1 – Reliability

Modeling and Simulationof Communication Systems and Networks

Chapter 1. Reliability

Prof. Jochen Seitz

Technische Universitat IlmenauCommunication Networks Research Lab

Summer 2010

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1 – Reliability

Outline

1 1.1 Definition of Reliability

2 1.2 Modeling Reliability

3 1.3 System Reliability1.3.1 Systems Without Redundancy1.3.2 Systems With Redundancy1.3.3 Comparing System Reliability

4 1.5 References

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1 – Reliability

1.1 Definition of Reliability

What does “Reliability” mean?

• Measure for the ability of a component / a system to sustainits functionality.

• Probability that a desired function can be fulfilled during aspecified time period under given working conditions.

• Reliability Parameters:• Reliability – R(t)• Failure Probability – F (t)• Failure Density Function – f (t)• Failure Rate – λ(t)

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1 – Reliability

1.1 Definition of Reliability

Typical Characteristics of the Failure Rate

t

l(t)

Phase 1 Phase 2 Phase 3

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1 – Reliability

1.1 Definition of Reliability

Important: Mean ValuesMean value of failure probability:

• Mean Time To Failure (MTTF)

MTTF = E [T ] =

∞∫0

t · f (t)dt

=

∞∫0

R(t)dt =

∞∫0

e−

t∫0

λ(τ)dτdt

• Mean Time Between Failures (MTBF)Special case: λ(t) = λ = constant

MTBF =

∞∫0

e−λtdt =1

λ

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1 – Reliability

1.2 Modeling Reliability

Exponential Distribution – R(t)R(t) = e−λt

l = 0,25l = 0,5l = 0,75l = 2

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1 – Reliability

1.2 Modeling Reliability

Exponential Distribution – F (t)F (t) = 1 − R(t) = 1 − e−λt

l = 0,25l = 0,5l = 0,75l = 2

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1 – Reliability

1.2 Modeling Reliability

Exponential Distribution – f (t)f (t) = −dR(t)

dt = λe−λt

l = 0,25l = 0,5l = 0,75l = 2

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1 – Reliability

1.2 Modeling Reliability

Exponential Distribution – λ(t)λ(t) = λ

l = 0,25l = 0,5l = 0,75l = 2

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1 – Reliability

1.2 Modeling Reliability

Other Distributions (I)

The Epxonential Distribution is only valid for phase 2, since λ isconstant.Other distribution functions used for phases 1 or 3:

• Weibull Distribution(generalization of the exponential distribution)

F (t) = 1 − e−( tλ

)k

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1 – Reliability

1.2 Modeling Reliability

Other Distributions (II)

• Gamma Distribution

F (t) =1

Γ(β)

λt∫0

xβ−1e−xdx

f (t) = λ(λt)β−1

Γ(β)e−λt

λ(t) =f (t)

1 − F (t)

Γ(β) =

∞∫0

xβ−1e−xdx

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1 – Reliability

1.3 System Reliability

Reliability of Systems (I)

• Communication systems and networks consist of differentcomponents that have different failure characteristics.

Network

Subnet

Router

SubnetA Subnet B

Subnet CSubnet D

Router 1

Router 2

Router 4

Router 6

Router 7

Router 5

Router 3

NetworkInterface

1

NetworkInterface

2Processor Memory

PowerSupply

…

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1 – Reliability

1.3 System Reliability

Reliability of Systems (II)• To improve the reliability of a system, components can be

redundantly built in:1 Hot Redundancy

There is no differentiation between the main and theredundant component.

2 Warm RedundancyThe redundant components are not as heavily loaded as themain component.

3 Cold Redundancy / Standby RedundancyThe redundant components are not loaded at all.

• Redundancy is the duplication of critical components of asystem with the intention of increasing reliability of thesystem.

• How can the system’s reliability be determined if thereliabilities of all components are known?

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1 – Reliability

1.3 System Reliability

1.3.1 Systems Without Redundancy

Systems Without Redundancy

System

Compo-nent1

Compo-nent2

Compo-nent3

Compo-nent4

Compo-nentn

………

• All components have to function to provide system reliability.• If all n components of the system are working independently

from each other, then

RS(t) =n∏

i=1

Ri (t)

λS(t) =n∑

i=1

λi (t)

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1 – Reliability

1.3 System Reliability

1.3.2 Systems With Redundancy

Systems With Hot Redundancy (I)

Hot 1-out-of-2-Redundancy

System

Compo-nent1

Compo-nent2

• Two components with the samefunctionality run in parallel.

• If both components have an equalconstant failure rate λ, then

RS(t) = 2R(t) − R2(t)

= 2e−λt − e−2λt

MTBFS =2

λ− 1

2λ

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1 – Reliability

1.3 System Reliability

1.3.2 Systems With Redundancy

Systems With Hot Redundancy (II)Hot 1-out-of-n-Redundancy

System

Compo-nent1

Compo-nent2

Compo-nent3

Compo-nent4

Compo-nentn

………

• n components with the same functionalityrun in parallel.

• If all components have an equal constantfailure rate λ, then

RS(t) =n∑

i=1

(ni

)R i (1 − R)n−i

= 1 − (1 − R)n

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1 – Reliability

1.3 System Reliability

1.3.3 Comparing System Reliability

Comparing Different Systems (I)

• Singular system: R(t) = e−λt

MTBF =1

λ

• System with hot1-out-of-n-Redundancy:

RS(t) = 2e−λt − e−2λt

MTBFS =2

λ− 1

2λ

• System with cold1-out-of-n-Redundancy:

RS(t) = e−λt + λte−λt

MTBFS =2

λ

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1 – Reliability

1.3 System Reliability

1.3.3 Comparing System Reliability

Comparing Different Systems (II)

singularsystem

double systemwith hot redundancy

double systemwith cold redundancy

2

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1 – Reliability

1.3 System Reliability

1.3.3 Comparing System Reliability

Comparing Different Systems (III)

Mean time to failure (with failure rate λ = 2):

• Singular system:MTBF = 0.5 (1)

• Double system with hot redundancy:

MTBF = 1 − 1

4= 0.75 (2)

• Double system with cold redundancy:

MTBF = 1 (3)

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1 – Reliability

1.5 References

References

A. Birolini.Reliability Engineering — Theory and Praxis.Springer, Berlin; Heidelberg; New York, 5th edition, 2007.ISBN 978-3540-49388-4.

Z. Enrico.An Introduction to the Basics of Reliability and Risk Analysis.Series on Quality, Reliability and Engineering Statistics. WorldScientific Publishing, London, 2007.ISBN 978-9812-70639-3.

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