- Email: [email protected]
The use of Burr type XII distribution on software reliability growth modelling
Microelectron.
R.&b.,
Vol. 37, No. 2, pp. 305-313,
1997
Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/97 $17.00 + .OO
Pergamon 00262714(95)00124-7
THE USE OF BURR TYPE XII DISTRIBUTION ON SOFTWARE RELIABILITY GROWTH MODELLING A. A. ABDEL-GHALY, Department
of Statistics,
Faculty
G. R. AL-DAYIAN
and F. H. AL-KASHKARI
of Science, King Abdulaziz University, Kingdom of Saudi Arabia (Receiwdfoor
publication
16 June
P.O.
Box
9028, Jeddah
21413,
1995)
Abstract-In this paper a new reliability growth model in which the operating time between successive failures is a continuous random variable is proposed. This model is based on Burr type XII distribution with one Parameter which is discussed in three versions: the order statistics, Bayesian and nonhomogeneous Poisson process. Several software reliability measures are obtained. The performance of all three versions of the suggested model is tested on two data sets. The versions perform with variable accuracy, which suggests that no universal “best” among the three versions of the model could be attained. Copyright ‘0 1996 Elsevier Science Ltd.
1. INTRODUCTION
There are several software reliability growth models in the literature that estimate the remaining error contained in the software. These models may be classified in two broad types. (1) Inter-failure time models in which the operating time between successive failures is a continuous random variable. (2) Discrete software reliability models where the interest is the number of faults or failures in a specified time interval.
errors detected by software testing, such as Duane model [3], non-homogeneous Poisson process for Jelinski-Moranda model [9] and non-homogeneous Poisson process for Littlewood model [3]. In 1991, Rayleigh model was introduced by AbdelGhaly [4]. He assumed that the system starts its life containing N faults. Each of these faults can cause system failure at any time T. Let X be the total time on test until the fault i is removed. The Xs are assumed to be independent identically distributed random variables with Rayleigh density function, given by
The inter-failure time models are our concern here. These models are roughly classified into the following three categories: 1.1. Order statistics models
otherwise.
(1)
Let Xcl,, Xc2,,. . . be order statistics of the time of occurrence of the lst, 2nd,. . . failures and T1, T,, . . . , be the inter-failure times. Then
In this category the failure times of a software reliability process are modeled as Order statistics of independent non-identically distributed random variablessuch as:Jelinski-Moranda model [1], Littlewood model [2], Weibull model [3] and Rayleigh model [4].
TI = -%I,, T
1.2. Bayesian model The parameters of the order statistics model are assumed to be random variables with a given prior density function. Different techniques are adapted and lead to different versions of the same model. These models are: Bayesian Jelinski-Moranda model [S], Bayesian Littlewood model [3], Littlewood-Verrall model [6], Keiller-Littlewood model [7] and Bayesian-Weibull model [S]. 1.3. Non-homogeneous
x > 0, (4 > 0)
=
x(i)
-
x(i-
I),
i =
1,2,. .
(2)
If a total time ‘c has elapsed and (i - 1) faults have been removed, then the probability density function of q (the inter-failure time needed for fault i to show itself) will be given in the following form f(fi 1A,)= 24(q-
1+ ti) exp[ -li((Ti-
1+ ti)'-Tfm
I)],
li > O, C3) where Ai = (N - i + 1)4
and ziml is the total elapsed time at the point (i - 1) = 1;:: tj. This result will be used in the new model which will be presented in this paper and it will be discussedin the three versions.
Poisson process
In this category, the testing time used as the unit of error detection period for describing the time dependent behavior of the cumulative number of 305
306
A. A. Abdel-Ghaly
et al
-
t
!
Failures
detected
and thus faults,
tied
in (0,Z)
X
t
next failure
(now) Fig.
2. THE
ORDER
STATISTICS
VERSION
(OSV)
In this version it is assumed that there are N software faults at the start of testing. Each of the N faults in the program will cause a failure after a time which is distributed as in (1) and independently of other faults, with a different failure rate q5j. When a failure occurs, there is an instantaneous removal of the faults which caused the failure, and no new faults are introduced during the debugging process. If total time 4-i has elapsed, and (i - 1) faults have been removed, the failure rate of the program is given by: N-it, ii = c#J1+. . . + t$N-i+l = (4) j:l $jT are the non-identical rates of where 41,...,4N-i+l the remaining faults. The 4s are regarded as realizations of random variables. Thus the occurrence of the N faults with which the program begins life can be regarded as i.i.d. random variables. When debugging starts (i.e. T = 0) each 4 has gamma distribution with shape parameter c(, i.e. 1‘(4 I4 = &
c!
FL
exp(-41,
4 > 0, (x > 0).
(5)
Bayes’ theorem can be used to describe how uncertainty about the occurrence rate of the faults (and hence the failure rate of the program) changes with time. Consider the occurrence rate, q5, of one of the N - i + 1 remaining faults at the epoch “now” in Fig. 1. f{4
1.
From (4), the failure rate of the program Ai, is a sum of the (N - i + 1) i.i.d. gamma distributed random variables with shape parameter (a(N - i + 1)) and scale parameter (1 + rf- i), with density function
x exp - (&(l + rf- ,)),
Ai > 0.
(7)
Multiplying (7) by (3), and integrating with respect to ii we get the p.d.f. of r given Km, as f(tilu,
N,zi+,)
= 2(N - i+ l)cc(ri-, + ti) (1 + r;ml)il(N-i+U x [I + (Tie1 + Qy-i+‘)+l
ti > 0. (8) This form can be obtained from the order statistics formulation of the model; if the detection times (and removal) of the fault are independent random variables, Xi, with density function given by f(xlu)
= 2ctx/[l + Xqa+l,
x > 0, (tl > Oh
(9)
which is a special case of Burr type XII distribution. The two parameters N and c( are estimated by maximizing the likelihood function to obtain their maximum likelihood estimates (MLEs) of N, CI. Such MLEs are used to predict the p.d.f., reliability function and its measures for the next failure time. The likelihood function based on the data of the first n failures is given
1this fault is not fixed in (0, li- ,)} = f{$
=
1no failure is caused by this fault in (0, ri- ,)}
(l+rf-i)“qY’exp(-f$(l+riz_,))
> 4>0.
Vu)
(6)
(10)
That is, q5 m gamma (c(, 1 + z:-~), where c-1
=
mexp(-$r’-,)r:l)$“-lexp(-~)dg s0
1 = (1 + t,Z_J’
The estimates &, fi of the parameters c(, N will be used to predict the p.d.f. of T,,, by substituting in (8)
fit,+1 Id = 2M - nw(~” + t,+ 1) (1 + ,y-nM x [l + (?” + t”+i)2](N~n)J+i’
and
t “+I > 0. (11)
Pr{no failure due to this fault in the interval 0 - T~-~} = exp(-@f-i).
