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INTRODUCTION TO STOCHASTIC PROCESSES
THE POISSON PROCESS
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assumed average cost characteristic function common distribution Consider contraction mapping counter counting process customers arrive cycle defined Definition delayed renewal process denote the number dispatch distributed random variables distribution F embedded Markov chain Example exists expected cost expected number exponential distribution finite given Hence implies independent and identically independent increments independent random variables initial interarrival interval inventory key renewal theorem Laplace transform lattice Lemma let Xn limiting probabilities Markov chain Markov decision process Markov renewal process minimize nonnegative null recurrent number of customers number of renewals obtain optimal policy period Poisson process positive recurrent probability density function problem process with rate process X(t Proof Proposition 3.4 prove queueing system random walk regenerative result follows Section semi-Markov process server stationary point stationary point process stationary policy stochastic process Suppose Theorem 5.8 total number transient transition probabilities Wald's equation Wiener process yields