Problems of Information Transmission, Vol. 37, No. 3, 2001, pp. 248–261. Translated from Problemy Peredachi Informatsii, No. 3, 2001, pp. 67–81.
Original Russian Text Copyright
2001 by Bocharov, D’Apice, Phong.
COMMUNICATION NETWORK THEORY
On a Retrial Single-Server Queueing System
with Finite Buﬀer and Poisson Flow
P. P. Bo charov, C. D’Apice, and N. H. Phong
Received May 18, 1999
Abstract—A retrial single-server queueing system with ﬁnite buﬀer is considered. The primary
incoming ﬂow is Poissonian. If the buﬀer is overﬂown, a call entering the system becomes a
repeat call and joins the group of repeat calls referred to as an orbit. The maximum number of
calls that can simultaneously be contained in the orbit is limited. A call from the orbit makes
new attempts to enter the system until a vacancy occurs. Time between repeat attempts for
each call is an exponentially distributed random variable. At the initial moment of service,
a type of a call is deﬁned: with probability a
it becomes a call of type i and its service time
in this case has distribution function B
(x),i= 1,K. For this system, the stationary joint
distribution of queues in the buﬀer and orbit is found. Numerical examples are given.
Models of retrial queues are an important part of queueing theory. These models arise because
of the necessity to allow for the retrial eﬀect in various networking systems, namely, telephone
networks, computing networks, etc. Therefore, much attention is paid to the analysis of such
models of queues [1–3].
However, in the great majority of works devoted to the analysis of models of retrial queues,
systems without buﬀers for primary calls are only considered. This is quite natural since account
of repeat calls for models of ﬁnite queues leads to a considerable increase in computations required.
On the other hand, allowance for the possibility of waiting for both primary and repeat calls (which
enter the system anew) greatly extends the domain of applicability of such models and makes it
possible to represent real information transmission processes in many networking systems more
In the present paper, we consider a single-server queueing system (QS) of the type M/HG
with a buﬀer of ﬁnite size and ﬁnite queue of repeat calls (in other words, ﬁnite orbit) where service
times have distribution function of the form B(t)=
(t), where the numbers a
deﬁne a probability distribution. A system of the type M/G/1/r with inﬁnite orbit and priority
of primary calls, where repeat calls may arrive directly at the server only, was considered in .
AsystemofthetypeMAP/G/1/r with Markov input ﬂow, ﬁnite orbit, and priority of primary
calls was studied in .
In Section 2, we give a detailed description of the M/HG
/1/r system analyzed in this paper.
In Section 3, we introduce a step Markov process and present diﬀerential equations that describe
the dynamics of the QS in a stationary regime. In Sections 4 and 5, we ﬁnd the solution of the
system of diﬀerential equations and, based on it, ﬁnd the steady state probabilities of the Markov
process. In Section 6, numerical examples are considered.
2001 MAIK “Nauka/Interperiodica”