1. QMD: Waiting-line analysis

2. Overview Terminology Characteristics of Waiting-lines
Operating characteristics
M/M/1 model – single channel
Example: video store
M/M/S model – multi-channel
Example: Multiplex theater
Psychology of waiting

3. You’ve Been There Before!
QM D Total Cost per Hours
You’ve Been There Before!
© 1995 Corel Corp.
Thank you for holding. Hello...are you there?
‘The other line always moves faster.’
‘If you change lines, the one you left will start to move faster than the one you’re in.’
MGT 301 S.K. Williams

4. Waiting Line System
Population
Service Facility
Waiting Line

5. QM D Total Cost per Hours
Waiting Line Examples
Situation Arrivals Servers Service Process
MGT 301 S.K. Williams

6. QM D Total Cost per Hours
Waiting Line Examples
Situation Arrivals Servers Service Process
Bank Customer Teller Deposit etc.
Doctor’s Patient Doctor Treatment office
Traffic Cars Light Controlled intersection passage
Assembly line Parts Workers Assembly
Tool crib Workers Clerks Check out/in tools
MGT 301 S.K. Williams

7. Waiting Line Terminology
QM D Total Cost per Hours
Waiting Line Terminology
Queue: Waiting line
Arrival: 1 person, machine, part, etc. that arrives and demands service
Queue discipline: Rules for determining the order that arrivals receive service
Channel: Number of servers
Phase: Number of steps in service
MGT 301 S.K. Williams

8. Arrival Characteristics of a Waiting Line System
Population
Service Facility
Waiting Line
Pattern of arrivals
random
scheduled
Behavior of the arrivals
join the queue, and wait until served
No balking; refuse to join the line
No reneging; leave the line

9. QM D Total Cost per Hours
Balking
Input source
Service facility
Waiting line
Service system
© 1995 Corel Corp.
Line was too long!
MGT 301 S.K. Williams 2

10. QM D Total Cost per Hours
Reneging
Input source
Service facility
Waiting line
Service system
© 1995 Corel Corp.
I give up!
MGT 301 S.K. Williams 2

11. Line Characteristics of a Waiting Line System - continued
Population
Service Facility
Waiting Line
Length of the queue
limited
unlimited
Service priority
FIFO
other

12. Waiting Line Arrangements
Service facilities
(a) Single line

13. Waiting Line Arrangements
Service facilities
(a) Single line
Service facilities
(b) Multiple lines

14. Waiting Line Arrangements
Service facilities
(a) Single line
Service facilities
(b) Multiple lines

15. Service Facility Characteristics of a Waiting Line System - continued
Population
Waiting Line
Number of channels
single
multiple
Number of phases in service system

16. Service Facility Arrangements

17. Service Facility Arrangements
(a) Single channel, single phase

18. Single-Channel, Single-Phase System
QM D Total Cost per Hours
Single-Channel, Single-Phase System
Arrivals
Served units
Service facility
Queue
Service system
Dock
Waiting ship line
Ships at sea
Ship unloading system
Empty ships
MGT 301 S.K. Williams 63

19. Service Facility Arrangements
(b) Single channel, multiple phase

20. Single-Channel, Multi-Phase System
QM D Total Cost per Hours
Single-Channel, Multi-Phase System
Arrivals
Served units
Service facility
Queue
Service system
Pick-up
Waiting cars
Cars in area
McDonald’s drive-through
Pay
Cars & food
MGT 301 S.K. Williams 64

21. Service Facility Arrangements
(c) Multiple channel, single phase

22. Multi-Channel, Single Phase System
QM D Total Cost per Hours
Multi-Channel, Single Phase System
Arrivals
Served units
Service facility
Queue
Service system
Example: Bank customers wait in single line for one of several tellers.
MGT 301 S.K. Williams 62

23. Decision Areas Arrival rates Number of service channels
Number of phases
Service time
Priority rule
Line arrangement

24. Performance Measures or Operating Characteristics
QM D Total Cost per Hours
Performance Measures or Operating Characteristics
Average queue length, Lq
Average number in system, Ls
Average queue time, Wq
Average time in system, Ws
System utilization, 
MGT 301 S.K. Williams

25. Types of Queuing Models
QM D Total Cost per Hours
Types of Queuing Models
Simple (M/M/1)
Example: Information booth at mall
Multi-channel (M/M/S)
Example: Airline ticket counter
Constant Service (M/D/1)
Example: Automated car wash
Limited Population
Example: Department with only 7 drills
MGT 301 S.K. Williams

26. Simple (M/M/1) Model Characteristics
QM D Total Cost per Hours
Simple (M/M/1) Model Characteristics
Type: Single-channel, single-phase system
Queue: Unlimited; single line
Queue discipline: FIFO (FCFS)
Service rate > arrival rate
MGT 301 S.K. Williams

27. Queuing Models  - ‘lambda’  - ‘mu’ arrival rate of the customers
service rate for a single server
If the service time is 15 minutes per customer, what is the service rate??

