Page 229 - A First Course In Stochastic Models

222 MARKOV CHAINS AND QUEUES

that simultaneously n 1 customers are present at station 1, n 2 customers at station
2, . . . , n K customers at station K. Define the customer-average probability

π j (n 1 , . . . , n K ) = the long-run fraction of arrivals at station j that see
n ℓ other customers present at station ℓ for ℓ = 1, . . . , K.
Note that in this definition n 1 + · · · + n K = M − 1.

K
Theorem 5.6.3 (arrival theorem) For any (n 1 , . . . , n K ) with ℓ=1 n ℓ = M −1,
π j (n 1 , . . . , n K ) = p M−1 (n 1 , . . . , n K ).
Proof By part (b) of Corollary 4.3.2,

the long-run average number of arrivals per time unit at station j that find
n ℓ other customers present at station ℓ for ℓ = 1, . . . , K

K

= µ i p ij p M (n 1 , . . . , n i + 1, . . . , n K )
i=1
for any (n 1 , . . . , n K ) with K n ℓ = M − 1. In particular,
ℓ=1
the long-run average number of arrivals per time unit at station j
K

= µ i p ij p M (m 1 , . . . , m i + 1, . . . , m K )
m∈I M−1 i=1
where m = (m 1 , . . . , m K ) and I M−1 = {m | m ≥ 0 and m 1 +. . .+m K = M −1}.
Thus
K

µ i p ij p M (n 1 , . . . , n i + 1, . . . , n K )
i=1
π j (n 1 , . . . , n K ) = .
K

µ i p ij p M (m 1 , . . . , m i + 1, . . . , m K )
m∈I M−1 i=1
By Theorem 5.6.2,
m k
K
π i π k
p M (m 1 , . . . , m i + 1, . . . , m K ) = C .
µ i µ k
k=1
Substituting this in the numerator and the denominator of the expression for
π j (n 1 , . . . , n K ) and cancelling out the common term K π i p ij , we find
i=1
K
n k
π k

π j (n 1 , . . . , n K ) = C M−1 .
µ k
k=1
for some constant C M−1 . The desired result now follows from Theorem 5.6.2.