Now, the user may be interested in obtaining the probability that the system will be running without
Burr type XII distribution failure up to time T. + i. The reliability function of T,, i is given by
where
(N-n)& &L+ 1lZ") =1+CT" +t,+1Y 1 1 + 7.’
The failure rate function of T,, i is
nUj-l,,-l
i= 1,2,...,n,
+Uj,n-17
[ 0,
j>n,
aO.n= 1, t/n and a,,, = 1.
(18)
By substituting from (15), it follows that the posterior density is given by
?I + L+1
r;(t,+IIT”) = 2(15 -n)&
aj.n=
(12)
[
307
1 + (% + t,+,Y
(13)
.I”*(& tiltI>. . >0 = C fi 2(i - (i - l)$)
3. BAYESIAN
VERSION
i=l
(BV)
As we have seen in the case of OSV, no explicit formulas can be obtained in estimating the parameters using the maximum likelihood method, and only numerical solutions may be obtained. So, to avoid this situation, the BV for the suggested model is proposed. In this model, Bayesian approach is used to get the probability density function of inter-failure time. We follow Littlewood and Sofer [S] in using their reparametrization technique applied to JM model. It is assumed that 1 = Nu, 4 = a, where 1 is taken to be the initial rate of occurrence of failures per unit time. Accordingly, the reparametrized density function corresponding to (8) is given by
X exp
-i$l
(A -
(i -
1M)CMl
+ (Tit1
+ ti)2)
i
- ln( 1 + z,Z_i)] f*(n, dltr,
I
, 1 > (n - l)f$
(19)
. . , t,) = zero otherwise,
where n-l c-1 = c Uj,“_l j=O (n
-j)!
’ (ln(1 + ~~))“-j+i(C~:~ ln(1 + rf))j+‘. (20) It can be shown, from (14) that the reliability function is given by expC-(i-n4)Cln(l
XeXp[-(I~-(i-l)~){lll(l+(7~-~+ti)2)
- ln(l + $,)}I,
ti > 0.
(14)
The reparametrized likelihood function corresponding to (10) is given by
w.,bit,, , t,) = fi [w
- (i - 114) l +‘;,’ +Jt,)’ 1 1
x exp[-(i
- (i - 1)4)[ln(l + (~~-i + ti)‘)
- ln(1 + T:- i)]]
1 1,
A,$>0
(16)
the posterior density function (which is known to be proportional to the product of the likelihood function and the prior) is then given by j*(& 4 1t,, . . . , t.) = t,)f(& $), L > (n - l)d and C is the CW,~lt,,..., normalizing constant. It can be shown that i; (x + i) = i i=O
=
- ln(l + ri)]];
1 > nq5 (21)
i 1; (n - l)f$ < A < n$. It may be observed that the current reliability immediately after occurrence of the nth failure (and removal of n faults) equals one (if (n - l)&~< 3, < nr$), i.e. that the program is now perfect and the last fault has been removed in the nth fix. The current predictive reliability function may be obtained from eqns (19) and (21), (see, for example [lo]), as follows
(15)
Assuming an improper non-informative uniform prior density of the form S(i4)=
m”+,I~>~)
+(~+t,+d*)
q”X”-i.
(17)
=
R(t,+,l1,~)f*(l,~lt,,...,t,)did~ ssti a m nQ = f*G, d~ltl>. . . >t,) d/i d4 ss0 O-lb+ m m + exp( -(A - nd)(ln(l + (z, + t,+ i)‘) ss0 nO - ln(1 + $)))f*(&
4lti , . . . , tn) dl d+.
Therefore
R*(r,+,It,,...,
t.)
=
C[A(t,+,)
+ c-l -B],
(22)
308
A. A. Abdel-Ghaly
et al.
where
and given by
Act,+ 11
P*(NIt,,
(n-j)! = jiYo ai.n (ln(l+(r,+t,+,)2))n~j+1(~~=l B=
i j=O
(l+zf))j+”
a. ~-__-. ..-(n - j)!j! “‘“(In(1 + 2,2))“~j+‘(C~=,
=
ln(1 + tf))j+l
/ t,) = 1 - CB.
4. NON-HOMOGENEOUS VERSION
POISSON
(N - n)! x exp[--pexp(-a$lln(l
and C is given in (20). If we let tn+l tend to infinite, we will obtain the predictive probability that the program is now perfect in the form: PO = R*(coIt,,.
, La)
(23)
+l(~~:,t,ti)z))], N > n, (u, p > 0).
The predictive p.d.f. of T.- 1 can be obtained eqns (11) and (26) as follows
from
f*(t, + L I i4 % L)
= f fitn+lIz,)P*(N-nlt,,...,t,,u),
PROCESS
(26)
t,,+l>O
N=”
(NHPPV)
Non-homogeneous Poisson process (NHPP) models are used to predict future failures. This approach is one of the best ways to measure software reliability and other measures. NHPP are also called counting models because the mean value function H(t) represents the cumulative number of faults exposed by time t. These models also have a failure rate (1) which is a function of time. The first model discussed in Section (2) is an OS model where N is the initial number of faults in the system (program). Suppose that x is of known value, and N has a Poisson distribution with parameter p and probability mass function
= 2u/l
G+t”+I (l+(r,+t,-,)*)“+’
xexp[~(1-(l+(~~~~+,,2~)l~
tn+l>o,(~d>oh
(27)
By integrating (27) with respect to tntl predictive c.d.f. of T,, , as follows
we get the
F*(t”+IIP,Td
(28) P(N) = $ ecu’,
N=0,1,2,...
(24)
Miller [I 1] suggested that any order statistics mixture model mixed over the Poisson IV, will be a non-homogeneous Poisson process. The posterior mass function of N can be obtained from eqns (10) and (24) as follows P*(N[t,,
. . . t,, cx)= CL(N, ait,,
, t,)P(N),
It follows, from eqn (28), that the predictive function is given by
R*(t”+l Ii4 co =exp
L -p-
1 (1 + 7;)
and the failure
.,t,)P(N)
then
5. THE
(25) It follows that the posterior distribution of the number of remaining faults is Poisson mass function,
(
rate function
JXt,+I) = 2v = f L(N,rlt,,.. N=”
1-
l+.r,2 ~-~ ( 1+(%+L+J2
il
))1 ’
(29)
where C-’
reliability
ANALYSIS
(hazard function) 7”
+
&+I
(1 + (7” + t,+ J2)n+ l. OF
PREDICTIVE
is (30)
QUALITY
The new software reliability growth model has been considered to measure and predict the reliability of software which has been studied in the three previous versions. The user usually needs to know which version would be most suitable in a particular context. Such comparison may be carried out with the assistance of many different statistical tools. Among these tools are the U-plot, Y-plot, measures of variability, the prequential likelihood ratio (PLR) [12]. The performance of all three versions of the new model can be achieved for real and generated data sets by using these statistical tools.
