Lecture-19: Queueing Networks
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چکیده
Corollary 1.1. Consider an M/M/s queue with Poisson(λ ) arrivals and each server having exponential service time exp(μ) service. If λ > sμ , then the output process in steady state is Poisson(λ ). 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 unity are the arrival instants of a Poisson process. Hence, by time reversibility, the time points at which X(t) increases by unity 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. Lemma 1.2. For an ergodic M/M/1 queue in steady state, the following are true.
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