Reliability analysis of a system using semi Markov process

Introduction

Many industrial systems under different operating conditions were analyzed in the past. The objective was to obtain the relevant indices reflecting the system's operational capabilities. Mostly, the semi-Markovian process was adopted for the system analysis. So many Researchers and Scientists are trying to improve the performance of industries using various reliability techniques. V.K.pathak [1] analyzed comparative study of reliability parameter of a system under different types of distribution functions. H. K. Saw, and V. K. Pathak [2] studied economic and performance evaluation of a stochastic model with hardware failure and human error, in this paper developed a stochastic Reliability model of single unit system in manufacturing plant. System has been investigated under the assumption that unit works in three different states: normal state, partial failed state and total failed state. Hemant Kumar Saw and H.L. Manker [3] develop a stochastic reliability model of one unit main system whith two associate units. Each associate unit also has one sub units. Jyotishree Ghosh et al. [5] A two non indentical unit parallel system with priority in repair. G. Taneja and R. Malhotra [6] studied cost-benefit analysis of a single unit system with scheduled maintenance and variation in demand. R. Bashir and J.P.S. Joorel [7] analyzed Reliability analysis of a three unit complex system working in seriesparallel configuration, S.M. Rizwan et al. [8] studied Reliability analysis of a three unit pumping system of a water pumping station is analyzed. In view of above study in this paper an attempt has been made by authors by incorporating the concept of semi-up mode obtained the reliability parameters of working system using semi Markov process and regenerative point technique.

System Description about the model

The system consists of three units namely one main unit A and two associate units B & C.

Here the main unit depends on associate unit B and C and the system is operable when the main unit and at least one associate unit is in operation and the system is semi operable when the main unit is failed and both associate are in operation. Associative units are essential in the operation of main unit A. As soon as a job arrives, all the units work with load.

It is assumed that only one job is taken for processing at a time. There is a single repairman who repairs the failed units on first come first served basis. Using regenerative point technique several system characteristics such as transition probabilities, mean sojourn times, availability and busy period of the repairman are evaluated. In the end the expected profit is also calculated.

3.

Assumptions used in the model 1. The system consists of one main unit and two associate units. 2. The main unit A works with the help of associate unit B and C. 3. There is a single repairman which repairs the failed units on priority basis. 4. After random period of time the whole system goes to preventive maintenance. 5. All units work as new after repair. 6. The failure rates of all the units are taken to be exponential whereas the repair time distributions are arbitrary. 7. Switching devices are perfect and instantaneous.

Symbols and Notations

It is easy to see that 𝑃 01 + 𝑃 02 + 𝑃 03 + 𝑃 05 = 1; 𝑃 20 + 𝑃 24 = 1;

Mean sojourn times

The main sojourn times (𝜇 𝑖 ) in the generative state "i" is defined as the time of stay in that state before transition to any other state. If T denotes the sojoun time in the regenerative state 'i', then :

We note that the Laplace-Stieltjes transform of 𝑄 𝑖𝑗 is equal to Laplace-Stieltjes transform of 𝑞 𝑖𝑗 i.e. It can so that 𝑚 01 + 𝑚 02 + 𝑚 03 + 𝑚 05 = 𝜇 0 ;

3.2 Main time to system failure Time to system failure can be regarded as the first passage time to the failed state. To obtain it we regard the down state as absorbing. Using the argument as for the regenerative process, we obtain the following recursive relations.

𝜋 0 (𝑡) = 𝑄 01 (𝑡) + 𝑄 02 (𝑡) $ 𝜋 2 (𝑡) + 𝑄 03 (𝑡) $ 𝜋 3 (𝑡) + 𝑄 05 (𝑡), (35)

𝜋 2 (𝑡) = 𝑄 20 (𝑡) $ 𝜋 0 (𝑡) + 𝑄 24 (𝑡),

𝜙 3 (𝑡) = 𝑄 30 (𝑡) $ 𝜋 0 (𝑡) + 𝑄 34 (𝑡).

Taking Laplace-Stieltjes Transforms (L.S.T.) on both side and solving we get, MTSF when the system starts from the state 𝑆 0 is given by

1-𝑃 01 𝑃 10 -𝑃 02 𝑃 20 -𝑃 03 𝑃 30

Availability analysis of the system

Let 𝑀 𝑖 (𝑡)(𝑖 = 0,1,2,3) denote the probability that system is initially in regenerative state 𝑆 𝑖 ∈ E is up at time t without passing through any other regenerative state or returning to itself through one or more non regenerative states.i.e. either it continues to remain in regenerative 𝑆 𝑖 or a non regenerative state including itself . By probabilistic arguments, we have the following recursive relations

Recursive relations giving point wise availability 𝐴 𝑖 (𝑡) given as follows:

𝐴 0 (𝑡) = 𝑀 0 (𝑡) + ∑ 𝑖=1,2,3,5 𝑞 0𝑖 (𝑡) © 𝐴 𝑖 (𝑡);

Taking Laplace stieltjes transformation (L.S.T.) of above equations and solving the steady state availability is given by

Taking Laplace stieltjes transformation (L.S.T.) of above equations and solving the steady state busy period of the repairman in preventive maintenance is given by

References

1. Pathak VK. Comparative study of reliability parameters of a system under different types of distribution functions. African Journal of Mathematics and Computer Science Research. 2014;7(4):47-54.
2. Saw HK, Pathak VK. Economic and performance evaluation of a stochastic model with hardware failure and human error. Bulletin of the Calcutta Mathematical Society. 2015;107(5):411-420.
3. Saw HK, Manker HL. Reliability modeling of a system with one main unit and two associate units along with a sub-unit. Communications in Mathematics and Applications. 2022;13(1).
4. Chaudhary A, Sharma S. A two non-identical unit parallel system with priority in repair. Reliability: Theory & Applications. 2023;3(74):113-122.
5. Ghosh J, Pawar D. Analysis of a triple-unit system with operational priority. Reliability: Theory & Applications. 2022;3(69):202210.
6. Taneja G, Malhotra R. Cost-benefit analysis of a single unit system with scheduled maintenance and variation in demand. Journal of Mathematics and Statistics. 2013;9(3):155-160.
7. Bashir R, Joorel JPS. Reliability analysis of a three-unit complex system working in series-parallel configuration. Indian Journal of Statistics and Applications. 2013;2(1-2):89-103.
8. Rizwan SM. Reliability analysis of a three-unit pumping system. International Journal of Engineering Trends and Technology. 2022;70(6):24-31.