1. Variability and Its Impact on Process Performance: Waiting Time Problems Chapter 7

2. A Somewhat Odd Service Process (Chapters 1-6)
Patient
Arrival
Time
Service 1 2 3 4 5 6 7 8 9 10 11 12 15 20 25 30 35 40 45 50 55
7:00
7:10
7:20
7:30
7:40
7:50
8:00

3. A More Realistic Service Process
Patient 1
Patient 3
Patient 5
Patient 7
Patient 9
Patient 11
Patient 2
Patient 4
Patient 6
Patient 8
Patient 10
Patient 12
Patient
Arrival
Time
Service 1 2 3 4 5 6 7 8 9 10 11 12 18 22 25 30 36 45 51 55
Time
7:00
7:10
7:20
7:30
7:40
7:50
8:00 3 2
Number of cases 1
2 min.
3 min.
4 min.
5 min.
6 min.
7 min.
Service times

4. Variability Leads to Waiting TimePt
Arrival
Time p 1 2 3 4 5 6 7 8 9 10 11 12 18 22 25 30 36 45 51 55
Service time
Wait time
7:00
7:10
7:20
7:30
7:40
7:50
8:00 5 4 3
Inventory
(Patients at lab) 2 1
7:00
7:10
7:20
7:30
7:40
7:50
8:00

5. Sources of Variability
Processing
Buffer

6. The “Memoryless” Exponential Function
Interarrival times follow an exponential distribution
Exponential interarrival times = Poisson arrival process
Coefficient of Variation (CV )

7. Quite possibly the worst run of three slides you will see this semester

8. Waiting Time Flow rate Average flow time T Inventory waiting Iq
In process Ip
Increasing
Variability
Inflow
Outflow
Entry to system
Begin Service
Departure
Time in queue Tq
Service Time p
Theoretical Flow Time
Total Flow Time T=Tq+p
Utilization
100%
A Waiting Time Formula
Service time factor
Utilization factor
Variability factor

9. Waiting Time for Multiple, Parallel Resources
Inventory
in service Ip
Inventory
waiting Iq
Inflow
Outflow
Flow rate
Entry to system
Begin Service
Departure
Time in queue Tq
Service Time p
Total Flow Time T=Tq+p
The Waiting Time Formula for Multiple (m) Servers

10. Summary of Queuing Analysis
Utilization
Flow unit
Server
Inventory
in service Ip
Time related measures
Outflow
Inflow
Inventory
waiting Iq
Inventory related measures
Entry to
system
Begin
Service
Departure
Waiting Time Tq
Service Time p
Flow Time T=Tq+p

11. 7. 2. E-mails arrive to My-Law. com from 8 a. m. to 6 p. m
7.2 s arrive to My-Law.com from 8 a.m. to 6 p.m. at a rate of 10 s per hour (cv=1). At each moment in time, there is exactly one lawyer “on call” waiting these s. It takes on average five minutes to respond with a standard deviation of four minutes.
What is the average time a customer must wait for a response?
How many s will a lawyer receive at the end of the day?
When not responding to s, the lawyer is encouraged to actively pursue cases that potentially could lead to large settlements. How much time during a ten hour shift can be devoted to this pursuit?
To increase responsiveness of the system, a template will be used for most responses. The standard deviation for writing the response now drops to 0.5 minutes but the average writing time is unchanged.
Now how much time can a lawyer spend pursuing cases?
How long does a customer have to wait for a response to an inquiry?

12. Service Levels in Waiting Systems
Fraction of customers who have to wait x seconds or less 1
90% of calls had to wait 25 seconds or less
Waiting times for those customers who do not get served immediately
0.8
0.6
0.4
Fraction of customers who get served without waiting at all
0.2 50
100
150
200
Target Wait Time (TWT)
Service Level = Probability{Waiting TimeTWT}
Example: Deutsche Bundesbahn Call Center - now (2003): 30% of calls answered within 20 seconds - target: 80% of calls answered within 20 seconds
Waiting time [seconds]

13. Data in Practical Call Center Setting
Number of customers
Distribution Function
Per 15 minutes
160
160 1 1
140
140
0.8
0.8
120
120
Exponential distribution
100
100
0.6
0.6 80 80
0.4
0.4
Empirical distribution 60 60
(individual points) 40 40
0.2
0.2 20 20
0:00:00
0:00:09
0:00:17
0:00:26
0:00:35
0:00:43
0:00:52
0:01:00
0:01:09
0:15
0:15
2:00
2:00
3:45
3:45
5:30
5:30
7:15
7:15
9:00
9:00
10:45
10:45
12:30
12:30
14:15
14:15
16:00
16:00
17:45
17:45
19:30
19:30
21:15
21:15
23:00
23:00
Time

14. The Power of Pooling Waiting Time Tq 70.00 m=1 60.00 50.00 40.00 m=2
Independent Resources
2x(m=1)
Waiting
Time Tq
70.00
m=1
60.00
50.00
40.00
- OR -
m=2
30.00
Pooled Resources
(m=2)
20.00
m=5
10.00
m=10
0.00
60%
65%
70%
75%
80%
85%
90%
95%
Utilization u

15. Priority Rules in Waiting Time Systems
Service times:
A: 9 minutes B: 10 minutes C: 4 minutes D: 8 minutes A C
4 min.
9 min. B D
19 min.
13 min. C A D
21 min.
23 min. B
Total wait time: =38 min
Total wait time: =51min
SPT
FCFS
EDD
Priority
WNIFOARB

16. 7.3 The airport branch of a car rental company maintains a fleet of 50 SUVs. The interarrival time between requests for an SUV is 2.4 hours with a standard deviation of 2.4 hours. Assume that, if all SUVs are rented, customers are willing to wait until an SUV is available. An SUV is rented, on average for 3 days, with a standard deviation of 1 day.
What is the average number of SUVs in the company lot?
What is the average time a customer has to wait to rent an SUV?
The company discovers that if it reduces its daily $80 rental price by $25, the average demand would increase to 12 rentals per day and the average rental duration will become 4 days. Should they go for it?
How would the waiting time change if the company decides to limit all SUV rentals to exactly 4 days? Assume that if such a restriction is imposed, the average interarrival time will increase to 3 hours as will the std deviation.