# D/M/1 queue

In queueing theory, a discipline within the mathematical theory of probability, a **D/M/1 queue** represents the queue length in a system having a single server, where arrivals occur at fixed regular intervals and job service requirements are random with an exponential distribution. The model name is written in Kendall's notation.^{[1]} Agner Krarup Erlang first published a solution to the stationary distribution of a D/M/1 and D/M/*k* queue, the model with *k* servers, in 1917 and 1920.^{[2]}^{[3]}

## Model definition

A D/M/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 deterministically at fixed times
*β*apart. - Service times are exponentially distributed (with rate parameter
*μ*). - 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.

## Stationary distribution

When *μβ* > 1, the queue has stationary distribution^{[4]}

where *δ* is the root of the equation *δ* = e^{-μβ(1 – δ)} with smallest absolute value.

### Idle times

The mean stationary idle time of the queue (period with 0 customers) is *β* – 1/*μ*, with variance (1 + *δ* − 2*μβδ*)/*μ*^{2}(1 – *δ*).^{[4]}

### Waiting times

The mean stationary waiting time of arriving jobs is (1/*μ*) *δ*/(1 – *δ*).^{[4]}

## 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. - ↑ Janssen, A. J. E. M.; Van Leeuwaarden, J. S. H. (2008). "Back to the roots of the M/D/s queue and the works of Erlang, Crommelin and Pollaczek" (PDF).
*Statistica Neerlandica*.**62**(3): 299. doi:10.1111/j.1467-9574.2008.00395.x. - 1 2 3 Jansson, B. (1966). "Choosing a Good Appointment System--A Study of Queues of the Type (D, M, 1)".
*Operations Research*.**14**(2): 292–312. doi:10.1287/opre.14.2.292. JSTOR 168256.