1. Queuing Models
Economic Analyses

2. 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

3. Example 1 Determining Optimal Number of Servers
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 = hours

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

5. Input values for  and  First time WQ < .008333 12 servers needed
First time WQ <
12 servers needed
Go to the MMk Worksheet

6. Example 2 Determining 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?

7. ANN
Ann 1/ = 6 min.
A = 60/6 = 10/hr.
LAnn = ?
Hourly Cost =
$6 + 4LAnn
 = 8/hr

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

9. BILL
Bill 1/ = 5 min.
A = 60/5 = 12/hr.
LBill = ?
Hourly Cost =
$10 + 4LBill
 = 8/hr

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

11. CHARLIE
Charlie 1/ = 4 min.
A = 60/5 = 15/hr.
LCharlie = ?
Hourly Cost =
$14 + 4LCharlie
 = 8/hr

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

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

14. Example 3 What 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?

15. Salary + Lanes + Wait Cost
System clerk, 1 lane
1/ = 2 min.
 = 60/2 = 30/hr.
Store
Total Hourly Cost
Salary + Lanes + Wait Cost
$ $20 + $25LQ
 = 24/hr.

16. System 1 -- 1 clerk, 1 lane LQ = 3.2 1/ = 2 min.  = 60/2 = 30/hr.
Store
Total Hourly Cost
Salary + Lanes + Wait Cost
$ $20 + $25(3.2) = $106.50
Total Hourly Cost
Salary + Lanes + Wait Cost
$ $20 + $25LQ
 = 24/hr.

17. Salary + Lanes + Wait Cost
System clerks, 1 lane
1 Service System
1/ = 1.25 min.
 = 60/1.25 = 48/hr.
Store
Total Hourly Cost
Salary + Lanes + Wait Cost
2($6.50) + $20 + $25LQ
 = 24/hr.

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

19. Salary + Lanes + Wait Cost
System clerks, 2 lanes
1/ = 2 min.
 = 60/2 = 30/hr.
Store
Store
Total Hourly Cost
Salary + Lanes + Wait Cost
2($6.50) + $40 + $25LQ
 = 24/hr.

20. System 3 -- 2 clerks, 2 lanes 1/ = 2 min.  = 60/2 = 30/hr. LQ = .152
Store
Store
Total Hourly Cost
Salary + Lanes + Wait Cost
2($6.50) + $40 + $25LQ
Total Hourly Cost
Salary + Lanes + Wait Cost
2($6.50) + $40 + $25(.152) = $56.80
 = 24/hr.

21. Optimal Best option -- System 2
System Total Hourly Cost = $106.50
System Total Hourly Cost = $
System Total Hourly Cost = $
Best option -- System 2

22. Example 4 Which 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, no line limit) or a Small Store ($200/day, 2 clerks – maximum of 3 in store)?

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

24. Small Store Lease Cost = $200/day = $200/10 = $20/hr.
2 Servers
Maximum
Queue Length = 1
Will not join the queue if there are
3 customers in the system
Will join system if 0,1,2 in the system
 = 30/hr.

25. Hourly Profit Analysis
Large Small
Arrival Rate  = e = 30(1-p3)
Hourly Revenue
$25(Arrival Rate) (25)(30)=$ $25e
Hourly Costs
Lease $ $20
Server $20(#Servers) $ $40
Waiting $8(Avg. in Store) $8Ll $8Ls
Net Hourly Profit ? ?

26. Large Store -- M/M/6 Ll

27. Small Store -- M/M/2/3 Ls p3
e = ( )(30) =

28. Hourly Profit Analysis
Large Small
Arrival Rate  = e =
Hourly Revenue
$25(Arrival Rate) $ $25e=$418
Hourly Costs
Lease $ $20
Server $20(#Servers) $ $40
Waiting $8(Avg. in Store) $ $17
Net Hourly Profit $ $341
Lease the
Large Store

29. Example 5 Which 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?

30. Machine 1

31. Machine 2

32. Machine 2 looks preferable
Machine Comparisons
Machine1 Machine 2
Prob (No Wait) -- P
Average Service Time 6 min min.
Average # in System
Average # in Queue
Average Time in System.1625 hr hr.
9.75 min min.
Average Time in Queue hr hr.
3.75 min min.
Machine 2 looks preferable

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