ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 3, pp. 289–298.
Pleiades Publishing, Inc., 2015.
Original Russian Text
A.N. Dudin, A.A. Nazarov, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 3, pp. 93–104.
COMMUNICATION NETWORK THEORY
The MMAP/M/R/0 Queueing System
with Reservation of Servers Operating
in a Random Environment
A. N. Dudin and A. A. Nazarov
Department of Probability Theory and Mathematical Statistics,
Tomsk State University, Tomsk, Russia
e-mail: email@example.com, firstname.lastname@example.org
Received April 4, 2014; in ﬁnal form, June 2, 2015
Abstract—We consider a multiserver queueing system without buﬀer, with customers of two
types, operating in a random environment. The system is fed by a marked Markovian arrival
process which depends on the environment state. Customers of the ﬁrst type have absolute
priority over customers of the second type. Instantaneous service rates are piecewise constant
with parameters depending on the customer type and the current environment state. The
system behavior is described by a continuous-time multivariate Markov chain. We present
a generator of this chain in a block-tridiagonal form. We brieﬂy describe the procedure for
ﬁnding a stationary probability distribution of system states and obtain formulas for the main
probabilistic characteristics of the system in terms of the stationary distribution. An algorithm
for computing the Laplace–Stieltjes transform of the sojourn time for an arbitrary customer of
the ﬁrst type is obtained.
Modern telecommunication systems and communication networks are complex technical facil-
ities aimed to provide data transmission over various distances with certain quality parameters.
At present, resources (channels and servers) of wireless communication networks are usually scarce;
this is due to the fact that available radio frequency spectrum is physically limited. To optimize the
use of the available radio spectrum, the cognitive radio technology can be applied, which attracts
attention of many researches; see, e.g., . In cognitive radio systems, unlicensed (secondary) net-
work users are allowed to temporarily “borrow” unused spectrum band, but the unlicensed users
must make no obstacles to licensed (primary) users; see, e.g., . The problem of optimizing the
joint access of primary and secondary users can be eﬃciently solved with the help of queueing
Presently, there are many works devoted to the analysis of cognitive radio systems; see, e.g., [2–5].
As a rule, models with two types of customers are considered in the literature, where customers of
the ﬁrst type have absolute priority over customers of the second. Servicing customers of the second
type is interrupted when ﬁrst-type customers arrive. To minimize the interruption probability,
threshold access to the system is used for secondary customers; i.e., a certain number of channels is
reserved for servicing primary customers only, and access of secondary customers is stopped when
the number of busy servers exceeds some threshold value; see, e.g., . The present paper also
assumes server reservation for customers of the ﬁrst type.
The work is supported by Tomsk State University Competitiveness Improvement Program.