M/D/1 queue

In queueing theory, a discipline within the mathematical theory of probability, an M/D/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times are fixed (deterministic). The model name is written in Kendall's notation. Agner Krarup Erlang first published on this model in 1909, starting the subject of queueing theory. An extension of this model with more than one server is the M/D/c queue. In queueing theory, a discipline within the mathematical theory of probability, an M/D/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times are fixed (deterministic). The model name is written in Kendall's notation. Agner Krarup Erlang first published on this model in 1909, starting the subject of queueing theory. An extension of this model with more than one server is the M/D/c queue. An M/D/1 queue is a stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of entities in the system, including any currently in service. The state space diagram for M/D/1 queue is as below: The transition probability matrix for an M/D/1 queue with arrival rate λ and service time 1, such that λ <1 (for stability of the queue) is given by P as below: P = ( a 0 a 1 a 2 a 3 . . . a 0 a 1 a 2 a 3 . . . 0 a 0 a 1 a 2 . . . 0 0 a 0 a 1 . . . . . . . . . . . . . . . . . . ) {displaystyle P={egin{pmatrix}a_{0}&a_{1}&a_{2}&a_{3}&...\a_{0}&a_{1}&a_{2}&a_{3}&...\0&a_{0}&a_{1}&a_{2}&...\0&0&a_{0}&a_{1}&...\...&...&...&...&...\end{pmatrix}}} ， a n = λ n n ! e − λ {displaystyle a_{n}={frac {lambda ^{n}}{n!}}e^{-lambda }} , n = 0,1,....