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

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

   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 1

Component 1 good

Component 2 good

Component 3 good

State 8

Component 1 fail

Component 2 fail

Component 3 fail

State 1

Component 1 good

Component 2 good

Component 3 good

State 8

Component 1 fail

Component 2 fail

Component 3 fail

Figure 2: Repairable system state diagram

Figure 2: Repairable system state diagram

The transition matrix is

1 2 3

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

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

   The steady state availability is

   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.