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Lecture 18: Queueing Networks

Proof. Let X(t) denote the number of customers in the system at time t. Since M/M/s process is a birth and death process, it follows from the previous proposition that {X(t), t ≥ 0} is time reversible. Now going forward in time, the time instants at which X(t) increases by 1 are the arrival instants of a Poisson process. Hence, by time reversibility, the time Points at which X(t) increases by 1 when we go backwards in time also constitutes a Poisson process. But these instants are exactly the departure instants of the forward process. Hence the result.