Reliability Engineering 12 (1985) 175-179

Reliability of k-to-l-out-of-n Systems

S. P. Jain and Krishna G o p a l

Department of Electrical Engineering, Regional Engineering College, Kurukshetra- 132119. India

(Received: 26 November, 1984)

A B S T R A C T

A k-to-l-out-o[:n: G system is one in which neither less than k nor more than l units out oJ n units are to Junction properly, (] the system is to Junction suceess[ully. The system [ails ![less than k or more than I units [unction simultaneously. Examples of this kind are jound in man), areas, e.g. multiprocessor, communication, transportation systems, ere. A simple expression based on the event space approach is presented. The expression obtained is computationally ejficient as it has fewer terms and each term has Jkwer multiplicants than the original expression.

P), qj R j, n

Rk l,n

S(x,n)

F ( x , n )

1 NOTATI ON

Reliability (unreliability) of unit j. Reliability of a j out of the first n: G system. Reliability of a system in which not less than k and not more than l out of n units are to function successfully for the system's Success.

Probability that exactly x units are successful out of the first n units. Probability that exactly x units are failed out of the first n units.

2 I N T R O D U C T I O N

Generally in any given system it is thought good if all its constituent units are functioning. In the usual k-out-of-n:G system, the number of

175 Reliability Engineering 0143-8174/85/$03-30 ~;~ Elsevier Applied Science Publishers Ltd, England, 1985. Printed in Great Britain

176 S. P. J a in , K r i s h n a G o p a l

successful units must be anything between k and n, both inclusive. If the reliability of its units improves, the system reliability also goes up. But there are situations where, due to the limitation of some member(s) of the complete system, we cannot afford to allow all n units to function simultaneously. In a power house we may have seven generating units and five to six units may be enough to feed the required power into the network. The seventh unit is simply in standby. If all the seven units are operating, the boilers may not be able to cope, thereby causing deterioration in unit performance and resulting in a failed state from the quality of the power supply point of view. It is therefore a 5-to-6-out-of- 7: G system.

Similarly, in a multiprocessor type of computer , 1 where some of the resources such as storage, buses, I /O units are shared between various processors, a large number of processors cannot be allowed to communicate with these resources simultaneously as this may result in traffic congestion on the buses and printers. If too few processors are working, then the efficiency of the system goes down. Thus a multiprocessor is a k-to-l-out-of-n:G system.

A k-to-l-out-of-n:G system (henceforth denoted by (k-l,n) system) is, moreover, a very general system. When I equals n, it becomes a k-out-of- n :G system; when k = l = 1, it is a parallel system; when k = l = n, a series system; and when k = l, a system wherein exactly k successes are required. This paper, therefore, proposes a method for determining a com- putationally efficient reliability expression for such systems. It is shown that, for the case of S-independent and identically distributed (iid)units, there exists an optimal value of number of total units n for a given set of values of k, / and p. These types of systems are termed non-coherent systems in reliability literature.

3 M A T H E M A T I C A L F O R M U L A T I O N

By the definition of a k l, n system: 1'2 l

Rk-,,, = E S(x, n) x : k

n - k

Rk_l.n = ~ F(x,n) x = n - [

(1)

(2)

Reliability of k-to-l-out-o/n systems 177

If k and l are much less than n, eqn (1) is preferred to eqn (2), and if k and l are close to n, then eqn (2) is preferred for reasons of ease in generation of combinations. 3

If the expression is written in the expanded form, there will be several sets of terms; terms in the successive sets will differ by one failure and each term will have n multiplicants. It is possible to combine some of the terms having the same number of multiplicants and differing in one failure only using the Boolean identity AB + ,4/) = i]. Such successive combinations lead to an expression that has fewer terms with each term having fewer multiplicants than the original expressions. The final simplified expressions corresponding to eqns (1) and (2) are given, respectively, as:

Rk_t, . = S(k, m) +

n - - m - - 1

2 i = 0

n - m - 1

+2 i = 0

n - m - 1

Rk_t, . = F(j, m) +

n -- rtl - - 1 +Z i = 0

where j = n - l, m = n - (l - An alternative approach

R k _ l , " ---- R k , n - - R t

Pro+ 1 + i S ( k - 1, m + i)

qm+ 1 + i S ( k + l +i, m + i)

p , . + l + g F ( j + l + i , m + i ) i = 0

q~+ l +iF(j - 1,m + i )

k). to determining Rk_l, n is

+ l , n

(3)

(4)

n - k - !

