ISSN 0005-1179, Automation and Remote Control, 2017, Vol. 78, No. 8, pp. 1361–1403.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
V.M. Vishnevskii, A.N. Dudin, 2017, published in Avtomatika i Telemekhanika, 2017, No. 8, pp. 3–59.
Queueing Systems with Correlated Arrival Flows
and Their Applications to Modeling
V. M. Vishnevskii
Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia
Belarusian State University, Minsk, Belarus
Received October 18, 2016
Abstract—We give a brief survey of literature devoted to studying queueing systems with
Markovian and batch Markovian arrival processes and their application to modeling telecom-
Keywords: queueing system, Markovian arrival process, stationary probability distribution,
wideband wireless networks.
Mathematical models of networks and queueing systems (QS) are widely used for study and
optimization of various technical, physical, economic, industrial, administrative, medical, military,
and other systems. The object of study in the theory of QS systems are situations when there is
a certain resource and a set of customers wishing to satisfy their need in this resource. Since the
resource is bounded, and the ﬂow of customers is random, some customers are rejected, or their
processing is delayed. The urge to reduce the number of these failures and the duration of delays
was the driving force behind the development of QS theory.
This development began in the works of a Danish mathematician and engineer A.K. Erlang
published in 1909–1922. These works were the result of his eﬀorts to solve design problems for
telephone networks. In these works, the ﬂow of customers arriving to a network node was modeled as
a Poisson ﬂow. This ﬂow is deﬁned as a stationary ordinary ﬂow without aftereﬀect, or as a recurrent
ﬂow with exponential distribution of interval durations between arrival moments. In addition,
Erlang presumed that the servicing time for a claim also has an exponential distribution. Under
these assumptions, since the exponential distribution has no “memory” the number of customers
in systems considered by Erlang was deﬁned by a one-dimensional Markov process, which makes it
easy to ﬁnd its stationary distribution. The assumption of an exponential form of the distribution
of interval durations between arrival moments appears rather strong and artiﬁcial. Nevertheless,
the theoretical results of Erlang agreed very well with the results of practical measurements in
real telephone networks. Later this phenomenon was explained in the works of A.Ya. Khinchin,
G.G. Ososkov, and B.I. Grigelionis who proved that under the assumption of uniform smallness
of ﬂow intensities a superposition of a large number of arbitrary recurrent ﬂows converges to a
stationary Poisson ﬂow. This result is a counterpart of the central limit theorem in probability
theory that states that under uniform smallness of random values their normalized sum in the
limit (as the number of terms tends to inﬁnity) converges in distribution to a random value with
standard normal distribution. Flows of customers arriving to a telephone station node represent