Page 364 - A First Course In Stochastic Models

THE M/G/1 QUEUE 359

d x
V (x) = λ V ∞ (x − y){1 − B(y)} dy, x > 0. (9.2.35)
′
∞
dx 0
Since V ∞ (x) = W q (x), these formulas follow from relations (8.4.2), (8.4.3) and
(8.4.4); take σ = 1 in these relations. Also, by (8.4.9), it holds under Assump-
tion 9.2.1 that

1 − V ∞ (x) ∼ γ e −δx as x → ∞, (9.2.36)
where γ and δ are given by (9.2.17).
Unlike the waiting-time distribution, the distribution of the work in system is
invariant among the so-called work-conserving queue disciplines. A queue disci-
pline is called work-conserving when the amount of time a customer is in service
is not affected by the queue discipline.

The maximum work in system during a busy period
Define the random variable V max as

V max = the maximum amount of work in system during a busy period.
A busy period is the time elapsed between the arrival epoch of a customer finding
the system empty and the next epoch at which the system becomes empty. The
following result holds:

1 V (K)
′
∞
P {V max > K} = , K > 0, (9.2.37)
λ V ∞ (K)
where V (x) is the derivative of V ∞ (x) for x > 0. To prove this result, we fix
′
∞
K > 0 and define the probability p K (x) for 0 < x < K by
p K (x) = the probability that the work process {V t } reaches the
level 0 before it exceeds the level K when the current
amount of work in system equals x.

It will be shown that
V ∞ (K − x)
p K (x) = , 0 < x < K. (9.2.38)
V ∞ (K)
The proof of this result is as follows. If the amount of work in the system is x < K
upon arrival of a new customer, the workload remains below the level K only if
the amount of work brought along by the customer is less than K − x. Thus, by
conditioning on what may happen in a very small time interval of length t = x,
we find
K−x
p K (x + x) = (1 − λ x)p K (x) + λ x p K (x + y)b(y) dy + o( x).
0