M/M/1 queue

In queueing theory, a discipline within the mathematical theory of probability, an M/M/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 have an exponential distribution. The model name is written in Kendall's notation. The model is the most elementary of queueing models and an attractive object of study as closed-form expressions can be obtained for many metrics of interest in this model. An extension of this model with more than one server is the M/M/c queue. An M/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. The model can be described as a continuous time Markov chain with transition rate matrix on the state space {0,1,2,3,...}. This is the same continuous time Markov chain as in a birth–death process. The state space diagram for this chain is as below. We can write a probability mass function dependent on t to describe the probability that the M/M/1 queue is in a particular state at a given time. We assume that the queue is initially in state i and write pk(t) for the probability of being in state k at time t. Then where ρ = λ / μ {displaystyle ho =lambda /mu } , a = 2 λ μ {displaystyle a=2{sqrt {lambda mu }}} and I k {displaystyle I_{k}} is the modified Bessel function of the first kind. Moments for the transient solution can be expressed as the sum of two monotone functions. The model is considered stable only if λ < μ. If, on average, arrivals happen faster than service completions the queue will grow indefinitely long and the system will not have a stationary distribution. The stationary distribution is the limiting distribution for large values of t.