Burr type XII distribution
309
Table I. Execution times in s between successive failures, Musa (1981) 1121 (read from left to right in rows) 3 138 325 36 97 148 0 44 44s 724 30 729 7s
1045
30 50 55 4 263 21 232 129 296 2323 143 1897 482 648
113 77 242 0 452 233 330 810 1755 2930 108 447 5509 5485
81 24 68 8 255 134 365 290 1064 1461 0 386
115 108 422 227 197 357 1222 300 1783 843 3110 446
100 1160
1864
9 88 180 65 193 193 543 529 860 12 1247 122 1071 4116
10
91 120 1146
2 670
10 176
112 26 600 457 816 369 529 828 33 865 875 1082 6150
58 79 31 16 160 707 1800 700 948 790
6 236
10 281 983 261 943 990 371
15 114 15 300 1351 748 379
1011 868 143s 245 22 3321
No. of observations = 136. Starting sample size = 35. Table 2. The analysis of data in Table 1 Version
osv BV NHPPV
iZ / I - c/ 4.7906
Rank
& 11 - $I I 1
Rank
U-Plot
Rank
Y-Plot
Rank
2
2.1655
2
0.3298 1% 0.5341 1% 0.5303
1
0.1173
3
12.5039
3
2.1829
3
2.0889
1
2.0245
1
5.1. Data analysis and results The analysis has been carried out for two different kinds of sets of inter-failure time data. The first kind is a real data set but the second is an artificial one. The reason for using these two types of data sets is for comparison purposes and decisions regarding the version’s preference for the new model. The following method will be used to accomplish the calculations for both data sets: assuming that each data set contains n inter-failure times, m( < n - 1) is chosen as a starting sample size. Using the results of Section 4, the MLEs are obtained for the sample and then used to direct In p.d.f., c.d.f. and rat, for the next failure-time. The sample is then increased by the observed tm+l, the process repeated for T, + 2, and so on. 5.2. Real data set (Musa system 1) This data set [12] is shown in Table 1 which contains 136 observations of the execution time in seconds between successive failures. The sample size at the beginning of the analysis is 35 observations to estimate the models’ parameters and to predict the 36th failure. The process is repeated for the next observation and so on. Table 2 summarizes the results concerning the quality of performance of the three versions of the new model. Figures 2(a), 3(a) and 4(a) show the U-Plot for
1%
3 2
20% 0.0846 N.S. 0.0843
N.S.
2
1
-In PL 7.2367 834.613 7.2527 853.163 7.2311
834.17
Rank 2 3
1
OSV, BV and NHPPV, respectively. The Kolmogorov distances are 0.3298, 0.5341 and 0.5303, respectively. The Kolmogorov distances of the U-Plot of the three versions are significant at 0.01 level. OSV, BV and NHPPV plots are everywhere above the line of unity slope. That is, there is evidence of optimistic prediction for the three versions of the new model. Figures 2(b), 3(b) and 4(b) show the Y-Plot for OSV, BV, NHPPV, respectively. The Kolmogorov distances are: 0.1173, 0.0846 and 0.0843, respectively. The Kolmogorov distance is significant for the first figure at the 0.20 level. However, the Kolmogorov distances for the second and third figures are nonsignificant at the 0.20 level. BV and NHPPV are more capable than OSV of capturing the trend in the data. The NHPPV of the new model is the least variable prediction system since it has the smallest median and rate variabilities. Thus the median plots for OSV and BV in Figs 2(c) and 3(c) show more noise than the median plots for NHPPV in Fig. 4(c). Table 3 gives the PLR of the three versions for the same data set, HHPPV outperforming OSV and BV. In summary, the BV exhibits the worst ranks of PL, U-Plot, the median and rate variabilities. NHPPV is performing best on these data. 5.3. Generated data set This data set contains 150 generated inter-failure times for Burr distribution with t( = 0.001 (Table 4).
A. A. Abdel-Ghaly
310
et al. Thousands m,
I / 08 09
/
06 07 05 01 0.3
,,J 1"
02 0; 0 b' 0
/'
/'
i/
i 02
04
0.6
08
1
0
(a) U-Plot
0.2
01
06
00
I
(b) Y-Plot
(c) Median plot
Fig. 2. OSV Plots for the data in Table 1. Billions vz08 01 0.4 0.2 0 0
02
04
08
OS
I
0
(a) f/-Plot
02
04
01
08
1
(b) Y-Plot
(c) Median plot
Fig. 3. BV Plot for the data in Table 1.
1 0.D 0.5 0.7 06 0.5 0.4 03 01 0.1 0 ;I// 0
(a) U-Plot
01
(b)
0..
0.5
0.4
i
Y-Plot
(c) Median plot
Fig. 4. NHPPV Plots for the data in Table 1
Table 3. The PLRs of the three versions at different sample sizes for the data in Table I n
PLR of OSV v BV
PLR of OSV v HNPPV
PLR of BV v NHPPV
50 65 80
1.2461
0.8994 0.842 0.7874 0.724
0.7218 0.5993
95 110
1.4049 1.5159 1.6161 1.6871
125 135
1.7143 1.7333
The sample size at the beginning observations.
0.6956 0.6697 0.6427
of the analysis is 50
Table 5 shows the summarized results of the three versions of the new model in this data set. All three versions show non-significant Y-plot distances. However, there are differences in noise and bias. Regarding the Kolmogorov distance of the U-plot of this model, it is significant at the 0.01 level. The detailed plots [Fig. 7(a)], are above the line of unity slope, suggesting that the NHPPV version
0.5194 0.448 0.4123 0.3906 0.3708
shows moderate optimistic predictions. Similarly, the BV [Fig. 7(a)] shows moderate optimistic predictions. However, the OSV shows pessimistic prediction since the detailed plots of this version are everywhere below the line slope [Fig. 6(a)]. The median plots [Figs 5(c), 6(c) and 7(c)] show that the predictions of NHPPV and OSV seem less optimistic than the BV. NHPPV has the smallest median and rate variabilities followed by osv.
