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Chapter 7 - Anvari.Net

By Esther Black,2014-11-18 09:25
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Chapter 7 - Anvari.Net

    Chapter Nine ? 167

Excel Problems

    1. A local bank has a single drive-in teller window. Customers arrive at the drive-in window

    following a Poisson distribution at an average rate of 20 per hour. Assume that service is

    provided at the drive-in window at a rate of 30 customers per hour and that service time

    follows an exponential distribution. Moreover, assume that the population size is infinite and

    the queue length is unrestricted. To determine the efficiency of operations, the bank wishes

    to examine several queue operating characteristics.

     a. What is the utilization rate of this service system?

     b. What is the average number of customers in line?

     c. What is the average time that each customer spends in the queue?

     d. What is the average time that each customer spends in the system?

     e. What is the probability that the drive-in window will be idle?

     Answer: a. 67%; b. 1.33; c. 4 minutes; d. 6 minutes; e. 33.33%

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    Chapter Nine ? 169

    2. A college cafeteria has a single check out lane manned by a single cashier. Students arrive at

    the cash register at the rate of 2 per minute during peak lunch hour. It takes about 20 seconds

    to check out each student’s tray. Assume that arrival rate follows a Poisson distribution and

    service time follows an exponential distribution. To determine the efficiency of operations,

    the cafeteria manager wishes to examine several queue operating characteristics.

     a. What is the utilization rate of this service system?

     b. What is the average number of students in line?

     c. What is the average time that each student spends in the queue?

     d. What is the average time that each student spends in the queue and being checked out?

     e. What is the probability that the cashier will be idle?

     Answer: a. 67%; b. 1.33; c. 0.67 minutes; d. 1 minute; e. 33.33%

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    Chapter Nine ? 171

    3. A small suburban post office is frequented by customers at the rate of 15 per hour. The post

    office has a single employee who typically processes each customer’s request at an average

    rate of 3 minutes per transaction. Assume that arrival rate follows a Poisson distribution and

    service time follows an exponential distribution. To determine the efficiency of operations,

    the postmaster wishes to examine several queue operating characteristics.

     a. What is the utilization rate of this service system?

     b. What is the average number of customers in line?

     c. What is the average time that each customer spends in the queue?

     d. What is the average time that each customer spends in the queue and being serviced?

     e. What is the probability that the post office employee will be idle?

     Answer: a. 75%; b. 2.25; c. 9 minutes; d. 12 minutes; e. 25%

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    Chapter Nine ? 173

    4. Refer to problem 3. The postmaster has been receiving complaints from customers about long waiting times. The postmaster is considering hiring an additional employee who will also be able to process each customer’s request at the same rate of 3 minutes per transaction. Assume that the post office pays its employees $12 per hour and that the cost of customer waiting time, in terms of customer dissatisfaction and goodwill, is $10 per hour. Assume that arrival rate follows a Poisson distribution and service time follows an exponential distribution. Should the postmaster hire an additional employee? Compare the total costs of this 2-employee system with the old single-server queuing system.

     Answer: Total cost with 1 employee = ($10)(3)+($12)(1)=$42/hour

     Total cost with 2 employees = ($10)(0.8727)+($12)(2)=$32.73/hour

    Postmaster should higher an additional employee.

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    5. A fast food restaurant currently has 2 cashiers. Upon arrival, customers form a single line and place their food order at the next available register. Assume that customers arrive at the rate of 35 per hour. It takes an average of 3 minutes to place and process each customer’s order. Assume that arrival rate follows a Poisson distribution and service time follows an exponential distribution. To determine the efficiency of operations, the cafeteria manager wishes to examine several queue operating characteristics. Based upon the analysis, the manager believes that if the average waiting time per customer in the system is greater than 5 minutes, then a request for an additional employee should be made. Should the manager make a request to hire an additional employee?

     Answer: Yes, current waiting time in the system is 12.8 minutes.

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