Page 76 - A First Course In Stochastic Models

A CONTROLLED QUEUE WITH REMOVABLE SERVER 67

during a busy period equals h N . By the renewal-reward theorem,

(h/2λ)N(N − 1) + K + h N
the long-run average cost per time unit =
N/λ + t N
with probability 1. To find the functions t n and h n , we need

a j = the probability that j orders arrive during the production time of
a single order.

Assume for ease that the production time has a probability density f (x). By con-
ditioning on the production time and noting that the number of orders arriving in
a fixed time y is Poisson distributed with mean λy, it follows that

j
∞ (λy)
−λy
a j = e f (y) dy, j = 0, 1, . . . .
0 j!
It is readily verified that
∞ ∞
2 2 2
ja j = λE(τ 1 ) and j a j = λ E(τ ) + λE(τ 1 ). (2.6.1)
1
j=1 j=1
We now derive recursion relations for the quantities t n and h n . Suppose that at
epoch 0 a production starts with n orders present. If the number of new orders
arriving during the production time of the first order is j, then the time to empty
the system equals the first production time plus the time to empty the system
starting with n − 1 + j orders present. Thus

∞

t n = E(τ 1 ) + t n−1+j a j , n = 1, 2, . . . ,
j=0
where t 0 = 0. Similarly, we derive a recursion relation for the h n . To do so, note
that relation (1.1.10) implies that the expected holding cost for new orders arriving
2
1
during the first production time τ 1 equals hλE(τ ). Hence
2 1
∞
1 2
h n = (n − 1)hE(τ 1 ) + hλE(τ ) + h n−1+j a j , n = 1, 2, . . . ,
1
2
j=0
where h 0 = 0. In a moment it will be shown that t n is linear in n and h n is
quadratic in n. Substituting these functional forms in the above recursion relations
and using (2.6.1), we find after some algebra that for n = 1, 2, . . . ,
nE(τ 1 )
t n = , (2.6.2)
1 − λE(τ 1 )