=Rk 'k+ ~ Pk+l+iF(l+i 'k+i) i = 0

n - l - 2

--(RI+I,I+I-F 2 p l+2+iF( l+i , l+l+i ) ) (5)

i = 0

All units are assigned serial identifiers (l, 2, 3, etc.) and, once assigned, identifiers are retained throughout and not altered.

178 S. P. Jain, Krishna Gopal

T A B L E 1 Comparison of Computational Effort

System k l n Ordinary equation (eqn ( 1 )) Proposed equation (eqn (3)) number . . . . . . . . . .

Number of Number ()/ Number o j Number o! multiplications additions multiplications addition~

1 6 8 10 3375 374 1 730 226 2 5 8 10 5643 626 2010 275 3 4 8 10 7533 836 1655 239 4 4 7 10 7128 791 2100 294 5 4 6 10 6048 671 2602 349 6 4 8 11 17490 1748 4597 569 7 4 8 12 38 478 3497 8 086 ~t~

For iid units, eqn (1) leads to the simplest possible expression l

i=k

If eqn (3) is applied to a (2 3, 51 system, the expression obtained is:

R2_3. 5 = 3(2 , 4) +psS(1 ,4 ) + q5S(3, 4)

=PlP2q3q4 + Plq2P3q4 + Plq2q3P4

+ qlPzP3q4 +qlP2q3P4 + q~q2P3P4

+Ps(Plq2q3q4 + qlP~q3q4 + qlq2P3q4 + qlq2q3P4)

+ q5(PlP2Paq4 +PlP2q3P4 +Plq2P3P4 + qlP2P3P4)

Table 1 gives the saving in computat ion effort using eqn (3) in comparison to eqn (1) for some arbitrarily chosen systems.

Table 2 gives the values of system reliability for various systems with iid units. It can be seen from Table 2 that, for a given set of values of k, t and p, there is an opt imum value ofn and for given values of k, I and n, there is an opt imum value of unit reliability p.

4 CONCLUSION

A new type of system, k-to-Lout-of-n: G system, has been discussed. The most economical expression for the reliability of such a system when the

Reliability o! k-to-l-out-~/~n systems

TABLE 2 Reliability of Some (k l, n) Systems for Different Values of Unit Reliability

179

System Value of system reliability./or. llUll7h~r

p=05 p=0.6 p=0"7 p=0"8 p=O.9

1 k =5 0-611 8 0.7865 0.803 0,6159 0.2637 l=8 n=10

2 k = 5 0.7868 0-8585 07376 0-4403 0.1108 /=9 n=12

3 k = 10 0.147 1 0-3759 0-5936 0.5408 0.181 8 / 12 n=15

units may be different has been given. The economy in expression is on account of suitable combina t ions and subsequent reduct ion of terms. The economy in number of mult ipl icat ions and addi t ions is as high as 79 '!/o and 70 O//o, respectively, in some cases. Thus the equat ions presented are computa t iona l ly efficient. Equat ion (6) gives the reliability for systems with lid units. This expression can be used to verify that : for a given set of k and / there is an opt imal value of n if p is known and vice-versa.

R E F E R E N C E S

1. Heidtmann, K. D. A class of noncoherent systems and their reliability analysis, l l th Int. Symp. Fault Tolerant Computing, FTCS-11, June 1981, Portland, USA.

2. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn, Wiley, New York, 1968.

3. Mifsud, C. J. Algorithm 154: combinations in lexicographic order, Comm. ACM, 6 (1963), p. 103.