a renewal process

The Poisson process insists the gaps between events are exponential. But many real cycles are not: a light bulb has a fairly predictable lifetime before it is replaced, a machine part wears for a typical span before failing, a bus runs a roughly fixed route time. A renewal process keeps the beautiful idea that the system 'starts fresh' at each event but drops the exponential straitjacket: the gaps are independent and identically distributed from ANY distribution you like.

Formally, let the interarrival times T1, T2, T3, ... be independent and identically distributed with some common distribution F (any positive distribution, not necessarily exponential). The arrival times are the partial sums Sn = T1 + ... + Tn, and the counting process N(t) counts how many renewals have occurred by time t. Each renewal is a clean restart — the process forgets everything and begins a new independent cycle — which is the namesake idea. The Poisson process is exactly the special case where F is exponential; that special case is unique in being memoryless mid-gap, but every renewal process is memoryless at the renewal epochs themselves.

Renewal theory is the natural framework for replacement, maintenance, reliability, and inventory: it answers 'how many replacements by time t?', 'what is the long-run replacement rate?', and 'what fraction of time is the machine down?'. The single most important long-run fact is the renewal-reward intuition: over a long horizon the average rate of events tends to 1 / E[T], the reciprocal of the mean gap — so a part lasting 5 years on average gets replaced at a long-run rate of 1/5 per year. The price you pay for the general gaps is that the count N(t) is no longer Poisson and exact formulas get harder, which is why so much of renewal theory is about long-run (asymptotic) behavior.