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MTBF Evaluation of Standby Sparing System by Means of Markov Model

Authors:
Gholam Reza Latif-Shabgahi at Shahid Beheshti University
Gholam Reza Latif-Shabgahi
Saeed Zaferanchi
Saeed Zaferanchi
Koorosh Aslansefat at University of Hull
Koorosh Aslansefat
Mehrdad Mohammadi at Power and Water University of Technology
Mehrdad Mohammadi
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Abstract

میانگین زمانی میان خرابی سیستم پارامتری است که صنعت پرکاربرد و با اهمیت است. از این پارامتر در سیستم‌های تشخیص و پیش‌بینی خرابی، در بخش صنایع و خدمات به منظور برنامه‌ریزی سیتماتیک نظام نگهداری و تعمیرات استفاده می‌گردد. از میان روش‌های تحلیلی ارزیابی میانگین میان خرابی، استفاده از مدل مارکوف به جهت قابلیت‌های مدلسازی مناسب‌تر است. در کنار این مزایا پیچیدگی مدل برای سیستم‌های بزرگ چالشی اساسی به شمار می‌رود. این موضوع سبب شده است که بسیاری از نرم افزارهای تجاری در زمینه قابلیت اطمینان قادر به ارزیابی میانگین بین خرابی سیستم‌های بزرگ و با پیچیدگی‌های دینامیکی نباشند. تحقیقات انجام شده در این زمینه نشان می‌دهد که تا کنون را بطه جامع و بسته‌ای برای ارزیابی پارامتر مذکور در سیستم‌های حمایت شده با رزرو ارائه نشده است. این مقاله به کمک روشی نوین به ارزیابی میانگین زمان میان خرابی سیستم پرداخته و فرمول‌هایی پارامتریک در این زمینه ارائه می‌نماید.
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MTBF




gh_latif@sbu.ac.ir
saeedzzm@yahoo.com
k_aslansefat@sbu.ac.ir
 mehrdadmohammadi@ace.sbu.ac.ir
–





 MTTFMTBF

    


 MTBF      





[1, 2]


MTBFMTTF

     SIL   

    MTBFMTTF 
 [3]



    MTBF    
 

[2, 3]







S1
R


MTTF
 MTTR 
MTBF
MTTFMTTR




MTBF
MTBF

MTBF




 






RAIDs


   



   

          
     PLC    
ECU

      
MTBF
MTBF
[4]
MTBF
  
[5] 
MTBFK-out-of – N
GM
[6]MTBF
 [7]



MTBF 
[8]

       MTBF



[9]
 MTBF"
"

MTBF

MTBF






MTTF[2]
MTBF
[2, 8]


  
     n   

       
12
T
n
P t P t P t P t
 
 
  
MTBF
𝑞𝑗


   
Sj
j Reliable States
R t P t

MTBF
 
 
0
0
j
j R States
jj
j R States
j R States
MTBF P t dt
P t dt q








A


   
.P t A P t

nm
Tnm
A
11 21 1
12 22 2
12
n
n
m m nm
T T T
t t t
T T T
t t t
A
T T T
t t t
 


 

 


 




 

 

     
nn
𝑞𝑛
MTBF




    
  
1
q
 
n
q
      
MTBF
 
 
12
1
2
0.
01
0
1
00 ..
10 1
T
n
T
n
P P A q q q
q
q
A q A
q
 

    
    
   
 
    
 






   
MTBF
MTBF
 

 P
S1
  S2

F







 
.P t t M P t 
P M

         
12
, , , T
p S S F
P t P t P t P t P t



1λ λ 0 0
01λ λ 0
0 0 1 λλ
0 0 0 1
tt
tt
Mtt



 


 

   P(0) 
nt


 
.0
n
P n t P P


 
.0P t A P
A


λ 0 0 0
λ λ 0 0
0λ λ 0
00λ0
T
M
At





 

A


1
2
3
4
1 0 0
0 0 0 .
0 0 0
1 0 0 0
0q
q
q
q


 

  

  

  

  


1
q

3
q



1 2 3 1
q q q
  
MTBF


1 2 3 3
MTBF q q q
  







       A   


 
 
00
0
0
00
A

 
 









MTBF


 
 
1
2
3
4
1 0 0
00
.
00
1 0 0
q
q
q
q

 
 









 


1
q

3
q



22
1 2 3
32
1
,,q q q
   
 
 
 
MTBF



22
1 2 3 3
32
MTBF q q q
 

  
MTBF
MTBF





  
P
S1

   S2  
F
        





A

3 0 0
2 0 0
0 2 0
0
3
0 0 0
A









1
2
3
4
1
0.
0
1
3 0 0
2 0 0
0 2 0
0 0 0
0
3
q
q
q
q













 


1
q

3
q



1 2 3
1 1 1
,,
32
q q q
 
  
MTBF

1 2 3 1 1 1 11
3 2 6
MTBF q q q
 
   

       




1
q

3
q



22
1 2 3
32
21
,,
62
q q q
   
 
 
 



MTBF
MTBF

MTBF




d
C



d
C






MTBF



1
d
C
MTBF



21
dd
CC
MTBF
 
  

