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Averaging for some simple constrained Markov processes

Genadot Alexandre

Probability and Mathematical Statistics

2019

Vol. 39, Fasc. 1

139--158

In this paper, a class of piecewise deterministic Markov processes with underlying fast dynamic is studied. By using a “penalty method”, an averaging result is obtained when the underlying dynamic is infinitely accelerated. The features of the averaged process, which is still a piecewise deterministic Markov process, are fully described.

Martingale approach for tail asymptotic problems in the generalized Jackson network

Miyazawa M.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 2

395--430

We study the tail asymptotic of the stationary joint queue length distribution for a generalized Jackson network (GJN for short), assuming its stability. For the two-station case, this problem has recently been solved in the logarithmic sense for the marginal stationary distributions under the setting that arrival processes and service times are of phase-type. In this paper, we study similar tail asymptotic problems on the stationary distribution, but problems and assumptions are different. First, the asymptotics are studied not only for the marginal distribution but also the stationary probabilities of state sets of small volumes. Second, the interarrival and service times are generally distributed and light tailed, but of phase-type in some cases. Third, we also study the case that there are more than two stations, although the asymptotic results are less complete. For them, we develop a martingale method, which has been recently applied to a single queue with many servers by the author.

Markov processes conditioned to never exit a subspace of the state space

Palmowski Z., Rolski T.

Probability and Mathematical Statistics

2004

Vol. 24, Fasc. 2

339--353

In this paper we study Markov processes never exiting (NE) a subspace A of the state space E or, in other words, Markov processes conditioned to stay in the subspace A. We show how the knowledge of the exact asymptotics of the tail distribution of the exit time helps to find the suitable exponential martingale, which, in turn, serves for the change of measure. Under the new probability measure the process is the sought for never exiting one the subspace A. We also find its extended generator and study relationships between the invariant measure (INE) and the quasi-stationary (QS) distribution. We analyze in detail the PDMP processes.