Page 1

Aggregation and Numerical Techniques for Passage

Time Calculations in Large semi-Markov Models

Marcel Christoph G¨ unther

mcg05@doc.ic.ac.uk

June 18, 2009

Marker: Dr. Jeremy Bradley

Second marker: Dr. William Knottenbelt

Department of Computing

Imperial College London

Page 2

2

Abstract

First-passage time densities and quantiles are important metrics in performance analysis. They

are used in the analysis of mobile communication systems, web servers, manufacturing systems

as well as for the analysis of the quality of service of hospitals and government organisations.

In this report we look at computational techniques for the first-passage time analysis on high-

level models that translate to Markov and semi-Markov processes. In particular we study exact

first-passage time analysis on semi-Markov processes. Previous studies have shown that it is

possible to analytically determine passage times by solving a large set of linear equations in

Laplace space. The set of linear equations arises from the state transition graph of the Markov

or semi-Markov process, which is usually derived from high-level models such as process algebras

or stochastic Petri nets. The difficulty in passage time analysis is that even simple high-level

models can produce large state transition graphs with several million states and transitions.

These are difficult to analyse on modern hardware, because of limitations in the size of main

memory. Whilst for Markov processes there exist several efficient techniques that allow the

analysis of large chains with more than 100 million states, in the semi-Markov domain such

techniques are still less developed. Consequently parallel passage time analyser tools currently

only work on semi-Markov models with fewer than 50 million states. This study extends existing

techniques and presents new approaches for state space reduction and faster first-passage time

computation on large semi-Markov processes. We show that intelligent state space partitioning

methods can reduce the amount of main memory needed for the evaluation of first-passage time

distributions in large semi-Markov processes by up to 99% and decrease the runtime by a factor

of up to 5 compared to existing semi-Markov passage time analyser tools. Finally we outline a

new passage time analysis tool chain that has the potential to solve semi-Markov processes with

more than 1 billion states on contemporary computer hardware.

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3

Acknowledgements

I would like to thank my supervisor Jeremy Bradley for all the support and guidance he has

given me throughout the project as well as for his enthusiam about my research which always

motivated me to carry on.

I would also like to thank Nicholas Dingle for giving me feedback on my experiments, provid-

ing SMARTA and helping me to overcome various technical problems I encountered during the

project. Likewise I would like to thank William Knottenbelt for his support and his feedback

on my written work.

Finally I would like to thank my friends and family, especially my parents, Netta, Marco, Steve

and Daniel whose birthday I forgot because of the write-up.

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4

Computers process what they are being fed. When rubbish goes in, rubbish comes out.

Trans.: EDV-Systeme verarbeiten, womit sie gef¨ uttert werden. Kommt Mist rein, kommt Mist raus.

—

Andr´ e Kostolany

Page 5

CONTENTS5

Contents

1 Introduction

1.1 Motivation

1.1.1

1.2 Current state of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3Project aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

8

8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Application of passage times in performance analysis . . . . . . . . . . . .

10

10

11

12

2Background

2.1 Semi-Markov Processes (SMPs) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2High-level modelling formalism for SMPs

2.2.1Petri nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.2Generalised stochastic Petri nets . . . . . . . . . . . . . . . . . . . . . . .

2.2.3 Semi-Markov stochastic Petri nets . . . . . . . . . . . . . . . . . . . . . .

2.2.4 SM-SPN models used in this study . . . . . . . . . . . . . . . . . . . . . .

2.3Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 Laplace transform inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4.1 Numerical Laplace transform inversion . . . . . . . . . . . . . . . . . . . .

2.5 Measures in SMP analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.1Transient and steady-state distribution

2.5.2Passage time analysis in semi-Markov models . . . . . . . . . . . . . . . .

2.6Numerical methods for first-passage time analysis . . . . . . . . . . . . . . . . . .

2.6.1 Iterative approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7 Exact state aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.8 Graph partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.8.1Graph Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.8.2 Partitioning metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.8.3Recursive bi-partitioning vs. k-way partitioning . . . . . . . . . . . . . . .

2.8.4Objective functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.8.5Flat vs. Multilevel hypergraph partitioning . . . . . . . . . . . . . . . . .

2.8.6Multilevel hypergraph partitioning . . . . . . . . . . . . . . . . . . . . . .

