Page 220 - A First Course In Stochastic Models

A PHASE METHOD 213

by using asymptotic expansions for f j as j → ∞ and 1 − W q (x) as x → ∞; see
Exercise 5.26.

Example 5.5.2 A finite-buffer storage problem
Data messages arrive at a transmission channel according to a Poisson process
with rate λ. The transmission channel has a buffer to store arriving messages. The
buffer has a finite capacity K > 0. An arriving message is only stored in the buffer
when its length does not exceed the unoccupied buffer capacity, otherwise the
whole message is rejected. Data are transmitted from the buffer at a constant rate
of σ > 0. The message lengths are independent of each other and are assumed to
have a continuous probability distribution function F(x). An important performance
measure is the long-run fraction of messages that are rejected. This model, which
is known as the M/G/1 queue with bounded sojourn time, is very useful. It also
applies to a finite-capacity production/inventory system in which production occurs
at a constant rate as long as the inventory is below its maximum level and the
demand process is a compound Poisson process, where demands occurring when
the system is out of stock are completely lost.
A possible approach to solving the model is to discretize the model; see
Exercise 9.9 for another approach. In the discretized model a message is repre-
sented by a batch consisting of a discrete number of data units. The probability of
a batch of size k is given by

b k ( ) = F(k ) − F((k − 1) ), k = 1, 2, . . .

with F(− ) = 0. The buffer only has room for K( ) data units, where
K
K( ) = .

It is assumed that the number is chosen such that K( ) is an integer. An
arriving message is only stored in the buffer when its batch size does not exceed
the number of unoccupied buffer places, otherwise the whole message is rejected.
The data units are transmitted one at a time at a constant rate of σ > 0. The key
step is now to take an exponential distribution with mean 1/µ( ) = /σ for the
transmission time of a data unit. This approach is motivated by Theorem 5.5.1.
A data unit leaves the buffer as soon as its transmission is completed. For the
discretized model, let
π (K) = the long-run fraction of messages that are rejected.

In view of Theorem 5.5.1 one might expect that π (K) is an excellent approxima-
tion to the rejection probability in the original model when is chosen sufficiently