D.1 Customers arrive at Rich Dunn’s Styling Shop at a
rate of 3 per hour, distributed in a Poisson fashion. Rich can perform
haircuts at a rate of 5 per hour, distributed exponentially.
a) Find the average number of customers waiting for haircuts.
b) Find the average number of customers in the shop.
c) Find the average time a customer waits until it is his or her
d) Find the average time a customer spends in the shop.
e) Find the percentage of time that Rich is busy.
λ = 3 customers/hour
µ = 5 customers/hour
D.3 Paul Fenster owns and manages a chili-dog and softdrink
stand near the Kean U. campus. While Paul can service 30
customers per hour on the average (m), he gets only 20 customers
per hour (l). Because Paul could wait on 50% more customers
than actually visit his stand, it doesn’t make sense to him that he
should have any waiting lines.
Paul hires you to examine the situation and to determine
some characteristics of his queue. After looking into the problem,
you find it follows the six conditions for a single-server waiting
line (as seen in Model A). What are your findings?
D.6 Calls arrive at Lynn Ann Fish’s hotel switchboard at
a rate of 2 per minute. The average time to handle each is 20 seconds.
There is only one switchboard operator at the current time.
The Poisson and exponential distributions appear to be relevant
in this situation.
a) What is the probability that the operator is busy?
b) What is the average time that a customer must wait before
reaching the operator?
c) What is the average number of calls waiting to be answered?
D.8 Virginia’s Ron McPherson Electronics Corporation
retains a service crew to repair machine breakdowns that occur
on average l = 3 per 8-hour workday (approximately Poisson in
nature). The crew can service an average of m = 8 machines per
workday, with a repair time distribution that resembles the exponential
a) What is the utilization rate of this service system?
b) What is the average downtime for a broken machine?
c) How many machines are waiting to be serviced at any given
d) What is the probability that more than 1 machine is in the
system? The probability that more than 2 are broken and
waiting to be repaired or being serviced? More than 3? More