Page 224 - A First Course In Stochastic Models

QUEUEING NETWORKS 217

we get for the process {X(t)} the equilibrium equations
k K

p(n) r j + p(n) µ j = p(n + e i − e j )µ i p ij + p(n − e j )r j
j=1 j:n j >0 j:n j >0 i=1
K

+ p(n + e j )µ j p j0 .
j=1
These equations are certainly satisfied by

n k
K
λ k λ k

p(n) = 1 − (5.6.4)
µ k µ k
k=1
when this product-form solution satisfies the partial balance equations
K K

p(n) r j = p(n + e j )µ j p j0 , (5.6.5)
j=1 j=1
K

p(n)µ j = p(n + e i − e j )µ i p ij + p(n − e j )r j , 1 ≤ j ≤ K. (5.6.6)
i=1
For the product-form solution (5.6.4) we have

−1
−1
λ i λ j λ j
p(n + e i − e j ) = p(n) and p(n − e j ) = p(n).
µ i µ j µ j
(5.6.7)
After substitution of (5.6.7) in (5.6.6), it remains to verify whether the relation
−1
−1

K
λ i λ j λ j

µ j = µ i p ij + r j (5.6.8)
µ i µ j µ j
i=1
holds for each j = 1, . . . , K. This is indeed true since the relation (5.6.8) coincides
with the traffic equation (5.6.1) after cancelling out common terms (verify). In a
similar way we can verify that (5.6.5) holds. Substituting p(n+e j ) = (λ j /µ j )p(n)
into (5.6.5), we get
K K

r j = λ j p j0 .
j=1 j=1
This relation is indeed true since it states that the rate of new customers enter-
ing the system equals the rate of customers leaving the system. This completes
the proof.