A manufacturing station operates in the following way. Items
to be processed arrive at the station as a Poisson process with
the mean time between arrivals equal to 5 minutes. Processing
is done on machines that require an average time of 12 minutes
per item. Three machines are provided to meet this load. Based
on this data, the probabilities at the right are computed. In
addition, we have that the expected number of items in the system
is 4.986.
Probability

Value

P(0)

0.056

P(1)

0.135

P(2)

0.162

P(3)

0.129

P(4)

0.104

P(5)

0.083

P(6)

0.066

a. What is the expected number in the queue?
b. What is the throughput time for the items (this is the expected
time between when the item enters the system to when it leaves
the system)?
c. What is the probability that an arriving item will have
to wait for processing?
d. Adjacent to the machines there is room for three waiting
items. When that space is full, items are moved to separate
storage to await processing. What is the probability that an
arriving item will be stored in this separate storage?
e. If we add another machine to aid in the processing, how
will the expected number of items actually being processed (not
in the queue) change? (up, down, or stay the same)
