Comparative study of reliability parameter of a system under different types of distribution functions

In this paper a two unit standby system with single repair facility has been considered. When a working unit fails, it is immediately taken over by standby unit and repair on the failed unit is started immediately. Taking two types of distribution, namely, Weibull and Erlangian, various system effectiveness measures such as MTSF, Availability and Busy Periods are compared and results are interpreted numerically. Regenerative Point Technique and Semi-Markov process have been employed in this paper to find the results. Results are supported with numerical data also. Failure time distributions are taken to be exponential whereas the repair times are particular. The result obtained from this can be applied to study complex system where small change in the value of one variable affects the system measures to a great extent.


INTRODUCTION
Reliability measures of a component for a two-component system with repair facility were obtained by several authors under different assumptions.Harris (1968), considered a two-unit parallel redundant system in which failure times of the components are dependent and distributed as bivariate exponential of Marshall and Olkin (1967), to derive the mean time to system failure using the supplementary variable technique for an arbitrary repair time distribution.Osaki (1980) extended the analysis to obtain the availability of the system by using a variant of Semi-Markov process with non-regeneration point technique.Jye-Chyi (1992) studied the effects of dependence on modeling system reliability via multivariate Weibull distribution.In reliability theory, the steady-state availability of a repairable system is an important feature.
In this paper, we have taken two different types of distributions namely Weibull and Erlangian to study the reliability measures such as mean time to system failure, steady-state availability, busy period of the repairman in repairing the failed units and profit analysis and compare them.Very few authors have attempted to compare the *Corresponding author.E-mail: vkpath21162@yahoo.co.in.Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Mathematics Subject Classification.53A04, 53A05.system effectiveness measures before now.The comparison can be helpful in studying the performance of complex system from reliability point of view and will be fruitful to system managers for evaluating the profit analysis of working systems.In the present work a twounit standby system with single repair facility is considered.As soon as the main unit fails, it is taken over by the standby unit and the failed unit is sent for repair.To improve reliability, the concept of preventive maintenance is also added.When both the units fail, the whole system is shut down to prevent any further losses and the system starts afresh.

Literature review
Earlier Gopalan and Waghmare (1992) worked on evaluating cost benefit analysis of single server n-unit system.Gupta (2002), Yong et al. (2012), Li and Zuo (2008) have calculated reliability measures of k out of n systems.Jane and Laih (2010) has provided dynamic algorithm for multistate two-terminal reliability.Whereas Lie et al. (1977) has provided calculation techniques for Availability.Marshall and Olkin (1967b), Nadarajah and Kotz (2005), Osaki (1970) provided excellent results on BVE Weibull and Markov Renewal Process which are the backbone of reliability literature.Paul and Chandrasekar (1997) have introduced the idea of dependent structure for failure and repair times.On the other hand, Pijnenburg et al. (1993) gave the idea of dependent parallel system.Rander et al. (1992) and Singh et al. (1986) discussed two unit cold standby concept considering various assumptions regarding failures, repairs, inspections and replacement.Yearout et al. (1986) provide excellent review on standby redundancy.

Assumptions used in the model
(a) The system consists of two main units along with an associate unit in which one main unit is kept on standby mode.(b) Whenever an operational unit fails, it is immediately taken over by standby unit.(c) There is a single repairman which repairs the failed unit on priority basis.(d) If both the main units fail the system shuts down.(e) After repair all units work as new.(f) After random period of time the whole system goes for preventive maintenance.(g) The failure rates of all the units are taken to be exponential whereas the repair time distributions are particular.

