Page 419 - A First Course In Stochastic Models

414 ALGORITHMIC ANALYSIS OF QUEUEING MODELS

the batch size X has a discrete probability distribution {β j , j ≥ 1} with mean β.
Denoting by µ the mean service time of a customer, it is assumed that the load
factor ρ = λβµ/c is smaller than 1. As before {p j , 0 ≤ j ≤ N + c} denotes the
limiting distribution of the number of customers present. For finite-buffer queues
with batch arrivals we must distinguish between these two cases:

(a) Partial rejection: an arriving batch whose size exceeds the remaining capacity
of the buffer is partially rejected by turning away only those customers in
excess of the remaining capacity.
(b) Complete rejection: an arriving batch whose size exceeds the remaining capac-
ity of the buffer is rejected in its entirety.

The emphasis of the discussion will be on the case of partial rejection. We first
(∞) X
derive an expression for the tail probability in the infinite-capacity M /G/c
N+c
(∞)
queue. Let {p } denote the time-average probabilities in the infinite-capacity
j
X
M /G/c queue. Then, by the PASTA property,
the long-run fraction of batches finding k other customers present upon arrival
(∞)
= p , k = 0, 1, . . . . (9.8.9)
k
Suppose that the customers are numbered as 1, 2, . . . in accordance with the order in
which the batches arrive and in accordance with the relative positions the customers
take within the same batch. Define for j = 0, 1, . . . ,
(∞)
π = the long-run fraction of customers who have j other customers in front
j
of them just after arrival (including customers from the same batch).
In Section 9.3.2 we have already shown that

the long-run fraction of customers taking the rth position in their batch
∞
1
= β j , r = 1, 2, . . . .
β
j=r
This result in conjunction with (9.8.9) gives
j
1 ∞
(∞) (∞)
π = p β s , j = 0, 1, . . . . (9.8.10)
j k
β
k=0 s=j−k+1
Hence, in the infinite-capacity model, the long-run fraction of customers having
N + c or more customers in front of them just after arrival is given by
∞ j ∞
(∞) 1 (∞)
= p β s . (9.8.11)
N+c β k
j=N+c k=0 s=j−k+1