Page 210 - A First Course In Stochastic Models

INSENSITIVITY 203

a well-developed theory for insensitivity is available; see Schassberger (1986) and
Whittle (1985). This theory will not be discussed here. In this section the insensitiv-
ity property for the Erlang loss model and the Engset loss model is made plausible
through a closed two-node network model. This model is also used to argue insen-
sitivity in a controlled loss model with several customer classes. Also, the M/G/1
queue with the processor-sharing discipline is discussed as an example of a stochas-
tic service system with no queueing and possessing the insensitivity property.

5.4.1 A Closed Two-node Network with Blocking

Consider a closed network model with two nodes in cyclic order. A fixed number
of M jobs move around in the network. If a job has completed service at one of the
nodes, it places a request for service at the other node. Node 1 is an infinite-server
node, that is, there is an ample number of servers at node 1. Node 2 is the only
node at which blocking can occur. A job that is accepted at node 2 is immediately
provided with a free server. Further, it is assumed that there are r different job
types h = 1, . . . , r with M h jobs of type h, where M 1 + · · · + M r = M. The
blocking protocol is as follows: if a job of type h arrives at node 2 when n 2 jobs
(h)
are already present at node 2, including n 2 jobs of type h, then the arriving job
of type h is accepted at node 2 with probability
(h)
A(n 2 )A h (n 2 ), h = 1, . . . , r (5.4.1)

for given functions A(.), A 1 (.), . . . , A r (.). An accepted job is immediately pro-
vided a free server and receives uninterrupted service at a constant rate. If a job
is rejected at node 2, it returns to node 1 and undergoes a complete new service
at node 1. The service time of a job of type h at node i has a general probability
distribution function with mean 1/µ ih for i = 1, 2 and h = 1, . . . , r. For each type
of job it is assumed that the service-time distribution for at least one of the nodes
has a positive density on some interval. The service requirements at the nodes are
assumed to be independent of each other.
(h) (h)
The system is said to be in state n = (n ) when there are n jobs present at
i i
(h) (h)
node i for i = 1, 2 and h = 1, . . . , r with n + n = M h for h = 1, . . . , r. Let
1 2
p(n) denote the limiting probability that the process is in state n at an arbitrary
(ℓ)
point in time. Also, for fixed job type ℓ, let π ( n) denote the limiting proba-
i
bility that a job of type ℓ arriving at node i finds the other jobs in state n with
(ℓ) (ℓ) (h) (h)
n + n = M ℓ − 1 and n + n = M h for h = ℓ. Assuming that each of the
1 2 1 2
service-time distributions is a mixture of Erlangian distributions with the same scale
parameters, Van Dijk and Tijms (1986) used rather elementary arguments to prove
(h)
that the probabilities p(n) and π ( n) depend on the service-time distributions
i
only through their means and are thus insensitive to the form of the service-time
distributions. Next, by deep mathematics, the insensitivity property for general
∗
∗ Also the so-called product-form solution applies to these probabilities. The product-form solution will
be discussed in detail in Section 5.6.