# Reliability and Availability for Non-Repairable & Repairable Systems using Markov Modelling

DOI : 10.17577/IJERTV9IS030082

Text Only Version

#### Reliability and Availability for Non-Repairable & Repairable Systems using Markov Modelling

1G.Saritha

Department of Mathematics Kakatiya University Warangal, TS, India

2M. Tirumala Devi and 3T. Sumathi Uma Maheswari

2,3Department of Mathematics Kakatiya University Warangal, TS, India

AbstractMarkov model provides great flexibility in modelling the timing of events. Markov analysis is a method of analysis that can be applied to both repairable and non- repairable types of systems. In this paper, Markov modelling technique is used to compute the reliability for non-repairable system and defined the mean time to failure of non-repairable systems with different failure rates. This technique is also used to compute the steady-state availability for repairable systems and to derived the mean time between failure of repairable systems with different failure rates and repair rates.

Keywords Markov model; Repairable and Non-repairable systems; MTTF; MTBF; Failure rate; Repair rate; Reliability;

INTRODUCTION

Markov analysis is the mathematical abstractions to model simple or complex concepts in quite easily computable form. The Markov analysis is also considered powerful modelling and analysis tool in solving reliability tribulations. Markov analysis is a tool for modelling complex system designs involving timings, sequence, repair, redundancy and fault- tolerance.

Mean Time To Failure (MTTF) describes the expected time to failure for a non-repairable system. MTTF is commonly refer to as the life time of any product or a device. MTTF can be mathematically calculated by

Mean Time Between Failure (MTBF) is the predicted time that passes between one previous failure of a mechanical/electrical system to the next failure during normal operation or, the time between one system breakdown and the next.

James Li [1] & [2] derived the reliability for a parallel redundant system with different failure rate & repair rate using Markov modelling and reliability comparative evaluation of active redundancy vs. Standby redundancy respectively. M.A. El-Damcese and N.S. Temraz [3] studied analysis for a parallel repairable system with different failure

al [6] estimated availability and reliability analysis of three elements parallel system with fuzzy failure and repair rate.

MATHEMATICAL MODEL:

1. 3-component non-repairable system with different failure rates

A non-repairable system has finite life time. The probability of a system failure will increase over time during system operation. A non-repairable system remains failed, after it failed once.

3-component system can have 23 8 distinct states. The failure rates are 1 , 2 and 3 . The transition rate diagram is given below.

State 2

Component 1 fail

Component 2 good

Component 3 good

State 5

Component 1 fail

Component 2 fail

Component 3 good

State 2

Component 1 fail

Component 2 good

Component 3 good

State 5

Component 1 fail

Component 2 fail

Component 3 good

State 1

Component 1 good

Component 2 good

Component 3 good

State 1

Component 1 good

Component 2 good

Component 3 good

State 3

Component 1 good

Component 2 fail

Component 3 good

State 3

Component 1 good

Component 2 fail

Component 3 good

State 8

Component 1 fail

Component 2 fail

Component 3 fail

State 8

Component 1 fail

Component 2 fail

Component 3 fail

State 6

Component 1 good

Component 2 fail

Component 3 fail

State 4

Component 1 good

Component 2 good

Component 3 fail

State 7

Component 1 fail

Component 2 good

Component 3 fail

State 4

Component 1 good

Component 2 good

Component 3 fail

State 7

Component 1 fail

Component 2 good

Component 3 fail

#### Figure 1: Non-repairable system state diagram

The probability of the system state one i.e., the probability of 3-component good at time is

Taking the Limit as t 0 , and the differential equation is

Solving the above equation by using the Laplace transformations, and taking assumption that the system good

modes. Jacob Cherian et al [4] has described reliability of a

standby system with repair. Garima Chopra [5] studied reliability measures of two dissimilar units parallel system using Gumbel-Hougaard family copula. M.A. El-Damcese et

at initial time t = 0, hence

then the following equation is obtained

P1 0 1

The probability of the system in state two at t t time is Taking the Limit as t 0 , and the differential equation is

Taking Laplace transform, the equation is

Using inverse Laplace transform, we get

The probability of the system in state six at t t time

Taking the Limit as

t 0 , and the differential equation is

Using inverse Laplace transform

The probability of the system in state three at t t time is Taking the Limit as t 0 , and the differential equation is

Taking Laplace transform, the equation is

Using inverse Laplace transform, we get

Taking Laplace transform, the equation is

Using inverse Laplace transform, we get

The probability of the system in state seven at t t time is

The probability of the system in state four at t t time is

Taking the Limit as t 0 , and the differential equation is

Taking the Limit as

t 0 , and the differential equation is

Taking Laplace transform, the equation is

Using inverse Laplace transform, we get

The probability of being in state five at t t time

Taking Laplace transform, the equation is

Using inverse Laplace transform, we get

Taking the Limit as

t 0 , and the differential equation is

Taking Laplace transform, the equation is

The reliability of the 3-component system, if at least one component should operate

Mean Time To Failure (MTTF) of the system

The reliability function in the exponential case

is failure rate and t is the period of time over which reliability is measured. The probability of failure is equal to .

