Reliability schemes in digital systems with asymmetrical failure mode

Meisel, W. S.

Schaeffer, P. C.

English

NASA Technical Reports Server

, 
8 .  
? . *  . c. r b  
* ,  * .  
i 
,* -s 
1 -  
1.  * 
, -  
4 
. r  . - .  - :. 
RELIABILITY IN DIGITAL SYSTEMS WITH ASYMMETRICAL 
FAILURE MODES : ' 
\ 
W. S .  Mkisel 
. - P. C. Schaeffer 
. .  
Electrical Engineering Department: 
Los Angeles, California 9Q007 
c 
-\ University of Southern Caliiornia 
I 
.. 
J f  
-. . 
This work was supported in par t  by the Ns t jona l  Acronautjcs 
. and Spa-ce Adniinistratjon under grant no. NG&05-018-044, 
supp. 1 .  , 
(ACCESSION NUMBER) 
. -  
https://ntrs.nasa.gov/search.jsp?R=19680020896 2020-03-12T10:19:07+00:00Z
. _  . .  
. .  
*. * 
I . RELIABILITY IN DIGITAL SYSTE.MS WITH -ASYMMETRICAL 
. .FAILURE MODES . . . . 
4 
,. 
W. S .  -Meisel  and P. C. Schaeffer 
> . I  
.ABSTRACT ? %  . -  
* 
- Most.present day r\eliability schemes using redundancy to mask  the 
. .  
failure of individual logic modules employ majority voting with the assump- 
d tion that the replicated modules have symmetr ical  failure charac ter is  tick. 
This paper presents an analysis, of such schemes'  when the modules exhibit 
I 
  
