Stochastic analysis of two unit system with increasing failure and repair rates

Introduction

A system may consist of a number of units and individual units have great importance in a system for its proper functioning. Parametric values of a system largely depend on the failure and repair rate of individual units. In this paper a stochastic analysis and sensitivity analysis of a two units (one having subunits in parallel and other having subunits in series) system is carried out. Kumar, J. & Malik, S. C. [1] have discussed the concept of preventive maintenance for a single unit system. Liu., R. [2] , Malik, S. C. [3] , Nakagawa, T. and Osaki, S. [4] have discussed reliability analysis of a one unit system with un-repairable spare units and its applications. Goel, P. & Singh J. [5] , Gupta, P., Singh, J. & Singh, I.P. [6] , Kumar, S. & Goel, P. [7] , Gupta, V. K. [8] , Chaudhary, Goel & Kumar [9] Sharma & Goel [10] , Ritikesh & Goel [11] and Goyal & Goel [12] have discussed behavior with perfect and imperfect switch-over of systems using various techniques. Most of researchers have assumed that failure rates of units are constant, whereas it is not practically, so, whereas a unit fails its failure rate increases generally on expert on each repair of a unit practically. Here in this paper both situations have been considered to model a tow unit system. Here for the sensitivity analysis we have considered two units system 'A' & 'B' in which unit 'A' having sub units in parallel hence if one or more sub units fail then the system works in reduced capacity and if the number of sub units failure is greater than a predefined number or else, then system is considered to be in the failed state. Unit 'B' have sub units connecting in series hence if any of its sub unit fail than the unit fail causing the whole system in failed state. Fuzzy concept is used to determine failure/working state of a unit. Taking failure rates exponential (constant), repair rates general & independent and taking into consideration various probabilities, a transition diagram of the system is developed to find Primary circuits, Secondary circuits & Tertiary circuits and Base state. Problem is solved using RPGT to determine system parameters. System behavior and sensitivity analysis is carried out and ell illustrated with the help of tables. Particular cases are taken for different repair and failure rates for sensitivity analysis of the system.

Assumptions and Notations:

The following assumptions and notations are taken: -1. A single repair facility is always available. 2. The distributions of failure times and repair times are exponential and general respectively and also different for different units. 3. Failures and repairs are statistically independent. 4. Repair is perfect and repaired system is as good as new one.

5. Nothing can fail when the system is in failed state. 6. The system is discussed for steady-state conditions. 7. Replacement of Un-repairable unit and repair facility is immediate. Ri (t): Reliability of the system at time t, given that the system entered the un-failed Regenerative state 'i' at t = 0. Vi (t): The expected no. of server visits for doing a job in (o,t] given that the system Entered regenerative state 'i' at t = 0. μi: Mean sojourn time spent in state i, before visiting any other states;

: The total un-conditional time spent before transiting to any other regenerative states, given that the system entered regenerative state 'i' at t=0. ni: Expected waiting time spent while doing a given job, given that the system entered regenerative state 'i' at t=0; ξ: Base state of the system. fj: Fuzziness measure of the j-state. α1/n: Constant failure rate of server A from full working state to reduce state. α 2/n: Constant failure rate of local server from reduce state to complete failure. α/n: Constant failure rate of remote server from full working to failed state. β1/m: Constant repair rate of unit A from full reduced state to full state. β2/m: Constant repair rate of unit 'A' from failed state to reduced state. β/m: Constant repair rate of unit 'B' from failed state to good state. m: Decreasing repairman's expert coefficient 0 < m < 1, so on subsequent repairs m decrease, so that repair rate will increase. n: Increasing repairman's expert coefficient 0 < n < 1, so on subsequent failures n increase, so that failure rate will increase.

Full Capacity Working State Reduced State Failed State

A/ /a: Unit in full capacity working / reduced state / failed state. B/b: Unit 'B' in full capacity working / failed state. Taking into consideration the above assumptions and notations the Transition Diagram of the system is given in Figure 1.

Primary, Secondary & Tertiary Circuits at various vertices

From the table 2, we see that at working state '0', there are maximum number of primary circuits, hence state '0' is the base state.

Primary, Secondary, Tertiary Circuits w. r. t. the Simple Paths (Base-State '0') Table 3:

Transition Probabilities

Table 4: qi,j(t) Pij = q*i,j(0)

MTSF (T0):

The regenerative un-failed states to which the system can transit (initial state '0'), before entering any failed state are: 'i' = 1 taking 'ξ' = '0'.

Availability of the System (A0):

The regenerative states at which the system is available are 'j' = 0, 1 and the regenerative states are 'i' = 0 & 1 taking 'ξ' = '0' the total fraction of time for which the system is available is given by

Busy Period of the Server: The regenerative states where server is busy are j = 1, 2, 3, 4 and regenerative states are 'i' = 0 & 1, taking ξ = '0', the total fraction of time for which the server remains busy is

Expected Fraction of Inspections by the repair man: The regenerative states where the repair man to do repair are j = 1, 3, 4 the regenerative states are i = 0 & 1, Taking 'ξ' = '0', the number of visit by the repair man is given by

Particular Cases: Particular results are carried out for 1 st failure and repair rate for ease of calculations.

MTSF (T0):

Effect of failure rate of different units on MTSF for m = n = 1, β1 = β2 = β = 0.8 Availability of the System (A0): -Effect of change of repair rate of units on availability of the system for α = α1 = α2 = 0.1 For optimum value of V0 failure rates of all units should be kept lowest and to have its value lower the failure rate of unit 'A' should be less than that of unit 'B'. For α = α1 = α2 = 0.1

MTSF (T0) Table

Expected Fraction of Server's Visits (V0): Effect of change of repair rate of units on expected fraction of server's visits Table Table 25 There is no significant improvement in the value of V0 on increasing the repair rates.

Conclusion:

We have made analysis for m = n = 1, but for very poorly designed system results may be obtained when m > 1 and n < 1. For good system m should be less than one and n greater than one for optimum values of system parameters. From the above discussion and analysis managements may decide to maintain repair/ rates of units in view of the importance of a particular parameter in need of hour.

References

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