garcia, Jean-Marie (2000). "Analytical Solution of Finite Capacity M/D/1 Queues". Journal of Applied Probability. 37 (4). Applied Probability Trust: 1092–1098. doi:10.1239/jap/1014843086. JSTOR 3215497.

[9] Garcia, Jean-Marie; Brun, Olivier; Gauchard, David (2002). "Transient Analytical Solution of M/D/1/N Queues". Journal of Applied Probability. 39 (4). Applied Probability Trust: 853–864. JSTOR 3216008.

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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
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