. 
- asymmetr ical  failure modes; that is, the probability that a module fails 
with a. 0 output is not equal to the probability tha,t i t  €ails with a 1 output. 
A 
._ _. -
Fail-sa-fe logic systems (discussed by other authors) fall in this category. 
A general  expression is presented which gives the reliability of a network 
consisting of n identical modules feeding a k-out-of-n voter. 
that a simple majority element does not always represent  the optimal 
choice. 
3 
It is shown 
Plots for selecting the optimal k for  n = 3 , 4 ,  5 when the individual 
"module reliability parameters  a r e  known a r e  pres  ented. Also included are  
graphs of network reliability in te rms  of individual module' reliability and 
the degree of asymmetry.  
. .  
- .  
i 
. .  
* .  . .  
. -  . .  
RELIABILITY I N  DIGITAL SYSTEMS WITH ASYMMETRICAL 
FAILURE M.ODES 
4 
W. S. .  Meisel and P. C. Schaeffer 
IN TR OD UC T ION 
Most reliability schemes for digital systems a s sume  that a given 
1 
module (gate, flip-flop, sub-sys tem) has an  equal probability of faiiing 
with output 0 as with output 1. 
schemes (deriving f rom Von Neumann [ 3 3 ) make such an  assumption. .* 
In particular,  some majority voting 
! 
The assumption is not a lways  valid; Mann has discussed: this point [: 1'3. 
. ' Fu r the r ,  one may intentionally design circui ts  to fail  a lmos t  
1) 
certailily in either the 0 or 1 state. 
logical syi tems [ 2 1. 
These have been termed"'fail-safe" 
This paper considers s y s t e n k  such as that i l lustrated in  figure 1 * 
. .  
where  the modules a r e  identical'. If there a r e  no failures,  the logic is 
and is 's0 chosen that a t  l eas t  one-knodule failure may be t'olerated. With 
. .  
symmetr ical  modnl e h i l u r e s  a the' best  choice is  a majority element, a 
r 
logic element whose output. is equal to the majority of i ts  inputs, Assuming ' . 
., the majority element and modules have sufficicntly high reliability, the 
a 
composite sys tern will bc m o r e  reliable than a single module. - 
: If the modulcs  have asymmetr ical  probabiljlics of failure,  i t  is 
not obvious what logic eleineat is besl. -For example, in thc exti-cmc case  
. -  . 
. .  . .  
clioice for the logic element than a majority gate. 
is concerned with the problems of choosing the optimal logic element given 
the failure probabilities of the modules and of deriving the system reliabil- 
The present  paper 
- -  ' . .  ity. .. . 
ANALYSIS - 
We assume that a module wi l l  . .  fail: stuck a t  0 or 1 and that the 
. _  
i 
.* . 
Ifollowing module parameters  a r e  available: 
._ . . - - . __  -. . . . . - -. _. - - - ", 
= the probability that the module will fail and 
. _ .  . -  ! -- fa i l  a t  0, . - . . - . - - - . Po 
P1 
. -. -_ - 
= the probability that the module w i l l  €ail and 
fail a t  1. 
3 
If a given collection of modules are tested for a given period of time, some 
wi l l  fa i l  a t  1 ,~sG~ne 'Wi l l~ fa i l  a t -0 ,  and some wil l  not fail  a t  all. . . -.o- "-.<--.. In these 
_ -  . t e rms  a 
number of modules failed as 0 
-. . number tested 
- - --__ - . _. - ._.. - - _ _  _ _ _ _  ~ _ _  - - - 
-2 
number modules failed a s  1 .: -_  pl--= -- 
number tested 
D 
I 
L 
Not all logical functions need be considered. Since the modules a r e  
identical, only symmetr ic  functions need be considered. 
output is to be identical with the moclul e outpnls, only positive functions 
need be considered. 
ti ons, only those of the iollon~ing type wi l l  bc  consjdcycd: 
Since the system 
F~urLIier, out of the class of posit ive syniinctric func- 
0 
.. .._ _ _  ~ . .... . . .  
.< 
. . .  . .  
. .  
.. 
* .  
. .  
' .. . . .  
. .  . .  
. .  
1 i f  k or more  of e n inputs are 1 
, 0 if not = .  { (1') 'k/n 
' T h e  function S . can be realized by a single threshold element with unity 
. .  k/n 
is an n-input OR gate, and S is an n- 
'1 I n  n l n  . 
weights and a threshold k. 
input AND gate. 
' .  . .  
The reliability of the output gate wil l  be considered unity for  the 
! 
sake of the computations which follow. This is a ma t t e r  of convenience' f 
. .  
1 since the reliability of this gate  enters  as a multiplicative factor and may 
be taken into account a t  a la te r  point. ' S  
The system w i l l  be considered to have failed when i t  no longer 
follo_ivs the behavior of the working modules, Then the system will fail' 
when k or  m o r e  modules fail as a 1 or n - k t l  o r  m o r e  modules fail  a s  a 0. 
Since these two events a r e  disjoint, we have for  a logic elemn'ent realizing 
'k/n . . 
. .  
. whcre 
. .  
P = probability of system failure . 
k/n 
. .  
. .  
. .  
~ . .. 
. .  
. '  . 
Pk/li* 
r l -  
v k l n  
The rcliabi3iLy of tlic systcni is 
-3  - 
.. 
. .  
\ 
Figures  2-8 a r e  plots of V through n’ =5 in the parameters  
k/n . 
. .  \ 
P = reliability of each module 
= 1 - Po- P1 
! 
i’ u = probability of fkiling a5  a zero  given a 
. fa i lure  has occurred 
’J PO 
. -  -  
. .  P p P *  
A 
By applying equation (31, the missing plots may be read off the existing 
may be read from the plot for  V by interpreting 
v3 / 3  1 / 3  
plot$; for  example, 
the horizontal axis as 
. .  
p1 . ’  
P I +  Po 
a1 = . 
These curves al low rapid estimation of the system reliability. 
The curves of figures 2-8 may be used to’determine when the system 
. -  
becomes l e s s  reliable than’a single module. Since P represen’ts the module 
I 
‘I reliability, the intercept of the c u r v e  for P = x with the horizontal !jne 
= x yields the value of a for which the system breaks down. If the . Vk/n 
reliability of the logic element S 
valuc @ ,  the curves of figures 2-8 a r e  correct  i f  the values on the vertical 
sca le  a r c  multipljecl by f.l . 
is not assunicd unity but takes sonic 
. k / n  
Given 11 133 odulcs and tlic ass ociatcd p and 1) which func ti on S 0 I ’  . k / n  
. -  . .  ._ . 
. _  
. -  
. .  
. .  - .  
. ,  
'1 
i '  :_ 
' . is optimal? F igures  .9:11 answer this'question for n = 3,4, 5. These * 
- . curves were  obtained by' numerical 'solution (on a general-purpose digital - - .  
. , .  . .  . 
: .  . 
e .  
< . REFERENCES 
c . .  
[ 1) Mann, W. C. "Restorative Processes  f o r  Redundant Computing 
Systems, " I n  Redundancy Techniques for  Computing Sys tenis, Wilcox 
and Mann (eds. ), Spartan Books, 1962. 
Mine, H. , and Y. Koga, "Basic Propert ies  and a Construction Method 
for a Fail-safe Logical System, I f  IEEE Trans.  on Electronic Compu- 
- ters, Volume EC-16, No. 3 (June 1967). 
Von Neumann, 'xProbablis tic Logics and the Synthesis of Reliable 
Or-ganisrns from Unreliable Components, '' Automata. Studies (edited 
[ 2 1  
\ -.- - .  
[ 3 )  
by C. E. Shannon and J. McCarthy), Princeton University - Press, 
1956. 4 . .  
. +  
4 
- - -  . . - . ?  
- .. - d 
.1. 
- _ _ - -  ..-. - - -  . - -  . 
. .  
3 
.-. 
_-I  ----- - --.- . - - ~ . -  -.-  . -. ----- - - -  - -  .- 
. .  
. _ I _ - -  _. 
, .  - ,  
I 
, & .  
. . _. - . . . .  - - _  - --. - .  
. .  
. .  
_ .  . '  
. .  . .  
- .  I . . .  
... . . _ _ _ _  .. - .. - ~.~ .- I.__. . -- . ,. 
. .  
. -  - .  . .  .. . 
. .  . .  
. .  
- .  
. .  
. . - .  . .  . .  
. . .  
. .  v vs' a . .  . b.3 
i. . 
.. 
1.0 
. .  



