Poisson processes and renewals

A homogeneous Poisson process starts with count zero and has independent stationary increments: counts on disjoint time intervals are independent, and an interval's count law depends only on its length. Over an interval of length t, the count has Poisson parameter lambda t. Lambda has units of events per time; the parameter is dimensionless. Both the mean and variance of that interval count equal lambda t. Specifying only an empirical average rate does not establish these stronger distributional and independence claims.

The process has isolated unit jumps almost surely. In a short interval h, a single arrival has probability lambda h plus a smaller-order remainder, while two or more arrivals have probability of order h squared. That short-time description is consistent with the Poisson count law. A workload that routinely submits groups of jobs simultaneously violates the simple unit-arrival model, even if its hourly mean happens to match. Nonhomogeneous Poisson models permit a deterministic changing rate but retain independent increments with integrated-rate parameters.

The first waiting time exceeds t precisely when there are no arrivals by t, giving the exponential survival function. Independent stationary increments then restart the same waiting-time law after each arrival. An exponential waiting time is memoryless: conditional on having already waited s, the probability of waiting at least another t is unchanged. This concerns the stochastic model, not a claim that a particular job grows no older while it waits or that a service system has no state.

Independent thinning gives another useful consistency check. Keep each Poisson event independently with probability q, without consulting other events or arrival times. The retained events form a Poisson process with rate q lambda; discarded events form an independent process with rate (1−q) lambda. This conclusion can fail if the keep rule depends on congestion or recent history. Superposition also preserves the Poisson family when the input processes are independent, but dependent streams need not combine into a Poisson process.

A renewal process starts at time zero and has independent identically distributed positive interarrival times T₁,T₂,… . The nth arrival occurs at their cumulative sum. Its count at time t is the largest n whose arrival time is no later than t. The strong law for these sums implies that the count divided by elapsed time approaches the reciprocal mean waiting time almost surely, provided the mean is finite and strictly positive. No exponential distribution is required for that long-run result.

A fresh start occurs at an arrival epoch because the future interarrival times are independent copies of the past ones. At an arbitrary observation time, the age and remaining length of the current interval generally matter. Long intervals are more likely to contain a randomly chosen observation time, producing the inspection paradox. Consequently a deterministic two-second spacing and exponential spacing of mean two seconds share rate one-half, yet have different residual-time distributions. Capacity calculations based on a rate alone omit burstiness, tails, and dependence that can dominate waiting-time performance.