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Practical Management Science

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter12: Queueing Models

Section12.5: Analytic Steady-state Queueing Models

Problem 9P

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Suppose that annually an average of l library patrons want to borrow a book. A patron borrows the book for an average of 1/μ years. Suppose we observe that the book is actually borrowed an average of R times per year. Explain how we can estimate λ, which is an unobservable quantity. (Hint: Let U be the expected number of times per year a patron wants to borrow the book and the book is out. Note that λ = R + U.)

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At the beginning of each week, a machine is in one of four conditions: 1 = excellent; 2 = good; 3 = average; 4 = bad. The weekly revenue earned by a machine in state 1, 2, 3, or 4 is 100, 90, 50, or 10, respectively. After observing the condition of the machine at the beginning of the week, the company has the option, for a cost of 200, of instantaneously replacing the machine with an excellent machine. The quality of the machine deteriorates over time, as shown in the file P10 41.xlsx. Four maintenance policies are under consideration: Policy 1: Never replace a machine. Policy 2: Immediately replace a bad machine. Policy 3: Immediately replace a bad or average machine. Policy 4: Immediately replace a bad, average, or good machine. Simulate each of these policies for 50 weeks (using at least 250 iterations each) to determine the policy that maximizes expected weekly profit. Assume that the machine at the beginning of week 1 is excellent.
A small grocery store has a single cashier lane where groceries line up to pay for their purchases. Interarrival time of customers is exponentially distributed with mean of 3 mins. Due to the variety in type and number of items purchased, transaction time is approximately exponentially distributed with mean 2.5 mins. a. Compute for the average number of customers in the cashier lane. b. Find the probability that there are at most 4 customers in the lane. c. What is the proportion of time that the cashier is idle? d. Find the average time a customer spends in the cashier, including queuing time and transaction time. e. Buying a POS machine will help reduce average transaction time to only 2 mins. (20% reduction) with the aid of a barcode reader - eliminating the need to type item codes manually. If implemented, find the percent reduction in the average time a customer spends in the cashier.

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