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Open and short circuit failure of hammock networks
Electronics Reliability & 3dricrominiaturlzation Pergamon Press 1963. Vol. 2, pp. 205-206. Printed in Great Britain
OPEN AND SHORT CIRCUIT FAILURE OF HAMMOCK NETWORKS D. V. B L A K E
Ship Division, National Physical Laborato~', Faggs Road, Feltham, Middlese× Abstract--An extension of 3,ioore and Shannon's work on nets of relay contacts is used to provide a rapid method of assessing the probabili~" of failure of a hammock net when the elements are liable to open and short-circuit failure. INTRODUCTION A COM.~ION method of using redundancy to improve reliability is to use a number of component s arranged as a "hammock" network whose overall properties are similar to those of its components. Failure of some of the components does not then necessarily cause failure of the whole netavork. This was first studied by Moore and Shannon a~ for nets of relay contacts, and they considered the probability of the net being closed in terms of the probability of the individual contacts being closed. It does not seem to be generally realized that this can be extended to determine the probability of failure of any network of passive elements for given probabilities of open and shortcircuit failure of its elements. MOORE AND SHANNON FORMULAE FOR RELAY NETS We consider the networks shown in Figs. 1-3, first as Moore and Shannon did, as networks of relay contacts, each contact having an independent probability p of being closed, and work out the probability of the network being closed. T h e probability of the first two networks being closed can easily be shown to be (see Appendix): hi(p) = 1--(I--p-°)2 = 2p2--p ~ h2(p) = [1--(1--p)212 ___ 4p2_4p3+p4. (Subscripts refer to networks 1, 2, 3.) The probability function for network 3 can be calculated from these by using a useful and littleused method suggested by Moore and Shannon.
Consider the contact d in Fig. 3, and calculate the probability function for the network with this contact replaced by an open circuit, hi(p) of network 1, and with the contact replaced by a short circuit, ho_(p) of network 2. The probability of this particular contact, X, being open is l - - p , and of being closed p. The overall probability function is then clearly, h3(p) = ( l - - p ) ht(p)+p.h.,.(p ). Substituting, = 2p"+2pa--5p4+2p 5. OPEN AND SHORT CIRCUIT FAILURE We now consider networks of other passive elements which either (a) work normally, (b) have probability R of being open circuit, (c) have probability S of being short circuit. For any individual element, S is the probability of short circuit and 1-S the probability of "not short circuit", i.e. either normal or open. We can interpret the h(p) formula in terms of "short circuit" and "not short circuit" instead of "closed" and "open ", and thus h(S) gives the probability of the net being short circuit. Similarly R is the probability of an element being "open circuit" and 1-R is the probability of it being "not open circuit". Thus h(1--R) is the probability, of the net being "not open circuit" and 1--h(1--R) the probability of it being "open circuit". The probabilities of the net being open and short circuit are independent because h(S) is the probability that a sub-set of elements will cause the network to be a short circuit regardless of what happens to the other elements. Clearly no other sub-set can cause it to be open. T h e probability of 2O5
206
D. V. B L A K E
failure is then the sum of the probabilities of open and short circuits. For net~vork 1, we have: Pshort
= lh(S) = 2S2--S ~
Popen
= 1--ht(1--R)=
X
because network 2 is the dual of 1
=
4R2--4R3+R
ho.(R)
)(
X
)(
)(
1
[
~
FIG. 1.
Ptallure -~- 2 S 2 + 4 R 2, if R and S are small. Similarly, for network 2: Pfailure -"- 4S~'+2R% For network 3:
Pshort
= h~(S) = 2 S 2 + 2 S 3 - - 5 S ~ + 2 S
Popen
= 1--hz(1--R ) = h : ( R ) since the net-
s
The The The is The
probability of one side being closed is p2. probability of one side being open is l--p2. probabiliw of both sides, i.e. the net, being open (l--p2) ~, probabili~' of the net being closed h~(p) =
1 --(1 ---p "~):.
work is self-dual =
Ptallure
2R2+2R3--5R4+2R~
-~- 2 S 2 + 2 R 2.
Thus network 3 is always better than either of the other two, but uses five elements instead of four; if the probability of an open circuit is small compared with that for a short circuit one should use circuit 1, if the probability of a short is small one should use circuit 2.
\/
CONCLUSION
W
Providing the probability of a network being closed can be found, it is easy to calculate the probability of failure of the network for any combination of open and short circuit failures of its elements. T h e alternative of considering all possible states of the network is very laborious, e.g. even with a simple network such as Fig. 3, there are 3 '~ = 243 possible states.
-
\I
T
FIG. 2. The The The The
probability of a contact being open is 1--p. probability of one end being open is (l--p)2. probability of one end being dosed is 1--(1--p)L probability of both ends, i.e. the net, being closed,
h~(p) = [1--O--p)']".
Acknowledgment--The work described above was carried out in the Applied Physics Division of the National Physical Laboratory on behalf of the Ministry of Aviation. This paper is published by permission of the Director of the Laboratory and with the approval of the Ministry of Aviation. REFERENCE 1. E. F. MOORE and C. E. SHANNO.X'. Reliable circuits using less reliable relays. Part 1: J. Franklin Inst. 262, 191 (1956); Part 2: J. Franklin Inst. 262, 281 (1956). APPENDIX The probability, h(p), of the nets of Figs. 1 and 2 being closed when the independent probabilities of each contact being closed is p.
