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Performability (Performance and Reliability)

Modeling

- Conducted by Meng-Lai Yin, Ph.D.
- Specialty Engineering, Network Centric Systems
- Raytheon Company, Fullerton, California
- 714-446-3080, mlyin_at_raytheon.com
- Department of Electrical and Computer Engineering
- California State Polytechnic University, Pomona
- 909-869-2535, myin_at_csupomona.edu

An Example

The purpose of this example is to showthe

existences of performance degradable systems

An email received on July 20, 2005 433PM

- We are experiencing problems with the AIX user

account file systems. We need to take the AIX

system off-line immediately to fix the problem.

We expect the AIX file systems to be off line for

approximately an hour and a half. We hope to

have the file systems back on-line by 600PM. - Sorry for any inconvenience.
- Sys Admin Team

The system is off completely

Later that day July 20, 2005 626PM

- All AIX file systems are back on-line except

wei_snoop which is in a rebuild stage. Wei_snoop

file system will be back on-line by 0600 tomorrow

morning. - Thanks,
- Sys Admin Team

The system is on a degraded mode

Observations

- The system can operate without the wei_snoop file

system

More and more systems become performance

degradable

Performance Degradable Systems

- Performance degradable systems have the

capability of continuing to operate failure-free

in the presence of certain faults or errors by

diminishing the level of quality of service 7.

Typical Scenario A system starts with all

components operational and performs at its

maximum capability. When a component fails, the

system will reconfigure itself and operate with

degraded performance, etc.

Reasons for Performability Modeling

- Two separate measures
- Traditional dependability analysis assumes no

performance degraded states. - Performance measures always are applied to fully

operational state. - Need an integrated, meaningful metric
- For performance degradable systems, where the

system can operate in many different states, how

do you address the systems performance with the

consideration of degraded performance situations? - Traditional metrics (performance, reliability,

availability. etc.) and the corresponding

modeling techniques cannot catch the overall

performance feature for performance degradable

systems.

The Beginning of Performability

- The term Performability was introduced almost

three decades ago 4, by Prof. J. F. Meyer.

John F. Meyer Address 4111 EECS Phone (734)

763-0037Fax (734) 763-1503 Professor Emeritus,

Electrical Engr Computer ScienceDegree Ph.D.,

U-Michigan

A Tribute to M. D. Beaudry

- Before Dr. John F. Meyer gave the name

performability to the world, several works

actually had already been devoted to address the

issue of providing appropriate metrics for

performance degradable systems. - In Particular, the work conducted by Danielle

Beaudry 1 has been referenced in many places. - In 1, she addressed the performance-related

reliability measures for gracefully degraded

systems (performance degradable systems ).

Objectives

- At the conclusion of this tutorial, a participant

will be able to - know the basic concepts about performability
- know how to
- conduct a basic dependability analysis using

Reliability Block Diagram (RBD) or Markov

techniques - conduct a basic performance analysis using

Queuing models - conduct a basic performability analysis

Approach

Outline

Part 2

Part 1

The Two Basic Questions

- What is performability?
- Why is performability needed?

- What is performability?
- Performability is a metric to evaluate the

performance over time - Modeling performability is modeling the effect of

reliability on performance.

Why is performability needed?

- The appearance of performance degradable systems
- Better system designs evaluations for systems

that considers both performance and dependability

Example 2

A Performance-Degradable, 3-processor System

The best performance occurs when all three

processors work correctly in parallel, as the

three processors share the workload. The system

is performance-degradable, meaning if one or two

are failed, the system continues working with

degraded performance.

The purpose of this example is to show that

traditional performance and dependability models

cannot tell the overall picture of the systems

operations.

Performance Model

The best performance occurs when all three

processors work correctly in parallel, as the

three processors share the workload. A typical

queuing model for performance assessment

Jobs arrival rate ? Processors service rates

?1, ?2 and ?3Solving this model yields

performance measures such as response time,

throughput, etc.

