IERG 3050 Week 8Output Data Analysis and Variance

Reduction Techniques

Bolei ZhouDepartment of Information Engineering

The Chinese University of Hong Kong

Announcement

• Homework 3 is due tomorrow• Homework 4 will be out by the end of today. No need to hand in, but

very relevant to Quiz 2• Quiz 2 will be on Nov. 6• No lectures next week (Time for your course project)• End of Oct. 2019: a 4-page check-point report for course project

Outline

• Output Data Analysis• Stochastic Processes• Ensemble Averages vs. Time Averages • Transient State and Steady State Behavior • Method of Replication• Transient Removal Methods• Terminating and Non-Terminating Simulations • Obtaining Specified Precision• Choosing Initial Conditions• Variance-Reduction Techniques: Common Random Numbers, Antithetic Variates, Control

Variates, Indirect Estimation, ConditioningReading: Chapters 9 and 11

Output Data Analysis

4

Output Data Analysis

5

Output Data Analysis

6

Review: Stochastic processes

Review: Random Variables

Review: Random Variables

Review: Random Variables

Review: Stochastic Processes

Review: Stochastic Processes

12

Some Simple Stochastic Processes

13

Example 1: I.I.D. Process

14

Example 2: “Fixed-Value” Process

15

Example 3: Autoregressive Process

16

Example 4: Hidden Markov Process

17

Example 5: M/M/1/1 Queueing System

18

Notations and Terms of Queueing Systems

Kendall notation: A/S/m/B/K/SD• A: interarrival time distribution• S: service time distribution• m: number of servers• B: system capacity: max number of users accommodated in the

system

Interesting Quantities

20

Interesting Quantities

21

Interesting Quantities

22

Ensemble Averages

23

Example 1: I.I.D. Process

24

Example 2: “Fixed-Value” Process

25

Example 3: Autoregressive Process

26

Estimating Ensemble Averages

27

Estimating Time Averages

28

Time Averages

29

Example 1: I.I.D. Process

30

Example 2: “Fixed-Value” Process

31

Example 3: Autoregressive Process

32

Time Averages

33

Transient state and steady-state behavior

Example 3: Autoregressive Process

35

Example 3: Autoregressive Process

36

Example 3: Autoregressive Process

37

Example 3: Autoregressive Process

38

Example 3: Autoregressive Process

39

Transient vs. Steady State

40

Method of Replication

41

Method of Replication

42

Method of Replication

43

Method of Replication

44

Transient Removal

45

Some Transient Removal Methods

46

Terminating and non-terminating simulations

Terminating and Non-Terminating Simulations

48

Terminating and Non-Terminating Simulations

49

Obtaining a specified precision

Obtaining a Specified Precision

51

Choosing initial conditions

Choosing Initial Conditions

53

Variance reduction techniques

Variance-Reduction Techniques

Common Random Numbers

Common Random Numbers

Common Random Numbers

Common Random Numbers

Antithetic variates

Antithetic Variates

Antithetic Variates

Antithetic Variates

Control Variates

Control Variates

Control Variates

Control Variates

Indirect Estimation

Indirection Estimation

Indirection Estimation

Conditioning

Conditioning

Conditioning

Thank you!

• Homework 3 is due tomorrow• Homework 4 will be out by the end of today. No need to hand in, but

very relevant to Quiz 2• Quiz 2 will be on Nov. 6• No lectures next week (Time for your course project)• End of Oct. 2019: a 4-page check-point report for course project