Technique based upon matrix representation and matrix collapsing calculates the probability of successfully completing manned missions and of returning the spacecrew safely to earth. This technique provides analytic expressions for each subsystem, making it possible to relate changes in subsystem reliability directly to mission success and crew safety

Austin, W. L.

English

NASA Technical Reports Server

December 1969	 Brief 69-10793 
NASA TECH BRIEF 
NASA Tech Briefs are issued to summarize specific innovations derived from the U.S. space program, 
to encourage their commercial application. Copies are available to the public at 15 cents each 
from the Clearinghouse for Federal Scientific and Technical Information-,Springfield, Virginia 22151. 
Estimating Reliability by Application of Matrix Representation 
In estimating the reliability of a manned spacecraft 
in performing a predetermined mission there are two 
probabilities to calculate: (1) the probability of mis-
sion success—the probability of completing the mission 
objectives with no loss of life, and (2) the probability 
of crew safety—the probability of returning the crew 
safely to earth with or without having accomplished 
the objectives of the mission. The large number of 
factors that must be taken into consideration when 
estimating these probabilities, such as the changing 
hardware configuration and the large number of 
components in a manned space vehicle, make compu-
tations complicated. A new and simplifying technique 
for performing this calculation has been developed, 
based upon matrix representation and matrix collaps-
ing. This technique provides analytic expressions of 
mission success and crew safety for each subsystem, 
making it possible to relate changes in subsystem 
reliability directly to mission success and crew safety. 
The reason for the development of the concepts of 
matrix representation and matrix collapsing is that 
ordinary Boolean techniques are quite cumbersome 
in cases where the configuration of a complicated 
subsystem is permitted to change with respect to time. 
These configuration changes may be due to changing 
functional or reliability requirements on the subsys-
tem. 
The functional and reliability requirements of any 
given subsystem, "i", for any phase, K, are repre-
sented in matrix form in order to facilitate deriving an 
analytic reliability function for the subsystem which 
is valid from time zero to the end of that particular 
phase. Even though there are three mutually exclu-
sive states to consider—mission success, abort enabling 
failure, and fatal failure—the only two matrices which 
need to be determined are the success matrix which 
contains all of the distinct success paths through the 
subsystem, and the abort enabling failure matrix 
which contains all of the distinct abort enabling failure 
paths through the subsystem. The probability of a fatal 
failure at the end of any phase K, is the probability of 
success at the end of phase K-I minus the probability 
of crew safety to the end of phase K, with the total 
probability of a fatal failure being the sum of all the 
phase fatal-failure probabilities. The problem of 
determining the unconditional success matrix can be 
solved by utilizing the concept of matrix collapsing. 
Given two success matrices, A and B, where A is 
valid for the first mission phase and B is valid for the 
next mission phase in which the subsystem occurs, 
every row of A is intersected with every row of B and 
all redundant paths are eliminated. This process is 
continued for all subsequent mission phases in which 
the subsystem occurs; this is the same asintersecting 
the Boolean phase models, but is a simpler way of 
doing it. The abort enabling failure ground rules are 
applied to the unconditional success matrix to deter-
mine the unconditional abort enabling failure matrix. 
In order to include the aborts, the unconditional suc-
cess matrix and the unconditional abort, enabling 
failure matrix, are collapsed with the subsystem func-
tional and reliability requirements abort matrix. Once 
the unconditional matrices are determined, a relia-
bility measure may be applied to them and the neces-
sary functions are then derived in order to determine 
the probabilities of mission success and crew safety. 
(continued overleaf) 
Res 
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Aeronautics and Space Administration. Neither the Unitel States
	 information contained in this document, or warrants that such use 
Government nor any person acting on behalf of the United States
	 will be free from privately owned rights.
https://ntrs.nasa.gov/search.jsp?R=19690000664 2020-03-16T18:33:53+00:00Z
Note: 
Requests for further information may be directed to 
Technology Utilization Officer 
Headquarters 
National Aeronautics 
and Space Administration 
Washington, D.C. 20546 
Reference: TSP69-10793
Patent status: 
No patent action is contemplated by NASA. 
Source: W. L. Austin of 
General Electric Company 
under contract to 
NASA Headquarters 
(HQN- 10246) 
Brief 69-10793	 Category 02


Estimating reliability by application of matrix representation

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