Symbols and Notations
0 E =State of the system at epoch t=0

TRANSITION PROBABILITIES AND MEAN SOJOURN TIMES
Using Markovian regenerative process, simple probabilistic considerations yield the following non zero transition probabilities (Figure 1): (1) And the mean sojourn times are given by:

Let
in the state be defined as the time that system continuous to be in state before transiting to any other states.If T denotes the Sojourn time in state , then time to system failure can be regarded as the first passage time to the failed state.To obtain it we regarded down state as absorbing states.Using argument as for the regenerative process, we obtain the following recursive relation for as follows: (18 Taking Laplace-Stieltje's transform and solving the subsequent matrix

Availability analysis
Let denote the probability that the system is up initially in regenerative state at epoch t without passing through any other regenerative state.It might return to itself through one or more non regenerative states so that either it continues to remain in regenerative state without visiting any regenerative state including itself by probability arguments.We observe that the entry to any of the state is a regenerative point.is defined as the probability that the system is up in state at epoch t.To obtain it, all possible consequences are considered: (1) Probability that the system initially up is is up at epoch t without transiting to any other regenerative state in E which is .
(2) Probability that the system transits to in E during (u,u+du) and then starting from it is up at epoch t which is Taking Laplace-transform of the above equations and writing in matrix form: q q q q q q q q q q q q We get the following expression for Availability:

BUSY PERIOD ANALYSIS Busy period repairman for performing normal repair
Let denote the probability that the repairman is busy initially with repair in regenerative state S 4 and remains busy at epoch t without transiting to any other state or returning to itself through one or more regenerative states.By probabilistic argument, we have:


Developing similar relationship as in availability for normal repair, we have to find the steady state, the fraction of time for which the repairman of busy with normal repair is given by:

Busy period repairman performing for shutdown repair
Similarly, to find the steady state the fraction of time for which the repairman of busy with shut down repair:

Busy period of repairman performing the preventive maintenance
Similarly, in the long run, for the fraction of time the repairman in busy with the preventive maintenance is given by: Pathak et al. 51

Case (i)
A random variable is said to have the Weibull distribution if its distribution is given, for some λ>0 ,α>0 by The failure rate function for Weibull distribution equals Then we have, When all repair time distribution are n-phases Erlangian distribution, that is, Density Function and Survival Function Busy period analysis:

Profit analysis
The profit analysis of the system can be carried out by considering the expected busy period of repairman in repair of the unit in [0,t].Therefore, G(t)= total revenue earned by the system in [0,t]-Expected repair cost in [0,t] where

RESULTS AND DISCUSSION
Case I: When distribution is taken to be Weibull as shown in Tables 1 to 3.
Case II: When distribution is taken to be Erlangian as shown in Tables 4 to 6.
An excellent work in this direction involving components of two-unit system was done by Gaver (1964) and Harris (1968) but the comparative study of various parameters was not taken into account by any of these authors.We have considered two distributions namely Erlangian and Weibull which are regarded as the best distributions for achieving optimum results.We have employed regenerative point technique for obtaining mean time to system failure, availability and busy period analysis which are helpful in performing the profit analysis for arbitrary repair time distribution.
However, the whole work could also have been viewed with the help of developing differential equations and taking Laplace-Transform thereof and Inverse Laplace-Transform after that and reliability analysis could also have been performed as well as performance evaluation could have been undertaken, which the authors plan to carry out in the next work.

Conclusion
It is observed that the mean time to system failure (MTSF) and availability of the system decreases rapidly with the increase of failure rates other parameters, when the distribution is taken to be Weibull.However, it is noted that when the distribution is assumed to be n-phase Erlangian, the mean time to system failure and availability of the system do not decrease so rapidly.Same can be predicted for profit analysis also.

t
Thus, the Weibull distribution is IFR (Increasing Failure Rate) when α≥0 and DFR (Decreasing Failure Rate)

Table 1 .
  & for fixed values of Pathak et al. 53 Variation in MTSF vis-a-vis failure rate of main unit.

Table 2 .
Variation in Availability vis-a-vis failure rate of main unit.

Table 3 .
Variation in Profit vis-a-vis increase in failure rate of main unit.

Table 4 .
Variation in MTSF vis-a-vis failure rate of main unit.

Table 5 .
Variation in Availability vis-a-vis failure rate of main unit.

Table 6 .
Variation in Profit vis-a-vis failure rate of main unit.