311
Burr type XII distribution Table 4. Generated inter-failure times (read from left to right in rows) 0 0
1 0
0 1
0 0
0 0
0 0
0 2 3
2 0 5
2 2 7
1 0 8
0
1
1
12
1
47 59 187 5 108 323 270 20,238 242
9 8 32 965 891 LO@) 2,985 18,555 13,569
4 2 1 7 184 55 241 646 482 6,483 4,979 4,899 35,716
143 270 68 126 833 4,146 7,023 13,129 1,763
1 12 2 52 133 25 65 26 I,26 1 2,776 22,544 68,456
32 97 257 396 638 3,553 3,853 6,619 39,960
0
0
0
0
2 0 4 4 23 13 108 39 1,548 1,290 176 3,784 4,282 199,659
0
0
3
1
0
5
5 25 51 53 381 26 707 317 2,713 13,892 8,912 5,564
8 3 7 229 70 36 361 759 1,106 1,153 17,026 12,843
3 5 158 3 47 262 467 82 3,996 1,437 629 67,437
No. of observations = 150. Starting sample size = 50.
Table 5. The analysis of data in Table 4 Version
$Z 11 - 2
osv
10.4302
BV
213.2631
1
Rank
i1 11 - A(
Rank
U-Plot
Rank
Y-Plot
Rank
-In PL
Rank
2
8.4899
3
0.2583 1% 0.1307
3
0.0834
3
3
2
2.6823 774.84 2.7662
2
1
771.37 2.5831
1
3
8.065
N.S. 2
NHPPV
9.3331
1
1
7.8744
0.1096 10%
0.0800
N.S.
2% 1
0.0556
N.S.
769.86
am
no c--d!0
0
0.2
0.4
0.8
0.8
a
1
(b) Y-Plot
IDammlmtm
(c) Median plot
Fig. 5. OSV Plots for the data in Table 4.
Millions 1
1 0.6
i”’ /
0.8 L
“I /
0.6 IL!? 0.4 0.6
,/“’
,,.. Y
0.4
0.2
0.2 0
0 0
0.2
0.4
0.6
(a) U-Plot
0 0
1
0
0.2
0.4
06
0.8
1
(b) Y-Plot Fig. 6. BV Plots for the data in Table 4.
(c) Median plot
312
A. A. Abdel-Ghaly
1
et al.
9
17 4
06
l/i
0
, /II
08 i 0.4
ii’
O.j,rLL-0
0.2
0.4
06
0.6
06
0
0.4 06 tI/
m
‘11 i. om400
1
(a) U-Plot
(b) Fig. 7. NHPPV
Table n
70 90 110 130 150
6. The PLR
Thousands
PLRs
of OSV
of the three v BV
Y-Plot
Plots
versions
for
(c) the data
at different
in Table
sample
PLR of OSV v HNPPV
1.4515 0.6344 0.08 168 0.0348 0.03 I I
The NHPPV has good PL performance (Table 6). This version can be recommended as the best for these data. 6. CONCLCSIONS
In this study a new reliability growth model in which the operating time between successive failures is a continuous random variable is considered. This model has been discussed in three versions, the order statistics, the Bayesian and the non-homogeneous Poisson process. Various interesting quantities for software reliability measurements have been derived from the three versions of the new model. Several statistical tools for the analysis of accuracy which help the users of software reliability models to choose the best prediction system for this data source are used to examine the performance of all three versions for real and simulated data sets, and compare the predictive accuracy of the prediction system for the three versions. Two main conclusions may be drawn from this study. First, there is no universal version which can be recommended as “best” in all contexts among the three versions of the new model. In other words, no single version that can be relied on for accurate predictions in all circumstances. Users are therefore advised to try more than one prediction system for a particular data source. Second, the three versions of the new model can be classified as optimistic and pessimistic groups by their performance: (i) the optimistic group consisting of all three versions (in a real data set). Although 5V and NHPPV (in a generated data set) are considered in this group,
1.1537 0.1604 0.0191 0.0085 0.0069
Median
plot
in Table
4
0
w
.( lrn
4.
sizes for the data PLR
of BV v NHPPV 0.7949 0.2528 0.2334 0.2439 0.2209
the OSV seems less optimistic than the other versions on the real data set. (ii) the pessimistic group includes the OSV (in a generated data set). It exhibits the largest rank of U-plot. Acknowledyement-The authors wish to express their deepest gratitude to Prof. E. K. Al-Hussaini for the time and effort he spent on reading this paper and adding his valuable remarks which have greatly contributed to the final form of this paper.
REFERENCES 1. Jelinski, K. and Moranda, P., Software reliability research. In Statistical Computer Performance Evaluation. Academic Press, London, U.K., 1972, pp. 465-484. 2. Littlewood, B., Stochastic reliability growth: a model for fault removal in computer programs and hardware design. IEEE Trans. Reliab., 1981, R-30 (4), 313-320. 3. Abdel-Ghaly, A. A., Analysis of predictive quality of software reliability models. Ph.D. thesis, City University, London, U.K., 1986. 4. Abdel-Ghaly, A. A., A new model for software and hardware fault’s removal process. In Economic and Business Review, Faculty of Commerce, Ain-shams University No. 1, in press, 1996. 5. Littlewood, B. and Sofer, A., A Bayesian modification to the Jelinski-Moranda reliability growth model, unpublished manuscript. The Centre for Software Reliability, The City University, London, U.K., 1981. 6. Littlewood. B. and Verrall. J. L.. Likelihood function of a debugging model for computer software reliability. IEEE Trans. Reliab., 1981, R-30 (2), 145-148. 7. Keiller, P. A., Littlewood, B., Miller, D. R. and Sofer, A., On the quality of software reliability prediction. NATO ASI Series. Vol. F3. Electronic System Effectiveness and L$e Cycle Costing, ed. by J. K. Skwirzynski. Springer, Heidelberg (1983). 8. Abdel-Ghaly, A. A. and Attiah, A. F., A Bayesian analysis of the Weibull model of Software reliability. Proc. 25th
Burr
type
Annual ConJerence I.S.S.R., Cairo University, 25 (1) pp. 1 S-34, 1990. 9. Goel, A. L., Software error detecting model with applications. J. Syst. Software, 1, pp. 2433249, 1980. 10. Aitchison, J. and Dunsmore, I. R., Statistical Prediction Analysis. Cambridge University Press, Cambridge, U.K., 1975.
XII
distribution
313
11. Miller, D. R., Exponential order statistics models of software reliability growth, Technical Report T-496184. George Washington University, Washington, U.S.A., 1984. 12. Abdel-Ghaly, A. A., Chan, P. Y. and Littlewood, B., Evaluation of computing software reliability predictions. IEEE Trans. Software Engng, 1986, SE12 (9), 950-967.
