Page 372 - A First Course In Stochastic Models

M/G/1 QUEUES WITH BOUNDED WAITING TIMES 367

finite storage space and telecommunication systems with a finite buffer for storing
incoming data.
An important characteristic of the finite-buffer M/G/1 queue is

π(K) = the long-run fraction of arrivals that cause a partial overflow.
The following result can be proved:
1 V (K)
′
π(K) = ∞ , (9.4.1)
λ V ∞ (K)
where V ∞ (x) is defined in Section 9.2.4. It is remarkable that π(K) is identical to
the probability P {V max > K}, where V max is the maximal buffer content during
a busy period in the infinite-buffer model; see relation (9.2.37). The proof of the
result (9.4.1) is based on the proportionality relation
V ∞ (x)
V K (x) = for 0 ≤ x ≤ K, (9.4.2)
V ∞ (K)
where V K (x) is defined by

(K)
V K (x) = lim P {V t ≤ x}
t→∞
(K)
with the random variable V t denoting the amount of work in the buffer at time t.
We defer the proof of (9.4.2) to later. First we sketch how the result (9.4.1) can be
obtained from the proportionality relation (9.4.2). A customer who finds an amount
of work x in the buffer upon arrival causes an overflow only if the customer brings
an amount of work larger than K −x. In statistical equilibrium the amount of work
in the buffer seen by an arrival has V K (x) as probability distribution function by
the PASTA property. Hence, by conditioning,

K
π(K) = {1 − B (K)}V K (0) + {1 − B(K − x)}v K (x) dx,
0
where v K (x) denotes the derivative of V K (x) for x > 0. Using (9.4.2), it is not
difficult to verify by partial integration that
1 K
π(K) = V ∞ (K) − V ∞ (K − x)b(x) dx .
V ∞ (K) 0
′
By (9.2.34) the term between brackets equals λ −1 V (K), proving (9.4.1).
∞
Assuming that the probability distribution function B(x) satisfies Assumption
9.2.1, it follows from (9.2.36) that
γ δ −δK
π(K) ∼ e as K → ∞,
λ
where γ and δ are given by (9.2.17).