Dependability Model

Problems with Traditional Measures

Pure performance measure too optimistic! Outage-

and-recovery behaviors are not considered Pure

dependability measure too conservative! Degraded

levels of performance are not considered (The

system is either working or failed)

Outline

Part 2

Part 1

Basics

DependabilityAnalysis

PerformanceAnalysis

Reliability, Availability, Dependability

- They are all probabilities.
- What are the differences?

Definition of Reliability The probability of an

item to perform a required function under given

conditions for a given time interval.

Definition of Availability "The probability of

an item to be in a state to perform a required

function at a given instant of time, assuming

that the external resources, if required, are

provided.

Differences

time

t0

?

Reliability the probability that the item

survive theduration t0, ?)

time

t0

?

Availability the probability that the item is

working at time ?, given that the item was

working at time t0.

Picture the Differences

1.0

Steady state availability

A typical reliability figure (without repair)

A typical availability figure (with repair)

Calculating Reliability Availability

- Let ? be the failure rate for a component, and ?

be the repair rate for that component. - Assume exponential distribution for the failures
- Then reliability can be calculated as R(t) e

-? t - SS (Steady-State) -Availability can be assessed

as - or

Dependability Umbrella term

Courtesy of prof. Trivedi

Dependability Analysis

Modeling Taxonomy

Simulation

Modeling

RBD

Non-State-Space Method

Analytic modeling

State-Space Method

Markov

- Approaches discussed here
- Reliability Block Diagrams
- Markov Models

Combinatorial Approach

- Consider
- a system of n components
- every component is either working or failed
- We can
- list out all the possible combinations
- calculate the probability for each combination
- sum up probabilities for all working conditions

Complexity Concerns

- How many possible combinations out of the n

components? - What can be done to manage the complexity?
- During model construction
- Need a more intelligent way to describe the

systems failure behavior - Series and parallel RBD (Reliability Block

Diagram) approach - During model solution
- Need more efficient and effective ways of

calculations, rather than counting individual

probabilities

Structured Combinatorial Approach

- Reliability block diagrams
- Integrate certain probability events into a

module, which contains the info - A probability of failure
- A failure rate
- A distribution of time to failure
- Steady-state and instantaneous unavailability
- Organize the modules in a structured way,

according to the effects of each modules failure - Statistical independence Assumption
- Failures independence
- Repairs independence

Series Systems

- Each component (block) is needed to make the

system work - If any one of the components fails, the system

fails - Example 3

The purpose of this example is to show how to

construct a simple series RBD model and solve it

using Excel

RDB Example for a Series System

- System Block Diagram for Example 3

Reliability Block Diagram Model Reliability

Calculation

- RBD for Example 3

Processor

Monitor

Keyboard

Let ?1 be the failure rate for Monitor Assume

exponential distribution for the failures,

thenRmonitor(t) e -?1 t Similarly,

Rprocessor(t) e -?2 t and Rkeyboardv(t) e

-?3 t

Rsystem (t) Rmonitor (t) Rprocessor (t)

Rkeyboard (t) e -?1 t e -?2 t e -?3 t

e (?1 t ?2 t ?3 t) e (?1?2?3) t

When exponential failure distribution is

assumed, the failure rate of a series system is

the sum of individual components failure rates

Excel Exercise 1

- Use Excel Spreadsheet to construct the above

Series RBD - Show the trend of reliability with regard to the

time factor - Show the relationship between reliability and the

failure rate

SS-Availability Calculation

Let ?1, ?2, ?3 be the failure rates and ?1, ?2,

?3 be the repair rates for the monitor, processor

and keyboard. Then

- ASS-Monitor
- ASS-processor
- ASS-keyboard

ASS-system-series

Hierarchical Composition/Decomposition

- Problem the size of the model grows with the

size of the system. - Issue Fidelity vs. Complexity

Hudson Professor of Electrical and Computer

Engineering Duke University Phone (919)

660-5269Fax (919) 660-5293Email

kst_at_ee.duke.edu

Trivedi

Parallel Systems

- A basic parallel system only one of the N

identical components is required for the system

to function - Example 4

Example 4 Basic Parallel System

- System Block Diagram

The purpose of example 4 is to show the parallel

RBD and the corresponding reliability/availability

calculations.