32 1
d d d
C C C
MTBF
 
   



1
Cold Standby 0
i
Sd
i
C
MTBF
"s"

          








MTBF



2
1
MTBF




22
23
1
MTBF
   
 
 
 

22
23
3 2 2 3
4
1
MTBF
   
 
  
 
 
  
     MBTF
"s"


1(k 1 i)
10
Cold Standby 1
ki
Sik
k
MTBF


MTBF



            N





MTBF


1
2
d
C
MTBF



21
23
dd
CC
MTBF
 
 

32 1
2 3 4
d d d
C C C
MTBF
  
   

MTBF


 
1
Hot Standby 01
i
Sd
i
C
MTBF Si





        







MTBF



2
12
MTBF




22
23
12
26
MTBF
   
 
 
 

22
23
3 2 2 3
4
12
26
62
24
MTBF
   
 
  
 
 
 
        
MTBF


 
1(k 1 i)
10
1
!
!
ki
Sik
k
i
MTBF k



 


MTBF
MTBF

MTBF


MTBF

MTBF
MTBF

      
MTBF          

        


MTBF



MTBF


MTBF

      
MTBF

         
MTBF


MTBF
 
MTBF

MTBF
MTBF


MTBF

MTBF


     MTBF

MTBF

MTBF

MTBF

        
        MTBF


MTBFMTBF
MTBF


MTBF


MTBF

   

       

