At a certain level of abstraction, computing and communication systems as well as banking, manufacturing and transport systems, can be described in terms of ‘jobs’ and ‘servers’, i.e. requests for service and devices that provide service. The jobs may be computing tasks, input/output commands, telephone calls, data packets. The servers may be processors, storage devices, communication channels, software modules. A model aimed at evaluating and predicting the performance of such a system has to capture the following essential aspects of its behaviour:

1. (a) The pattern of demand, i.e. the the manner in which jobs arrive into the system and the nature of services that they require.

2. (b) The competition for service, i.e. the effect of admission, queueing and routing policies on performance.

3. This chapter is devoted to (a). It introduces tools and results that are used when modelling the arrivals and services of jobs.

Renewal processes

Consider a phenomenon which takes place first at time 0 and thereafter keeps occurring, at random intervals, ad infinitum. Denote the consecutive instants of occurrence by Tn (n = 0,1,…; T0 0), and let Sn Tn — Tn-1 (n = 1,2,…) be the intervals between them. Assume that the random variables Sn are independent and identically distributed.