Renewal processes: expected renewals and interarrival times
A renewal process models events that “restart” a system: replacing lightbulbs, machine repairs, component swaps,
software restarts, or any repeated cycle where the time between events is random but identically distributed.
The key object is the counting process \(N(t)\), the number of renewals that occur by time \(t\).
1) Interarrival times and renewal times
Let \(X_1,X_2,\dots\) be i.i.d. interarrival times with:
mean \(E[X_i]=\mu\) and variance \(\mathrm{Var}(X_i)=\sigma^2\).
Define the \(n\)-th renewal time:
\[
S_n = X_1 + X_2 + \cdots + X_n.
\]
Then the renewal counting process is:
\[
N(t) = \max\{n:\, S_n \le t\}.
\]
\(N(t)\) is a step function: it increases by 1 at each renewal time \(S_1,S_2,\dots\). That is exactly what the animation draws.
2) The long-run rate and the renewal theorem
One of the most important results is that renewals occur at an asymptotic rate \(1/\mu\). Informally, if each cycle lasts
\(\mu\) time units on average, then after a long time \(t\), you expect roughly \(t/\mu\) renewals.
\[
\frac{N(t)}{t}\to \frac{1}{\mu}\quad (t\to\infty),
\qquad
\Rightarrow\quad
E[N(t)] \approx \frac{t}{\mu}.
\]
3) A refined approximation for \(E[N(t)]\)
Beyond the first-order term \(t/\mu\), a common refinement (for many “non-lattice” continuous interarrival distributions)
adds a constant correction that depends on variability:
\[
E[N(t)] \approx \frac{t}{\mu}+\frac{1}{2}\Big(\frac{\sigma^2}{\mu^2}-1\Big).
\]
Higher variance \(\sigma^2\) (more variability in cycle lengths) changes the expected count by shifting the renewal function.
This correction matters most when \(t\) is not extremely large.
4) Special case: exponential interarrivals (Poisson process)
If interarrivals are exponential with mean \(\mu\), then the renewal process is exactly a Poisson process with rate \(1/\mu\):
\[
X_i\sim \mathrm{Exp}\!\Big(\frac{1}{\mu}\Big)
\quad\Rightarrow\quad
N(t)\sim \mathrm{Poisson}\!\Big(\frac{t}{\mu}\Big),
\qquad
E[N(t)]=\frac{t}{\mu}\ \text{(exact)}.
\]
5) Inspection paradox (university preview)
A classic “gotcha” in renewal theory is the inspection paradox: if you observe a system at a random time,
the interval you land in tends to be longer than a typical interarrival interval. Intuitively, longer intervals occupy
more of the timeline and are more likely to be observed. This affects waiting-time questions and residual life.
6) What this calculator does
- Computes the long-run approximation \(E[N(t)]\approx t/\mu\) and the refined correction term.
- Uses the exact formula \(E[N(t)]=t/\mu\) when interarrivals are exponential (Poisson case).
- Estimates \(E[N(t)]\) by Monte Carlo simulation and shows a sample renewal timeline animation.
For a “Custom” distribution, simulation needs an explicit sampling model. This tool uses a moment-matched gamma distribution to generate positive interarrivals consistent with \(\mu\) and \(\sigma^2\).