28. Simple (M/M/1) Model Equations
QM D Total Cost per Hours
Simple (M/M/1) Model Equations
Average number of customers in the system
Average number of customers in the queue
MGT 301 S.K. Williams

29. Simple (M/M/1) Model Equations
Average time customers are in the system
Average time customers are in the queue

30. Simple (M/M/1) Model Equations
Utilization factor – percent of time that the single server is busy; also the probability that a customer will wait

31. Example: Video Store
The manager of a video store is interested in providing good service. On a Friday or Saturday night, on average 30 customers per hour arrive at the counter to check out a video. The customers are served at an average rate of 35 customers per hour from a single cash register.

32. Example: Video Store
Determine the operating characteristics for the video store.
Average number of customers in line
Average number of customers in the system
Average wait time in line
Average time in the system
Server utilization

33. Average number of customers in line

34. Average number of customers in the system

35. Average wait time in line

36. Average time in the system

37. Server utilization

38. Example: Video Store
What service rate would be required to have customers average only 5 minutes in the system?

39. What service rate would be required to have customers average only 5 minutes in the system?
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40. Multichannel (M/M/S) Model Characteristics
QM D Total Cost per Hours
Multichannel (M/M/S) Model Characteristics
Type: Multichannel system
Queue discipline: FIFO (FCFS)
 Service rates > arrival rate
MGT 301 S.K. Williams

41. Multichannel M/M/S Equations
QM D Total Cost per Hours
Multichannel M/M/S Equations
Probability of zero customers in the system:
Average number of customers in the system:
Average number of customers waiting for service:
MGT 301 S.K. Williams

42. Multichannel M/M/S Equations
QM D Total Cost per Hours
Multichannel M/M/S Equations
Average time a unit spends in the system:
Average time a person or unit spends in the queue
MGT 301 S.K. Williams

43. Example: Movie Theater
A multiplex movie theater has 3 concession clerks serving customers on a first come, first served basis. The service time per customer is exponentially distributed with an average of 2 minutes per customer. Concession customers arrive at a rate of 81 customers per hour. 10 minutes of previews run in the lobby. If the average time in the concession area exceed 10 minutes, customers become dissatisfied.

44. Example: Movie theater
What is the average number of customers in the concession area?

45. What is the average number of customers in the concession area?
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46. Multiplex Movie Theater
The distribution is reducing the length of the previews to 8 minutes. How many servers will be required to so that the average wait is no longer than 8 minutes?

47. QM D Total Cost per Hours
Total cost per Hour= waiting cost + service cost
TC = CwLs + KCs
Cw – cost of waiting per hour per customer
Ls – number of customers in the system
K – number of servers
Cs- service cost per hour per server
MGT 301 S.K. Williams

48. Total Cost: Video Store
Considering adding another cashier for Friday and Saturday nights.
Part-time cashier will cost $6.00 per hour
Customers waiting time (cost of goodwill, etc) is $10.00 per hour.
Is it cost effective to hire the cashier?

49. Total Cost: Video Store M/M/1
TC = CwLs + KC
Cw = $10/hour
Ls = 6
K = 1
CS = $6.00/hr
TC = 10(6) (6)
= $66/hour

50. Total Cost: Video Store M/M/2
TC = CwL + KC
Cw = $10/hour
Ls = 1.05
K = 2
Cs = $6.00/hr
TC = 10(1.05) + 2(6)
= $22.50/hr

51. Remember:  &  Are Rates
QM D Total Cost per Hours
Remember:  &  Are Rates
 = Mean number of arrivals per time period
e.g., 3 units/hour
 = Mean number of people or items served per time period
e.g., 4 units/hour
1/ = 15 minutes/unit
If average service time is 15 minutes, then μ is 4 customers/hour
© T/Maker Co.
MGT 301 S.K. Williams

52. The Psychology of Waiting
Maister’s (1985) Propositions of customer satisfaction in queues
Unoccupied time feels longer than occupied time
Pre-process wait feels longer than in-process waits.
Anxiety makes waits feel longer

53. The Psychology of Waiting
Uncertain waits seem longer than certain waits
Unexplained waits seem longer than explained waits
Unfair waits seem longer than equitable waits

54. Psychology of Waiting
More valuable the service, the longer people will wait.
Solo waiting feels longer than group waiting.