M/M/1 queue

A M/M/1 queue an exponential distribution for the arrival and customer service. An M/M/1 queue is represented as follows:


In the majority of cases, the customer in a service is included in the number of costumers in the queue.

The number of costumers in the queue is modeled by the following continuous time Markov chain:


The stationary probabilities exist because the chain is irreducible. Let p(n) be the probability that the number of clients in the queue N(t)=n when t goes to infinity. The equilibrium equations gives the following system:


If we pose ρ=λ/μ then we find p(n)=ρnp(0), which implies:


From this we deduce that the queue is stable if ρ <1. That is, the average processing time of a customer is strictly less than the average arrival time of a customer (the average time between 2 customer arrivals). The queue is unstable if ρ≥1, in which case the clients accumulate infinitely in the queue.

All performance parameters are calculated in stationary mode in the case where the queue is stable. If we apply the law of Little and performance measures to M/M/1 queue (and more generally to M/M/C queues ):

  • For M/M/1 queues:


  • For M/M/C queues: