Page 211 - A First Course In Stochastic Models

204 MARKOV CHAINS AND QUEUES

service-time distributions can be concluded by a continuity argument. This argu-
ment is based on the fact that the class of mixtures of Erlangian distributions with
the same scale parameters is dense in the class of all probability distributions on
the non-negative axis; see Hordijk and Schassberger (1982) and Whitt (1980). In
Section 5.5 we give an elementary proof that any service-time distribution can
be arbitrarily closely approximated by a mixture of Erlangian distributions with
the same scale parameters. Taking for granted the insensitivity property of the
closed two-node network model, we give two applications of loss systems with the
insensitivity property.

Example 5.4.1 Insensitivity for a finite-source model with grading

Let us consider a finite-source model with grading. Such a model is an extension
of the Engset model discussed in Section 5.2.2. In the Engset model a newly
generated message is only blocked when all c servers are occupied. In the grading
model a newly generated message hunts for a free server among K servers that are
randomly chosen from the c servers, with K fixed. The message is blocked when
no free server is found among the K chosen servers. The closed two-node model
with a single job type applies (r = 1). The blocking protocol indeed allows for the
representation (5.4.1). This follows by taking

n 2 c (h)
A(n 2 ) = 1 − and A h (n 2 ) = 1, h = 1, . . . , r
K K
n
with the convention ( ) = 0 for n < m. Thus we can conclude that the time-
m
average and customer-average probabilities in the grading model are insensitive
to both the form of the think-time distribution and the form of the service-time
distribution. The Engset model is a special case of the grading model with K = c.
Thus we also have insensitivity for the Engset model. By letting the number of
sources tend to infinity and the thinking rate to zero, the input process becomes
a Poisson process. It will now intuitively be clear that the Erlang loss model has
the insensitivity property. However, a rigorous proof of this fact requires deep
mathematics.

Example 5.4.2 A loss model with competing customers

Messages of types 1 and 2 arrive at a communication system according to two
independent Poisson processes with the respective rates λ 1 and λ 2 . The communi-
cation system has c identical service channels for handling the messages but there
is no buffer to temporarily store messages which find all channels occupied. Each
channel can handle only one message at a time. The transmission times of the mes-
sages are independent of each other and the transmission times of messages of the
same type j have a general probability distribution with mean 1/µ j for j = 1, 2.
The following admission rule for arriving messages is used. Messages of type 1
are always accepted whenever a free service channel is available. However, for a