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Principle Seminar of the Department of Probability Theory, Moscow State University
December 3, 2014 16:45, Moscow, MSU, auditorium 12-24
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On a weak convergence of distribution of one-dimensional piecewise linear Markov process to the stationary one
G. A. Zverkina Moscow State University of Railway Communications
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Abstract:
The piecewise linear Markov processes (PLMPs) were invented by
B.V.Gnedenko and I.N.Kovalenko in 1966 as one of the convenient devices
for research of behavior of queueing systems. In queueing theory the
research of convergence and limit distributions of PLMPs describing
queueing systems has an important role; many researches are devoted to
this. However the exact algorithm of calculation of constants in
convergence rate estimation isn't always known (for various type of rate
estimation and in various metrics). Usually, for the many queueing
systems the convergence rate depends on stationary distribution.
Possibly, for the first time stationary distribution of PLMP for
queueing system with nonexponentially distributed service was described
by B.A.Sevastjanov (1957), but still stationary distribution of PLMPs
wasn't described in a general view. We’ll tell about necessary and
sufficient conditions of week convergence of one-dimensional PLMPs to
the stationary distribution, we’ll describe this stationary
distribution, also we’ll give an example of use of the stationary
distribution of PLMP for an estimate of convergence rate of distribution
of some queueing system to the stationary one in a total variation
metrics.
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