Optimal Allocation of Fault Detectors - IEEE Xplore

8 downloads 0 Views 430KB Size Report
Key Words-Detector allocation, Optimization, Markov model, 0-1 in-t ... Results useful to: Reliability and maintainability engineers and theoreticians assumption ...
IEEE TRANSACTIONS ON RELIABILITY, VOL. R-27, NO. 5, DECEMBER 1978

360

Optimal Allocation of Fault Detectors I. Takami T. Inagaki, Student Member IEEE ,B E. Sakino K. Inoue, Member IEEE Key Words-Detector allocation, Optimization, Markov model, 0-1 in-

teger program.

v(k)

t

P,3(t) P

QO

Reader AidsPurpose: Widen state of the art Special math needed: Markov process

Results useful to: Reliability and maintainability engineers and theoreticians

Summary and Conclusions-A Markov model is given for a class of series systems which have fault detectors to find component failures. The optimal allocation of fault detectors is determined. This problem is a nonlinear 0-1 integer programming (0-1 IP) problem. The problem is solved easily because the nonlinearity is of a special type. An illustrative example is given.

1. INTRODUCTION

T r

C Cs

value of v when 1i ai = k prime, implies dldt system states: o ... operating, iF ... component i is failed but is not announced, iR component i is in repair. probability that the system is in state ,B at time steady-state probability for

PF(t)

I-Pa, steady-state unavailability of the system

planning interval of system operation probability that a detector operates properly (see

assumption Sa), viz, reliability of a detector

as csumtio

of detector ah a,vzeiblt

cost of the detector allocated to component loss per unit time caused by system failure alctdt

opnn

3. A PROBABILISTIC MODEL

Assumptions-

1. The system is series. Thus system failure coincides with component failure.

2. Failures of components are s-independent. 3. While the system is failed, no component can fail. 4. System failure is self-announcing. System unavailability is an important measure in esti5. The detector is not perfect. Three states are possible. mating the s-expected loss caused by system failure. In a) The detector operates properly, viz. produces an models repair begins immediately any usual Markov after alarm immediately when its component fails: otherwise component fails, i.e. every component has a detector which monitors the state of the component and produces it does not produce an alarm. b) The detector fails in mode an immediate unerring alarm when the component fails. 1, viz. produces no alarm even when its component fails. This assumption does not hold in some practical cases c) The detector fails in mode 2, viz. produces a false alarm because allocating a detector to every component is ex- when its component is operating. pensive. Thus obtaining a cost-effective allocation policy The occurrence of failure mode 2 is not inconvenient of detectors is a subject of practical concern. because we can distinguish false alarms from correct We give a Markov model which holds for any allocation alarms under assumption 4. Thus we ignore failure mode policy of detectors. Though we restrict our attention to 2 in the following discussions. In proper operation, repair a class of series system where functioning components begins at once when a component with a detector fails. suspend operation while the system is failed [1, p 194], Probability of proper operation, viz. reliability of a deapplicability of our models to series-parallel systems is tector, is assumed to be known and same for every debriefly discussed. Generalization of our model is discussed. tector. In failure mode 1, we need identification time, viz. time for finding a failed component, even if the failed 2. NOTATION component has a detector. 6. Failures of components which do not have detectors are not announced at all (but see assumption 4). We need N number of components in the system identification time, and repair begins after finding the Xi,Ai csum over i from 1 to N it,, constant failure and repair rates of component ifailed component. 1 if a detector is allocated to component i 7. The s-expected time for checking component states a, otherwise is same for every component. 8. The identification time is exponentially distributed ... (ar, an allocation (constant identification rate). The number of components CE *a SN), k number of detectors allocated whose states should be checked in finding a failed component increases as the number of allocated detectors, c4(k) an allocation satisfying Li ca = k v constant identification rate k, decreases. The s-expected identification time does not

to0

0018-9529/78/1200-0360 $00.75 © 1978 IEEE

TAKAMI ET AL: OPTIMAL ALLOCATION OF FAULT DETECTORS

361

vanish even if every component has a detector, because r < 1 (see assumption 5). Thus v is a function of k and r. The functional relation is known. We denote v(k) instead of v(k, r) because r is a given constant. 9. The failure rate Xi and the repair rate pti are constant. The state transition diagram is partially illustrated in Fig. 1. For an arbitrary allocation a, balance equations are given in (1)-(5).

Problem Select a to minimize + iY2 i [l4L ( -a )/v] i This problem is a nonlinear 0-1 IP problem since v is a function of Ei ai.

4.2 Solution Method

The proposed method of solution consists of two steps.

Step 1: Solve N- I subproblems which are linear 0-1 IP problems. Subproblem k: For each k, I . k < N-1, select cx(k) which gives J(a(k)) min {Y2aiCi +

t U1)

A

CSTEYiX [I4,li + (1-air)lv(k)]}

(8)

subject to the constraint Yi ai = k. The above subproblems can be solved by hand calculation. In Subproblem k, for example, we only have to t calculate coefficients of ai's, i.e. Ci - Cs T r X/v(k), and set k a-'s to be equal to 1 whose coefficients are smaller than those of the remaining N-k as's. The resulting a(k) is an optimal solution for Subproblem k. Step 2: Select a(k) which gives

J*-=min {J(a(O)), J(a(1)), ..., J(a(N- 1)),

Figure 1.

P'(t) P22F(t)

Xi PO(t)

= -i

=

+

1i /Ui PiR(t)

(1 -air) [XiPo(t)-XPiF(t)], for all i

( 1)

Po(t) + Xi [PiR(t) + (I-air) PF(t)] 1 PO(0) = 1, PAF(O) = PiR(O) = 0, for all i The steady-state unavailability is QO = LiAi [ Il[i + (I -+air)(v])

J(a(N))

CsT-iXi [l/wi+l/v(0)]

Y-iCi + CsTEi2i [i/,ai + (I -r)lv(N)]

The minimizing a(k) is the optimal allocation for the original problem and J* gives the minimum of J. (4) Remark. If detectors are perfect, viz. each operates prop-

(5) erly (see assumption 5-a), N- 1 detectors are necessary and sufficient for immediate identification of a failed component. In this case, (8) is replaced by

(6)

1 [ +ii/i If we further assume that « ,iand Xi