## Quick Reference

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 1*μ* and *σ*^{2} denoting the mean and variance of the service time distribution and with *ρ*=*λ**μ,* 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*.

**From:**
queue
in
A Dictionary of Statistics »

*Subjects:*
Probability and Statistics.

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