An example of a continuous-time Markov chain (see Markov process). The properties of queues are much studied by analysts of stochastic processes. Three components of a queuing system are the inter-arrival times (a Markov process (M), a more general process (G), or a pre-determined process (D), the service times (also Markovian, general, or predetermined), and the number of servers (one or more). The standard nomenclature for queues describes them as M/M/1, M/D/1, M/G/1, or G/M/k queues as appropriate.
The basic quantities of interest are the expected values of the number in the queue, the number waiting (i.e. not being served) in the queue, the queueing time, and the waiting time (the sum of the queueing and service times).
For an M/G/1 queue, with arrival rate λ and with 1μ and σ2 denoting the mean and variance of the service time distribution and with ρ=λμ, then the expected value of the number in the system (queueing or being served) is n̄ given by the Pollaczek–Khintchine formula A useful general result, with t̄ denoting the average time spent in the system, is Little's formula which states that λt̄=n̄.
Subjects: Probability and Statistics.