.. 
- .  . .  
, 
4 
. .  . . . . . . .  ... . . . . . . . .  
. . .  . .  . . . . . .  
c .  
. .  . . .  . .  
. .  . .  . .  
1 .  ' 
- .  . 
. . _  . .  . .  
4 
. .  
. . .  . . . .  . - .  1 . . .  
. .  
- .  
. .  
. . .  
I-- - - . . . .  .I ..... --', .. 
. .  .. 
. . . .  . . . .  .. 
.. 
. . .  ... .. . . . . . . . . . . . . . . . . . .  
-. . 
.. -. . 
. .. 
. .  
. . .  . .  
i' 
1 .  
- .  0.3 . .  . .  
3 
5 
. .  
. .. 
.. * .  
. .  . .  . '  
I 
r . *  . . .  
\ - -  _ -  . . 0 I .  
. .  
4 
.. 
- . -  
* FIVE INPUT CASE 
. * Po, Pi TXRESHOLD: BOUNDARIES . 0.6 I I f 1 
.. 
. . . - I  - - - -  
. .  
0.2 
... . -_ 
' 0.1 
0 
, *  
. .  , 
. .- 
. .  
A 
v =v 
*% 
.. . .. . 
I. 
. < -  5 . 
. .  
.. - . 
- .  
. .  . .  . .  
. -. . 
. . .. _._. .. - -*_c-- ..-..
.. . . -4  . . . - .  - .  
. .  . .  - . -. .. . 
. .  
. .  
. - -  i. _ _  
- ---___.______-_.-AI- 1 
:. I 
- .  
.. 
. .  
. .  
.. 
\ .. 
I '  
. .  . .  
. . .  . . .  
. . . .  
. .  
. . .  . .  
~ ,002 
- . - ._^__ - 
. .  
. . . . . - . . - .___..._. 
-002 .00 4 .o Q.6 .001 .OI 0 
. .  
. .  
. .  
. .  
r l  
i . _  
. .  -. 
. .  . - .  
... - 
- .  
. 
. .  
.. . 
+ 
. ~. 
. .  
* .  
. ' r  
. 


Reliability in digital systems with asymmetrical failure modes

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server