1. The circuit in Fig. 3 has substantial advantages as compared with that in Fig. 1, since with a considerably reduced number of mutual control circ...

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OF T H E S T A N D B Y THE PROBABILITY

MEASUREMENT D. Z h o l k o v e r

and

OF S I G N A L

TRANSDUCERS

OF O B T A I N I N G

RESULTS A. I.

Yakovlev

UDC681.2.001.1

It is known that every measurement is generally preceded by the conversion of the measured quantity,by means of devices known as transducers of signals,to a form convenient for measurement. The probability of obtaining a correct measurement result is P = PcPm,

where Pc is the probability of a correct transformation of the measured quantity, Pm is the probability of a correct measurement of the converted quantity (c and m are independent events). Let us e x a m i n e the method of increasing Pc, and therefore P, by providing the signal transducers with a standby equipment operated by their mutual control circuit. Amorig the various mutual control circuits used for regulating the standby equipment of t e c h n i c a l devices, increasing attention is now being paid to circuits with the s o - c a l l e d quorum elements (QE) [1]. In essence such circuits are of the self-arranging type which change their structure when failures occur, thus e l i m i n a t i n g the consequences of those failures. The functional effectiveness of QE circuits with mutual control devices*L(whichare i d e a l l y reliable) is examined in [1]. Let us analyze the functional effectiveness of QE circuits by taking into consideration the actual r e l i a b i l i t y characteristics of the mutual control and switching (MCS) devices. The logic of quorum elements is designed in such a way that the decision about the failure of one of the controlled members is a c c e p t e d only by comparing it with the remaining controlled members (CM). It is natural that the principle of operation of these systems presupposes the utilization of at least three controlled members. In view of the fact that the quorum elements can be used for controlling standby technical equipment, the evaluation of their effect on the functional effectiveness of standby systems is of considerable interest. Figure 1 shows an e x a m p l e of a circuit with a single quorum element. If a l l three controlled members (in our case standby signal converters SC1, SC2, and SC 3) are in working order, as well as the MCS devices, the signals at the output of the comparing devices CDI_ ~, CDs_ 1, and CD2_ 3 will be equal to zero (the controlled members are connected to the working outputs) and the system as a whole will be in working order. If one of the controlled members should fail, two comparing devices will transmit signals to the two inputs of an AND gate A 1, A s, or A s, which control the disconnection of the faulty CM. The operated gate sends a signal to the corresponding control converter (CC 1, CC 2, or CC s) which disconnects the controlled m e m b e r from the working input, and the system continues to function. It can be easily seen that the failure of two controlled members with a functioning MCS device leads to the disconnection of the entire system. By using the theorem of c o m p l e t e probability it is possible to show that Pc = kop~ + k13p 2 (1 - - p) + ke3p (1 - - p)2,

(1)

where p is the probability of a correct conversion of the measured quantity by one of the signal converters; k 0, k I, and k z are conditional probabilities of obtaining correct results by means of a system with a correct conversion of the measured quantity and respectively three, two, and one converters. For simplicity of treatment let us assume that all the components of the same type in Fig. 1 are equally reliable. Let us also assume that the failures of various circuit elements are independent events and occur singly. A necessary condition for the advisability of applying a quorum e l e m e n t consists of meeting the inequality Pc > P.

(2)

Translated ~ o m I z m e r i t e l ' n a y a Tekhnika, No. 9, pp. 14-17, September, 1967. Original a r t i c l e submitmd February 9, 1966.

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r

MCS

!

i

I I

!

_ u

__

m,,, ,( I

I

mI

/ / ///

~

ii! I)I

.A

L-__

Fig. 2 Fig. 1 Let us examine (Fig. 2) a family of curves Pc = f(P) and compare them with straight line Pc* = P, which represents the effectiveness of a system with a single signal c o n verter. It will be seen from (2) that the utilization of a system consisting of standby converters and MCS devices is useless if the curve of Pc = f(P) lies below line 1 in the range of possible values of probability p.

g) o,

In case of an ideal MCS device the conditional probabilities are k0 = kl = 1 and k s = O, and according to [1] we obtain

f Fig. 3

Pc = 3P'~q- 2P8'

(3)

which corresponds to curve 2. With a reduction in the working reliability of MCS devices, curve 2 is displaced to the right into the position of curve 3, and the range in ~rhich the points of curve Pc = f(P) are above line 1 is correspondingly reduced. At the rate of a further reduction in the working reliability of MCS devices, curve 3 is transformed into curve 4 which is completely in the range below line 1. In fact, by solving simuhaneously (1) and the straightline equation Pc* = P, we obtain

(4)

AoP~ -~- Atp 2 -]- A2p = O, where Ao=ko~3kl+3k2;

Al=3kx--6k~;

Ao.=3k2~l.

