Regenerative Process

A stochastic process $\{X_t\}$ is regenerative if there exist random times $0\le S_0<S_1<S_2<\cdots$ (the regeneration times) such that the process's future behavior after each $S_n$, i.e. $\{X_{S_n+t}{\}}_{t\ge 0}$, is independent of the past before $S_n$ and has the same probability distribution for every $n$. The intervals $[S_n,S_{n+1})$ are called regeneration cycles, and the sequence of cycle lengths $\{S_{n+1}-S_n\}$ forms an i.i.d. sequence, exactly the interarrival times of an embedded renewal process. This structure is what makes regenerative processes analytically tractable despite arbitrarily complex within-cycle dynamics: the renewal-reward theorem gives the long-run average of any quantity of interest as (expected total reward accumulated per cycle) divided by (expected cycle length), reducing an infinite-horizon question to a single-cycle calculation.

• A Markov chain returning to a fixed recurrent state $i$ is a regenerative process with regeneration times exactly the successive visit times to $i$, this is the deep structural reason recurrent Markov chains have well-defined, computable long-run stationary distributions in the first place.
• Queueing systems are a workhorse application: the moments when a queue empties completely (goes idle) are natural regeneration points, letting analysts compute long-run average waiting time or queue length by studying the behavior of just one busy-idle cycle and applying the renewal-reward theorem.
• The regenerative structure is a powerful tool for simulation as well as theory, since successive cycles are i.i.d., independent replicates of a simulated system's behavior can be obtained just by running past several regeneration points, giving a principled way to construct confidence intervals for long-run simulation output.