RDB example Parallel System

- Reliability Block Diagram

RDB using Hierarchical Composition/Decomposition

The Highest level (overall system level)

Computer

Computer

or

1 of 2

1 of 2

Usually indicate two different components

On the Computer level

Monitor

Processor

Keyboard

Reliability Calculation

- The Unreliability of the parallel system can be

computed as the probability that all N components

fail. - Assume all N components are having the same

failure rate ?, and the probability that a

component is failed at time t is Pfail(t) - Rparallel(t) 1- ?i1 to N Pfail(t)

Independence Assumption

- Where in the above equation that the independence

assumption is made? - Just to remind you

- Failure/Repair Dependencies are often assumed
- RBD usually does not handle the dependency such

as - Event-dependent failure
- Shared repair

Availability Calculation

- ASS-Monitor
- ASS-processor
- ASS-keyboard

ASS-system-parallel

Excel Exercise 2

- ?monitor 1? 10-4 failures per hour
- ?processor 1? 10-5 failures per hour
- ?keyboard 4? 10-4 failures per hour
- ? 2 repair per hour for all components (MTTR30

minutes) - For series system, ASS is
- For parallel system (with 12 redundancy), ASS is

Parallel/Series System Example 5

Processor 1

Keyboard 1

Monitor 1

Bus 1

Bus 2

Computer 2

Keyboard 1

Monitor 1

What is the corresponding RBD ?

The purpose of Example 5 is to demonstrate a

simple design process using RBD

Corresponding RBD

Assuming Buses are perfect

Monitor

Processor

Keyboard

Keyboard

Monitor

Processor

Compare to the RBD below, which one has better

reliability?

Monitor

Processor

Keyboard

Monitor

Processor

Keyboard

Modeling Steps

- Model construction
- Model parameterization
- Model solution
- Result interpretation
- Model validation

N Modular Redundancy

- K of N System
- K of the total of N identical modules are

required to function, K ? N - TMR (Triple Modular Redundancy) is a famous

example, where K is 2 and N is 3

Example 6 RBD for TMR

Module 1

Voter

Module 2

Module3

Module 3

Single point of failure

Module2

Voter

Module1

The purpose of example 6 is to 1. introduce

TMR 2. show how to model a TMR component. 3.

show the impacts of single-point-of-failure

2 3

TMR Reliability

Module3

Module2

Voter

Module1

- Cases for the TMR to be working
- all of the 3 modules are working
- any 2 modules are working, and 1 module is

failed - Look at it from another way
- Cases for the TMR to be failed
- all 3 modules are failed
- any one module is working, however, the rest 2

are not working - Remember, the voter is a Single-Point-Of-Failure

2 3

one Module voter TMR System

0.999 0.999 0.999997 0.998997005

From this chart, you can see the effect that a

single point of failure made ismuch more

significant than that of a component with

redundancy

Dependability Analysis Markov Modeling

Modeling Taxonomy

- Approaches (Discussed here)
- Reliability Block Diagrams
- RBD for Series Systems
- RBD for Parallel Systems
- Markov Models

- Why Markov ?
- Who is Markov?

- What is Markov ?

- How to construct a Markov model?

- How to solve a Markov model ?

- What are the issues to be considered ?

Model Selection

- There are wide range of models available, each

has its strength and weakness.

- Combinatorial models (reliability block diagrams,

fault trees) are straightforward and easy to

understand.

- However, it is not easy to model non-independent

behavior using combinatorial models. - Markov model can model the state changes

Who was Markov?

- Andrei A. Markov graduated from Saint Petersburg

University in 1878 and subsequently became a

professor there. - His early work dealt mainly in number theory and

analysis, etc. - Markov is particularly remembered for his study

of Markov chains. - These chains are sequences of random variables in

which the future behavior is determined by the

present state but is independent of the way of

how the present state is reached.

