The Poisson process assumes exponential inter-arrivals, which implies the memoryless property: the time until the next event is independent of how long you have already waited. Many real systems violate this. A neuron has a refractory period — it cannot fire again immediately after a spike. A machine component is more likely to fail the older it gets. A human takes at least some minimum time between heartbeats.
Renewal processes generalize the HPP by allowing arbitrary inter-arrival distributions while preserving the i.i.d. structure.
A renewal process is a point process whose inter-arrival times X₁, X₂, ... are i.i.d. non-negative random variables with distribution F and mean μ = E[X] < ∞.
tₙ = X₁ + X₂ + ... + Xₙ
N(t) = max{n : tₙ ≤ t}
The HPP is the special case F = Exponential(λ) with μ = 1/λ.
The conditional intensity of a renewal process depends only on the time since the last event, not on the full history:
λ*(t) = h(t − t_{N(t-)})
where h(x) = f(x) / (1 − F(x)) is the hazard function of F.
| Distribution | Parameters | Mean μ | Hazard h(x) | Character |
|---|---|---|---|---|
| Exponential(λ) | λ | 1/λ | λ (constant) | Memoryless (Poisson) |
| Gamma(k, β) | k, β | k/β | Increasing for k>1, decreasing for k<1 | Regular (k>1) or bursty (k<1) |
| Weibull(k, β) | k, β | β·Γ(1+1/k) | Increasing for k>1, decreasing for k<1 | Aging (k>1) or decreasing hazard (k<1) |
| Log-normal(μ, σ) | μ, σ | exp(μ+σ²/2) | Non-monotone | Heavy-tailed variability |
| Pareto(α, x_m) | α, x_m | αx_m/(α-1) | Decreasing | Heavy-tailed, very bursty |
The coefficient of variation CV = σ/μ quantifies regularity:
CV = 1: exponential (Poisson), Fano factor = 1CV < 1: more regular than Poisson (e.g., Gamma with k > 1)CV > 1: more variable/bursty than Poisson (e.g., log-normal, Pareto)The renewal function M(t) = E[N(t)] satisfies the renewal equation:
M(t) = F(t) + ∫₀ᵗ M(t − s) dF(s)
This integral equation says: the expected count by time t is the probability of at least one event by t, plus the expected count in the residual time after the first event.
Elementary renewal theorem: As t → ∞,
M(t) / t → 1/μ
The long-run rate is simply the reciprocal of the mean inter-arrival time, regardless of the distribution shape.
Variance of N(t): For large t,
Var[N(t)] ≈ σ²/μ³ · t = CV² · (1/μ) · t
The Fano factor F = Var/Mean → CV² for large t. This is a key diagnostic: F ≠ 1 signals non-Poisson behavior.
Simulating a renewal process is straightforward — just sample i.i.d. inter-arrivals:
def simulate_renewal(dist, T, **params):
times = []
t = 0.0
while True:
t += dist.rvs(**params)
if t > T:
break
times.append(t)
return np.array(times)
# Gamma renewal process (more regular than Poisson)
from scipy.stats import gamma
events = simulate_renewal(gamma, T=100.0, a=5.0, scale=0.5)
The scale parameter is 1/β, so a=5, scale=0.5 gives mean = 2.5 and CV = 1/√5 ≈ 0.45.
See code/05_renewal_processes.py for simulation with multiple distributions and Fano factor estimation.
Blackwell’s renewal theorem (non-lattice case): For h > 0 small,
M(t, t+h] → h/μ as t → ∞
The rate of renewals converges to 1/μ locally.
The inspection paradox: If you observe the process at a random time t, the length of the inter-arrival interval containing t (the “current life”) has a longer expected length than a typical inter-arrival interval. This is a size-biased sampling effect: longer intervals are more likely to contain the observation time.
For an exponential distribution, there is no paradox (the distribution of the current life equals Exponential(λ)). For all other distributions, the current life is stochastically larger than X.
An alternating renewal process alternates between ON and OFF periods with i.i.d. durations drawn from F_ON and F_OFF. The long-run fraction of time spent ON is:
p_ON = μ_ON / (μ_ON + μ_OFF)
This models machine availability (up/down cycles), sleep-wake cycles, and server ON/OFF patterns.
F; the HPP is the exponential special case.F → CV² for large t — a useful summary of dispersion compared to Poisson.CV < 1 → regular, CV = 1 → Poisson, CV > 1 → bursty.F until the cumulative sum exceeds T.| ← Chapter 4 | Table of Contents | Chapter 6: The Hawkes Process → |