On reliability of a certain elementary safety system

Mwroelectron Rehab, Vol 26, No 2, pp 271 273,1986 Printed m Great Britain

0026-2714/8653 00+ 00 Pergamon Journals Ltd

O N RELIABILITY OF A CERTAIN ELEMENTARY SAFETY SYSTEM JACEK M CZAPLICKI Mining Mechamzatmn Institute, Silesian Technical University, ul.Pstrowskiego 2,44-101 Gllwlce, Poland and BOLESLAWKOPOCIIqSKI Institute of Mathematics, Wroctaw Umversity, P1 Grunwaldzk, 2/4,50-384 Wroctaw, Poland (Recewed for pubhcanon 9 August 1985)

Abstract--The paper deals with the determination of rellabihty of an elementary safety system which consists of two identical elements used simultaneously (hot reserve) whereas for correct system operation is enough when one out of it is good The operation time is so short that is assumed it is of zero length Knowing the rehablhty of element, rehablhty of system is found

1. INTRODUCTION One of the most Interesting chapters of the reliablhty theory in which intensive development may be observed in Poland now, is a part concerning safety problems New theoretical models are being developed [1, 2], several new terms are being defined [3] and gradation of failure with regard to possible consequences of their appearance has been introduced [3,4]. The central term considered in this class of problems is a catastrophic failure. It is a random event and its occurrence in an object means serious consequences as a result --serious with respect to economical losses. Frequently this type of failure is accompanied by failure or damage to cooperating objects, destruction of an environment or occurrence of accidents to staff Usually it is assumed that catastrophic failure of object means the end of its life. But it is not a rule. In technical systems for which catastrophic failure means danger to men's health and life and very serious economical losses, special protecting subsystems are built These subsystems are called safety systems and a goal of their construction is to diminish the probablhty of an occurrence of such a type of failure One of the practical directions of development of the rehablhty of safety is formulating of construction

principles for such types of system from one side, emplrlCo-theoretlcal analysis of hitherto existing solutions in this field from the other. Series safety systems have been constructed earlier so the're was no possibility of assessing their quality from a reliability point of view. This paper deals with rehability estimation of a certain elementary protective subsystem used for years in mining winders. Taking into consideration the rehabllity model constructed here, unpublished up to now it seems the paper possesses a cognitive advantage. 2. F O R M U L A T I N G OF A PROBLEM

There is given a two-demented system (Fig. 1) in a standstill state The system is an element of a certmn object and it operates when the object is in danger. This operation is so short in time that it can be assumed the operation time is zero length. A result of the operation is of 1~) type, meaning that the system either fulfils its task or does not. F o r the fulfilling it is enough when one element of a system acts correctly. The second element identical to the first one is a hot reserve. The system fails when both elements are down. O u r problem is to find the reliabihty of the system if reliability of the consisting element is known. Let us form the above as a mathematical model. There are given two sequences of random variables {X~'),) = 1, 2,. }, z = 1,2 which represent work time of considered elements There are r a n d o m variables stochastically independent, integer and positive. We assume they are Identically distributed, i.e. X~ 'l a= X , where 6 means equality of random variables distribution. This c o m m o n distribution will be marked with PR = P ( X = k),

Fig 1 Structure of system

k = 1,2 .....

The introduced work times of elements generate two 271

272

JACEK M CZAPLICKIand BOLESLAWKOPOCIIqSKI X j I,)

X(al)

el -

-

xT

xi~;

02

7"Ix) T

•

OperotlOn

•

RenewGk

Fig 2 Renewal fluxes in a system and corresponding random variables renewal fluxes w.~m-X~')+- . +X(d ), n = 1,2, . }, = 1, 2 The point of our interest is a r a n d o m m o m e n t T when coincidence event appears for the first time This means catastrophic failure of system The dlstrlburton of r a n d o m variable T m a y be defined by equahty of events { T = k} = lexist such m and n that

T a k i n g into consideration (1) we o b t a i n ra--1

Pk(m) = ~ Pk-~(m--J)Pj+6k, mP,. j=l k

+

~

Pk_,.(j--rn)pl ,

s(.'= s~'}

k = 1,2 . . . .

Let us denote

Pk = P(T = k),

(3)

j=r4+ 1

m=1,2,

.

whereas from (2)

k = 1,2,.

We would like to get a distribution {PR, k = 1,2 .... G e n e r a l solution of the p r o b l e m will be given by means of certain auxiliary expressions which fulfil a set of recurring equations In the particular case when X is geometrically distributed the desirable distrib u t i o n will be given in explicit form

PR = ~ PmPk(m)•

(4)

m=l

E q u a t i o n s (3) allow us to calculate Pk(m) one by one since this set can be s h o w n as follows PI(1) = px, Pz(1)=PI(1)p2,

P2(2)=PI(1}pl+pz,

P3(1) = Pz(1)pz + PE(2)P3, P3(2) = P2(1)pa + P t ( 1 ) P 3 ,

3. RELIABILITY OF THE SYSTEM

Let us suppose that in a particular m o m e n t of time a renewal appeared in the first flux In the second flux time till the nearest renewal equals m, m = 1, 2,.. F o r such a condition let us m a r k time to the nearest catastrophic failure of the system by T(m). Let T(0) = 0 additionally (Fig. 2). D e n o t i n g by

