Questions > OPERATIONS 101 Practice 3_Questions. Practice problems on Queuing Theory.

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Practice problems on Queuing theory.

Problem 1

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Five executives share the services of a typing pool. The average number of jobs submitted for typing is six per day per executive and a job takes 15 minutes on average to type. Evaluate the situation, by first describing the performance measures of possible interest; and computing appropriate performance measures. A day consists of 8 hours. Assume that the arrivals over any period of time are Poisson distributed and that the service time per job is exponentially distributed.

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Problem 2 Kristen offers lunch specials on weekends. (This has nothing to do with the cookie business). She manages the weekend business by herself. On a typical weekend she receives on the average about 6 orders per hour. It takes Kristen on the average 7.5 minutes to prepare and box an order (this time is the service time and does not include the waiting time of the customer.) Kristen’s profit per order is $3. Assume that the arrival process of orders follows a Poisson distribution and that the time to fill an order is exponentially distributed.

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(a) What is the arrival rate and what is the service rate in this example? (b) What is the average time that a customer has to wait (from placing an order to getting it filled)? (c) Based on talking to customers, Kristen realizes that the waiting times her customers experience could have a long term impact on her business. She believes that if the average waiting time (from placing an order to getting it filled) of a customer is x minutes, the impact on the net present value (NPV) of her business is (- 1000 x)$. That is if she reduces the average waiting time by one minute, somehow, the NPV increases by $1000. Kristen has an option of investing in some resource that will help her reduce the “prepare and box” time from 7.5 minutes per order on an average to 6 minutes per order on an average. The time is still exponentially distributed. What is the maximum amount Kristen should be willing to pay for this resource (assume there is no maintenance costs etc.)?

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Problem 3 There are two streams of customers arriving independently at a supermarket: “Express” customers arrive at a rate 70 per hour, and stay on average 10 minutes within the shop. Each of them spend on average $25. “Regular” customers arrive at a rate 40 per hour. They stay within the shop for 30 minutes, spending $80 (on average). (a) What is the average number of “express” customers in the supermarket, at a given moment of time? What is the expected total number of customers at the supermarket, at a given moment of time?

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(b) What is the average revenue per hour?

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Problem 4 You manage a car wash shop with one washing machine. On a typical day, you receive on average 8 cars per hour. It takes on average 6 minutes to wash a car (this is the service time only). Assume that the inter-arrival times are exponentially distributed, and that the time to wash a car is also exponentially distributed.

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(a) What is the arrival rate and service rate for the shop? (b) What is the average time that a customer has to wait before actually being served? (c) What is the probability that a car has to wait in line when entering the system? (d) If the profits are $15 per car, how long it takes this business to generate a $90 profit?

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Problem 5 In Fall 1994, the Levi Strauss company introduced its "Personal Pair Jeans" program.

A sales clerk at the store attends the customer throughout the fitting process. A touch screen software system leads the salesclerk through the fitting process and it requires no special computer skills to operate. The manager of the store thinks there should be 5, 6 or 7 sales clerks in the store.

An issue confronting Levi's is staffing and product costing of its Personal Pair services. It has been observed that in a typical store, on the average, twelve customers require the Personal Pair services per hour. Assume that each customer purchases only one pair of jeans. It takes a sales clerk on the average 20 minutes to serve a customer.

The clerk is paid $5 per hour. The cost of a customer waiting to be served has been estimated to be $8 per hour.

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(a) Under the usual assumptions, determine how many sales clerks should be employed. Justify this number based on performance measures. Assume that there are no constraints with regard to space, computers and other equipment.

Assume a single line, first come first served service, and that customers will wait for sales clerks to be free (and not just go away upon finding every clerk occupied).

(b) Based on your staffing plan, determine the number of comfortable chairs to place in the waiting area of the Personal Pair services for customers to lounge in while waiting for their turn to be fitted. (Hint: You may find it necessary to use more than just the average number of customers in the queue.)

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(c) Based on your answer to (a), estimate the average cost of selling a pair of jeans.

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Problem 6

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Five executives share the services of a typing pool. The average number of jobs submitted for typing is six per day per executive and a job takes 15 minutes on average to type. A day consists of 8 hours. Say whether it is better to have one or two typists in the typing pool based on some relevant performance measures.

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Problem 7 The arrival process of customers to a service facility is Poisson with an average arrival rate of 4 customers per hour. The service facility has a single employee that serves the customers. This person takes on average 10 minutes to serve one customer (assume this time is exponentially distributed).

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(a) What is the service rate? (b) What is the average number of customer in the system? (c) What is the average time spent by customers in the system? (d) What is the fraction of time that an employee is idle?

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Problem 8

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The rate of calls to a call center facility is 100 calls per hour. Suppose that every time you call you are told (by an automatic system) how many customers are currently in the system. Given you past experience with this call center, you know that on average there are 25 customers in the system every time you call. Suppose you about to make a call to this call center, how much time do you expect to be on the phone?

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Problem 9 BizTravel.com is a travel web-site that recently announced “BizTravel Guarantee” putting money behind customer service guarantees.

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a. One of the items in the BizTravel Guarantee states “Your customer service email not responded to within two hours we'll give you a $10 payment”.

Customers currently send emails to service@biztravel.com. The email server of BizTravel equally distributes these emails to the specific address of each of the five Customer Service Reps (CSR). For example, one-fifth of the emails are directed to the mailbox of CSR1@biztravel.com, another one-fifth to CSR2@biztravel.com, and so on. Collaborative Inc. has developed a collaborative software for Customer Relationship Management (CRM) which allows the firm to keep all customer service requests in a central mailbox and dynamically route the emails based on agent availability. Do you think the software from Collaborative, Inc. will help BizTravel meet its customer guarantee better? Explain. b. Another service guarantee offered is “Your phone call not answered within 90 seconds - we'll give you a $10 payment”. Peak arrival rate of calls to BizTravel is during the lunch hour from 12noon to 1pm and averages one customer every minute. A transaction takes on average 5 minutes to service.

The manager decides to schedule 6 agents during this period. Do you expect the BizTravel to have to pay out any money? Why? Show all calculations.

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