13

13

14

14

15

16

17

17

19

19

21

21

21

22

22

24

26

26

28

29

29

29

29

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

3Partitioning the SMP state space

3.1SMP transition matrix partitioners . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.1Row striping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.2Graph partitioner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.3Hypergraph partitioner. . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

33

33

33

34

Page 6

6CONTENTS

3.1.4

Aggregation of partitions

3.2.1 Partition sorting strategies

3.2.2 Transition matrix predictor . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.3 Quality of partitionings . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Next-Best-State-Search (NBSS) partitioner . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

34

34

35

36

36

3.2

4State-by-state aggregation of partitions

4.1State aggregation techniques

4.1.1 Fewest-Paths-First aggregation . . . . . . . . . . . . . . . . . . . . . . . .

4.1.2 Exact-Fewest-Paths-First aggregation . . . . . . . . . . . . . . . . . . . .

4.2Transition matrix fill-in during aggregation of partition

4.3 Partial aggregation of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.1Cheap state aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 Implementation of state-by-state aggregation . . . . . . . . . . . . . . . . . . . .

4.4.1 Data structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

41

41

41

43

44

44

46

46

47

47

47

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

5Atomic aggregation of entire partitions

5.1Aggregation techniques

5.1.1Restricted FPTA aggregator

5.1.2Discrete event simulation aggregator . . . . . . . . . . . . . . . . . . . . .

5.1.3 RFPTA with extra vanishing state . . . . . . . . . . . . . . . . . . . . . .

5.2 Barrier partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1 Passage time computation on barrier partitionings . . . . . . . . . . . . .

5.2.2 Balanced barrier partitioner . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3K-way barrier partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.1 K-way barrier partitioner . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4 Implementation of atomic partition aggregation . . . . . . . . . . . . . . . . . . .

5.4.1 Performance RFPTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4.2 Performance of the barrier strategies . . . . . . . . . . . . . . . . . . . . .

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

49

50

52

52

55

57

58

60

62

63

64

65

65

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

6Applying new techniques for faster FPTA calculation

6.1FPTA techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1.1Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1.2Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2Path truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2.1 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2.2 Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3 Parallelisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

66

66

67

67

68

69

71

72

7Evaluation, conclusion and further work

7.1 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3.1Building the billion state semi-Markov response time analyser . . . . . . .

73

73

73

74

74

A Models studied

A.1 Voting model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A.2 Web-content authoring (web-server) model . . . . . . . . . . . . . . . . . . . . . .

A.3 Courier model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

75

76

78

Page 7

CONTENTS7

B Additional diagrams for barrier partitioning discussion 79

C Additional diagrams for FPTA performance discussion80

Bibliography87

Page 8

81. INTRODUCTION

CHAPTER 1

Introduction

1.1Motivation

Whenever we time processes we would like to know the worst-case time to complete the job.

This notion of time until completion is captured by response time distributions. In particular

the cumulative density function of response time distributions are of interest since they allow

us to make statements such as: ”In 90% of all cases the job is completed after x seconds”.

Such intervals are also known as response time quantiles or percentiles.

metric is preferable to average reponse times, as these fail to give an intuition of the worst-

case scenario. Reponse time quantiles are widely used in the analysis of network latencies, web

servers, manufacturing systems as well as for the analysis of the quality of service of hospitals and

government organisations to name a few areas of application. Response time analysis can also

be performed on models such as Markov and semi-Markov processes. In this case we talk about

first-passage time distributions, as response time analysis in the Markovian domain corresponds

to evaluating the distribution over the time it takes to reach a set of target states from a set of

source states in the transition graph of the chain.

This performance

1.1.1 Application of passage times in performance analysis

In this section we give two brief examples of applications of response time quantiles, one real-

world example and one example that illustrates the passage time analysis on a semi-Markov

model that has been generated from a semi-Markov stochastic Petri net (see sect. 2.2.3).

The first example was drawn from a report of the U.S. department of homeland security[31].

The report investigates the performance of the national fire services. The measure of interest is

the distribution of the time it takes from the point a call is received by the emergency call center

until a fire-engine arrives at the scene. The 90thpercentile in this case is less than 11 minutes

(see fig. 1.1). The report further investigates regional and seasonal differences in response time.

Clearly such investigations are useful especially when introducing new regulation or procedures

to public services or in industry, as they provide an objective measure on how the quality of

service compares to earlier years.

