Renewal Theory

Renewal theory studies event counts generated by iid interarrival times. Let $S_1,S_2,\dots$ be positive iid waiting times and

$$T_n=\sum _{i=1}^n{S}_i,N(t)=\max \{n:T_n\le t\}.$$

Here $S_i$ is the waiting time before event $i$, $T_n$ is the time of the $n$th event, and $N(t)$ is the number of events observed by time horizon $t$. The process renews because each new waiting time has the same distribution and does not depend on the past.

The elementary renewal theorem gives the long-run rate:

$$\frac{\mathbb{E}[N(t)]}{t}\to \frac{1}{\mathbb{E}[S_1]}.$$

The left side is the expected count per unit time. The right side is the reciprocal of the mean waiting time, so events become less frequent when average interarrival times are longer.

This is a counting-process version of the law of large numbers. Markov renewal processes add state-dependent transition and holding-time structure.

Worked simulation

The code simulates many renewal processes with Gamma-distributed interarrival times and compares the observed count rate to the theorem’s $1/\mathbb{E}[S]$ prediction.

import numpy as np
 
rng = np.random.default_rng(20260711)
inter = rng.gamma(shape=2.0, scale=3.0, size=(20000, 400))
times = inter.cumsum(axis=1)
T = 1000
counts = (times <= T).sum(axis=1)
print("mean_interarrival", round(inter.mean(), 4), "theory_rate", round(1 / 6, 4))
print("mean_count_by_1000", round(counts.mean(), 3),
      "count_per_time", round(counts.mean() / T, 4))

Observed output:

mean_interarrival 6.0013 theory_rate 0.1667
mean_count_by_1000 166.289 count_per_time 0.1663

The simulated mean interarrival time is 6.0013, close to the modeled mean 6, so the theoretical renewal rate is about 0.1667. The observed count per unit time, 0.1663, is close to $1/\mathbb{E}[S]$.

Object	Role
Interarrival time $S_i$	Time between event $i-1$ and event $i$.
Arrival time $T_n$	Sum of the first $n$ waiting times.
Count $N(t)$	Number of arrivals observed by horizon $t$.
Long-run rate $1/\mathbb{E}[S]$	Expected event frequency after transient effects average out.

Caveats

The renewal assumption fails when events change future risk, when waiting times depend on covariates, or when the system does not reset after an event. A Poisson process is only the exponential-waiting-time special case and is also a continuous-time Markov chain.

References

• Renewal theory
• SciPy statistics reference

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