Markovian Property

- Markovian property

Given the present state, the future is

independent of the past.

Definition of Markov Process A stochastic

process X(t) t ? T is called a Markov process

if for any t0 lt t1 lt ... tn lt t, the conditional

distribution of X(t) for given values of X(t0),

X(t1), ...X(tn) depends only on X(tn).

A simple Markov Chain

- A continuous-time, discrete-state Markov process

Pure-birth process (Poisson process if l0 l1

l2 )

Example 7

Non-identical p1 and p2 p1 has failure rate l1,

repair rate m1 p2 has failure rate l2, repair

rate m2

1, 2

Both p1 and p2 are working

p1 is working, p2 is failed

1

p2 is working, p1 is failed

2

0

Both p1 and p2 are failed

What can be solved ?

- Basically, the probability of each state.

- The transient solution is the probability at a

certain point of time t.

- The steady-state solution is the steady-state

probability (t ? ?).

- Others

Analytical Solution on a 2-state Model

l

W

F

m

Analytical Solution -cont. 1

Solving (1) and (2) obtains Pw(s) 1/(s(lm))

m/(s(s(lm)) PF(s) l/(s(s(lm))

Analytical Solution- cont. 2

- use the Inverse Laplace transform
- pw(t) m/(lm) l/(lm) e-(lm)t
- pF (t) l/(lm) - l/(lm) e-(lm)t
- note that when t goes to infinity, the above has

the steady state solution (recall the steady

state availability)

A Simple way of Solving Steady-State irreducible

Markov chains

A Markov chain is irreducible if every state can

be reached from every other state.

- Name the probability for each state
- List out the balance equations
- Add one more equation that the sum of all states

prob. is one. - You have the choice of deleting one balance

equation

Example 8 RBD Markov Approaches Comparison

- TMR with a perfect voter

RBD

Markov

(Only failures transitions are shown here. When

identical modules are assumed, the model can be

further reduced.)

Reduced Markov Model for TMR

Solve availability using RBD

Availability Prob. All 3 modules are working

Prob.Any two modules are working and one is

failed

Discussion

- Why do they both reach the same results?
- Due to the assumptions of
- Independent repairs
- Independent failures
- Exponential distribution
- Whats the implication?
- If the above assumptions were made, choose the

easier way

State Explosion Problem

The largeness problem can be handled in 2 ways

tolerated or avoided

Largeness Avoidance and Hierarchical Models

Large models can be avoided by using hierarchical

model composition or decomposition

References 1 P.J. Courtois , Decomposability -

Queueing and Computer System Applications,

Academic Press, INC. 1977. 2 R.A. Sahner, K.S.

Trivedi, Reliability Modeling Using SHARPE,

IEEE Trans. On Reliability, R-36, 2. June

1987. 3 R.A. Sahner, K.S. Trivedi, Antonio

Puliafito, Performance and Reliability Analysis

of Computer Systems, Kluwer Academic Publishers,

1996.

Example 9

The purpose of thisexample is to demonstrate the

Hierarchical Decomposition Method.

Required 1 processor, 2 memory, 1 network

A Demonstration of Hierarchical Decomposition

Modeling

Reliability Block Diagram

Memory 1

Processor 1

Network

Memory 2

Processor 2

Memory 3

1 of 2

2 of 3

1 of 1

Need to consider the effects of software errors,

and other dependencies

The Markov model with 6 components will have 26

64 states

Example 9 Continue

Highest level

ProcessorSubsystem

MemorySubsystem

NetworkSubsystem

1 of 2

2 of 3

1 of 1

Sub-System Level

NetworkSubsystem

ProcessorSubsystem

A small-size model is handled every

time. Flexibility vs. Complexity

MemorySubsystem

Outline

Part 2

Part 1

Introduction

ReliabilityModeling

PerformanceModeling

A Simple Performance Modeling Mechanism Task

Graph

Each task can be assigned a value ti to represent

the time the task takes

tA

tB

tC

tD

Then you can calculate the time to complete all

tasks as

tA max (tB , tC ) tD

Independent Parallel Tasks

- Let FA(t), FB(t), FC(t), FD(t) be the

distribution functions for the time each task

takes - Tasks B and C are executed in parallel. Denote

the probability distribution that both of them

are finished by time t as FBC (t) - If B and C were independent tasks (no sharing

resources), then FBC (t) FB(t)FC(t)

