- Email: [email protected]
A new model of software reliability prediction
Microelectron. Reliab., Vol. 23, No, 6, pp. 1009-1010, 1983. Printed in Great Britain.
A NEW MODEL
OF SOFTWARE
0026-2716/8353.00+ .00 © 1983 Pergamon Press Ltd.
RELIABILITY PREDICTION
K. K. GOVlL Electrical Engineering Department, C.T.A.E., M.L.S. University, Udaipur (Raj), India 313001
(Receivedfor publication24 March1983) Abstract--To start with, a new generalized structure of basic equations of software reliability has been defined. In this new structure, the software reliability has been defined in terms of execution time T and operational time t. Then, a new model of software reliability has been proposed and fitted into the new structure. l. I N T R O D U C T I O N
Recent developments in software reliability theory [1, 2] define the reliability of a program as "the probability that no execution failures occur in a sequence of say m executions". In the past many software reliability models have been presented and tested. A widely accepted theory of software reliability at present is that of Musa [3, 4]. The model is based on execution time (CPU). The validity of execution time theory has been discussed in detail by Musa [4]. It is necessary for software reliability modelling to talk about two different times as follows: execution time T, which counts from the beginning of test till the present; operational time interval t, which represents execution time projected from the present into the future on the basis that no further correction is performed. What is of interest is the reliability R(T, t), i.e. after execution time T and for the subsequent operational time interval t. 2. T H E N E W S T R U C T U R E O F S O F T W A R E RELIABILITY E Q U A T I O N S
The basic equations which can be used for all the models are suggested as follows: dN(T)
Z(T,t)= dT .g(t)
(1)
R(T,t)=exp(-flZ(T,t)dt )
(2,
MTTF(T) = Io
R(T, t)dt.
(3)
Although not necessary, sometimes a quantity To is defined, which is M T T F at the start of testing, i.e. mean time to first failure. In other words T O = MTTF(0) = I ~° R(0, t)dt. Jo
(4)
3. A NEW MODEL OF SOFTWARE RELIABILITY
The structure of basic equations proposed in the previous section is a generalized one. In this section we define a new model of software reliability and fit it into the new structure, For this new model we have
In this section we define a new structure of software reliability equations. Through this structure it becomes possible to incorporate execution time as a generalized oft) = 1 (5) parameter of software reliability modelling rather and than just a part of Musa's model. Let us define the following, dN(T) K Z(T,t)= d T = 1+aT" (6) T execution time, which counts from the beginning of test till the present t operation time (interval), which represents execution Performing the integration, time projected from the present into the future on the basis that no further correction is performed N(T)=(K)In(I+aT) (7) Z(T, t) failure occurrence rate as a function of execution time T and operational time t N(T) s-expected value of the net number of faults detected and and corrected during execution time T (note that N(O) = O) N(oo) s-expected value of total number of faults initially 1+aT present in the software package (this is nothing but MTTF(T) = ~ (9) the limiting value of N(T) as T --. oo ; one generally expects N ( ~ ) to be finite) and R(T,t) reliability of the software package after execution 1 T° K (10) time T and for an operational time t O(t) a function of operational time t (in some models it is equal to 1). Equation (8) predicts that reliability increases as 1009
1010
K.K. GovlL
execution time increases, but exponentially decreases with o p e r a t i o n time between any two successive errors. L e t f ( T , t) d e n o t e the probability density function as f(r,t)
t
-
l+aT
exp
-
. (11)
To conclude, a new generalized structure of basic equations of software reliability has been defined. Also, a new model of software reliability is proposed and the equations fitted in the new structure.
REFERENCES
1. E. C. Nelson, A statistical basis for software reliability assessment, TRW-ss-73-03. 2. T. Thayer, M. Lipow and E. C. Nelson, Software reliability study, RADC-TR-76-238 (also available as TR-ss76-O3). 3. J. D. Musa, A theory of software reliability and its applications, IEEE Trans. Sc?J~ware Engng SE-I. 312 327 (1975). 4. J. D. Musa, Validity of execution time theory of software reliability, IEEE Trans. Reliab. R-28, 181-189 (1979).
