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ONE AN OVERVIEW
TWO MARKOV CHAINS
THREE PRODUCTFORM NETWORKS
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approximations argument arrival process arrival rate Arrival Theorem Assume average number bandit busy cycle calculate closed network Consider converges corresponds customer in queue customer of class defined Definition Denote discussed distributed with rate dynamic programming equal equations ergodic evolution Example Exercise exponentially distributed fact Figure finds finite follows formula function given Hint implies interarrival invariant distribution irreducible Jackson network jump Little's result M/M/l queue Markov chain minimize Norton's Theorem number of customers observed optimal policy Phase-type distributions point process Poisson process priority queue probability problem process with rate product-form proof Proposition prove quasi-reversible queues queue length queueing system queues in tandem random variables rate matrix rate X renewal process resp reversed second queue sequence served service completion service rate shows sojourn stationary stationary policy stationary process stoch stochastic tion typical customer verify waiting cost