Coefficient As represents the difference of the t-it.t angle tangents of curve Pc = f(P) and straight line Pc* = P at point p = 0.

= A2

(dRcl ~-~P/~=o

- - 1.

For the circuit under consideration (Fig. 1),condition k z < 1/3 is virtually always met. Therefore, A 2 < 0 and curve Pc = f(P) at first runs below line Pc* = P' The utilization of this circuit is useful in the case when curve Pc = f(p) crosses straight line Pc* = P in the range of 0 < p - < l . This condition is met when (4) has the real roots." pl=0;

0~p~.~l;

0~pa~

(5)

It will be seen from (5) that the utilization of standby equipment and of mutual control is useful if A0<0;

Al>0andAs<0,

(6)

and discriminant (4) is positive, i.e., if 9k 2 ~ 1 2 k o k ~ - 4k0-]- 1 2 k l - 12k2.

(7)

If (7) is not met, curve Pc = f(P) will have in the range under consideration only one intersection point with straight line Pc* = P, and this point will coincide with the origin of coordinates (roots 1~ and P3 in this case a r e c o m -

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plex conjugates). When expression (6) becomes an equality, curve Pc = f(P) has one point of contact with line Pc* = P approaching it from below (P2 = P3). Let us note that, in the case of ideal MCS devices, P2 = 0.5 and P3 = 1, i.e., for converters with p ~ 1 it is not advisable to use mutual control circuits which have a real reliability. Moreover, the application of iteration [1] increases substantially the complexity of the system (this complexity rises with respect to the level of iteration as a geometrical progression). A circuit with one quorum e l e m e n t is at least three times more complicated than the original circuit, and after a single iteration, i.e., after the transition to a second level with the application of three quorum elements (as shown in Fig. 1 where CM l, CM 2, and CM s are replaced by QEl, QFe, and QE3) the circuit becomes nine times more complicated, etc. It is possible to state that, for any quorum element's iteration level, the value of Pc cannot exceed P3 (Fig. 2). Thus, it will be seen from (6) and (7) that the application of the circuit shown in Fig. 1 with CM1, CM 2, and CM S is not always advisable. The raising of Pc by means of iteration considerably complicates the circuit and has limited possibilities, The main deficiency of the circuit with CM z, CM z, and CM3 (Fig. 1) consists of the fact that w h e n o n e of the signal converters fails the circuit is switched into a condition in which two operating converters are connected in series. It is expedient to produce a standby control logic which, in the course of mutual control, disconnects two signal converters (the faulty and a serviceable one) when one converter fails. Such a circuit possesses in principle a higher functional effectiveness. Figure 3 shows a circuit which continues to function if, with an ideal MCS circuit, one of the signal converters fails (or the converters are in working order and the signals at the outputs of CDI_ 2, C ~ - 3 , and CD3-1 are equal to zero). In this case two signal converters are disconnected and there remains connected to the working output a single signal converter which is in working order. In deriving a formula for Pc of the circuit in Fig. 3 let us use the Erlang recurrence system of differential equations which represents the process of "pure loss and reproduction." For ideal MCS devices these equations have the form

dPo dt = - - 3 ~ 1 P ~ dP~ - dt

--

~,IP~

+

(8) 3~lPo

where P0 and P2 are the probabilities of obtaining with this system a correct conversion of the measured quantity for m = 0 and m = 2 respectively (m is the number of disconnected converters), By integrating (8) we obtain P0=p~;

3 P~= ~-p(1--p2),

(9)

t

--; ~ldt where

p = e

0

is the probability of converting correctly the measured quantity by means of a single converter,

The probability of converting correctly the measured quantity by means of the entire circuit is 3 Pc = Po + P~ = p3 + ,~- p (1__ p2).

