In queueing theory, a discipline within the mathematical theory of probability, an **M/D/c queue** represents the queue length in a system having *c* servers, 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]} The model is an extension of the M/D/1 queue which has only a single server.

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Model definition

An M/D/*c* 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*).
*c* servers serve customers 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.

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Waiting time distribution

Erlang showed that when *ρ* = (*λ* *D*)/*c* < 1, the waiting time distribution has distribution F(*y*) given by^{[4]}

- $F(y)=\int _{0}^{\infty }F(x+y-D){\frac {\lambda ^{c}x^{c-1)){(c-1)!))e^{-\lambda x}{\text{d))x,\quad y\geq 0\quad c\in \mathbb {N} .$

Crommelin showed that, writing *P*_{n} for the stationary probability of a system with *n* or fewer customers,
^{[5]}

- $\mathbb {P} (W\leq x)=\sum _{n=0}^{c-1}P_{n}\sum _{k=1}^{m}{\frac {(-\lambda (x-kD))^{(k+1)c-1-n)){((K+1)c-1-n)!))e^{\lambda (x-kD)},\quad mD\leq x<(m+1)D.$