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In queueing theory, a discipline within the mathematical theory of probability, an 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

An 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]

πi={0 when i=0(1δ)δi1 when i>0

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 − 2μβδ)/μ2(1 – δ).[4]

Waiting times

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

References

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