Page 204 - A First Course In Stochastic Models

LOSS MODELS 197

the rate at which the process leaves the set of states {i, i + 1, . . . , c} to the rate at
which the process enters this set, we obtain the recursive equation

iµp i = (M − i + 1)αp i−1 , i = 1, . . . , c.

This recursive equation allows for the explicit solution (verify):

M i M−i

p (1 − p)
i
p i = , i = 0, 1, . . . , c, (5.2.3)
c
M k M−k
p (1 − p)
k
k=0
where p is given by
1/µ
p = .
1/µ + 1/α
The distribution (5.2.3) is a truncated binomial distribution. To compute the fraction
of service requests that are lost, we need the customer-average probabilities

π i = the long-run fraction of service requests that
find i busy channels upon arrival, i = 0, 1, . . . , c.

The π i are found by noting that

π i = (the long-run average number of service requests that are
generated per time unit and find i busy channels upon

arrival) (the long-run average number of service requests
that are generated per time unit).
In state i, service requests are generated at a rate (M − i)α. Thus the arrival rate
of service requests that see i busy channels equals (M − i)αp i . Hence

(M − i)αp i
π i = c , i = 0, 1, . . . , c.
k=0 (M − k)αp k
It next follows from (5.2.3) that
M − 1 i M−1−i

p (1 − p)
i
π i = , i = 0, 1, . . . , c. (5.2.4)
M − 1
c
k M−1−k
p (1 − p)
k
k=0
It is a remarkable finding that the distribution {π i } is the same as the distribution
{p i } except that M is replaced by M−1. In other words, the equilibrium distribution
of the state just prior to the arrival epochs of new service requests is the same as