ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 4, pp. 364–377.
Pleiades Publishing, Inc., 2011.
Original Russian Text
S.A. Dudin, O.S. Dudina, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 4, pp. 68–83.
COMMUNICATION NETWORK THEORY
Call Center Operation Model as a MAP/PH/N/R − N
System with Impatient Customers
S. A. Dudin and O. S. Dudina
Belarusian State University, Minsk
Received February 22, 2011; in ﬁnal form, September 22, 2011
Abstract—We analyze a multiserver queueing system with a ﬁnite buﬀer and impatient cus-
tomers. The arrival customer ﬂow is assumed to be Markovian. Service times of each server
are phase-type distributed. If all servers are busy and a new arrival occurs, it enters the buﬀer
with a probability depending on the total number of customers in the system and waits for
service, or leaves the system with the complementary probability. A waiting customer may
become impatient and abandon the system. We give an algorithm for ﬁnding the stationary
distribution of system states and derive formulas for basic performance characteristics. We ﬁnd
Laplace–Stieltjes transforms for sojourn and waiting times. Numeric examples are given.
A call center is a centralized oﬃce used to receive and transmit information obtained as telephone
requests. Call centers play an important role in the world of today. According to , 500 major
world companies have at least one call center for interacting with their customers.
Permanently growing requirement for call centers in commerce, banking, telecommunications, as
well as their galloping development, causes great interest among researches. Presently, of importance
is not only development of technologies, consisting in using cutting-edge facilities with maximum
productivity, but also ensuring eﬃcient operation of the centers, which is directly related to devel-
oping new design engineering methods and providing adequate mathematical modeling methods for
precise evaluation of their time-probabilistic characteristics. Adequate mathematical modeling for
call centers may lead to a considerable growth of their economical eﬃciency, since accurate fore-
casting at the designing stage may considerably reduce maintenance costs. Methods of queueing
theory are widely used for call center modeling, description, and optimization.
As is stated in , 60–80% costs of a call center are due to training and upkeep of the personnel;
therefore, optimization of a call center operation by choosing an optimal number of operators is an
important aspect. By treating a call center as a mathematical model, one can solve a wide scope
of optimization problems, including ﬁnding the optimal number of operators required for serving
customers with a given grade of service.
The simplest call center model is an M/M/N queue (Erlang C system). This model assumes
that a customer arrival ﬂow is Poissonian, service times are exponentially distributed, and denial of
service and customer impatience are not taken into account. Another simple call center model is an
M/M/N/0 queue (Erlang B system) with no buﬀer. In , an M/M/N/R − N queue with a ﬁnite
buﬀer and impatient customers (Erlang A system) is analyzed. In , performance characteristics
are found for an Erlang A system with an arbitrary distribution of patience times of customers.
Dependence between characteristics of this model and its parameters is investigated in . The
possibility of customer retrials is considered in . In , a queueing system with two types of