R.&b.,
Vol. 37, No. 2, pp. 305-313,
1997
Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/97 $17.00 + .OO
Pergamon 00262714(95)00124-7
THE USE OF BURR TYPE XII DISTRIBUTION ON SOFTWARE RELIABILITY GROWTH MODELLING A. A. ABDEL-GHALY, Department
of Statistics,
Faculty
G. R. AL-DAYIAN
and F. H. AL-KASHKARI
of Science, King Abdulaziz University, Kingdom of Saudi Arabia (Receiwdfoor
publication
16 June
P.O.
Box
9028, Jeddah
21413,
1995)
Abstract-In this paper a new reliability growth model in which the operating time between successive failures is a continuous random variable is proposed. This model is based on Burr type XII distribution with one Parameter which is discussed in three versions: the order statistics, Bayesian and nonhomogeneous Poisson process. Several software reliability measures are obtained. The performance of all three versions of the suggested model is tested on two data sets. The versions perform with variable accuracy, which suggests that no universal “best” among the three versions of the model could be attained. Copyright ‘0 1996 Elsevier Science Ltd.
1. INTRODUCTION
There are several software reliability growth models in the literature that estimate the remaining error contained in the software. These models may be classified in two broad types. (1) Inter-failure time models in which the operating time between successive failures is a continuous random variable. (2) Discrete software reliability models where the interest is the number of faults or failures in a specified time interval.
errors detected by software testing, such as Duane model [3], non-homogeneous Poisson process for Jelinski-Moranda model [9] and non-homogeneous Poisson process for Littlewood model [3]. In 1991, Rayleigh model was introduced by AbdelGhaly [4]. He assumed that the system starts its life containing N faults. Each of these faults can cause system failure at any time T. Let X be the total time on test until the fault i is removed. The Xs are assumed to be independent identically distributed random variables with Rayleigh density function, given by
The inter-failure time models are our concern here. These models are roughly classified into the following three categories: 1.1. Order statistics models
otherwise.
(1)
Let Xcl,, Xc2,,. . . be order statistics of the time of occurrence of the lst, 2nd,. . . failures and T1, T,, . . . , be the inter-failure times. Then
In this category the failure times of a software reliability process are modeled as Order statistics of independent non-identically distributed random variablessuch as:Jelinski-Moranda model [1], Littlewood model [2], Weibull model [3] and Rayleigh model [4].
TI = -%I,, T
1.2. Bayesian model The parameters of the order statistics model are assumed to be random variables with a given prior density function. Different techniques are adapted and lead to different versions of the same model. These models are: Bayesian Jelinski-Moranda model [S], Bayesian Littlewood model [3], Littlewood-Verrall model [6], Keiller-Littlewood model [7] and Bayesian-Weibull model [S]. 1.3. Non-homogeneous
x > 0, (4 > 0)
=
x(i)
-
x(i-
I),
i =
1,2,. .
(2)
If a total time ‘c has elapsed and (i - 1) faults have been removed, then the probability density function of q (the inter-failure time needed for fault i to show itself) will be given in the following form f(fi 1A,)= 24(q-
1+ ti) exp[ -li((Ti-
1+ ti)'-Tfm
I)],
li > O, C3) where Ai = (N - i + 1)4
and ziml is the total elapsed time at the point (i - 1) = 1;:: tj. This result will be used in the new model which will be presented in this paper and it will be discussedin the three versions.
Poisson process
In this category, the testing time used as the unit of error detection period for describing the time dependent behavior of the cumulative number of 305
306
A. A. Abdel-Ghaly
et al
-
t
!
Failures
detected
and thus faults,
tied
in (0,Z)
X
t
next failure
(now) Fig.
2. THE
ORDER
STATISTICS
VERSION
(OSV)
In this version it is assumed that there are N software faults at the start of testing. Each of the N faults in the program will cause a failure after a time which is distributed as in (1) and independently of other faults, with a different failure rate q5j. When a failure occurs, there is an instantaneous removal of the faults which caused the failure, and no new faults are introduced during the debugging process. If total time 4-i has elapsed, and (i - 1) faults have been removed, the failure rate of the program is given by: N-it, ii = c#J1+. . . + t$N-i+l = (4) j:l $jT are the non-identical rates of where 41,...,4N-i+l the remaining faults. The 4s are regarded as realizations of random variables. Thus the occurrence of the N faults with which the program begins life can be regarded as i.i.d. random variables. When debugging starts (i.e. T = 0) each 4 has gamma distribution with shape parameter c(, i.e. 1‘(4 I4 = &
c!
FL
exp(-41,
4 > 0, (x > 0).
(5)
Bayes’ theorem can be used to describe how uncertainty about the occurrence rate of the faults (and hence the failure rate of the program) changes with time. Consider the occurrence rate, q5, of one of the N - i + 1 remaining faults at the epoch “now” in Fig. 1. f{4
1.
From (4), the failure rate of the program Ai, is a sum of the (N - i + 1) i.i.d. gamma distributed random variables with shape parameter (a(N - i + 1)) and scale parameter (1 + rf- i), with density function
x exp - (&(l + rf- ,)),
Ai > 0.
(7)
Multiplying (7) by (3), and integrating with respect to ii we get the p.d.f. of r given Km, as f(tilu,
N,zi+,)
= 2(N - i+ l)cc(ri-, + ti) (1 + r;ml)il(N-i+U x [I + (Tie1 + Qy-i+‘)+l
ti > 0. (8) This form can be obtained from the order statistics formulation of the model; if the detection times (and removal) of the fault are independent random variables, Xi, with density function given by f(xlu)
= 2ctx/[l + Xqa+l,
x > 0, (tl > Oh
(9)
which is a special case of Burr type XII distribution. The two parameters N and c( are estimated by maximizing the likelihood function to obtain their maximum likelihood estimates (MLEs) of N, CI. Such MLEs are used to predict the p.d.f., reliability function and its measures for the next failure time. The likelihood function based on the data of the first n failures is given
1this fault is not fixed in (0, li- ,)} = f{$
=
1no failure is caused by this fault in (0, ri- ,)}
(l+rf-i)“qY’exp(-f$(l+riz_,))
> 4>0.
Vu)
(6)
(10)
That is, q5 m gamma (c(, 1 + z:-~), where c-1
=
mexp(-$r’-,)r:l)$“-lexp(-~)dg s0
1 = (1 + t,Z_J’
The estimates &, fi of the parameters c(, N will be used to predict the p.d.f. of T,,, by substituting in (8)
fit,+1 Id = 2M - nw(~” + t,+ 1) (1 + ,y-nM x [l + (?” + t”+i)2](N~n)J+i’
and
t “+I > 0. (11)
Pr{no failure due to this fault in the interval 0 - T~-~} = exp(-@f-i).