Page 83

C. ADDITIONAL DIAGRAMS FOR FPTA PERFORMANCE DISCUSSION83

0

100000

5e+10

1e+11

1.5e+11

2e+11

2.5e+11

3e+11

200000 300000

Nof states in web-server model (FPTA with precision 1e-16)

400000 500000 600000 700000 800000 900000 1e+06

Absolute number of complex multiplications for FPTA

NoBarrier FPTA

NoBarrierCheap FPTA

NoBarrierTruncated FPTA

NoBarrierCheapTruncated FPTA

Barrier FPTA

BarrierCheap FPTA

BarrierTruncated FPTA

BarrierCheapTruncated FPTA

Figure C.4:

0

2e+10

4e+10

6e+10

8e+10

1e+11

1.2e+11

1.4e+11

1.6e+11

1.8e+11

200000 400000 600000 800000 1e+06

Absolute number of complex multiplications for FPTA

Nof states in voting model (FPTA with precision 1e-16)

NoBarrier FPTA

NoBarrierCheap FPTA

NoBarrierTruncated FPTA

NoBarrierCheapTruncated FPTA

Barrier FPTA

BarrierCheap FPTA

BarrierTruncated FPTA

BarrierCheapTruncated FPTA

Figure C.5:

Page 84

84C. ADDITIONAL DIAGRAMS FOR FPTA PERFORMANCE DISCUSSION

Voting model

Untruncated

Truncated

Number of states

NoBarrier

NoBarrierCheap

Barrier

BarrierCheap

NoBarrier

NoBarrierCheap

Barrier

BarrierCheap

100000

100%

84%

70%

61%

61%

35%

54%

44%

250000

100%

49%

64%

49%

45%

24%

39%

30%

500000

100%

49%

63%

49%

35%

19%

30%

24%

1100000

100%

190%

95%

113%

30%

51%

36%

43%

Web-server model

Untruncated

Truncated

Number of states

NoBarrier

NoBarrierCheap

Barrier

BarrierCheap

NoBarrier

NoBarrierCheap

Barrier

BarrierCheap

100000

100%

83%

58%

49%

36%

30%

26%

22%

250000

100%

84%

57%

48%

33%

28%

24%

20%

500000

100%

84%

56%

48%

31%

26%

22%

19%

1000000

100%

84%

56%

48%

26%

23%

18%

16%

Table C.1: Table contains numerical data used to plot figs. 6.1 and C.1. To get this data we simply divided the number of of complex multiplicationneeded for the FPTA with a particular technique by the number of complex multiplications needed for the NoBarrier method.

Page 85

BIBLIOGRAPHY85

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Page 88

88INDEX

Index

M[t >, 15

M → M?, 15

s-point, 19

t-point, 19

Atomic partition aggregation, 49

Balanced barrier partitioning, 58

Barrier partitioning, 55

Boundary-cut, 28

Cheap states, 44

Convolution, 18

Discrete event simulation (DES) aggregator, 52

Edge-cut, 28

Enhanced-Fewest-Paths-First (EFPF) sort, 35

Equilibrium state, 21

Euler inversion, 19

Exact state-by-state aggregation, 24

Exact-Fewest-Paths-First(EFPF) aggregation,

41

Exponential order, 18

Extra vanishing state, 53

Fewest-Paths-First (FPF) sort, 35

Fewest-Paths-First(FPF) aggregation, 41

First-passage time distribution, 21

Flat graph, 29

Fully connected, 35

Gain, 29

Generalised stochastic Petri net (GSPN), 15

Graph partitioner, 33

Hyperedge-cut, 29

Hypergraph, 27

Hypergraph partitioner, 34

Intermediate state, 33

Inverse Laplace transform, 19

k-way barrier partitioning, 60

k-way partitioning, 26

Kernel, 13

Kolmogorov–Smirnov (K–S), 66

Laguerre inversion, 19

Laplace transform, 18

Look-Ahead-N-Steps, 35

Marking, 15

Max-way barrier partitioning, 62

Negligably small Laplace transform sample, 67

Net-enabling function, 15

Next-Best-State-Search (NBSS) partitioner, 34

Partition entry state, 52

Partition exit state, 53

Partition transient path, 49

Partitionwise observations, 32

Place-Transition net, 14

Predecessor states, 24

rthtransition first-passage time, 23

Restricted first-passage time analysis (RFPTA),

49

Restricted FPT aggregator, 50

Reverse RFPTA, 51

Row striping partitioner, 33

Semi-Markov process (SMP), 13

Sojourn time, 13

Sparse matrix, 26

State-space, 15

Steady-state distribution, 21

Stochastic Petri nets, 16

Sub-matrix, 41

Successor states, 24

Tangible marking, 16

Total volume of communication, 28

Transient distribution, 21

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Transition graph, 14

Transition matrix, 14

Transition matrix fill-in, 26

Vaninishing state, 52

Vanishing marking, 16