International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
INTERNATIONAL JOURNAL OF COMP...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
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  1. 1. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING & ISSN 0976 - 6375(Online), Volume 4, Issue 5, September - October (2013), © IAEME TECHNOLOGY (IJCET) ISSN 0976 – 6367(Print) ISSN 0976 – 6375(Online) Volume 4, Issue 5, September – October (2013), pp. 277-284 © IAEME: www.iaeme.com/ijcet.asp Journal Impact Factor (2013): 6.1302 (Calculated by GISI) www.jifactor.com IJCET ©IAEME SPRT -TWO STEP APPROACH: HLSRGM Dr. R. Satya Prasad1, Shaheen2, G. Krishna Mohan3 1 (Associate Professor, Dept. of Computer Science & Engg., Acharya Nagrjuna University Nagarjuna Nagar, Guntur, Andhrapradesh, India) 2 (Asst. Professor, IPE, Osmaina university campus, Hyderabad) 3 (Reader, Dept. of Computer Science, P.B.Siddhartha College, Vijayawada, Andhrapradesh, India) ABSTRACT In Classical Hypothesis testing volumes of data is to be collected and then the conclusions are drawn, which may need more time. But, Sequential Analysis of Statistical science could be adopted in order to decide upon the reliability / unreliability of the developed software very quickly. The procedure adopted for this is, Sequential Probability Ratio Test (SPRT). It is designed for continuous monitoring. The likelihood based SPRT proposed by Wald is very general and it can be used for many different probability distributions. The parameters are estimated using two step approach. In the present paper, the HLSRGM (Half Logistic Software Reliability Growth model) is used on five sets of existing software reliability data and analyzed the results. Keywords: HLSRGM, Two step approach, Decision lines, Software testing, Software failure data. I. INTRODUCTION Wald's procedure is particularly relevant if the data is collected sequentially. Sequential Analysis is different from Classical Hypothesis Testing were the number of cases tested or collected is fixed at the beginning of the experiment. In Classical Hypothesis Testing the data collection is executed without analysis and consideration of the data. After all data is collected the analysis is done and conclusions are drawn. However, in Sequential Analysis every case is analyzed directly after being collected, the data collected upto that moment is then compared with certain threshold values, incorporating the new information obtained from the freshly collected case. This approach allows one to draw conclusions during the data collection, and a final conclusion can possibly be reached at a much earlier stage as is the case in Classical Hypothesis Testing. The advantages of Sequential Analysis are easy to see. As data collection can be terminated after fewer cases and 277
  2. 2. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 5, September - October (2013), © IAEME decisions taken earlier, the savings in terms of human life and misery, and financial savings, might be considerable. In the analysis of software failure data we often deal with either Time Between Failures or failure count in a given time interval. If it is further assumed that the average number of recorded failures in a given time interval is directly proportional to the length of the interval and the random number of failure occurrences in the interval is explained by a Poisson process then we know that the probability equation of the stochastic process representing the failure occurrences is given by a Homogeneous Poisson Process with the expression e − λt ( λ t ) P  N (t ) = n =   n! n (1.1) Stieber (1997) observes that if classical testing strategies are used, the application of software reliability growth models may be difficult and reliability predictions can be misleading. However, he observes that statistical methods can be successfully applied to the failure data. He demonstrated his observation by applying the well-known sequential probability ratio test (SPRT) of Wald (1947) for a software failure data to detect unreliable software components and compare the reliability of different software versions. In this paper we consider popular SRGM HLSRGM and adopt the principle of Stieber (1997) in detecting unreliable software components in order to accept or reject the developed software. The theory proposed by Stieber (1997) is presented in Section 2 for a ready reference. Extension of this theory to the SRGM – HLSRGM is presented in Section 3. Application of the decision rule to detect unreliable software with respect to the proposed SRGM is given in Section 4. Analysis of the application of the SPRT on five data sets and conclusions drawn are given in Section 5 and 6 respectively. II. WALD'S SEQUENTIAL TEST FOR A POISSON PROCESS The sequential probability ratio test (SPRT) was developed by A.Wald at Columbia University in 1943. Due to its usefulness in development work on military and naval equipment it was classified as ‘Restricted’ by the Espionage Act (Wald, 1947). A big advantage of sequential tests is that they require fewer observations (time) on the average than fixed sample size tests. SPRTs are widely used for statistical quality control in manufacturing processes. An SPRT for homogeneous Poisson processes is described below. Let {N(t),t ≥0} be a homogeneous Poisson process with rate ‘λ’. In our case, N(t)= number of failures up to time ‘ t’ and ‘ λ’ is the failure rate (failures per unit time ). Suppose that we put a system on test (for example a software system, where testing is done according to a usage profile and no faults are corrected) and that we want to estimate its failure rate ‘ λ’. We can not expect to estimate ‘ λ’ precisely. But we want to reject the system with a high probability if our data suggest that the failure rate is larger than λ1 and accept it with a high probability, if it’s smaller than λ0. As always with statistical tests, there is some risk to get the wrong answers. So we have to specify two (small) numbers ‘α’ and ‘β’, where ‘α’ is the probability of falsely rejecting the system. That is rejecting the system even if λ ≤ λ0. This is the "producer’s" risk. β is the probability of falsely accepting the system .That is accepting the system even if λ ≥ λ1. This is the “consumer’s” risk. With specified choices of λ0 and λ1 such that 0 < λ0 < λ1, the probability of finding N(t) failures in the time span (0,t ) with λ1,λ0 as the failure rates are respectively given by 278
  3. 3. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 5, September - October (2013), © IAEME e− λ1t [ λ1t ] Q1 = N (t ) (2.1) N (t )! Q0 = e− λ0t [ λ0t ] N (t ) (2.2) N (t )! The ratio Q1 at any time ’t’ is considered as a measure of deciding the truth towards λ0 or λ1 Q0 , given a sequence of time instants say t1 < t2 < t3 < ........ < tK and the corresponding realizations N (t ) λ  Q Q N (t1 ), N (t2 ),........N (tK ) of N(t). Simplification of 1 gives 1 = exp(λ0 − λ1 )t +  1  Q0 Q0  λ0  The decision rule of SPRT is to decide in favor of λ1 , in favor of λ0 or to continue by Q observing the number of failures at a later time than 't' according as 1 is greater than or equal to a Q0 constant say A, less than or equal to a constant say B or in between the constants A and B. That is, we decide the given software product as unreliable, reliable or continue the test process with one more observation in failure data, according as Q1 ≥A Q0 (2.3) Q1 ≤B Q0 (2.4) B< Q1 <A Q0 (2.5) The approximate values of the constants A and B are taken as A ≅ 1− β α , B≅ β 1−α Where ‘ α ’ and ‘ β ’ are the risk probabilities as defined earlier. A simplified version of the above decision processes is to reject the system as unreliable if N(t) falls for the first time above the line NU ( t ) = a.t + b2 (2.6) To accept the system to be reliable if N(t) falls for the first time below the line N L ( t ) = a.t − b1 (2.7) To continue the test with one more observation on (t, N(t)) as the random graph of [t, N(t)] is between the two linear boundaries given by equations (2.6) and (2.7) where 279
