Queueing theory

Definition

Let $A$ and $B$ be probability distributions on $(0, \infty)$ and let $c \ge 1$ be an integer. A single-station queue $A/B/c$, in Kendall's notation, is the following model:

1. customers arrive at times $0 < \tau_1 < \tau_2 < \cdots$ whose gaps $\tau_{n+1} - \tau_n$ and first time $\tau_1$ are independent and identically distributed with law $A$;
2. customer $n$ needs a service time $\sigma_n$, where $\sigma_1, \sigma_2, \ldots$ are independent with law $B$ and independent of the arrivals;
3. there are $c$ servers and an unlimited waiting room, and customers are served in order of arrival by the first free server.

The queue length $Q(t)$ is the number of customers in the system at time $t$, waiting or in service. Queueing theory studies such models and their networks.