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Process Analysis and Design Operations StrategyProcess AnalysisManufacturing Service System Design Matrix Service Blueprinting Service Classifications Waiting Line Analysis Services 62

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Irwin/McGraw-Hill Queues (Waiting Lines) People waiting to be served/machines waiting to be overhauled Cant have inventory in services! The issue is the trade-off between cost of service and the cost of waiting 63

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Irwin/McGraw-Hill Suggestions for Managing Queues 1. Determine an acceptable waiting time 2. Divert customers attention while waiting 3. Inform your customers of what to expect 4. Keep other employees out of sight 5. Segment customers 6. Train employees to be friendly 7. Encourage customers during slack periods 8. Adopt a long-term perspective 63

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Irwin/McGraw-Hill Suggestions for Managing Queues For example, can control arrivals by: Restricting the line (short line) - e.g., Wendys drive-thru Post business hours Establishing specific hours for specific customers or price - e.g., run specials; increase price for adult haircuts Can also provide faster (or slower) servers, machines, layouts, set-up times, etc. 63

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Irwin/McGraw-Hill Components of the Queuing Phenomenon Customer Arrivals Server(s) Waiting Line Servicing System Exit Population Source? Infinite or Finite Service Rate? Constant or Variable Infinite Queue Length FCFS Balking or Reneging? 64

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Irwin/McGraw-Hill Queuing Components Arrivals from a finite population- Limited size customer pool (important distinction, as probabilities change after each customer) FCFS (First come, first served)- most common priority rule, but not only one - can have emergencies first, best customers first, triage, etc. Balking (look and then leave) vs. Reneging (wait for awhile and then decide to leave) 63

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Irwin/McGraw-Hill Queuing Components (contd) Poisson Distribution: - The Poisson is the most common distribution used in queuing theory for arrivals - The Poisson distribution is discrete, as the number of arrivals must be an integer - The probability of n arrivals within a T minute period = P T (n)=(λT) n( e -λT )/n! 63

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Irwin/McGraw-Hill Queuing Components (contd) Service Rates: - The capacity of the server (in units per time pd) (e.g., a service rate of 12 completions per hour) - Use the Exponential distribution when service times are random (as opposed to constant), where μ = avg # of customers served per time pd, and the probability the service time will be less than or equal to a time of length t: P = 1 – e -µt. 63

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Irwin/McGraw-Hill Line Structures Single Channel Multichannel Single Phase Multiphase One-person barber shop Car wash Hospital admissions Bank tellers windows 65 (Also, Mixed; See text pages )

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Irwin/McGraw-Hill Properties of Waiting Line Models ModelLayout Source PopulationService Pattern 1Single channelInfiniteExponential 2Single channelInfiniteConstant 3MultichannelInfiniteExponential 4Single or MultiFiniteExponential These four models share the following characteristics: · Single phase · Poisson arrival · FCFS · Unlimited queue length 66 (See formulas for each model in text on page 253)

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Irwin/McGraw-Hill Waiting Line Models Characteristics of a waiting line model: L q = Average number in line L s = Average number in system W q = Average time in line W s = Average time in system ρ = Utilization of Server P n = Probability of exactly n in system 67

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Irwin/McGraw-Hill Characteris- tics of Waiting Lines Definition Model 1 (single channel, exp. service rate) Model 2 (single channel, constant service rate) LqLq Average number in line 2 /[ ( - )] 2 /[2 ( - )] LsLs Avg. number in system /( - ) L q + ( / ) WqWq Average time in line /[ ( - )] or L q / /[2 ( - )] WsWs Average time in system 1/( - ) or W q +(1/ ) (rho) Utilization of server / - PnPn Prob.of exactly n in system [1-( / )]( / ) n - PoPo Prob. of exactly zero in system 1 - ( / ) - Where = Arrival rate (e.g., = 2 would represent a mean arrival rate of 2 per minute), and = Service rate or avg number of customers served per time period (e.g., 1 per 3 mins =20 per hour). and must be in the same units. (corrected 5/3/05)

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Irwin/McGraw-Hill Example: Model 1 (worked problem) Drive-up window at a fast food restaurant. Customers arrive at the rate of 25 per hour. The employee can serve one customer every two minutes. Assume Poisson arrival and Exponential service rates. A) What is the average utilization of the employee? B) What is the average number of customers in line? C) What is the average number of customers in the system? D) What is the average waiting time in line? E) What is the average waiting time in the system? F) What is the probability that exactly two cars will be in the system? WP1

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Irwin/McGraw-Hill Example: Model 1 (worked problem) A) What is the average utilization of the employee? B) What is the average number of customers in line? WP2

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Irwin/McGraw-Hill Example: Model 1 (worked problem) C) What is the average number of customers in the system? D) What is the average waiting time in line? WP3

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Irwin/McGraw-Hill Example: Model 1 (worked problem) E) What is the average waiting time in the system? F) What is the probability that exactly two cars will be in the system (one being served and the other waiting in line)? WP4

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Irwin/McGraw-Hill Example: Model 2 (worked problem) An automated pizza vending machine heats and dispenses a slice of pizza in 4 minutes. Customers arrive at a rate of one every 6 minutes with the arrival rate exhibiting a Poisson distribution. Determine: A) The average number of customers in line. B) The average waiting time in the system. WP5

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Irwin/McGraw-Hill Example: Model 2 (worked problem) A) The average number of customers in line. B) The average waiting time in the system. = + = hrs+ 1 15/hr =.1333 hrs = 1 8 mins = ( - ) = (10) (2)(15)(15-10) = W q = ( - ) = )(-10) = hrs= ( 4 mins WP6 WsWs W q LqLq

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Irwin/McGraw-Hill Approximating Customer Waiting Time 70 A quick and dirty method has been developed to compute average waiting time for multiple servers (see text, pp ). No assumptions about the underlying distributions are required! All that is needed are 4 numbers – the average and standard deviation of the inter-arrival time and service time.

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Irwin/McGraw-Hill Utilization and Time in System 70

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