Reference no: EM132281619

Exercises -

Exercise 1 - The Edge Convenience Store has approximately 300 customers shopping in its store between 9 A.M. and 5 P.M. on Saturdays. In deciding how many cash registers to keep open each Saturday, Edge's manager considers two factors: customer waiting time (and the associated waiting cost) and the service costs of employing additional checkout clerks. Checkout clerks are paid an average of $10 per hour. When only one is on duty, the waiting time per customer is about 10 minutes (or 1/6 hour); when two clerks are on duty, the average checkout time is 6 minutes per person; 4 minutes when three clerks are working; and 3 minutes when four clerks are on duty.

Edge's management has conducted customer satisfaction surveys and has been able to estimate that the store suffers approximately $13 in lost sales and goodwill for every hour of customer time spent waiting in checkout lines. Using the information provided, determine the optimal number of clerks to have on duty each Saturday to minimize the store's total expected cost

Exercise 2 - Rockwell Electronics Corporation retains a service crew to repair machine breakdowns that occur on an average of λ = 3 per day (approximately Poisson in nature). The crew can service an average of μ = 8 machines per day, 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 machine that is broken?

(c) How many machines are waiting to be serviced at any given time?

(d) What is the probability that more than one machine is in the system? What is the probability that more than two machines are broken and waiting to be repaired or being serviced? More than three? More than four?μ

Exercise 3 - A computer processes jobs on a first-come, first-served basis in a time-sharing environment. The jobs have Poisson arrival rates, with an average of six minutes between arrivals. The objective in processing these jobs is that they spend no more than eight minutes, on average, in the system. How fast does the computer have to process jobs, on average, to meet this objective?

Exercise 4 - The wheat harvesting season in the U.S. Midwest is short, and most farmers deliver their truckloads of wheat to a giant central storage bin within a two-week span. Because of this, wheat-filled trucks waiting to unload and return to the fields have been known to back up for a block at the receiving bin. The central bin is owned cooperatively, and it is to every farmer's benefit to make the unloading/ storage process as efficient as possible. The cost of grain deterioration caused by unloading delays and the cost of truck rental and idle driver time are significant concerns to the cooperative members. Although farmers have difficulty quantifying crop damage, it is easy to assign a waiting and unloading cost for truck and driver of $18 per hour. The storage bin is open and operated 16 hours per day, seven days per week during the harvest season and is capable of unloading 35 trucks per hour, according to an exponential distribution. Full trucks arrive all day long (during the hours the bin is open), at a rate of about 30 per hour, following a Poisson pattern.

To help the cooperative get a handle on the problem of lost time while trucks are waiting in line or unloading at the bin, find the

(a) Average number of trucks in the unloading system.

(b) Average time per truck in the system.

(c) Utilization rate for the bin area.

(d) Probability that there are more than three trucks in the system at any given time.

(e) Total daily cost to the farmers of having their trucks tied up in the unloading process.

(f) The cooperative, as mentioned, uses the storage bin only two weeks per year. Farmers estimate that enlarging the bin would cut unloading costs by 50% next year. It will cost $9,000 to do so during the off-season. Would it be worth the cooperative's while to enlarge the storage area?