Now, the user may be interested in obtaining the probability that the system will be running without
Burr type XII distribution failure up to time T. + i. The reliability function of T,, i is given by
where
(N-n)& &L+ 1lZ") =1+CT" +t,+1Y 1 1 + 7.’
The failure rate function of T,, i is
nUj-l,,-l
i= 1,2,...,n,
+Uj,n-17
[ 0,
j>n,
aO.n= 1, t/n and a,,, = 1.
(18)
By substituting from (15), it follows that the posterior density is given by
?I + L+1
r;(t,+IIT”) = 2(15 -n)&
aj.n=
(12)
[
307
1 + (% + t,+,Y
(13)
.I”*(& tiltI>. . >0 = C fi 2(i - (i - l)$)
3. BAYESIAN
VERSION
i=l
(BV)
As we have seen in the case of OSV, no explicit formulas can be obtained in estimating the parameters using the maximum likelihood method, and only numerical solutions may be obtained. So, to avoid this situation, the BV for the suggested model is proposed. In this model, Bayesian approach is used to get the probability density function of inter-failure time. We follow Littlewood and Sofer [S] in using their reparametrization technique applied to JM model. It is assumed that 1 = Nu, 4 = a, where 1 is taken to be the initial rate of occurrence of failures per unit time. Accordingly, the reparametrized density function corresponding to (8) is given by
X exp
-i$l
(A -
(i -
1M)CMl
+ (Tit1
+ ti)2)
i
- ln( 1 + z,Z_i)] f*(n, dltr,
I
, 1 > (n - l)f$
(19)
. . , t,) = zero otherwise,
where n-l c-1 = c Uj,“_l j=O (n
-j)!
’ (ln(1 + ~~))“-j+i(C~:~ ln(1 + rf))j+‘. (20) It can be shown, from (14) that the reliability function is given by expC-(i-n4)Cln(l
XeXp[-(I~-(i-l)~){lll(l+(7~-~+ti)2)
- ln(l + $,)}I,
ti > 0.
(14)
The reparametrized likelihood function corresponding to (10) is given by
w.,bit,, , t,) = fi [w
- (i - 114) l +‘;,’ +Jt,)’ 1 1
x exp[-(i
- (i - 1)4)[ln(l + (~~-i + ti)‘)
- ln(1 + T:- i)]]
1 1,
A,$>0
(16)
the posterior density function (which is known to be proportional to the product of the likelihood function and the prior) is then given by j*(& 4 1t,, . . . , t.) = t,)f(& $), L > (n - l)d and C is the CW,~lt,,..., normalizing constant. It can be shown that i; (x + i) = i i=O
=
- ln(l + ri)]];
1 > nq5 (21)
i 1; (n - l)f$ < A < n$. It may be observed that the current reliability immediately after occurrence of the nth failure (and removal of n faults) equals one (if (n - l)&~< 3, < nr$), i.e. that the program is now perfect and the last fault has been removed in the nth fix. The current predictive reliability function may be obtained from eqns (19) and (21), (see, for example [lo]), as follows
(15)
Assuming an improper non-informative uniform prior density of the form S(i4)=
m”+,I~>~)
+(~+t,+d*)
q”X”-i.
(17)
=
R(t,+,l1,~)f*(l,~lt,,...,t,)did~ ssti a m nQ = f*G, d~ltl>. . . >t,) d/i d4 ss0 O-lb+ m m + exp( -(A - nd)(ln(l + (z, + t,+ i)‘) ss0 nO - ln(1 + $)))f*(&
4lti , . . . , tn) dl d+.
Therefore
R*(r,+,It,,...,
t.)
=
C[A(t,+,)
+ c-l -B],
(22)
308
A. A. Abdel-Ghaly
et al.
where
and given by
Act,+ 11
P*(NIt,,
(n-j)! = jiYo ai.n (ln(l+(r,+t,+,)2))n~j+1(~~=l B=
i j=O
(l+zf))j+”
a. ~-__-. ..-(n - j)!j! “‘“(In(1 + 2,2))“~j+‘(C~=,
=
ln(1 + tf))j+l
/ t,) = 1 - CB.
4. NON-HOMOGENEOUS VERSION
POISSON
(N - n)! x exp[--pexp(-a$lln(l
and C is given in (20). If we let tn+l tend to infinite, we will obtain the predictive probability that the program is now perfect in the form: PO = R*(coIt,,.
, La)
(23)
+l(~~:,t,ti)z))], N > n, (u, p > 0).
The predictive p.d.f. of T.- 1 can be obtained eqns (11) and (26) as follows
from
f*(t, + L I i4 % L)
= f fitn+lIz,)P*(N-nlt,,...,t,,u),
PROCESS
(26)
t,,+l>O
N=”
(NHPPV)
Non-homogeneous Poisson process (NHPP) models are used to predict future failures. This approach is one of the best ways to measure software reliability and other measures. NHPP are also called counting models because the mean value function H(t) represents the cumulative number of faults exposed by time t. These models also have a failure rate (1) which is a function of time. The first model discussed in Section (2) is an OS model where N is the initial number of faults in the system (program). Suppose that x is of known value, and N has a Poisson distribution with parameter p and probability mass function
= 2u/l
G+t”+I (l+(r,+t,-,)*)“+’
xexp[~(1-(l+(~~~~+,,2~)l~
tn+l>o,(~d>oh
(27)
By integrating (27) with respect to tntl predictive c.d.f. of T,, , as follows
we get the
F*(t”+IIP,Td
(28) P(N) = $ ecu’,
N=0,1,2,...
(24)
Miller [I 1] suggested that any order statistics mixture model mixed over the Poisson IV, will be a non-homogeneous Poisson process. The posterior mass function of N can be obtained from eqns (10) and (24) as follows P*(N[t,,
. . . t,, cx)= CL(N, ait,,
, t,)P(N),
It follows, from eqn (28), that the predictive function is given by
R*(t”+l Ii4 co =exp
L -p-
1 (1 + 7;)
and the failure
.,t,)P(N)
then
5. THE
(25) It follows that the posterior distribution of the number of remaining faults is Poisson mass function,
(
rate function
JXt,+I) = 2v = f L(N,rlt,,.. N=”
1-
l+.r,2 ~-~ ( 1+(%+L+J2
il
))1 ’
(29)
where C-’
reliability
ANALYSIS
(hazard function) 7”
+
&+I
(1 + (7” + t,+ J2)n+ l. OF
PREDICTIVE
is (30)
QUALITY
The new software reliability growth model has been considered to measure and predict the reliability of software which has been studied in the three previous versions. The user usually needs to know which version would be most suitable in a particular context. Such comparison may be carried out with the assistance of many different statistical tools. Among these tools are the U-plot, Y-plot, measures of variability, the prequential likelihood ratio (PLR) [12]. The performance of all three versions of the new model can be achieved for real and generated data sets by using these statistical tools.
