Problems of Information Transmission, Vol. 41, No. 3, 2005, pp. 243–253. Translated from Problemy Peredachi Informatsii, No. 3, 2005, pp. 64–75.
Original Russian Text Copyright
2005 by Tikhonenko.
COMMUNICATION NETWORK THEORY
Generalized Erlang Problem for Service Systems
with Finite Total Capacity
O. M. Tikhonenko
Jan Dlugosz University, Czestochowa, Poland
Received July 23, 2004; in ﬁnal form, April 24, 2005
Abstract—A multiserver on-demand system is considered in which each call has three inter-
dependent random characteristics: the required number of servers, capacity, and service time.
The total capacity of calls and the total number of servers in the system are limited. The
type of a call is deﬁned by the number of servers required for its service. We ﬁnd a stationary
distribution of the number of calls in the system, as well as the loss probability for a call of
In queueing theory and its applications to circuit switching networks and integrated services
digital networks, the Erlang problem  plays an important role. Presently, various generalizations
of this problem are known, which are due to the possibility of simultaneous servicing a call by
several servers [2, 3] and to the fact that real-world calls have diﬀerent information capacity (size)
and hence require diﬀerent memory to be stored while their stay in the system, the total memory
resources being limited [4–6]. Here the required number of servers, capacity, and service time are,
in general, dependent random variables.
In the present paper we obtain relations taking this dependence into account, which generalize
some results of [3,6].
Consider the system that diﬀers from the classical M/G/n/0 system in the following properties.
1. Each call, independently of its arrival time and characteristics of other calls, is simultaneously
served by m (m ≤ n) servers with probability q
= 1. In the sequel, following , we refer
to a call simultaneously served by m servers as an m-call, or a call of type m.
2. Independently of other calls and its arrival time, each m-call is characterized by the random
(note that the random variable ζ
is not necessarily discrete) and service time ξ
The distribution functions
are given. Denote by η(t) the number of calls in the system at time t, and denote by σ(t) the total
capacity, i.e., the total sum of capacities of calls that are in the system at this instant.
3. The total capacity in the considered system is bounded by a quantity V>0, which is called
the memory capacity.
Denote by L
(x, ∞) the distribution function of the capacity of an m-call, and denote
(∞,t) the distribution function of its service time. If at the arrival time τ of an
m-call there are less than m free servers in the system, the call is lost and has no eﬀect on further
2005 Pleiades Publishing, Inc.