Two independent events

Serial Tasks

- When 2 tasks are executed serially, the

distribution function for the time until the

second job finishes is the convolution of the 2

distributions F1(t)?F2(t) - The overall distribution function for the time to

finish all tasks in task graph is - FA?FBC ?FD

Contention for Resources

- The model above assumes no contention for

resources - In real world applications, limited resources

must be shared. Hence resource contention is

expected. - Queuing model is useful in modeling this kind of

systems

Queuing Network

- A queuing network consists of service centers and

customers (often called jobs) - A service center consists of one or more servers

and one or more queues to hold customers waiting

for service.

? customers arrival rate

? service rate

Interarrival time

- Interarrival time the time between successive

customer arrivals - Arrival process the process that determines the

interarrival times - It is common to assume that interarrival times

are exponentially distributed random variables.

In this case, the arrival process is a Poisson

process

Service Time

- The service time depends on how much work the

customer needs and how fast the server is able to

perform the work - Service times are commonly assumed to be

independent, identically distributed (iid) random

variables.

A Queuing Model Example

Server Center 1

Server Center 2

cpu1

Disk1

cpu2

Disk2

queue

Arrivingcustomers

cpu3

servers

Server Center 3

Queuing delay

Service time

Response time

Terminology

Measures of Interests

- queue length
- response time
- throughput the number of customers served per

unit of time - utilization the fraction of time that a service

center is busy

A relationship between throughput and response

time the Littles law.

The Littles Law

- The mean number of jobs in a queuing system in

the steady state is equal to the product of the

arrival rate and the mean response time

The average number of customers in a queuing

system

The average response time

? the average arrival rate of customers

admitted to the system

Notation

- Queuing Model is usually described as X/Y/Z/K/L/D

(M denotes the exponential distribution, E for

Erlang, H for hyperexponential, D for

deterministic and G for general)

X Arrival process Y Service processZ Number

of servers at the service center K Buffer

sizeL Population size D The queuing discipline

K, L and D are often omitted, which means K, L

are ? and D is FCFS

M/M/1 Queue

- M/M/1 queue
- The first M means the arrival process is

exponential distributed - The second M means the service process is

exponential distributed - 1 means the number of servers is 1
- Assuming buffer size and population size are

infinity - First-Come First-Served discipline is applied

?

?

This example shows how to solve the M/M/1 queue,

and many other aspects with regard to Queuing

models.

Solving M/M/1 queue

- Solved for
- The steady-state probability in each state
- Server Utilization
- The expected number of customers in the system
- The average response time

Construct the corresponding Markov Chain

A typical birth-death process

Birth-Death Process

Birth-Death Process is a Markov chain where the

transitions can occur only between adjacent

states. You can solve the Markov chain

analytically by solving the set of balance

equations.

Solving M/M/1 Queue

called the traffic intensity

Define the ratio

When ? lt 1 (meaning ?lt?), the system is called

stable, and the steady state probabilities can

be determined by

M/M/1 Queue Property

The mean of number of customers in the system

EN

M/M/1 Queue Property Average Response Time

The Littles Formula

From previous discussion

Average Response Time

Example 10 How does Fast Lane work in

Disneyland?

