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Arithmetico-Geometric Process Model for a Single Unit Cold Standby System

C.Manmatheswara Reddy1* and B. Venkata Ramudu2†

1Principal, Sri Balaji P.G. College, Anantapur-515001 (AP) INDIA.

2Assistant Professor, Dept. of Statistics, SSBN Degree & P.G. College, Anantapur-515001 (AP) INDIA.

Research Article

Abstract: This paper studies a cold standby repairable system consisting of two identical components namely component, component 2 and one repairman is studied. Assume that each component after repair is not ‘as good as new’ and also the successive working times form a decreasing arithmetic-geometric process, the successive repair time’s form an increasing arithemetico-geometric process and both the processes are exposing to Weibull failure law. Under these assumptions we study an optimal replacement policy N in which we replace the system when the number of failures of component 1 reaches N. We determine an optimal repair replacement policy N* such that the long run average cost per unit time is minimized. We derive an explicit expression of the long-run average cost and the corresponding optimal replacement policy N* can be determined analytically. Numerical results are provided to support the theoretical results.

Key Words: Convolution, Geometric process, Monotone Processes, Repair replacement policy, Renewal Process.

1 Introduction

No one disputes the need for articles to be reliable organizations such as airlines, the military and public utilities are aware of the costs of unreliability.   Manufacturers often suffer high costs of failure under warranty.   Argument and misleading begin when we try to quantify reliability values, or try to assign financial or other benefit values to levels of reliability.  As systems become more complex and require new technologies and methodologies for effective operations, the development of more effective and cost-saving maintenance models and their appropriate utilization are needed to maximize the performance of such systems. In this paper, we consider a system which is subject to failures.  These failures are attributed to many sources.  Such as designer’s negligence, defective components, natural component deterioration, and operator mistakes, etc.  Upon failure of the system, two types of actions are usually taken.  One action is to replace the failed system by a new identical one, and the other is to repair the system so that the repaired system operates satisfactorily as required.  In general, a repair can bring the state of failed system to a level which is somewhere between new and prior to repair.  Once a repair action is taken, the system may be renewed (good-as-new state)  or may return to the state just prior to the failure initiating the repair action (bad-as-old state).  It is also conceivable that the system may be in some intermediate state upon repair.  The first is referred to as a perfect repair, the second is a minimal repair, and the last is called an imperfect repair.  Replacing a failed system with a new one is regarded as a perfect repair.  Therefore, the perfect and minimal repairs are considered to be two special cases of the imperfect repair action.  In an imperfect repair model suggested by Brown and Pro schan (11), it is assumed that the repair is perfect with probability p and minimal with probability q = 1-p. If the items are stochastically deteriorating, the successive operating periods after repair will become shorter and shorter, where as the mean lengths of the repair periods will be increasing.  Thus, an alternative approach is to use some kind of monotone process.  As a first order approximation, Lam(23) studied the geometric process replacement model in which the successive operating periods{ Xn, n=1,2,….,} of an item form a non-increasing geometric process and the consecutive repair periods { Yn, n=1,2,….,} constitute a non-decreasing geometric process. Lam considered the geometric process replacement model when the repair times are not negligible.  He assumed that the length of the nth operating period Xn and the length of the nth period Yn are distributed as an-1 X1 and bn-1 Y1 respectively.  Where a<1 and b>1, and that X1, Y1, X2, Y2,…… is an independent sequence.  Thus the operating (repair) intervals are stochastically decreasing (increasing) at a geometric rate.  Then two types of replacement strategies were considered:

1. Replace the system if its total operating time attains a certain predetermined time T
2. Replace the system at the Nth failure (where again N is fixed in advance).

Zhang(71) generalized Lam’s(23) work by introducing a bivariate policy   (T, N) under which the system is replaced at working age T or at the time of Nth failure whichever occurs first.  Stadje and Zuckerman (62) considered the monotone process replacement models when the repair times are not negligible. The above various researchers pointed to a one – component repairable system.  However, in practical application, for improving the reliability or enhancing the availability of the system, the standby techniques are usually used.  An example of cold stand by component is a spare light bulb in an over Head projector.          Many fielded systems use cold-standby redundancy as an effective system design strategy. Cold-standby means that the redundant units cannot fail while they are waiting.  Space exploration and satellite systems achieve high reliability by using cold standby redundancy for non repairable systems.  Space intertial reference units are required to accurately monitor critical information for extended mission times without opportunities for repair.  Many other systems use cold-standby redundancy as an effective strategy to achieve high reliability including textile manufacturing systems and carbon recovery systems used in fertilizer plants. Zhang (72) applied monotone process replacement model to a two-identical component cold-standby repairable system with one repairman.  Assume that each component after repair is not ‘as good as new’ by using the geometric process.  We consider a policy N based on the number of repairs of component1.  The problem is to determine the optimal policy N* such that long-run expected reward per unit time is maximized.  The explicit expression of the long-run expected reward per unit time is derived.  Also, the optimal number of repairs N* is found by a numerical method. In the next section, we develop the model and optimal solution for a cold standby deteriorating repairable system with an Arithmetico geometric process.  Using this process, we derive explicit expressions for optimal number of failures (N) by maximizing the long-run expected reward per unit time and also we provide the results leading to optimal procedure.

In modeling of these deteriorating repairable systems we state the definitions relating to stochastic order and arithmetico geometric process which are given in

Lam[     ]

Schematic representation of the model under study is as follows.