 State Component 1 Component 2 Component 3 System state 1 good good good success 2 failed good good success 3 good failed good success 4 good good failed success 5 failed failed good success 6 good failed failed success 7 failed good failed success 8 failed failed failed down

Table.1

If all the three components have the same failure rate, we get

2. 3-component repairable system with different failure rates and repair rates

A repairable sytem is a system which, after failure, can be restored to a functioning condition by some maintenance action other than replacement of the entire system.

3-component system can have distinct states. The failure rates are and the repair rates are The transition rate diagram is given below.

 State 2 State 5 Component 1 fail Component 1 fail Component 2 good Component 2 fail Component 3 good Component 3 good

State 1

Component 1 good

Component 2 good

Component 3 good

 State 3 State 6 Component 1 good Component 1 good Component 2 fail Component 2 fail Component 3 good Component 3 fail

State 8

Component 1 fail

Component 2 fail

Component 3 fail

State 1

Component 1 good

Component 2 good

Component 3 good

 State 3 State 6 Component 1 good Component 1 good Component 2 fail Component 2 fail Component 3 good Component 3 fail

State 8

Component 1 fail

Component 2 fail

Component 3 fail

 State 4 State 7 Component 1 good Component 1 fail Component 2 good Component 2 good Component 3 fail Component 3 fail

#### Figure 2: Repairable system state diagram

The transition matrix is

 State Compon ent 1 Compone nt 2 Compone nt 3 System state 1 2 3 4 5 6 7 8
 State Compon ent 1 Compone nt 2 Compone nt 3 System state 1 2 3 4 5 6 7 8

1 2 3

 1 2 2 1 0 2 0 3 0 0 3 0 0 1 3 0 0 1 0 0 2 0 0 3 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2 1 0 0 0 0 0 0 0 0 1
 1 2 2 1 0 2 0 3 0 0 3 0 0 1 3 0 0 1 0 0 2 0 0 3 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2 1 0 0 0 0 0 0 0 0 1

1

2 3 0

0 0

3 2 0

0 0 0

0 0 0

3 1 3

From the transition matrix, the probability of the system in state one at t t time is

Table.2

From table.2 the probability of success is

The probability of the system in state two at t t time is

Taking the Limit as t 0 , and the differential equation is

Integrating above equation

and MTTF is

The boundary conditions are at all other conditions.

P1 0 1& P8 1 , zero

Where T1 ,T2 &T5

and five respectively.

are the expected time in state one, two

The probability of the system in state six at t t time is

The probability of the system in state three at t t time is Taking the Limit as t 0 , and the differential equation is

Integrating above equation

Where T1 ,T3 &T6 are the expected time in state one, three

Taking the Limit as t 0 , and the differential equation is

Integrating the above equation

Substituting T value in T we get

and six respectively. 6 3

The probability of the system in state four at t t time is Taking the Limit as t 0 , and the differential equation is

Integrating above equation

Substituting T3 value in T6 we get

The probability of the system in state seven at t t time is

Where T ,T &T

are the expected time in state one, four

Taking the Limit as t 0 , and the differential equation is

1 4 7

and seven respectively.

The probability of the system in state five at t t time is Taking the Limit as t 0 , and the differential equation is

Integrating above equation

Substituting T5 value in T2 we get

Substituting T2 value in T5 we get

Integrating the above equation

Substituting T7 value in T4

Substituting T4 value in T7 we get

The probability of the system in state eight at t t time is

Taking the Limit as t 0 , and the differential equation is

Substituting T5 ,T6 &T7 values in above equation

The MTBF is the sum of the expected time in state one, two, three, four, five, six and seven.

Substituting T1 value in above equation, we get

If all the 3-components have the same failure rate and repair rate, we get

 0.02 0.06 0.001 40 0.987867 1041.975 0.02 0.06 40 0.903527 155.3059 0.02 0.06 0.017 40 0.838035 104.8716 0.02 0.06 0.025 40 0.787066 87.61905 0.02 0.06 0.033 40 0.747312 79.27058 0.02 0.06 0.041 40 0.716239 74.52798 0.02 0.06 0.049 40 0.691902 71.57095 0.02 0.06 0.057 40 0.672804 69.61199 0.02 0.06 0.065 40 0.657792 68.25818
 0.02 0.06 0.001 40 0.987867 1041.975 0.02 0.06 0.009 40 0.903527 155.3059 0.02 0.06 0.017 40 0.838035 104.8716 0.02 0.06 0.025 40 0.787066 87.61905 0.02 0.06 0.033 40 0.747312 79.27058 0.02 0.06 0.041 40 0.716239 74.52798 0.02 0.06 0.049 40 0.691902 71.57095 0.02 0.06 0.057 40 0.672804 69.61199 0.02 0.06 0.065 40 0.657792 68.25818

Mean time to repair (MTTR) is a basic measure of the maintainability of repairable items. It represents the average time required to repair a failed component or device.

