Modeling and Simulation of Communication Systems and Networks Chapter 1. Reliability

Transcription

1 Modeling and Simulation of Communication Systems and Networks Chapter 1. Reliability Prof. Jochen Seitz Technische Universität Ilmenau Communication Networks Research Lab Summer / 20

2 Outline Definition of Reliability Modeling Reliability System Reliability Systems Without Redundancy Systems With Redundancy Comparing System Reliability References 2 / 20

3 1.1 Definition of Reliability What does Reliability mean? Measure for the ability of a component / a system to sustain its functionality. Probability that a desired function can be fulfilled during a specified time period under given working conditions. Reliability Parameters: Reliability R(t) Failure Probability F (t) Failure Density Function f (t) Failure Rate λ(t) 3 / 20

4 1.1 Definition of Reliability Typical Characteristics of the Failure Rate l(t) Phase 1 Phase 2 Phase 3 t 4 / 20

5 1.1 Definition of Reliability Important: Mean Values Mean value of failure probability: Mean Time To Failure (MTTF) MTTF = E[T ] = t f (t)dt = 0 R(t)dt = e t λ(τ)dτ 0 dt Mean Time Between Failures (MTBF) Special case: λ(t) = λ = constant 0 MTBF = 0 0 e λt dt = 1 λ 5 / 20

6 1.2 Modeling Reliability Exponential Distribution R(t) R(t) = e λt l = 0,25 l = 0,5 l = 0,75 l = 2 6 / 20

7 1.2 Modeling Reliability Exponential Distribution F (t) F (t) = 1 R(t) = 1 e λt l = 0,25 l = 0,5 l = 0,75 l = 2 7 / 20

8 1.2 Modeling Reliability Exponential Distribution f (t) f (t) = dr(t) dt = λe λt l = 0,25 l = 0,5 l = 0,75 l = 2 8 / 20

9 1.2 Modeling Reliability Exponential Distribution λ(t) λ(t) = λ l = 0,25 l = 0,5 l = 0,75 l = 2 9 / 20

10 1.2 Modeling Reliability Other Distributions (I) The Epxonential Distribution is only valid for phase 2, since λ is constant. Other distribution functions used for phases 1 or 3: Weibull Distribution (generalization of the exponential distribution) F (t) = 1 e ( t λ )k 10 / 20

11 1.2 Modeling Reliability Other Distributions (II) Gamma Distribution F (t) = 1 Γ(β) λt f (t) = λ (λt)β 1 Γ(β) f (t) λ(t) = 1 F (t) Γ(β) = 0 0 x β 1 e x dx e λt x β 1 e x dx 11 / 20

12 1.3 System Reliability Reliability of Systems (I) Communication systems and networks consist of different components that have different failure characteristics. Network Subnet A Subnet C Subnet B Subnet D Subnet Router 1 Router 2 Router 3 Router 4 Router 6 Router 7 Router 5 Router Network Interface 1 Processor Memory Power Supply Network Interface 2 12 / 20

13 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 the redundant component. 2 Warm Redundancy The redundant components are not as heavily loaded as the main component. 3 Cold Redundancy / Standby Redundancy The redundant components are not loaded at all. Redundancy is the duplication of critical components of a system with the intention of increasing reliability of the system. How can the system s reliability be determined if the reliabilities of all components are known? 13 / 20

14 1.3 System Reliability Systems Without Redundancy Systems Without Redundancy System n All components have to function to provide system reliability. If all n components of the system are working independently from each other, then n R S (t) = R i (t) λ S (t) = i=1 n λ i (t) i=1 14 / 20

15 1.3 System Reliability Systems With Redundancy Systems With Hot Redundancy (I) Hot 1-out-of-2-Redundancy System 1 2 Two components with the same functionality run in parallel. If both components have an equal constant failure rate λ, then R S (t) = 2R(t) R 2 (t) = 2e λt e 2λt MTBF S = 2 λ 1 2λ 15 / 20

16 1.3 System Reliability Systems With Redundancy Systems With Hot Redundancy (II) Hot 1-out-of-n-Redundancy System n n components with the same functionality run in parallel. If all components have an equal constant failure rate λ, then R S (t) = n ( n i i=1 = 1 (1 R) n ) R i (1 R) n i 16 / 20

17 1.3 System Reliability Comparing System Reliability Comparing Different Systems (I) Singular system: R(t) = e λt MTBF = 1 λ System with hot 1-out-of-n-Redundancy: System with cold 1-out-of-n-Redundancy: R S (t) = 2e λt e 2λt MTBF S = 2 λ 1 2λ R S (t) = e λt + λte λt MTBF S = 2 λ 17 / 20

18 1.3 System Reliability Comparing System Reliability Comparing Different Systems (II) double system with cold redundancy 2 double system with hot redundancy singular system 18 / 20

19 1.3 System Reliability 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 = = 0.75 (2) Double system with cold redundancy: MTBF = 1 (3) 19 / 20

20 1.5 References References A. Birolini. Reliability Engineering Theory and Praxis. Springer, Berlin; Heidelberg; New York, 5th edition, ISBN Z. Enrico. An Introduction to the Basics of Reliability and Risk Analysis. Series on Quality, Reliability and Engineering Statistics. World Scientific Publishing, London, ISBN / 20