MTBF
MTBF

MTBF


 MTBF      



MTBF  
MTBF
 

MTBF

MTBF

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A Novel Approach for Reliability and Safety Evaluation of Control Systems with Dynamic Fault Tree
Thesis
  • Dec 2014
Reliability and Safety are important attributes for critical applications such as toxic, chemical, and nuclear process control, traffic control in aviation and railway transportation, servers and databases of business enterprises. Due to the role of reliability and safety in the evaluation of such systems, a number of methods are provided. Dynamic Fault Tree (DFT) is a graphical and logical model presenting the casual relationship between the failure of systems' components and non-desire failure of system named "Top event". In the last two decades, some researches have been carried out on DFT solutions for developing and extending its modelling abilities such as priority and sequence dependencies, spare, repair, functional dependencies and considering the non-exponential failure rate. The literature suffers from the lack of a hierarchical Semi-Markov-based approach for DFT solution. Moreover, since most of industrial fault tolerant systems have reconfigurable architecture a need for a dynamic gate to model such a behavior is apparent. In the existing research works no dynamic gate for reconfiguration was presented. This thesis aims to represent new dynamic gates for modelling "reconfigurable TMR architecture supported with spare" and "reconfigurable parallel system supported with cold spare" in systems with reconfiguration capability. In addition a hierarchical method for reliability and safety evaluation of systems based on their DFT are given. This method uses semi-Markov theorem. In this thesis several example are provided to shows the ability, validity and accuracy of this approach. This approach is also implemented in MATALB environment and it can be used as a sub-tool to DFT's reliability and safety evaluation tool. Finally, the results of this approach can be used to evaluate system MTBF and sensitivity.
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The Interval Estimation of MTBF Based on Markov Chain Monte Carlo Method
Article
  • Jun 2013
The distribution of time between failures of numerical control (NC) system follows the Weibull distribution, thus it's estimation of Mean Time Between Failures (MTBF) in reliability engineering is of significance. But there are great difficulties in interval estimation of MTBF using traditional method for Weibull distribution. After the introduction of the approximate estimation, the Markov chain Monte Carlo (MCMC) method is proposed. Combined with the specific characteristics of two-parameter Weibull distribution, Markov chain is established to calculate the interval estimation of MTBF, which solves the problems effectively. And MCMC is more accurate than that of engineering approximation. By analyzing various results in different conditions of MCMC transition kernel, the paper proves that MCMC is a good method for solving interval estimation of Weibull distribution parameters, which has systematic solution process and good adaptability. It greatly enhanced the robustness, effectiveness and accuracy of the calculation.
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Fault-Tolerant Design
Chapter
  • Mar 2013
Along with cost and performance, dependability is the third critical criterion upon which system-related decisions are made. Dependability evaluation is important, because it helps identifying aspects of the system which are critical for its dependability. Such aspects can be, for example, component reliability, fault coverage, or maintenance strategy. Once the critical points are identified, design engineers can focus on their improvements early in the product development stage. In this chapter, we introduce common dependability measures, such as failure rate, mean time to failure, mean time to repair, mean time between failures, and fault coverage. We consider combinatorial dependability models such as reliability block diagrams, fault trees, and reliability graphs. We also study stochastic dependability models such as Markov chains, which make possible the analysis of more complex scenarios. Finally, we show how these models can be used for evaluating system reliability, availability, and safety.
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Fault-Tolerant Design
Chapter
  • Mar 2013
Hardware redundancy impacts size, weight, power consumption, and cost of a system. In some applications, it is preferable to use extra time rather than extra hardware to tolerate faults. In this chapter, we describe time redundancy techniques for detection and correction of transient faults. We also show how time redundancy can be combined with some encoding scheme to handle permanent faults. We consider four encoding schemes: alternating logic, recomputing with shifted operands, recomputing with swapped operands, and recomputing with duplication with comparison.
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Analysis of MTBF/MTTR for logistics service system
Conference Paper
  • Oct 2013
MTBF/MTTR are important parameters for measuring the reliability of systems. Shorter MTBF/MTTR normally means high service quality. However, a reasonable level of reliability is usually not defined. In this paper, the calculation method of MTBF/MTTR is presented. This paper also provided the simulation model and simulation experiment oriented service systems for the quantitative analysis of the relation between MTBF/MTTR and performance metrics of service systems. In the paper, the technique is developed for defining the rational level of reliability, and a practical case is used to demonstrate the aforementioned approaches.
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Reliability Engineering: Theory and Practice
Book
  • Jan 2014
Basic concepts, quality and reliability assurance of complex equipment and systems -- Reliability analysis during the design phase -- Qualifications tests for components and assemblies -- Maintainability analysis -- Design guidelines for reliability, maintainability and software quality -- Reliability and availability of repairable systems -- Statistical quality control and reliability tests -- Quality and reliability assurance during the production phase -- Appendixes: Terms and definitions. Definition and realization of quality and reliability requirements. Checklist for design reviews. Requirements for quality data reporting systems. Basic probability theory. Basic stochastic process theory. Basic mathematical statistics -- Tables and charts. Acronyms. References
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Availability of continuous service and computing long-run MTBF and reliability for Markov systems
Article
  • Jul 2001
  • PROBAB ENG INFORM SC
Steady-state availability has long been a popular descriptor of effectiveness for repairable systems because it captures both the operability and repairability aspects of the system. A related measure of effectiveness is the availability of continuous service, which is particularly relevant for safety critical applications. In this article, two different measures of this quantity are described for a repairable system whose state is described by an ergodic finite-state-space continuous-time Markov chain. Using these ideas, formulas for computing system long-run mean time between failures and the long-run system reliability function are derived.
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Prognostic modeling options for remaining useful life estimation by industry
Article
  • Dec 2011
  • MECH SYST SIGNAL PR
Over recent years a significant amount of research has been undertaken to develop prognostic models that can be used to predict the remaining useful life of engineering assets. Implementations by industry have only had limited success. By design, models are subject to specific assumptions and approximations, some of which are mathematical, while others relate to practical implementation issues such as the amount of data required to validate and verify a proposed model. Therefore, appropriate model selection for successful practical implementation requires not only a mathematical understanding of each model type, but also an appreciation of how a particular business intends to utilise a model and its outputs.This paper discusses business issues that need to be considered when selecting an appropriate modelling approach for trial. It also presents classification tables and process flow diagrams to assist industry and research personnel select appropriate prognostic models for predicting the remaining useful life of engineering assets within their specific business environment. The paper then explores the strengths and weaknesses of the main prognostics model classes to establish what makes them better suited to certain applications than to others and summarises how each have been applied to engineering prognostics. Consequently, this paper should provide a starting point for young researchers first considering options for remaining useful life prediction. The models described in this paper are Knowledge-based (expert and fuzzy), Life expectancy (stochastic and statistical), Artificial Neural Networks, and Physical models.
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MTBF for K-out-of-N: G systems with M failure modes
Article
  • Jan 2008
This paper presents a continuous time Markov chain (CTMC) model to obtain closed form expressions of the mean time between system failures (MTBF) for K-out-of-N :G systems subject to M exponential failure modes and repairs. The results are obtained by using explicit matrix representations of the systems.
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A methodology for analyzing the transient availability and survivability of a system with repairable components
Article
  • Jan 2007
  • APPL MATH COMPUT
In this paper we present a method for transient analysis of availability and survivability of a system with the identical components and identical repairmen. The considered system is supposed to consist of series of k-out-of-n or parallel components. We employed the Markov models, eigen vectors and eigenvalues for analyzing the transient availability and survivability of the system. The method is implemented through an algorithm which is tested in MATLAB programming environment. The new method enjoys a stronger mathematical foundation and more flexibility for analyzing the transient availability and survivability of the system.
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Bounds on MTBF of Systems Subjected to Periodic Maintenance
Article
  • Oct 2006
Mean time between failures (MTBF) is a common reliability measure used to assess the failure behavior of repairable systems. In order to increase MTBF, in most systems, it is a common practice to perform preventive maintenance activities at periodic intervals. In this paper: We first discuss the validity of a commonly used equation for computing MTBF of systems subjected to periodic maintenance.) For complex systems where this equation is valid, we propose a simple and better approximation than the exponential approximation proposed in a recent paper. In addition, we prove that for systems with increasing failure rate on average (IFRA) distributions, the exponential approximation proposed in a recent paper always underestimates the MTBF; hence, it is a lower bound at best.) The proposed approximation and bounds are applicable for a wide range of systems because systems which contain components with exponential or any increasing failure rate (IFR) distribution (viz., Weibull with beta>1, gamma, Gumbel, s-normal, and uniform) follow an IFRA distribution. As a special case, the proposed bounds & approximations provide better results for systems that contain only exponential failure distributions
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