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.{{Cite journal | last1 = Kendall | first1 = D. G. | author-link1 = David George Kendall| title = Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain | doi = 10.1214/aoms/1177728975 | jstor = 2236285| journal = The Annals of Mathematical Statistics | volume = 24 | issue = 3 | pages = 338 | year = 1953| url = http://projecteuclid.org/euclid.aoms/1177728975 | doi-access = free }} 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.{{Cite journal | last1 = Kingman | first1 = J. F. C. | author-link1 = John Kingman | title = The first Erlang century—and the next | journal = Queueing Systems | volume = 63 | pages = 3–4 | year = 2009 | doi = 10.1007/s11134-009-9147-4}}{{Cite journal | last1 = Janssen | first1 = A. J. E. M. | last2 = Van Leeuwaarden | first2 = J. S. H. | doi = 10.1111/j.1467-9574.2008.00395.x | title = Back to the roots of the M/D/s queue and the works of Erlang, Crommelin and Pollaczek | journal = Statistica Neerlandica | volume = 62 | issue = 3 | pages = 299 | year = 2008 | url = http://alexandria.tue.nl/repository/books/641261.pdf| doi-access = free }}
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{{Cite journal | last1 = Jansson | first1 = B.| title = Choosing a Good Appointment System--A Study of Queues of the Type (D, M, 1) | doi = 10.1287/opre.14.2.292 | journal = Operations Research | volume = 14 | issue = 2 | pages = 292–312 | year = 1966 | jstor = 168256}}
::
0 & \text{ when } i=0\\
(1-\delta)\delta^{i-1} &\text{ when } i>0
\end{cases}
where δ is the root of the equation δ = e-μβ(1 – δ) with smallest absolute value.