ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 3, pp. 270–281.
⃝ Pleiades Publishing, Inc., 2009.
Original Russian Text
⃝ S.A. Dudin, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 3, pp. 85–97.
COMMUNICATION NETWORK THEORY
The MAP// Retrial Queueing System
with Time-Phased Batch Arrivals
S. A. Dudin
Belarusian State University, Minsk
Received November 12, 2008; in ﬁnal form, April 25, 2009
Abstract—We consider a retrial queueing system with batch arrival of customers. Unlike
standard batch arrival, where a whole batch enters the system simultaneously, we assume that
customers of a batch (session) arrive one by one in exponentially distributed time intervals.
Service time is exponentially distributed. The batch arrival ﬂow is MAP. The number of
customers in a session is geometrically distributed. The number of sessions that can enter the
system simultaneously is a control parameter. We analyze the joint probability distribution of
the number of sessions and customers in the system using the techniques of multidimensional
asymptotically quasi-Toeplitz Markov chains.
Queueing systems well describe operation of telecommunication network channels and servers,
so they are widely covered in the literature. In many queueing systems it is assumed that customers
arrive in groups of arbitrary sizes. Usually it is assumed that when a batch arrives, all customers
of the batch enter the system simultaneously, and the decision whether to accept the batch or not
is based on comparing the batch size with the number of vacancies in the system.
However, a peculiar property of many modern networks is that customers also arrive in batches
but arrival of customers within a batch is not simultaneous. We refer to such time-phased batches
as sessions. The ﬁrst customer of a session arrives at the session arrival moment, and the others
arrive one by one in arbitrary time intervals. The size of a session is arbitrary and is not known
at its arrival moment. This situation is typical, for example, in modeling the transmission of video
and audio data. It is also described in  in the framework of modeling the scheme of alternative
packet overﬂow routing in IP networks.
In , performance characteristics of this routing scheme in IP networks were obtained by
simulation. In , a ﬁnite-buﬀer Markovian queueing system was introduced and analytically
studied. Results of this paper can be used to compute characteristics of this routing scheme, as
well as of other existing systems with time-phased batch arrivals.
In  it is assumed that session acceptance to the system is restricted by means of tokens. Each
accepted session is given a token, which can be interpreted as a timer supposed to run for a certain
time interval. Arrival of each customer of a session under service resets time at the corresponding
timer to the initial value. If the time at the timer is over and a new customer of this session has
not arrived, the session is assumed to have arrived completely, and the timer becomes free. The
total number of tokens (timers) in a system deﬁnes the maximum number of sessions that can be in
the system simultaneously. A customer can easily be attributed to a particular session, e.g., with
the help of his IP address. The number of tokens is a very important control parameter. If there
are few timers in the system, the server will be underloaded, and therefore the system will lose