Cycle 1               Cycle 2                               Cycle 3                             Cycle 4

Component 1

Component 2

Cycle 1                                        Cycle 2                                Cycle 3

: working state,            : cold stanby state,              : repair state,

: waiting for repair state      4.1 A possible course of the system

2 Model:

In this section, we develop a model for optimal replacement policy for cold standby system specializing to arithmetico geometric process by maximizing long-run expected reward per unit time with the following assumptions.

Assumptions:

Now we make the following assumptions about the arithmetico geometric process model for a cold standby repairable system.

1. At the beginning, two components both are new, and component 1 is working and component 2 is under cold standby.
2. When the two components in the system are both good, one is working and the other is under cold standby.  The repairman will repair the working one as soon as it fails.  At the same time, the standby one begins to work.  When the repair of the failed one is completed it either begins to work again or comes under cold standby.  If one fails while the other is still under repair,  the system breaks down.
3. Assume that each component after repair is not as good as new. The time interval between the completion of the (n-1)th repair and the completion of the nth  repair on complement i  is called the nth cycle of component i where i = 1,2; n=1,2………,
4. Let Xn(i) and Yn(i) be respectively the working time and the repair time of component  i  in the nth cycle i=1,2;  n=1,2,….. Assume that the distributions of Xn(i) and Yn(i)  (i=1,2) are respectively               and       where 0<a, b<1
5. A sequence { ; i=1,2; n=1,2,…..} and a sequence {  i=1,2; n=1,2,…..} are independent.
1. The system will be replaced sometime by a new and identical one and the replacement time is negligible.
1. A component in the system cannot produce the working reward during cold standby and no cost is incurred during waiting for repair.
2. Repair cost rate of two components are both Cr the working reward rate of two components are both Cw and the replacement cost of the system C.

For this model, a methodology for obtaining optimal number of failures (N), which maximizes the long run, expected reward per unit time is discussed below.

3. Optimal Solution:

When the number of repairs of component 1 reaches N, then component 2 is either under a working state or under a waiting for repair state in the Nth cycle. Naturally, the former works until failure in the Nth cycle, the later is not repaired any more in the Nth cycle, while component1 works until failure in the (N+1)th cycle. Thus the renewal point under policy N is found. Let T1 be the first replacement time of the system under policy N. Let Tn (n≥2) the time between the (n-1)th replacement and the nth replacement of the system under policy N. Obviously, {T1,T2……}forms a renewal process, the inter arrival time between two consecutive replacements is called a renewal cycle.

Let C(N) be the long run expected reward per unit time of the system under policy N. Thus according to renewal reward theorem, we have

(3.1)

Let W be the length of a renewal cycle of the system under Policy N, then

W =     (3.2)

Where, the first term, the second term, the third term and the fourth term are respectively the length of working age, the length of repair time, the length of time of waiting for repair and the length of cold standby state of component1 under policy N.

Where I is an indicator function such that

1, if an event A occurs

IA                    =

0, if an event A does not   occurs.

Now our problem is to evaluate E[W]. According to the assumptions of the model and definition of convolutions, the distribution functions of

and      are

And   Respectively

Thus                  (3.3)

According to equation (3.1) and (3.3)

(3.4)

Where       ,

,

In the next section, we determine the optimal replacement policy N* by analytical or numerical methods such that C(N*) is maximized.

4 Empirical Results and Conclusions

For given fixed values of l, m, C, Cr ,Cw the optimal replacement policy N* is calculated as follows:

Let µ=100,a=0.9,b=0.85,  l=25, d1=1.15, d2-0.8,Cr=2, Cw=200, C=4000

Table: 4.1 N Vs C(N)

 N C(N) N C(N) 2 -5319.172363 15 3.610524 3 -169.474091 16 3.124773 4 -40.985847 17 2.695149 5 -9.611918 18 2.318578 6 1.211678 19 1.990622 7 5.287989 20 1.706354 8 6.670899 21 1.460830 9 6.874488 22 1.249351 10 6.549132 23 1.067587 11 5.998919 24 0.911632 12 5.372939 25 0.778005 13 4.745433 26 0.663638 14 4.152574

Graph: 4.1 N Vs C (N)

Conclusions:

• From the table and graph (4.1) , it can be observed that the working reward rate is maximum at N*=9. Thus, when the number of failures N*=9 then the system should be replaced.
• This can also be modeled as an improved system by certain modification of the hypothetical values.

References:

1. Borwn, M., and Proschan, F., “Imperfect Repair”, Journal of Applied       Probability, Vol.20, 1983, PP 851-859.
2. Lam Yeh., “Geometric Processes and Replacement Problems”, Acta Mathematicae Applicatae Sinica, Vol.4,         1988 a, pp 366-377.
3. Lam Yeh., “A Note on the Optimal Replacement Problem”, Advanced Applied Probability, Vol.20, 1988 b, pp 479-482.
4. Stadje, W., and Zuckerman, D., “Optimal Strategies for some Repair        Replacement Models”, Advanced Applied Probability, Vol.22, 1990, pp        641-656.
5. Zhang, Y.L., “A Bivariate Optimal Replacement Policy for a Repairable System”, Journal of Applied Probability, Vol.31, 1994 pp 1123-1127.
6. Zhang, Y.L., “An Optimal Geometric Process Model for a Cold Standby Repairable System”,          Reliability Engineering and System Safety, Vol.No.63, 1999, pp 107-110.