Construct a M/M/1 queue Solve for the average

response time, as a function of the inter-arrival

time Control the people flow to assure the

response time (refer to Excel)

Intermission

References

- 1M.Danielle Beaudry, Performance-Related

Reliability Measures for Computing Systems, IEEE

Transactions on Computer, Vol. C-27, No. 6, June

1978. - 2 G. Ciardo and A.S. Miner, SMART Simulation

and Markovian analyzer for reliability and

timing, Proceeding of IEEE International

Computer Performance and Dependability Symposium

(IPDS96), September 1996. - 3G. Clark, T. Courtney, D. Daly, D. Deavours,

S. Derisavi, J. M. Doyle, W. H. Sanders, and P.

Webster. (01CLA01), The Möbius Modeling Tool,

Proceedings of the 9th International Workshop on

Petri Nets and Performance Models, Aachen,

Germany, September 11-14, 2001, pp. 241-250. - 4 John F. Meyer, On Evaluating the

Performability of Degradable Computer Systems,

IEEE Transactions on Computers, Vol. C-29, No.8,

August 1980. - 5 John F. Meyer, Performability A

retrospective and some pointers to the future,

Performance Evaluation 14 (1992) 139-156. - 6 John F. Meyer, William H. Sanders,

Specification and Construction of Performability

Models, Chapter 9 in the book Performability

Modeling Techniques and Tools 7. - 7 Performability Modeling Techniques and

Tools, Edited by B.R. Haverkort, R. Marie, G.

Rubino,K. Trivedi, John Wiley Sons, Inc. 2001.

ISBN 0-471-49195-0. - 8 Andrew L. Reibman, Modeling the Effect of

Reliability on Performance, IEEE Transactions on

Reliability, Vol. 39, No.3, Aug. 1990. - 9 Robin A. Sahner, Kishort S. Trivedi, Antonio

Puliafito, Performance and Reliability Analysis

of Computer Systems, Kluwer Academic Publishers,

1996, ISBN 0-7923-9650-2. - 10 William H. Sanders and John F. Meyer,

Stochastic Activity Networks Formal Definitions

and Concepts, in E. Brinksma, H. Hermanns, and

J. P. Katoen (Eds.), Lectures on Formal Methods

and Performance Analysis, First EEF/Euro Summer

School on Trends in Computer Science, Berg en

Dal, The Netherlands, July 3-7, 2000, Revised

Lectures, Lecture Notes in Computer Science no.

2090, pp. 315-343. Berlin Springer, 2001. - 11 Ann Tai, John F. Meyer, and Algirdas

Avizienis, Software Performability from Concepts

to Applications, Kluwer Academic Publishers,

1996, ISBN 0-923-9670-7. - 12 Kishor S. Trivedi, Probability and

Statistics with Reliability, Queuing and Computer

Science Applications, Wiley, 2002. ISBN

0-471-33341-7. - 13 Meng-Lai Yin, Hierarchical-Compositional

Performability Modeling for Fault-Tolerance

Multiprocessor Systems, Ph.D. Dissertation,

University of California, Irvine, 1995, UMI

Dissertation Services. - 14 Meng-Lai Yin, Douglas Blough, Lubomir Bic,

A Dependability Analysis for Systems with Global

Spares, IEEE Transactions on Computers, Sep.

2000, pp. 958-963.

A Special birth-death process Poisson Process

- ?(n)? for all n ? 0
- ?(n)0 for all n ? 0
- Pure birth process
- Definition of the Poisson process
- The counting process N(t), t ? 0 is said to be

a Poisson process having rate ?, ?gt0, if - N(0)0
- The process has independent increments
- The number of events in any interval of length t

is Poisson distributed with mean ?t.

Poisson Process

- For all s, t ?0
- PN(ts)-N(s) n
- EN(t) ? t
- Poisson properties
- Inter-arrival times are exponentially distributed
- Memoryless property the prob. of occurrence of

an event is independent of how many events have

occurred in the past and the time since the last

event

M/M/2 queue with Heterogeneous Servers

- The service rates of the two processors are not

identical

- Constructing the model
- Represent a state with 2 numbers (n, m)
- Where n is the number of jobs at server 1, m is

the number of jobs at server 2

Solving the model for steady-state prob.

Two Servers in Tandem

?1

?2

?

- Constructing the model
- Represent a state with 2 numbers (n, m)
- Where n is the number of jobs at server 1, m is

the number of jobs at server 2

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