X
FIG 3
OPEN AND SHORT CIRCUIT FAILURE OF HAMMOCK NETWORKS D. V. B L A K E
Ship Division, National Physical Laborato~', Faggs Road, Feltham, Middlese× Abstract--An extension of 3,ioore and Shannon's work on nets of relay contacts is used to provide a rapid method of assessing the probabili~" of failure of a hammock net when the elements are liable to open and short-circuit failure. INTRODUCTION A COM.~ION method of using redundancy to improve reliability is to use a number of component s arranged as a "hammock" network whose overall properties are similar to those of its components. Failure of some of the components does not then necessarily cause failure of the whole netavork. This was first studied by Moore and Shannon a~ for nets of relay contacts, and they considered the probability of the net being closed in terms of the probability of the individual contacts being closed. It does not seem to be generally realized that this can be extended to determine the probability of failure of any network of passive elements for given probabilities of open and shortcircuit failure of its elements. MOORE AND SHANNON FORMULAE FOR RELAY NETS We consider the networks shown in Figs. 1-3, first as Moore and Shannon did, as networks of relay contacts, each contact having an independent probability p of being closed, and work out the probability of the network being closed. T h e probability of the first two networks being closed can easily be shown to be (see Appendix): hi(p) = 1--(I--p-°)2 = 2p2--p ~ h2(p) = [1--(1--p)212 ___ 4p2_4p3+p4. (Subscripts refer to networks 1, 2, 3.) The probability function for network 3 can be calculated from these by using a useful and littleused method suggested by Moore and Shannon.
Consider the contact d in Fig. 3, and calculate the probability function for the network with this contact replaced by an open circuit, hi(p) of network 1, and with the contact replaced by a short circuit, ho_(p) of network 2. The probability of this particular contact, X, being open is l - - p , and of being closed p. The overall probability function is then clearly, h3(p) = ( l - - p ) ht(p)+p.h.,.(p ). Substituting, = 2p"+2pa--5p4+2p 5. OPEN AND SHORT CIRCUIT FAILURE We now consider networks of other passive elements which either (a) work normally, (b) have probability R of being open circuit, (c) have probability S of being short circuit. For any individual element, S is the probability of short circuit and 1-S the probability of "not short circuit", i.e. either normal or open. We can interpret the h(p) formula in terms of "short circuit" and "not short circuit" instead of "closed" and "open ", and thus h(S) gives the probability of the net being short circuit. Similarly R is the probability of an element being "open circuit" and 1-R is the probability of it being "not open circuit". Thus h(1--R) is the probability, of the net being "not open circuit" and 1--h(1--R) the probability of it being "open circuit". The probabilities of the net being open and short circuit are independent because h(S) is the probability that a sub-set of elements will cause the network to be a short circuit regardless of what happens to the other elements. Clearly no other sub-set can cause it to be open. T h e probability of 2O5
206
D. V. B L A K E
failure is then the sum of the probabilities of open and short circuits. For net~vork 1, we have: Pshort
= lh(S) = 2S2--S ~
Popen
= 1--ht(1--R)=
X
because network 2 is the dual of 1
=
4R2--4R3+R
ho.(R)
)(
X
)(
)(
1
[
~
FIG. 1.
Ptallure -~- 2 S 2 + 4 R 2, if R and S are small. Similarly, for network 2: Pfailure -"- 4S~'+2R% For network 3:
Pshort
= h~(S) = 2 S 2 + 2 S 3 - - 5 S ~ + 2 S
Popen
= 1--hz(1--R ) = h : ( R ) since the net-
s
The The The is The
probability of one side being closed is p2. probability of one side being open is l--p2. probabiliw of both sides, i.e. the net, being open (l--p2) ~, probabili~' of the net being closed h~(p) =
1 --(1 ---p "~):.
work is self-dual =
Ptallure
2R2+2R3--5R4+2R~
-~- 2 S 2 + 2 R 2.
Thus network 3 is always better than either of the other two, but uses five elements instead of four; if the probability of an open circuit is small compared with that for a short circuit one should use circuit 1, if the probability of a short is small one should use circuit 2.
\/
CONCLUSION
W
Providing the probability of a network being closed can be found, it is easy to calculate the probability of failure of the network for any combination of open and short circuit failures of its elements. T h e alternative of considering all possible states of the network is very laborious, e.g. even with a simple network such as Fig. 3, there are 3 '~ = 243 possible states.
-
\I
T
FIG. 2. The The The The
probability of a contact being open is 1--p. probability of one end being open is (l--p)2. probability of one end being dosed is 1--(1--p)L probability of both ends, i.e. the net, being closed,
h~(p) = [1--O--p)']".
Acknowledgment--The work described above was carried out in the Applied Physics Division of the National Physical Laboratory on behalf of the Ministry of Aviation. This paper is published by permission of the Director of the Laboratory and with the approval of the Ministry of Aviation. REFERENCE 1. E. F. MOORE and C. E. SHANNO.X'. Reliable circuits using less reliable relays. Part 1: J. Franklin Inst. 262, 191 (1956); Part 2: J. Franklin Inst. 262, 281 (1956). APPENDIX The probability, h(p), of the nets of Figs. 1 and 2 being closed when the independent probabilities of each contact being closed is p.
X
FIG 3