Page 324 - A First Course In Stochastic Models

ASYMPTOTIC EXPANSIONS 319

and

h(x) = the expected holding costs incurred until the workload is zero
when the current workload is x and the server is working.

The functions t(x) and h(x) are given by
x h 2 λxµ 2
t(x) = and h(x) = x + (8.2.9)
1 − λµ 1 2(1 − λµ 1 ) 1 − λµ 1
for x ≥ 0. The proof is as follows. By conditioning on the number of arrivals
during a time x, it follows that

∞ n
−λx (λx)
t(x) = x + e t n , x ≥ 0,
n!
n=1
where t n is defined as the expected amount of time needed to empty the system
when service is begun with n batches (= customers) present. Let us also define h n
as the expected holding cost incurred during the time needed to empty the system
when service is begun with n batches present. Then, using relation (1.1.8),

∞ n n
h 2 −λx (λx) kx
h(x) = x + e h x − µ 1 + h n
2 n! n + 1
n=1 k=1
∞ n
h 2 λ 2 −λx (λx)
= x + h x µ 1 + e h n , x ≥ 0.
2 2 n!
n=1
The formula t n = nµ 1 /(1−λµ 1 ) was obtained in Section 2.6. Substituting this into
the above relation for t(x) gives the first relation in (8.2.9). By the same arguments
as used in Section 2.6 to obtain t n , we find

n
1
h n = {h 1 + (n − k)t 1 hµ 1 } = nh 1 + hµ 1 n(n − 1)t 1 .
2
k=1
Substituting this into the relation for h(x) gives

h 2 λ 2 1 2
h(x) = x + h x µ 1 + λxh 1 + hµ 1 (λx) t 1 , x ≥ 0.
2 2 2
Integrating both sides of this equation over the probability density f (x) of the
∞

batch size and noting that h 1 = h(x)f (x) dx, we find an explicit expression
0
for h 1 and next we obtain the second relation in (8.2.9). The details are left to the
reader.