Page 212 - A First Course In Stochastic Models

INSENSITIVITY 205

given control parameter L, messages of type 2 are only accepted when less than
L messages of type 2 are present and not all of the channels are occupied. Such a
control rule is used to increase the throughput of accepted messages. What is the
optimal value of L?
To compute the average throughput for a given L-policy, it is no restriction to
assume exponentially distributed transmission times. The reason is that the long-run
average throughput is insensitive to the form of the transmission time distributions.
The average throughput is the difference between the average arrival rate λ 1 +
λ 2 and the average number of messages lost per time unit. To argue that the
loss probabilities for both types of messages are insensitive to the form of the
transmission-time distribution, consider the finite-source variant of the model with
Poisson input. Messages of type j are generated by M j identical sources for j =
1, 2, where the think time of the sources has a probability density. A source can
only start a think time when it has no message in transmission at the communication
system. The sources act independently of each other. This finite-source model can
be seen as a cyclic closed two-node network model, where a fixed number of type 1
jobs, M 1 , and a fixed number of type 2 jobs, M 2 , move around in the network.
Node 1 is an infinite-server node, while node 2 is a blocking node with c servers.
In the two-node closed network, take the blocking protocol (5.4.1) with

1, n 2 < c,
A(n 2 ) =
0, n 2 = c,
and
(2)
(1) (2) 1 for n < L,
A 1 (n ) = 1, A 2 (n ) = 2
2 2 0 otherwise.
The closed two-node network with this blocking protocol behaves identically to
the finite-source model. Thus the finite-source model has the insensitivity property.
This result provides a simple but heuristic argument that the controlled loss model
with Poisson input also has the insensitivity property. In general, insensitivity holds
for a wide class of loss networks; see Kelly (1991) and Ross (1995).
Let us now assume exponentially distributed transmission times for the loss
model controlled by an L-policy. Define

X j (t) = the number of channels occupied by type j messages at time t

for j = 1, 2. The stochastic process {(X 1 (t), X 2 (t))} is a continuous-time Markov
chain with state space

I = {(i 1 , i 2 ) | 0 ≤ i 1 + i 2 ≤ c, i 1 ≥ 0, 0 ≤ i 2 ≤ L}.

Its transition rate diagram is given in Figure 5.4.1. By equating the rate out of state
(i 1 , i 2 ) to the rate into state (i 1 , i 2 ), we obtain the equilibrium equations for the