Example 7.4

Finished goods come out of the production process on an average every 12 minutes, then find the average production in a week, considering 24 hours utilization every day and 7 days a week.

Solution 7.4

Pure birth process

In general a pure birth process occurs in which ONLY arrivals are allowed. Remember that the exponential distribution will be used to describe the inter-arrival time in a pure birth process.
In a pure birth process we start with 0 number of customers at  and no queue is considered in the system and we consider arrivals happen at a particular rate which is .

So let us define the following

: Probability of no arrivals during a time period of t
: Average arrival rate of number of customer per unit time

Then we have:

Now for a sufficient small interval time, h > 0, we would have

,further more we can prove that the following also holds and is true which is , which is Poisson and the expected value is  .

Pure death process

In general a pure death process occurs in which ONLY departures are allowed. Remember that the exponential distribution will be used to describe the inter departure time in a pure death process.
In a pure death process we start with N number of customers at  and no new arrivals are considered in the system and we consider departure occurs at a particular rate which is  .

So let us define the following
p₀(t): Probability of no departures during a time period of t
: Average departure rate of number of customer per unit time

Using similarly we can prove that

where.Now remember that the following equations we have, is the truncated Possion distribution.