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Module C8 Queuing Economic/Cost Models

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ECONOMIC ANALYSES Each problem is different Examples –To determine the minimum number of servers to meet some service criterion (e.g. an average of < 4 minutes in the queue) -- trial and error with M/M/k systems –To compare 2 or more situations -- Consider the total (hourly) cost for each system and choose the minimum

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Example 1 Determining Optimal Number of Servers Customers arrive to an electronics store at random at an average rate of 100 per hour. Service times are random but average 5 min. How many servers should be hired so that the average time of a customer waits for service is less than 30 seconds? –30 seconds =.5 minutes =.00833 hours

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GOAL: Average time in the queue, W Q <.00833hrs. ……….. How many servers? Arrival rate = 100/hr. Average service time 1/ = 5 min. = 5/60 hr. = 60/5 = 12/hr.

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.003999 First time W Q <.008333 12 servers needed Input values for and

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Example 2 Determining Which Server to Hire Customers arrive at random to a store at night at an average rate of 8 per hour. The company places a value of $4 per hour per customer in the store. Service times are random. The average service time that depends on the server. ServerSalary Average Service Time – Ann$ 6/hr.6 min. – Bill$ 10/hr.5 min. – Charlie$ 14/hr.4 min. Which server should be hired?

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Ann 1/ = 6 min. A = 60/6 = 10/hr. Hourly Cost = $6 + 4L Ann L Ann = 8/hr L Ann = 4 L Ann Hourly Cost = $6 + $4(4) = $22 4

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Bill 1/ = 5 min. B = 60/5 = 12/hr. Hourly Cost = $10 + 4L Bill L Bill = 8/hr L Bill = 2 Hourly Cost = $10 + $4(2) = $18 2

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Charlie 1/ = 4 min. C = 60/4 = 15/hr. Hourly Cost = $14 + 4L Charlie L Charlie Hourly Cost = $14 + $4(1.14) = $18.56 1.14 = 8/hr L Charlie = 1.14

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Optimal Ann--- Total Hourly Cost = $22 Bill--- Total Hourly Cost = $18 Charlie--- Total Hourly Cost = $18.56 Hire Bill

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Example 3 What Kind of Line to Have.A fast food operation is considering opening a drive-up window food service operation. Customers arrive at random at an average rate of 24 per hour. Three systems are being considered. Customer waiting time is valued at $25 per hour. Each clerk makes $6.50 per hour. Each drive-thru lane costs $20 per hour to operate. Which system should be used?

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Option 1 -- 1 clerk, 1 lane Store = 24/hr. 1/ = 2 min. = 60/2 = 30/hr. Total Hourly Cost Salary + Lanes + Wait Cost $6.50 + $20 + $25L Q L Q = 3.2 Total Hourly Cost Salary + Lanes + Wait Cost $6.50 + $20 + $25(3.2) = $106.50

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Option 2 -- 2 clerks, 1 lane = 24/hr. 1 Service System 1/ = 1.25 min. = 60/1.25 = 48/hr. Total Hourly Cost Salary + Lanes + Wait Cost 2($6.50) + $20 + $25L Q Store L Q =.5 Total Hourly Cost Salary + Lanes + Wait Cost 2($6.50) + $20 + $25(.5) = $45.50

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Option 3 -- 2 clerks, 2 lanes Store = 24/hr. Total Hourly Cost Salary + Lanes + Wait Cost 2($6.50) + $40 + $25L Q Store 1/ = 2 min. = 60/2 = 30/hr. L Q =.152 Total Hourly Cost Salary + Lanes + Wait Cost 2($6.50) + $40 + $25(.152) = $56.80

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Optimal Option 1--- Total Hourly Cost = $106.50 Option 2--- Total Hourly Cost = $ 45.50 Option 3--- Total Hourly Cost = $ 58.80 Best Option 2

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Example 4 Which Store to Lease Customers are expected to arrive to a store location at random at an average rate of 30 per hour. The store will be open 10 hours per day. Service times can be considered random. The average sale grosses $25. Clerks are paid $20/hr. including all benefits. The cost of having a customer in the store is estimated to be $8 per customer per hour. Clerk Service Rate = 10 customers/hr. Should they lease a Large Store or Small Store?

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Large Store = 30/hr. … 6 Servers Unlimited Queue Length All customers get served! Lease Cost = $1000/day = $1000/10 = $100/hr.

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Small Store = 30/hr. 2 Servers Maximum Queue Length = 1 Lease Cost = $200/day = $200/10 = $20/hr. Will join system if 0 or 1 in the queue Will not join the queue if there is 1 customer in the queue

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Hourly Profit Analysis Large Small Arrival Rate = 30 e = 30(1-p 3 ) Revenue $25(Arrival Rate) (25)(30)=$750 $25 e Costs Lease $100$20 Server $20(#Servers) (20)(6) =$120 (20)(2) =$40 Waiting $8(Avg. in Store) =$8L $8L Net Profit ? ?

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Large Store -- M/M/6 3.099143 L

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Small Store -- M/M/2/3 L p3p3 e = (1-.44262)(30) = 16.7213

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Hourly Profit Analysis Large Small Arrival Rate = 30 e = 30(1-p 3 ) Revenue $25(Arrival Rate) (25)(30)=$750 $25 e Costs Lease $100$20 Server $20(#Servers) (20)(6) =$120 (20)(2) =$40 Waiting $8(Avg. in Store) =$8L $8L Net Profit ? ? 30 16.7213 $750 $418 $100 $ 20 $120 $ 40 $8(3.099143) $ 25$8(2.11475) $ 17 $750-$100-$120-$25 $505 $418-$20-$40-$17 $341 Lease the large store

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Example 5 Which Machine is Preferable Jobs arrive randomly to an assembly plant at an average rate of 5 per hour. Service times do not follow an exponential distribution. Two machines are being considered –(1) Mean service time of 6 min. ( = 60/6 = 10/hr.) standard deviation of 3 min. ( = 3/60 =.05 hr.) –(2) Mean service time of 6.25 min.( = 60/6.25 = 9.6/hr.) standard deviation of.6 min. ( =.6/60 =.01 hr.) Which of the two designs seems preferable?

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Machine 1

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Machine 2

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Machine Comparisons Machine1 Machine 2 Prob (No Wait) -- P 0.5000.4792 Average Service Time 6 min. 6.25 min. Average # in System.8125.8065 Average # in Queue.3125.2857 Average Time in System.1625 hr..1613 hr. 9.75 min. 9.68 min. Average Time in Queue.0625 hr..0571 hr. 3.75 min. 3.43 min.

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Module C8 Review List Components of System Develop a model Use templates to get parameter estimates Select “optimal” design

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