A facility produces items according to a Poisson process with rate 20 per hour.

Question

A facility produces items according to a Poisson process with rate 20 per hour. However, it has shelf space for only 10 items and so it shuts down production whenever 10 items are present. Customers arrive at the facility according to a Poisson process with rate 5 per hour. Each customer wants one item and will immediately depart either with the item or empty handed if there is no item available. Find the average number of items on the shelf. (Hint: This model is mathematically equivalent to the M/M/1/GD/10/infinity queue . The produced items constitute the arrivals to the queue, and the arriving customers constitute the services. )

Explanation / Answer

Solution:

The proportion of customers that go away empty-handed is equal to P0, the proportion of time there are no items on the shelves.,
P0= [1-/]/[1-(/)^k+1]
Here given values are = 20, = 5, k= 10
= [1-5/20]/[1-(20/5)^10+1]
= 0.75

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