Module C8
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Module C8
Queuing
Economic/Cost Models
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
Example 1Determining 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
GOAL:
Average time in the queue, WQ < .00833hrs.
………..
How many servers?
Arrival rate = 100/hr.
Average service time1/ = 5 min. = 5/60 hr.
= 60/5 = 12/hr.
.003999
First time WQ < .008333
12 servers needed
Input values for and
Example 2Determining 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.
Server Salary Average Service Time
– Ann $ 6/hr. 6 min.
– Bill $ 10/hr. 5 min.
– Charlie $ 14/hr. 4 min.
• Which server should be hired?
Ann1/ = 6 min.
A = 60/6 = 10/hr.
Hourly Cost =$6 + 4LAnn
LAnn
= 8/hr
LAnn = 4
LAnn Hourly Cost =$6 + $4(4) = $22
4
Bill1/ = 5 min.
B = 60/5 = 12/hr.
Hourly Cost =$10 + 4LBill
LBill LBill
= 8/hr
LBill = 2
Hourly Cost =$10 + $4(2) = $18
2
Charlie1/ = 4 min.
C = 60/4 = 15/hr.
Hourly Cost =$14 + 4LCharlie
LCharlie LCharlie Hourly Cost =$14 + $4(1.14) = $18.56
1.14
= 8/hr
LCharlie = 1.14
Optimal
• Ann --- Total Hourly Cost = $22
• Bill --- Total Hourly Cost = $18
• Charlie --- Total Hourly Cost = $18.56
HireBill
Example 3What 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?
Option 1 -- 1 clerk, 1 lane
Store
= 24/hr.
1/ = 2 min. = 60/2 = 30/hr.
Total Hourly CostSalary + Lanes + Wait Cost
$6.50 + $20 + $25LQ
LQ = 3.2
Total Hourly CostSalary + Lanes + Wait Cost
$6.50 + $20 + $25(3.2) = $106.50
Option 2 -- 2 clerks, 1 lane
= 24/hr.
1 Service System1/ = 1.25 min.
= 60/1.25 = 48/hr.
Total Hourly CostSalary + Lanes + Wait Cost
2($6.50) + $20 + $25LQ Store
LQ = .5
Total Hourly CostSalary + Lanes + Wait Cost
2($6.50) + $20 + $25(.5) = $45.50
Option 3 -- 2 clerks, 2 lanes
Store
= 24/hr.
Total Hourly CostSalary + Lanes + Wait Cost
2($6.50) + $40 + $25LQ
Store
1/ = 2 min. = 60/2 = 30/hr.
LQ = .152
Total Hourly CostSalary + Lanes + Wait Cost
2($6.50) + $40 + $25(.152) = $56.80
Optimal
• Option 1 --- Total Hourly Cost = $106.50
• Option 2 --- Total Hourly Cost = $ 45.50
• Option 3 --- Total Hourly Cost = $ 58.80
BestOption 2
Example 4Which 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?
Large Store
= 30/hr.
…
6Servers
UnlimitedQueueLength
All customersget served!
Lease Cost = $1000/day= $1000/10 = $100/hr.
Small Store
= 30/hr.
2Servers
MaximumQueue
Length = 1
Lease Cost = $200/day= $200/10 = $20/hr.
Will join system if0 or 1 in the queue
Will not join thequeue if there is
1 customer in the queue
Hourly Profit Analysis
Large Small
Arrival Rate = 30 e = 30(1-p3)
Revenue
$25(Arrival Rate) (25)(30)=$750 $25e
Costs
Lease $100 $20
Server $20(#Servers) (20)(6) =$120 (20)(2) =$40
Waiting $8(Avg. in Store) =$8L $8L
Net Profit ? ?
Large Store -- M/M/6
3.099143
L
Small Store -- M/M/2/3
Lp3
e = (1-.44262)(30) = 16.7213
Hourly Profit Analysis
Large Small
Arrival Rate = 30 e = 30(1-p3)
Revenue
$25(Arrival Rate) (25)(30)=$750 $25e
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
Example 5Which 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?
Machine 1
Machine 2
Machine Comparisons
Machine1 Machine 2
Prob (No Wait) -- P0 .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.
Module C8 Review
• List Components of System
• Develop a model
• Use templates to get parameter estimates
• Select “optimal” design
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