Chapter 9

Chapter 12

8-1 Bechtold Construction is in the process of installing power lines to a large housing development. Steve Bechtold wants to minimize the total length of wire used, which will minimize his costs. The housing development is shown as a network. Each house has been numbered, and the distances between houses are given in hundreds of feet. a. What is the required length of power line required? b. What is the recommended route for the lines?

House 7 is currently being demolished and will be removed from the system.

c. With that change, what will be the requirement for power lines and what will the route be?

8-2 The Rockwell Electronics Corporation retains a service crew to repair machine breakdowns that occur on an average of 3 per day (approximately Poisson in nature). The crew can service an average of 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? e. What is the probability that more than two are broken and waiting to be repaired or being serviced? f. What is the probability that more than three are in the system? g. What is the probability that more than four are in the system?

8-3 Mike Dreskin manages a large Los Angeles movie theater complex called Cinema I, II, III, and IV. Each of the four auditoriums plays a different film; the schedule is set so that starting times are staggered to avoid the large crowds that would occur if all four movies started at the same time. The theater has a single ticket booth and a cashier who can maintain an average service rate of 225 movie patrons per hour. Service times are assumed to follow an exponential distribution. Arrivals on a typically active day are Poisson distributed and average 210 per hour. To determine the efficiency of the current ticket operation, Mike wishes to examine several queue operating characteristics. a. Find the average number of moviegoers waiting in line to purchase a ticket. b. What percentage of the time is the cashier busy? c. What is the average time that a customer spends in the system? d. What is the average time spent waiting in line to get to the ticket window? e. What is the probability that there are two or more people in the system? f. What is the probability that there are more than four people in the system? g. What is the probability that there is no one in the system? h. What are two things Mike could to reduce the time to get a ticket?

8-4 The wheat harvesting season in the American 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, the cost of truck rental, and idle driver time are significant concerns to the coop members. Although farmers have difficulty quantifying crop damage, it is easy to assign a waiting and unloading cost for truck and driver of $58 per hour. The storage bin is open and operated 16 hours per day, 7 days per week, during the harvest season and is capable of unloading 32 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.

The cooperative 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 offseason. f. Would it be worth the cooperative’s while to enlarge the storage area?

8-5 Juhn and Sons Wholesale Fruit Distributors employ one worker whose job is to load fruit on outgoing company trucks. Trucks arrive at the loading gate at an average of 26 per day, or

3.25 per hour, according to a Poisson distribution. The worker loads them at a rate of 4 per hour, following approximately the exponential distribution in service times. Determine the operating characteristics of this loading gate problem [the utilization rate, time and number in the system and in the queue]. What is the probability that there will be three or more trucks either being loaded or waiting? Discuss the results of your queuing model computation.

Juhn believes that adding a second fruit loader will substantially improve the firm’s efficiency. He estimates that a two-person crew, still acting like a single-server system, at the loading gate will double the loading rate from 4 trucks per hour to 8 trucks per hour. Analyze the effect on the queue of such a change and compare the results with those found in (a) above [the utilization rate, time and number in the system and in the queue].

Truck drivers working for Juhn and Sons are paid a salary of $30 per hour on average. Fruit loaders receive about $18 per hour. Truck drivers waiting in the queue or at the loading gate are drawing a salary but are productively idle and unable to generate revenue during that time. What would be the hourly cost savings to the firm associated with employing two loaders instead of one?

Juhn and Sons are considering building a second platform or gate to speed the process of loading their fruit trucks. This, they think, will be even more efficient than simply hiring another loader to help out the first platform and will not require any more loaders that it will to put both loaders on one dock. Assume that workers at each platform will be able to load 4 trucks per hour each and that trucks will continue to arrive at the rate of 3.25 per hour. Find the waiting line’s new operating conditions [the utilization rate, time and number in the system and in the queue]. Is this new approach indeed speedier than the other two already considered?

8-6 Customers arrive at an automated coffee vending machine at a rate of 3 per minute, following a Poisson distribution. The coffee machine dispenses a cup of coffee in exactly 15 seconds. a. What is the average number of people waiting in line? b. What is the average number in the system? c. How long does the average person wait in line before receiving service?

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