(10)

In order to compare the effectiveness of the circuits in Fig. 3 and Fig, 1 let us determine the difference between (10) and (3) 3 APe = ~ - ( 1 - - p ) 2 This difference is positive, which indicates that in principle the circuit in Fig. 3 is preferable. Moreover, the circuit in Fig. 3 contains a smaller number of functional elements, which also helps to raise the reliability of its operation. For determining Pc in the circuit of Fig. 3 by taking into consideration the actual characteristics of the MCS devices let us derive an Erlang system of recurrent differential equations.

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dP o

dt

--

3(Ll+~)po

dP, -- - - 2~,IPI q- 3 (@x -~- ~,~) Po dt dP,

(11)

~zPs+3r~lPo+2ElpPl

dt

where k s is the intensity of the MCS device failures which leads to a complex disconnection of the member and of the mutual control circuit; q is the conditional probability of an MCS device failure which entails the disconnection of one signal transducer (the faulty one); r is the conditional probability of two signaltransducers and the MCS device being disconnected for a failure of one signal transducer (conditional probability of a correct "reject" result); p is the conditional probability of a signal transducer disconnecting itself (by self-disconnection of a signal transducer we understand its failures which do not affect the operation of the remaining circuit even if the switching devices do not operate). By integrating (11) and assuming that the failure of circuit elements occurs in the simplest manner for initial conditions of Pair = 0 = 1, P l l t = 0 = P2lt = 0.= O, we obtain

Po = aop3 P1 = 3al (1 - - aop) p2

I

P~=p[(1--a0p ~)as~6pal(1-p)]

J

(12)

where

qL1 ~ ~,~ . ao=:e -3~'2t; a l =

Lx~3Ls '

a~=

3rL1 - - 6Xlalp 2L 1 ~ 3 L ~

The probability of converting correctly the measured quantity by means of the entire circuit is represented by the sum:

Pc ----Po + P1 + P~ = aop~ "q- 3a, (1 - - top) p~ + P [(1 - - aop2) a2 + 6pax (1 - - p)].

(13)

By solving (13) simultaneously with the equation of straight line Pc *= p, it is possible to show that function Pc(p) in the c i r c u i t of Fig. 3 exceeds the probability of correctly converting the measured quantity by means of one converter in the range of 0 < p_

P,c =

.~ Crop,,-., . [(1 - -

(14)

p) p] " ,

m=0

where n is the number of standby converters (in our case n-< 3), m is the number of converters which have failed, C m is the number of combinations of m elements out of n. The condition of the advisability of using these circuits consists of meeting the inequality

Pc > Rc.

(15)

It should also be noted that Rc in (14) is at a m a x i m u m for n = ncr. The value of ncr can be c a l c u l a t e d from the formula In I In [ ( 1 - P)P] l ner=

Inl

In [p + ( 1 - - P )

Pl I

In[p+(1--p)p]--ln[(1--p)p]

If ncrm 3, the condition for the advisability of using quorum elements is determined by the inequality 2

c~p 3-m [(1 m==O

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p) p]" < Pc.

If nor < 3, then it is rounded off to the nearest integer, and the value of n is selected for which the maximum probability Rc.max is ensured for converting correctly the measured quantity. The condition for the advisability of using quorum elements in this case is represented by inequality Rc, max<: Pc, CONCLUSIONS t. The circuit in Fig. 3 has substantial advantages as compared with that in Fig. 1, since with a considerably reduced number of mutual control circuit components it provides a higher probability of obtaining a correct conversion of the measured quantity. 2. The application of iteration for increasing the probability of correctly converting the measured quantity by means of the circuits in question (Figs. 1 and 3) is not promising, since it increases substantially the complexity of the system and in principle cannot provide a value of Pc exceeding P3 or p~ (Fig, 2). 3. For a sufficiently large probability of the converters disconnecting themselves the application of a mutual control device can be inadvisable. In order to decide on the advisability of applying a MCS device it is necessary to use criterion (15). LITERATURE 1,

2.

CITED

Suren, Zarubezhnaya radio61ektronika, No, 5 (1965). B. V. Gnedenko, Yu. K. Belyaev, and A. D. Solov'ev, Mathematical Methods and the Theory of Reliability [in Russian], Izd. Nauka, Moscow (1965).

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