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.;D.3 Paul Fenster owns and manages a chili- dog and soft-drink 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. What is the probability that the operator is busy? What is the average time that a customer must wait before reaching the operator? 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?= 3 per 8- hour workday (approximately Poisson in nature). The crew can service an average of ?= 8 machines per workday, with a repair time distribution that resembles the exponential distribution. What is the utilization rate of this service system? What is the average downtime for a broken machine? How many machines are waiting to be serviced at any given time? 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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