A NEW MODEL
OF SOFTWARE
0026-2716/8353.00+ .00 © 1983 Pergamon Press Ltd.
RELIABILITY PREDICTION
K. K. GOVlL Electrical Engineering Department, C.T.A.E., M.L.S. University, Udaipur (Raj), India 313001
(Receivedfor publication24 March1983) Abstract--To start with, a new generalized structure of basic equations of software reliability has been defined. In this new structure, the software reliability has been defined in terms of execution time T and operational time t. Then, a new model of software reliability has been proposed and fitted into the new structure. l. I N T R O D U C T I O N
Recent developments in software reliability theory [1, 2] define the reliability of a program as "the probability that no execution failures occur in a sequence of say m executions". In the past many software reliability models have been presented and tested. A widely accepted theory of software reliability at present is that of Musa [3, 4]. The model is based on execution time (CPU). The validity of execution time theory has been discussed in detail by Musa [4]. It is necessary for software reliability modelling to talk about two different times as follows: execution time T, which counts from the beginning of test till the present; operational time interval t, which represents execution time projected from the present into the future on the basis that no further correction is performed. What is of interest is the reliability R(T, t), i.e. after execution time T and for the subsequent operational time interval t. 2. T H E N E W S T R U C T U R E O F S O F T W A R E RELIABILITY E Q U A T I O N S
The basic equations which can be used for all the models are suggested as follows: dN(T)
Z(T,t)= dT .g(t)
(1)
R(T,t)=exp(-flZ(T,t)dt )
(2,
MTTF(T) = Io
R(T, t)dt.
(3)
Although not necessary, sometimes a quantity To is defined, which is M T T F at the start of testing, i.e. mean time to first failure. In other words T O = MTTF(0) = I ~° R(0, t)dt. Jo
(4)
3. A NEW MODEL OF SOFTWARE RELIABILITY
The structure of basic equations proposed in the previous section is a generalized one. In this section we define a new model of software reliability and fit it into the new structure, For this new model we have
In this section we define a new structure of software reliability equations. Through this structure it becomes possible to incorporate execution time as a generalized oft) = 1 (5) parameter of software reliability modelling rather and than just a part of Musa's model. Let us define the following, dN(T) K Z(T,t)= d T = 1+aT" (6) T execution time, which counts from the beginning of test till the present t operation time (interval), which represents execution Performing the integration, time projected from the present into the future on the basis that no further correction is performed N(T)=(K)In(I+aT) (7) Z(T, t) failure occurrence rate as a function of execution time T and operational time t N(T) s-expected value of the net number of faults detected and and corrected during execution time T (note that N(O) = O) N(oo) s-expected value of total number of faults initially 1+aT present in the software package (this is nothing but MTTF(T) = ~ (9) the limiting value of N(T) as T --. oo ; one generally expects N ( ~ ) to be finite) and R(T,t) reliability of the software package after execution 1 T° K (10) time T and for an operational time t O(t) a function of operational time t (in some models it is equal to 1). Equation (8) predicts that reliability increases as 1009
1010
K.K. GovlL
execution time increases, but exponentially decreases with o p e r a t i o n time between any two successive errors. L e t f ( T , t) d e n o t e the probability density function as f(r,t)
t
-
l+aT
exp
-
. (11)
To conclude, a new generalized structure of basic equations of software reliability has been defined. Also, a new model of software reliability is proposed and the equations fitted in the new structure.
REFERENCES
1. E. C. Nelson, A statistical basis for software reliability assessment, TRW-ss-73-03. 2. T. Thayer, M. Lipow and E. C. Nelson, Software reliability study, RADC-TR-76-238 (also available as TR-ss76-O3). 3. J. D. Musa, A theory of software reliability and its applications, IEEE Trans. Sc?J~ware Engng SE-I. 312 327 (1975). 4. J. D. Musa, Validity of execution time theory of software reliability, IEEE Trans. Reliab. R-28, 181-189 (1979).