## Technical Report, Issue 178 |

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arbitrarily chosen arrival Billingsley 1965 Borel set clearing cycle clearing instant reaches constant construct the stationary continuous-review coordinate space cumulative number defined desired result discrete-time E2 Ek entering the system ergodic theorem Doob establishing an ergodic existence of regeneration finite-dimensional distributions following theorem shows hence W(u integer interarrival-time process xn interarrival-time sequence invariant o-algebra last clearing instant limiting average limiting distribution limiting-average waiting Markov process measure-preserving transformation underlying mechanism that intermittently metrically transitive non-stationary number of arrivals number of units OPERATIONS RESEARCH position of arrival Pr{W probability distribution probability measure probability space fi,A,P Proof prove an ergodic quantity enters random variable regeneration points Renyi Rosenblatt 1971 sample point stationary and limiting stationary distribution stationary process stationary version stationary waiting-time process Stidham stochastic clearing systems stochastic input process stochastic process Suppose system in 0,t theorem Doob 1953 Tp is ergodic unit averages UNIT INPUT PROCESSES