Page 63 - A First Course In Stochastic Models

54 RENEWAL-REWARD PROCESSES

The continuous-time process {I (t)} and the discrete-time process {I n } are both
regenerative. The arrival epochs occurring when the workstation is idle are regen-
eration epochs for the two processes. Why? Let us say that a cycle starts each
time an arriving job finds the workstation idle. The long-run fraction of time the
workstation is busy is equal to the expected amount of time the workstation is
busy during one cycle divided by the expected length of one cycle. The expected
length of the busy period in one cycle equals β. Since the Poisson arrival process
is memoryless, the expected length of the idle period during one cycle equals the
mean interarrival time 1/λ. Hence, with probability 1,
the long-run fraction of time the workstation is busy
β
= . (2.4.1)
β + 1/λ
The long-run fraction of jobs that are lost equals the expected number of jobs lost
during one cycle divided by the expected number of jobs arriving during one cycle.
Since the arrival process is a Poisson process, the expected number of (lost) arrivals
during the busy period in one cycle equals λ × E(processing time of a job) = λβ.
Hence, with probability 1,

the long-run fraction of jobs that are lost
λβ
= . (2.4.2)
1 + λβ
Thus, we obtain from (2.4.1) and (2.4.2) the remarkable result

the long-run fraction of arrivals finding the workstation busy
= the long-run fraction of time the workstation is busy. (2.4.3)
Incidentally, it is interesting to note that in this loss system the long-run fraction
of lost jobs is insensitive to the form of the distribution function of the processing
time but needs only the first moment of this distribution. This simple loss system
is a special case of Erlang’s loss model to be discussed in Chapter 5.

Example 2.4.2 An inventory model

Consider a single-product inventory system in which customers asking for the
product arrive according to a Poisson process with rate λ. Each customer asks
for one unit of the product. Each demand which cannot be satisfied directly from
stock on hand is lost. Opportunities to replenish the inventory occur according to
a Poisson process with rate µ. This process is assumed to be independent of the
demand process. For technical reasons a replenishment can only be made when
the inventory is zero. The inventory on hand is raised to the level Q each time a
replenishment is done. What is the long-run fraction of time the system is out of
stock? What is the long-run fraction of demand that is lost?