Page 388 - A First Course In Stochastic Models

MULTI-SERVER QUEUES WITH POISSON INPUT 383

where q i = lim j→∞ P (L + = i). It remains to find the limiting probabilities q i .
j
These limiting probabilities can be obtained by a simple up- and downcrossing
argument: the long-run fraction of customers finding k other customers in queue
upon arrival equals the long-run fraction of customers leaving k other customers
behind in queue when entering service. This holds for any integer k ≥ 0. For
k = 0 we also have that the long-run fraction of arrivals finding k other customers
in queue equals the long-run fraction of arrivals who find k + c other customers
in the system. This latter fraction equals the time-average probability p c+k by the
PASTA property. Hence we find
c

q i = p c+i for i = 1, 2, . . . and q 0 = p j .
j=0
Interchanging the order of summation in (9.6.10), the result (9.6.9) now follows.

Asymptotic expansion
It is also possible to give an asymptotic expansion for 1 − W q (x):

1 − W q (x) ∼ γ e −λ(τ−1)x as x → ∞, (9.6.11)
where
σ
γ = c−1
(τ − 1)τ
with τ and σ as in (9.6.3) and (9.6.4). To prove this result, we fix u with 0 ≤ u < D
and let x run through (k − 1)D + u for k = 1, 2, . . . . Defining
r j
−λ(D−u) [λ(D − u)]
b r (u) = Q r−j e for r = 0, 1, . . . ,
j!
j=0
we have by (9.6.9) that

1 − W q (x) = 1 − b kc−1 (u) for x = (k − 1)D + u.
r
∞
Next consider the generating function B u (z) = (1 − b r (u))z . Since the
r=0
generating function of the convolution of two discrete sequences is the product of
the generating functions of the separate sequences, it follows that
1 λ(D−u)(z−1)
B u (z) = − Q(z)e ,
1 − z
j c+j
∞
where Q(z) = j=0 Q j z . Since Q j = k=0 p k , we find after some algebra that
c−1
k c
e λD(1−z) p k (z − z )
c−1
−c
z k c k=0
Q(z) = P (z) − p k (z − z ) = c λD(1−z) ,
1 − z (1 − z)(1 − z e )
k=0