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. 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. 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. 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. 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. When μβ > 1, the queue has stationary distribution where δ is the root of the equation δ = e-μβ(1 – δ) with smallest absolute value. The mean stationary idle time of the queue (period with 0 customers) is β – 1/μ, with variance (1 + δ − 2μβδ)/μ2(1 – δ). The mean stationary waiting time of arriving jobs is (1/μ) δ/(1 – δ).