Burr type XII distribution
309
Table I. Execution times in s between successive failures, Musa (1981) 1121 (read from left to right in rows) 3 138 325 36 97 148 0 44 44s 724 30 729 7s
1045
30 50 55 4 263 21 232 129 296 2323 143 1897 482 648
113 77 242 0 452 233 330 810 1755 2930 108 447 5509 5485
81 24 68 8 255 134 365 290 1064 1461 0 386
115 108 422 227 197 357 1222 300 1783 843 3110 446
100 1160
1864
9 88 180 65 193 193 543 529 860 12 1247 122 1071 4116
10
91 120 1146
2 670
10 176
112 26 600 457 816 369 529 828 33 865 875 1082 6150
58 79 31 16 160 707 1800 700 948 790
6 236
10 281 983 261 943 990 371
15 114 15 300 1351 748 379
1011 868 143s 245 22 3321
No. of observations = 136. Starting sample size = 35. Table 2. The analysis of data in Table 1 Version
osv BV NHPPV
iZ / I - c/ 4.7906
Rank
& 11 - $I I 1
Rank
U-Plot
Rank
Y-Plot
Rank
2
2.1655
2
0.3298 1% 0.5341 1% 0.5303
1
0.1173
3
12.5039
3
2.1829
3
2.0889
1
2.0245
1
5.1. Data analysis and results The analysis has been carried out for two different kinds of sets of inter-failure time data. The first kind is a real data set but the second is an artificial one. The reason for using these two types of data sets is for comparison purposes and decisions regarding the version’s preference for the new model. The following method will be used to accomplish the calculations for both data sets: assuming that each data set contains n inter-failure times, m( < n - 1) is chosen as a starting sample size. Using the results of Section 4, the MLEs are obtained for the sample and then used to direct In p.d.f., c.d.f. and rat, for the next failure-time. The sample is then increased by the observed tm+l, the process repeated for T, + 2, and so on. 5.2. Real data set (Musa system 1) This data set [12] is shown in Table 1 which contains 136 observations of the execution time in seconds between successive failures. The sample size at the beginning of the analysis is 35 observations to estimate the models’ parameters and to predict the 36th failure. The process is repeated for the next observation and so on. Table 2 summarizes the results concerning the quality of performance of the three versions of the new model. Figures 2(a), 3(a) and 4(a) show the U-Plot for
1%
3 2
20% 0.0846 N.S. 0.0843
N.S.
2
1
-In PL 7.2367 834.613 7.2527 853.163 7.2311
834.17
Rank 2 3
1
OSV, BV and NHPPV, respectively. The Kolmogorov distances are 0.3298, 0.5341 and 0.5303, respectively. The Kolmogorov distances of the U-Plot of the three versions are significant at 0.01 level. OSV, BV and NHPPV plots are everywhere above the line of unity slope. That is, there is evidence of optimistic prediction for the three versions of the new model. Figures 2(b), 3(b) and 4(b) show the Y-Plot for OSV, BV, NHPPV, respectively. The Kolmogorov distances are: 0.1173, 0.0846 and 0.0843, respectively. The Kolmogorov distance is significant for the first figure at the 0.20 level. However, the Kolmogorov distances for the second and third figures are nonsignificant at the 0.20 level. BV and NHPPV are more capable than OSV of capturing the trend in the data. The NHPPV of the new model is the least variable prediction system since it has the smallest median and rate variabilities. Thus the median plots for OSV and BV in Figs 2(c) and 3(c) show more noise than the median plots for NHPPV in Fig. 4(c). Table 3 gives the PLR of the three versions for the same data set, HHPPV outperforming OSV and BV. In summary, the BV exhibits the worst ranks of PL, U-Plot, the median and rate variabilities. NHPPV is performing best on these data. 5.3. Generated data set This data set contains 150 generated inter-failure times for Burr distribution with t( = 0.001 (Table 4).
A. A. Abdel-Ghaly
310
et al. Thousands m,
I / 08 09
/
06 07 05 01 0.3
,,J 1"
02 0; 0 b' 0
/'
/'
i/
i 02
04
0.6
08
1
0
(a) U-Plot
0.2
01
06
00
I
(b) Y-Plot
(c) Median plot
Fig. 2. OSV Plots for the data in Table 1. Billions vz08 01 0.4 0.2 0 0
02
04
08
OS
I
0
(a) f/-Plot
02
04
01
08
1
(b) Y-Plot
(c) Median plot
Fig. 3. BV Plot for the data in Table 1.
1 0.D 0.5 0.7 06 0.5 0.4 03 01 0.1 0 ;I// 0
(a) U-Plot
01
(b)
0..
0.5
0.4
i
Y-Plot
(c) Median plot
Fig. 4. NHPPV Plots for the data in Table 1
Table 3. The PLRs of the three versions at different sample sizes for the data in Table I n
PLR of OSV v BV
PLR of OSV v HNPPV
PLR of BV v NHPPV
50 65 80
1.2461
0.8994 0.842 0.7874 0.724
0.7218 0.5993
95 110
1.4049 1.5159 1.6161 1.6871
125 135
1.7143 1.7333
The sample size at the beginning observations.
0.6956 0.6697 0.6427
of the analysis is 50
Table 5 shows the summarized results of the three versions of the new model in this data set. All three versions show non-significant Y-plot distances. However, there are differences in noise and bias. Regarding the Kolmogorov distance of the U-plot of this model, it is significant at the 0.01 level. The detailed plots [Fig. 7(a)], are above the line of unity slope, suggesting that the NHPPV version
0.5194 0.448 0.4123 0.3906 0.3708
shows moderate optimistic predictions. Similarly, the BV [Fig. 7(a)] shows moderate optimistic predictions. However, the OSV shows pessimistic prediction since the detailed plots of this version are everywhere below the line slope [Fig. 6(a)]. The median plots [Figs 5(c), 6(c) and 7(c)] show that the predictions of NHPPV and OSV seem less optimistic than the BV. NHPPV has the smallest median and rate variabilities followed by osv.
