Renewal Theorem

For a renewal process with i.i.d. interarrival times having finite mean $\mu =\mathbb{E}[T_i]$, the elementary renewal theorem states that $\frac{N(t)}{t}\to \frac{1}{\mu }$ almost surely (and in expectation, $\frac{\mathbb{E}[N(t)]}{t}\to \frac{1}{\mu }$) as $t\to \infty$, the long-run renewal rate is exactly the reciprocal of the mean interarrival time. Blackwell's renewal theorem strengthens this into a statement about a fixed-length window: if the interarrival distribution is non-lattice (not concentrated on multiples of some fixed spacing), then $\mathbb{E}[N(t+h)-N(t)]\to h/\mu$ as $t\to \infty$, for any fixed window length $h$, the expected number of renewals in any window of length $h$, far enough into the process's history, converges to $h/\mu$, independent of exactly where the window sits. The renewal theorem underlies the derivation of long-run average formulas throughout queueing theory and reliability engineering, wherever a renewal-reward argument is used to compute a long-run rate.

• The non-lattice condition in Blackwell's theorem is not a technicality that can be ignored, if interarrival times are concentrated on multiples of a fixed spacing (e.g. always exactly 1, 2, or 3 hours), renewals can cluster at specific phases relative to the starting time forever, and the clean convergence to $h/\mu$ genuinely fails without this assumption.
• The renewal theorem's rate interpretation, $1/\mu$, gives a memorable and broadly reusable rule of thumb: whenever some event repeats with i.i.d. inter-event times, its long-run frequency is simply the reciprocal of its average waiting time, regardless of the shape of that waiting-time distribution.
• This result underlies the renewal-reward theorem's long-run average formula (long-run average reward per unit time equals expected reward per cycle divided by expected cycle length), a widely applied technique throughout applied probability for computing steady-state performance metrics of systems that regenerate periodically.