ISSN 0032-9460, Problems of Information Transmission, 2008, Vol. 44, No. 4, pp. 352–369.
Pleiades Publishing, Inc., 2008.
Original Russian Text
L.G. Afanas’eva, E.E. Bashtova, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 4, pp. 72–91.
COMMUNICATION NETWORK THEORY
Limit Theorems for Queueing Systems
with Doubly Stochastic Poisson Arrivals
(Heavy Traﬃc Conditions)
L. G. Afanas’eva and E. E. Bashtova
Lomonosov Moscow State University,
Faculty of Mathematics and Mechanics, Probability Theory Chair
Received January 18, 2008; in ﬁnal form, June 5, 2008
Abstract—We consider a single-server queueing system with a doubly stochastic Poisson ar-
rival ﬂow under heavy traﬃc conditions. We prove the convergence of the limiting stationary
or periodic distribution to the exponential distribution. In a scheme of series, we consider the
C-convergence of the waiting time process to a diﬀusion process with constant coeﬃcients and
reﬂection at the zero boundary. Examples of computation of the diﬀusion coeﬃcient in terms
of characteristics of the arrival ﬂow and service time are given.
A large part of studies in queueing theory is made under the assumption that arrivals form a
Poisson ﬂow. This is due to not only the relative simplicity of the corresponding mathematical
models but also to the adequacy of this assumption to a wide range of applied problems. The point
is that observed processes are often sums of a large number of independent processes, which, as is
shown in [1–4, etc.], are well approximated by Poisson processes under rather general assumptions.
Furthermore, when describing real-world objects, there arise models where the initial ﬂow, passing
through successive systems, loses some of its calls. As follows from [5, etc.], this thinning ﬂow also
tends to a Poisson ﬂow.
In recent years, there is a growing interest to more complex arrival ﬂows, in particular, to a
doubly stochastic Poisson process (DSPP). This is mainly due to the existence of problems of
practical importance where the assumption that the arrival ﬂow is Poisson aﬀects performance
characteristics of a system (for instance, underrates the average number of calls in the system).
Finding explicit forms for characteristics of queueing systems with complex arrival ﬂows (such as
DSPP) is possible for a narrow class of models only, so investigations mainly follow two directions.
The ﬁrst is ﬁnding estimates for these characteristics. The second is analysis of conditions of heavy
traﬃc (there are many calls in the system) and low traﬃc (the server is most often idle).
The present paper analyzes queueing systems with DSPP arrival ﬂows under heavy traﬃc.
Proofs of the main statements are based on results of Borovkov, who developed the general theory
of limiting behavior of queueing processes under very general conditions on an arrival ﬂow, service
times, and structure of the system. This generality makes applying the obtained results to particular
models rather diﬃcult, since one has to check a large number of conditions an interpret them in
terms that are convenient for applications. We study a nonstationary doubly stochastic Poisson
Supported in part by the Russian Foundation for Basic Research, project no. 07-01-00362.