A new approach is used to obtain the transient probabilities of the M/M/1 queueing system. The first step of this approach deals with the generating function of the transient probabilities of the uniformized Markov chain associated with this queue. The second step consists of the inversion of this generating function. A new analytical expression of the transient probabilities of the M/M/1 queue is then obtained.
A new approach is used to obtain the transient probabilities of the M/M/1 queueing system. The first step of this approach deals with the generating function of the transient probabilities of the uniformized Markov chain associated with this queue. The second step consists of the inversion of this generating function. A new analytical expression of the transient probabilities of the M/M/1 queue is then obtained.
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ABSTRACTWe consider the question of whether a function of a finite-state Markov chain is also Markovian, that is whether the chain is lumpable with respect to the partition determined by the function. We explore how an initial distribution with respect to which the chain is lumpable may differ from a pseudo-stationary initial distribution. Our results give insight into Peng's condition under which the Chapman-Kolmogorov equation implies that the lumped chain is Markovian. We illustrate these ideas by treating the question of whether the absorption time of a finite-state absorbing Markov chain is geometric.
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