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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

turn.

d) Find the average time a customer spends in the shop.

e) Find the percentage of time that Rich is busy.

Arrival rate

λ = 3 customers/hour

Service rate

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

distribution.

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

time?

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

than 4?

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