Pk(m) = P(T(m) = k),

k = 0, 1,.

for m = 0, 1 , . . we have

P3(3) = P2(2)pl + PI(1)P2 +P3 and so on. Particular case. Suppose work time distribution of element is geometric one

Pk=pqk-1, k = 1,2, . ; 0 < p < l ,

It IS obvious the r a n d o m variable geometrically distributed is w i t h o u t c o n t a g i o n (memory less property, it m e a n s P(X > k + n I X > k) = P(X > n) for k, n = 1,2,.... T h u s r a n d o m variables T may be defined as follows

G(O) = 6k, o, where 6k, . = 1 for k = n and 6k..=O for k ~ n IS K r o n e c k e r delta We have also Pk(m) = 0 for k = 1,2, . , m - 1 ; m = 2 , 3 , .becauseT(m)>.m F r o m the model conditions we o b t a i n t h a t distributions of r a n d o m variables T(m) for m > 0 fulfil equations

T(m)

~ (m+T(X-m) (X+ T(m-X)

If X-m>~O, If X - m < O

(1)

and

T ~

/

XI11)

If

X]I) IS a renewal m o m e n t in flux

{S(.2),n = 1,2,

( X]*)+ T'

},

otherwise

(6)

where: T' & T a n d T' and X ] 1) are stochastically independent. F r o m the renewal theory one can find (see [5] p 951 that renewal probability in a flux is c o n s t a n t and equals p Therefore

Pk = P(X~1) = k, exists such n that S(.2) = X] 1)) + P(X]I) + T' = k, does not exist such n t h a t S(.2) = X] 1)) =

T & mln(X~ILX'~ ')

k--1

+ T(max (X(1~', X ? )) - min (X~'(X(12)))

& T(X)

p + q = 1.(5)

(pqk-~)p+ ~ p q ' - ~ P ( k - O ( 1 - p ) , (2)

t=l

(7)

Safety system

q:::

~

273

Rr(£) For k

:

104

R r (k)~- Z~ ~- 0 9 9 5

0007

I I0 4

k

Fig 3 Surwval functions Rx and R T and increment function A whereas for k = 1 the second component of the right side of the above equation equals zero. Solution of the set of equations (7) is Pk = P2( 1 - p 2 ) k - 1 ,

k = 1,2,

This IS geometric distribution of appropriately modified probablhty of the renewal.

1. Let us make some remarks resulting from analysis of different types of criteria of system goodness assessment We devote our attentton to the particular case. The causes of it are results of the theorehco-emplrlcal investigations reahzed in the Mining Mechanization Instttute of Silesian Technical University, Ghwlce, Poland concerning among other things the particular system used in mining winders of the type descrtbed here. It was no ground to reject the hypothesis stating that the work time is geometrically distributed with the parameter p = 0.0007. Consider the expected value F o r the element E ( X ) = p - l , for the system E ( T ) = p-2. It is easy to notice that rejecting triteness limiting cases i.e. p = 0 and p = 1 where it is no sense using the second element because etther both are down (first case) or entirely rehable (second case), for

This function increases for k in a range p(1 + p)k< l, decreasing for k in a range p ( l + p ) k > l and can possess two maximal values in points k + 1 and k + 2 If p(1 + p ) k = 1 Thus such k always exists beginning from which function A will be decreasing one. If p is small and k big then (1 __p)k ~_ e x p ( - k p ) .

4. R E M A R K S

E(T)>E(X)

A = R T ( k ) - R x (k) = (1--p2)k--(1--p)/"

p e (0, 1).

Thus o,~ E = E ( T ) / E ( X ) = p - 1 is an illustration of profit of using the second element according to the mean value. Let us take survival functions for element R x (k) = P ( X > k), for system R r (k ) = P( T > k ) when k = 0, 1..... Increment of survival function value due to application of second element

For big k counting In teens or more function R(k) may be shown as continuous not as stepwlse. An example plot of RT(k), R x (k) and A for p = 0 0007 is shown in Fig 3 2. Analysing the above rehablhty factors it may be stated that application of the second element as a hot reserve is advantageous solution increasing significantly the reliabdity of the safety system. 3 This paper deals with the part of the theory which corresponds to the exploitation practice of mining winders directly Further theoretical investigations more general than presented here are being carried out They concern counter problems and alternatlng processes theory

REFERENCES

l. S BrodzlfiSkl and J M Czaphckl, Rehablhty of system operator-mining hoisting installation Proceedmgs oJ Wmter School "86 (to appear m Pohsh) 2 S Brodzlfiskl and J M. Czapllckl, Rehabdlty to safety assessment of hydrauhc disc brakes in winders Proceedings of 21st International Conference on SaJety m Mines Research lnstuutes, Sydney, 21-25 October 1985 3. J M Czaphckl and L Dzlembata, Survival function estimation of partly renewal system Exploit Problems Machines (to appear) 4 J JaSwlfiskl and K Wa~yfiska-Flok, Rellabdlty of system with the functional surplus in the safety aspect Exploit Problems Machines 1-2 (1984) (in Pohsh) 5 B Kopoofiskl, Introduction to Renewal and Rehabdlty Theory PWN, Warsaw (1973) (m Pohsh)