311
Burr type XII distribution Table 4. Generated inter-failure times (read from left to right in rows) 0 0
1 0
0 1
0 0
0 0
0 0
0 2 3
2 0 5
2 2 7
1 0 8
0
1
1
12
1
47 59 187 5 108 323 270 20,238 242
9 8 32 965 891 LO@) 2,985 18,555 13,569
4 2 1 7 184 55 241 646 482 6,483 4,979 4,899 35,716
143 270 68 126 833 4,146 7,023 13,129 1,763
1 12 2 52 133 25 65 26 I,26 1 2,776 22,544 68,456
32 97 257 396 638 3,553 3,853 6,619 39,960
0
0
0
0
2 0 4 4 23 13 108 39 1,548 1,290 176 3,784 4,282 199,659
0
0
3
1
0
5
5 25 51 53 381 26 707 317 2,713 13,892 8,912 5,564
8 3 7 229 70 36 361 759 1,106 1,153 17,026 12,843
3 5 158 3 47 262 467 82 3,996 1,437 629 67,437
No. of observations = 150. Starting sample size = 50.
Table 5. The analysis of data in Table 4 Version
$Z 11 - 2
osv
10.4302
BV
213.2631
1
Rank
i1 11 - A(
Rank
U-Plot
Rank
Y-Plot
Rank
-In PL
Rank
2
8.4899
3
0.2583 1% 0.1307
3
0.0834
3
3
2
2.6823 774.84 2.7662
2
1
771.37 2.5831
1
3
8.065
N.S. 2
NHPPV
9.3331
1
1
7.8744
0.1096 10%
0.0800
N.S.
2% 1
0.0556
N.S.
769.86
am
no c--d!0
0
0.2
0.4
0.8
0.8
a
1
(b) Y-Plot
IDammlmtm
(c) Median plot
Fig. 5. OSV Plots for the data in Table 4.
Millions 1
1 0.6
i”’ /
0.8 L
“I /
0.6 IL!? 0.4 0.6
,/“’
,,.. Y
0.4
0.2
0.2 0
0 0
0.2
0.4
0.6
(a) U-Plot
0 0
1
0
0.2
0.4
06
0.8
1
(b) Y-Plot Fig. 6. BV Plots for the data in Table 4.
(c) Median plot
312
A. A. Abdel-Ghaly
1
et al.
9
17 4
06
l/i
0
, /II
08 i 0.4
ii’
O.j,rLL-0
0.2
0.4
06
0.6
06
0
0.4 06 tI/
m
‘11 i. om400
1
(a) U-Plot
(b) Fig. 7. NHPPV
Table n
70 90 110 130 150
6. The PLR
Thousands
PLRs
of OSV
of the three v BV
Y-Plot
Plots
versions
for
(c) the data
at different
in Table
sample
PLR of OSV v HNPPV
1.4515 0.6344 0.08 168 0.0348 0.03 I I
The NHPPV has good PL performance (Table 6). This version can be recommended as the best for these data. 6. CONCLCSIONS
In this study a new reliability growth model in which the operating time between successive failures is a continuous random variable is considered. This model has been discussed in three versions, the order statistics, the Bayesian and the non-homogeneous Poisson process. Various interesting quantities for software reliability measurements have been derived from the three versions of the new model. Several statistical tools for the analysis of accuracy which help the users of software reliability models to choose the best prediction system for this data source are used to examine the performance of all three versions for real and simulated data sets, and compare the predictive accuracy of the prediction system for the three versions. Two main conclusions may be drawn from this study. First, there is no universal version which can be recommended as “best” in all contexts among the three versions of the new model. In other words, no single version that can be relied on for accurate predictions in all circumstances. Users are therefore advised to try more than one prediction system for a particular data source. Second, the three versions of the new model can be classified as optimistic and pessimistic groups by their performance: (i) the optimistic group consisting of all three versions (in a real data set). Although 5V and NHPPV (in a generated data set) are considered in this group,
1.1537 0.1604 0.0191 0.0085 0.0069
Median
plot
in Table
4
0
w
.( lrn
4.
sizes for the data PLR
of BV v NHPPV 0.7949 0.2528 0.2334 0.2439 0.2209
the OSV seems less optimistic than the other versions on the real data set. (ii) the pessimistic group includes the OSV (in a generated data set). It exhibits the largest rank of U-plot. Acknowledyement-The authors wish to express their deepest gratitude to Prof. E. K. Al-Hussaini for the time and effort he spent on reading this paper and adding his valuable remarks which have greatly contributed to the final form of this paper.
REFERENCES 1. Jelinski, K. and Moranda, P., Software reliability research. In Statistical Computer Performance Evaluation. Academic Press, London, U.K., 1972, pp. 465-484. 2. Littlewood, B., Stochastic reliability growth: a model for fault removal in computer programs and hardware design. IEEE Trans. Reliab., 1981, R-30 (4), 313-320. 3. Abdel-Ghaly, A. A., Analysis of predictive quality of software reliability models. Ph.D. thesis, City University, London, U.K., 1986. 4. Abdel-Ghaly, A. A., A new model for software and hardware fault’s removal process. In Economic and Business Review, Faculty of Commerce, Ain-shams University No. 1, in press, 1996. 5. Littlewood, B. and Sofer, A., A Bayesian modification to the Jelinski-Moranda reliability growth model, unpublished manuscript. The Centre for Software Reliability, The City University, London, U.K., 1981. 6. Littlewood. B. and Verrall. J. L.. Likelihood function of a debugging model for computer software reliability. IEEE Trans. Reliab., 1981, R-30 (2), 145-148. 7. Keiller, P. A., Littlewood, B., Miller, D. R. and Sofer, A., On the quality of software reliability prediction. NATO ASI Series. Vol. F3. Electronic System Effectiveness and L$e Cycle Costing, ed. by J. K. Skwirzynski. Springer, Heidelberg (1983). 8. Abdel-Ghaly, A. A. and Attiah, A. F., A Bayesian analysis of the Weibull model of Software reliability. Proc. 25th
Burr
type
Annual ConJerence I.S.S.R., Cairo University, 25 (1) pp. 1 S-34, 1990. 9. Goel, A. L., Software error detecting model with applications. J. Syst. Software, 1, pp. 2433249, 1980. 10. Aitchison, J. and Dunsmore, I. R., Statistical Prediction Analysis. Cambridge University Press, Cambridge, U.K., 1975.
XII
distribution
313
11. Miller, D. R., Exponential order statistics models of software reliability growth, Technical Report T-496184. George Washington University, Washington, U.S.A., 1984. 12. Abdel-Ghaly, A. A., Chan, P. Y. and Littlewood, B., Evaluation of computing software reliability predictions. IEEE Trans. Software Engng, 1986, SE12 (9), 950-967.