Problems of Information Transmission, Vol. 37, No. 4, 2001, pp. 397–406. Translated from Problemy Peredachi Informatsii, No. 4, 2001, pp. 130–140.
Original Russian Text Copyright
2001 by Bocharov,D
Apice, Manzo, P hong.
COMMUNICATION NETWORK THEORY
On a Retrial Single-Server Queueing System
with Finite Buﬀer and Multivariate Poisson Flow
Received November 23, 2000
Abstract—We consider a single-server queueing system with a ﬁnite buﬀer, K input Poisson
ﬂows of intensities λ
, and distribution functions B
(x) of service times for calls of the ith type,
1,K. If the buﬀer is overﬂowed, an arriving call is sent to the orbit and becomes a repeat
call. In a random time, which has exponential distribution, the call makes an attempt to reenter
the buﬀer or server, if the latter is free. The maximum number of calls in the orbit is limited;
if the orbit is overﬂowed, an arriving call is lost. We ﬁnd the relation between steady-state
distributions of this system and a system with one Poisson ﬂow of intensity λ =
type i of a call is chosen with probability λ
/λ at the beginning of its service. A numerical
example is given.
Mushroom growth of modern communication facilities necessitates the use of new mathematical
models in performance analysis of designed and existing information computer networks. A natural
apparatus for analytical modeling of network systems is the queueing theory apparatus [1, 2].
One of the diﬃcult problems related to construction of more adequate queueing models for
network systems is allowance for the retrial factor [3, 4]. Especially complicated is the problem of
analysis of retrial queueing systems in the case where the input (primary) ﬂow is multivariate.
There are but few works devoted to the analysis of retrial queueing systems with multivariate
Poisson ﬂow; moreover, the main object analyzed is a single-server system without a buﬀer [5–10].
In , an exponential queueing system of this type with K-dimensional Poisson ﬂow is studied
where exponentially distributed service times are diﬀerent for diﬀerent types of calls. The main
result of  is the construction of a system of equations for ﬁnding the average number of repeat
calls of each type. Explicit formulas for these characteristics in the case K = 2 were obtained in 
for arbitrary service time distributions. Under the same assumptions on service times for calls of a
K-dimensional batch Poisson ﬂow, results of  were extended in . In [8–11], an approach to the
analysis of an analogous model with ordinary K-dimensional ﬂow was proposed, which is diﬀerent
from the approaches used in [4, 6] and is the development of the approach of [12, 13] applied to
retrial systems. The content of the approach of [8–11] is the following: ﬁrst, a relation between
stationary characteristics of the multivariate model and the corresponding model with one input
ﬂow is established and then the latter (simpler) model is analyzed. Note that in [9, 11] the case
of an impersistent model is also considered where a call joins the orbit with a certain probability,
which can be diﬀerent from one. In [10, 11], a random idle time for the server on completion of
servicing a call is also allowed for.
In the present paper, the approach of [8–11] is extended also to the case where a system contains
a ﬁnite buﬀer with waiting places for calls that entered the system, both primary and repeat. It is
shown that the analysis of the model with ﬁnite buﬀer and K-dimensional Poisson ﬂow for any
service discipline is reduced to the analysis of a model with one Poisson ﬂow where the type of a
call to be served is chosen using a polynomial trial.
2001 MAIK “Nauka/Interperiodica”