Page 368 - A First Course In Stochastic Models

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THE M /G/1 QUEUE 363
∞ j
β
Assumption 9.3.1 (a) The convergence radius R of G(z) = j=1 j z is larger

∞ st
than 1. Moreover, e {1 − B(t)} dt < ∞ for some s > 0.
0
∞ st

(b) lim s→B 0 e {1 − B(t)} dt = ∞, where B is the supremum over all s with
∞

st
e {1 − B(t)} dt] < ∞.
0
G(x) = 1 + B/λ for some number R 0 with 1 < R 0 ≤ R.
(c) lim x→R 0
Under this assumption we obtain from Theorem C.1 in Appendix C that
−j
p j ∼ στ as j → ∞, (9.3.4)
where τ is the unique solution to the equation
λα(τ){1 − G(τ)} = 1 − τ (9.3.5)

on (1, R 0 ) and the constant σ is given by
−1
′
(1 − τ)G (τ)
′
σ = (1 − ρ)(1 − τ) λα (τ){1 − G(τ)} − + 1 . (9.3.6)
1 − G(τ)
A formula for the average queue size

∞
The long-run average number of customers in queue is L q = (j − 1)p j .
j=1

′ ∞
Using the relation P (1) = jp j , we obtain after some algebra from (9.3.3)
j=1
that
2
1 2 ρ 2 ρ E(X )
L q = (1 + c ) + − 1 ,
S
2 1 − ρ 2(1 − ρ) E(X)
where X denotes the batch size. Note that the first part of the expression for L q
gives the average queue size in the standard M/G/1 queue, while the second part
reflects the additional effect of the batch size. The formula for L q implies directly a
formula for the long-run average delay in queue per customer. By Little’s formula
L q = λβW q .
9.3.2 The Waiting-Time Probabilities
The concept of waiting-time distribution is more subtle for the case of batch arrivals
than for the case of single arrivals. Let us assume that customers from each arrival
group are numbered as 1, 2, . . . . Service to customers from the same arrival group
is given in the order in which those customers are numbered. For customers from
different batches the service is in order of arrival. Define the random variable D n as
the delay in queue of the customer who receives the nth service. In the batch-arrival
queue, lim n→∞ P {D n ≤ x} need not exist. To see this, consider the particular case