Module C8

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