Renewal processes

A renewal process is a stochastic point process {N(t) = number of occurrences by time t} which describes the appearance of a sequence of instant random events where the times between occurrences (e.g., called interarrival times in queueing theory) are a sequence of independent and identically distributed (i.i.d.) random variables. It is common to write the interoccurrence distribution function as F(t) and its density (if it exists) as f(t), with expected value 1/μ. The Poisson process represents a particularly important renewal process in which the intervals between occurrences are identically exponentially distributed (Cox, 1960; Cox and Isham, 1980; Feller, 1966; Smith, 1955).

The so-called renewal equation for the process expectation (or renewal function) H(t) = Ev[N(t)], plays a fundamental role in all renewal problems:

The derivative of H(t), h(t) = dH/dt, is often called the intensity function and has a simple interpretation: h(t)dtis the approximate probability of an...