Queuing ModelsQueuing Models

Economic Analyses

ECONOMIC ANALYSESECONOMIC 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 1Example 1Determining Optimal Number of Determining Optimal Number of

ServersServers• Customers arrive according to a Poisson

process to an electronics store at random at an average rate of 100 per hour.

• Service times are exponential and 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

1212 servers needed

Input values for and

Go to the MMkWorksheet

Example 2Example 2Determining Which Server to HireDetermining Which Server to Hire• Customers arrive according to a Poisson process

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 exponential and 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

= 8/hr

LAnn = ?

ANNANN

Ann1/ = 6 min.

A = 60/6 = 10/hr.

Hourly Cost =$6 + 4LAnn

LAnn

= 8/hr

LAnn = 4

Hourly Cost =$6 + $4(4) = $22

Bill1/ = 5 min.

A = 60/5 = 12/hr.

Hourly Cost =$10 + 4LBill

= 8/hr

LBill = ?

BILLBILL

Bill1/ = 5 min.

B = 60/5 = 12/hr.

Hourly Cost =$10 + 4LBill

LBill LBill

= 8/hr

LBill = 2

Hourly Cost =$10 + $4(2) = $18

Charlie1/ = 4 min.

A = 60/5 = 15/hr.

Hourly Cost =$14 + 4LCharlie

= 8/hr

LCharlie = ?

CHARLIECHARLIE

Charlie1/ = 4 min.

C = 60/4 = 15/hr.

Hourly Cost =$14 + 4LCharlie

LCharlie LCharlie1.14

= 8/hr

LCharlie = 1.14

Hourly Cost =$14 + $4(1.14) = $18.56

OptimalOptimal

• Ann --- Total Hourly Cost = $22

• Bill --- Total Hourly Cost = $18

• Charlie --- Total Hourly Cost = $18.56

HireBillBill

Example 3Example 3What Kind of Line to HaveWhat Kind of Line to Have

• A fast food restaurant will be opening a drive-up window food service operation whose service distribution is exponential.

• Customers arrive according to a Poisson process at an average rate of 24/hr. Three systems are being considered.

• Customer waiting time is valued at $25/hr. • Each clerk makes $6.50/hr. • Each drive-thru lane costs $20/hr. to operate

Which of the following systems should be used?Which of the following systems should be used?

System 1 -- 1 clerk, 1 laneSystem 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

System 1 -- 1 clerk, 1 laneSystem 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

System 2 -- 2 clerks, 1 laneSystem 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

System 2 -- 2 clerks, 1 laneSystem 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

System 3 -- 2 clerks, 2 lanesSystem 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.

System 3 -- 2 clerks, 2 lanesSystem 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

OptimalOptimal

• System 1 --- Total Hourly Cost = $106.50

• System 2 --- Total Hourly Cost = $ 45.50

• System 3 --- Total Hourly Cost = $ 58.80

Best option -- System 2

Example 4Example 4Which Store to LeaseWhich Store to Lease

• Customers are expected to arrive by a Poisson process to a store location at an average rate of 30/hr.

• The store will be open 10 hours per day. • 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. (Exponential)

Should they lease a Large Store ($1000/day, 6 clerks, Should they lease a Large Store ($1000/day, 6 clerks, no line limit) or a Small Store ($200/day, 2 clerks – no line limit) or a Small Store ($200/day, 2 clerks – maximum of 3 in store)?maximum of 3 in 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 if0,1,2 in the system

Will not join thequeue if there are

3 customers in the system

Hourly Profit AnalysisHourly Profit Analysis

Large Small

Arrival Rate = 30 e = 30(1-p3)

Hourly RevenueHourly Revenue

$25(Arrival Rate) (25)(30)=$750 $25e

Hourly CostsHourly Costs

LeaseLease $100 $20

ServerServer $20(#Servers) $120 $40

WaitingWaiting $8(Avg. in Store) $8L $8L

Net Hourly ProfitNet Hourly Profit ?? ??

Large Store -- M/M/6 Large Store -- M/M/6

3.099143

L

Small Store -- M/M/2/3Small Store -- M/M/2/3

Lp3

e = (1-.44262)(30) = 16.7213

Hourly Profit AnalysisHourly Profit Analysis

Large Small

Arrival Rate = 30 e = 16.7213

Hourly RevenueHourly Revenue

$25(Arrival Rate) $750 $25e=$418

Hourly CostsHourly Costs

LeaseLease $100 $20

ServerServer $20(#Servers) $120 $40

WaitingWaiting $8(Avg. in Store) $25 $17

Net Hourly ProfitNet Hourly Profit $505$505 $341 $341

Lease theLease theLarge StoreLarge Store

Example 5Example 5Which Machine is PreferableWhich Machine is Preferable

• Jobs arrive according to a Poisson process to an assembly plant at an average of 5/hr.

• 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.); std.

dev. of .6 min. ( = .6/60 = .01 hr.)

Which of the two M/G/1designs is preferable?Which of the two M/G/1designs is preferable?

Machine 1Machine 1

Machine 2Machine 2

Machine ComparisonsMachine Comparisons

Machine1 Machine1 Machine 2Machine 2

Prob (No Wait) -- P0 .5000.5000 .4792

Average Service Time 6 min.6 min. 6.25 min.

Average # in System .8125 .80658065

Average # in Queue .3125 .2857.2857

Average Time in System.1625 hr. .1613 hr..1613 hr.

9.75 min. 9.68 min.9.68 min.

Average Time in Queue .0625 hr. .0571 hr..0571 hr.

3.75 min. 3.43 min.3.43 min.

Machine 2 looks preferable

ReviewReview

• List Components of System

• Develop a model

• Use templates to get parameter estimates

• Select “optimal” design