Where

If the repair rate t is constant and is equal to then

On simplification, we get

Numerical results:

1. component non-repairable system with different failure rates

 0.001 0.02 0.05 40 0.988642 1041.831 0.009 0.02 0.05 40 0.909736 155.7947 0.017 0.02 0.05 40 0.848541 105.835 0.025 0.02 0.05 40 0.800984 88.94737 0.033 0.02 0.05 40 0.763956 80.88555 0.041 0.02 0.05 40 0.735073 76.37223 0.049 0.02 0.05 40 0.712504 73.60144 0.057 0.02 0.05 40 0.694843 71.79583 0.065 0.02 0.05 40 0.681005 70.5698
 0.03 0.001 0.04 40 0.986559 1030.164 0.03 0.009 0.04 40 0.893242 144.128 0.03 0.017 0.04 40 0.820961 94.16836 0.03 0.025 0.04 40 0.764867 77.2807 0.03 0.033 0.04 40 0.721257 69.21889 0.03 0.041 0.04 40 0.687296 64.70556 0.03 0.049 0.04 40 0.660808 61.93477 0.03 0.057 0.04 40 0.640123 60.12916 0.03 0.065 0.04 40 0.623952 58.90313

In this case the repair rates

1 2 3 , then

1 , 2 & 3 are constants and

 0.01 0.02 0.05 10 0.99638 145 0.01 0.02 0.05 20 0.978602 145 0.01 0.02 0.05 30 0.945621 145 0.01 0.02 0.05 40 0.901213 145 0.01 0.02 0.05 50 0.849698 145 0.01 0.02 0.05 60 0.794635 145 0.01 0.02 0.05 70 0.738605 145 0.01 0.02 0.05 80 0.683349 145 0.01 0.02 0.05 90 0.629983 145
 0.01 0.02 0.05 10 0.99638 145 0.01 0.02 0.05 20 0.978602 145 0.01 0.02 0.05 30 0.945621 145 0.01 0.02 0.05 40 0.901213 145 0.01 0.02 0.05 50 0.849698 145 0.01 0.02 0.05 60 0.794635 145 0.01 0.02 0.05 70 0.738605 145 0.01 0.02 0.05 80 0.683349 145 0.01 0.02 0.05 90 0.629983 145

1. component repairable system with same failure rate and repair rates

 0.02 0.1 0.996169 866.6667 0.04 0.1 0.983087 193.75 0.06 0.1 0.964798 91.35802 0.08 0.1 0.944299 56.51042 0.1 0.1 0.923077 40 0.12 0.1 0.901863 30.63272 0.14 0.1 0.881027 24.68416 0.16 0.1 0.860756 20.60547 0.18 0.1 0.841142 17.64975

CONCLUSION:

Reliability and MTTF have been derived for the non- repairable parallel redundant system with different failure rate and MTBF has been derived for repairable systems with different failure rate and repair rate. And also steady- state availability has been computed for repairable system with same failure rate and repair rate. It is observed from the computations that the reliability and MTTF decreases as

1 , 2 & 3 increases and reliability decreases as t increases for the non repairable system and as

increases steady state availability & MTBF decreases,

as increases steady- state availability & MTBF increases for the repairable system.

REFERENCES:

1. James Li, Reliability calculation of a parallel redundant system with different failure rate & repair rate using Markov modelling, Journal of Reliability and Statistical Studies, Vol.9, Issue 1(2016):1-10.

2. James Li, Reliability comparative evaluation of active redundancy vs. Standby redundancy, International Journal of Mathematical, Engineering and Management Sciences, Vol.1, No.3, 122-129, 2016.

3. M.A. El-Damcese and N.S. Temraz, Analysis for a parallel repairable system with different failure modes, Journal of Reliability and Statistical Studies, Vol.5, Issue 1(2012):95-106.

4. Jacob Cherian, Madhu Jain and G.C. Sharma, Reliability of a standby system with repair, Indian Journal Pure Applied Mathematics, 18(12):1061-1068, December 1987.

5. Garima Chopra, Reliability Measures of two dissimilar units parallel system using Gumbel-Hougaard family copula, International Journal of Mathematical, Engineering and Management Science, Vol.4, No.1, 116-130, 2019.

 0.05 0.1 0.974359 126.6667 0.05 0.12 0.981706 149.0667 0.05 0.14 0.98647 173.6 0.05 0.16 0.989704 200.2667 0.05 0.18 0.99198 229.0667 0.05 0.2 0.993631 260 0.05 0.22 0.994857 293.0667 0.05 0.24 0.995787 328.2667 0.05 0.26 0.996506 365.6
 0.05 0.1 0.974359 126.6667 0.05 0.12 0.981706 149.0667 0.05 0.14 0.98647 173.6 0.05 0.16 0.989704 200.2667 0.05 0.18 0.99198 229.0667 0.05 0.2 0.993631 260 0.05 0.22 0.994857 293.0667 0.05 0.24 0.995787 328.2667 0.05 0.26 0.996506 365.6

6. M.A. El-Damcese, F. Abbas and E. El-Ghamry, Availability and reliability analysis of three elements parallel system with fuzzy failure and repair rate, Journal of Reliability and Statistical Studies, Vol.75, Issue 1(2014):125-142.