  4. 4. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 5, September - October (2013), © IAEME a= λ1 − λ0 λ  log  1   λ0  (2.8) 1 − α  log    β  b1 = λ  log  1   λ0  (2.9) 1 − β  log   α   b2 =  λ1  log    λ0  (2.10) The parameters α , β , λ0 and λ1 can be chosen in several ways. One way suggested by Stieber λ.log ( q ) λ.log ( q ) λ1 q −1 q −1 λ0 If λ0 and λ1 are chosen in this way, the slope of NU (t) and NL (t) equals λ. The other two ways of choosing λ0 and λ1 are from past projects and from part of the data to compare the reliability of different functional areas. (1997) is λ0 = III. , λ1 = q where q = HLSRGM One simple class of finite failure NHPP model is the HLSRGM, assuming that the failure intensity is proportional to the number of faults remaining in the software describing an exponential failure curve. It has two parameters. Where, ‘a’ is the expected total number of faults in the code and ‘b’ is the shape factor defined as, the rate at which the failure rate decreases. The cumulative 1 − e−bt distribution function of the model is: F ( t ) = .The expected number of faults at time‘t’ is 1 + e−bt ( ( denoted by m (t ) = IV. a (1 − e − bt ) (1 + e ) − bt ) ) , a > 0, b > 0, t ≥ 0 . SEQUENTIAL TEST FOR SRGMS In Section II, for the Poisson process we know that the expected value of N(t) = λt called the average number of failures experienced in time 't' .This is also called the mean value function of the Poisson process. On the other hand if we consider a Poisson process with a general function m(t) as its mean value function the probability equation of a such a process is P[ N (t) = Y ] [ m(t)] = y .e−m(t ) , y = 0,1,2, −− − − y! Depending on the forms of m(t) we get various Poisson processes called NHPP. 280
  5. 5. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 5, September - October (2013), © IAEME We may write Q1 = Q0 = e − m1 (t ) .[ m1 (t )] N (t ) N (t )! e − m0 (t ) .[ m0 (t )] N (t ) N (t )! Where, m1 (t ) , m0 (t ) are values of the mean value function at specified sets of its parameters indicating reliable software and unreliable software respectively. Let P , P be values of the NHPP at 0 1 two specifications of b say b0 , b1 where ( b0 < b1 ) respectively. It can be shown that for our models m ( t ) at b1 is greater than that at b0 . Symbolically m0 ( t ) < m1 ( t ) . Then the SPRT procedure is as follows: Accept the system to be reliable if i.e., e − m1 (t ) .[ m1 (t )] Q1 ≤B Q0 N (t ) e − m0 (t ) .[ m0 (t )] N (t ) ≤B  β  log   + m1 (t ) − m0 (t )  1−α  i.e., N (t ) ≤ log m1 (t ) − log m0 (t ) Decide the system to be unreliable and reject if (4.1) Q1 ≥A Q0 1− β  log   + m1 (t ) − m0 (t )  α  i.e., N (t ) ≥ log m1 (t ) − log m0 (t ) (4.2) Continue the test procedure as long as  β   1− β  log  log   + m1 (t ) − m0 (t )  + m1 (t ) − m0 (t )  1−α   α  < N (t ) < log m1 (t ) − log m0 (t ) log m1 (t ) − log m0 (t ) (4.3) Substituting the appropriate expressions of the respective mean value function – m(t) of HLSRGM, we get the respective decision rules and are given in followings lines Acceptance region:  2 a ( e − b0t − e − b1t )  β    log  +  − t b −b  1 − α   1 + e − b0t + e − b1t + e ( 0 1 )    (4.4) N (t ) ≤  1 − e − b1t   1 + e − b0t   log  − b1t   − b0 t    1 + e   1 − e   281
  6. 6. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 5, September - October (2013), © IAEME Rejection region:  1− β log   α N (t ) ≥  2a e − b0t − e − b1t    +  − t b −b  1 + e − b0t + e − b1t + e ( 0 1 )    ( ) (4.5)  1 − e − b1t   1 + e − b0t   log  − b1t   − b0 t    1 + e   1 − e   Continuation region: −b0t −b1t   β   2a e − e log  +  −b t −bt −t(b −b )    1−α  1+ e 0 + e 1 + e 0 1    ( 1  1− e−bt   1+ e−b0t  log  −b1t   −b0t   1+ e   1− e  ) −b0t −bt 1   1− β   2a e − e log  +  −b t −b t −t(b −b )    α  1+ e 0 + e 1 + e 0 1    ( < N(t) < 1  1− e−bt   1+ e−b0t  log  −b1t   −b0t   1+ e   1− e  ) (4.6) It may be noted that in the above model the decision rules are exclusively based on the strength of the sequential procedure (α,β ) and the values of the respective mean value functions namely, m0 (t ) , m1 (t ) . If the mean value function is linear in ‘t’ passing through origin, that is, m(t) = λt the decision rules become decision lines as described by Stieber (1997). In that sense equations (4.1), (4.2), (4.3) can be regarded as generalizations to the decision procedure of Stieber (1997). The applications of these results for live software failure data are presented with analysis in Section 5. V. SPRT ANALYSIS OF LIVE DATA SETS The developed SPRT methodology is for a software failure data which is of the form [t, N(t)]. Where, N(t) is the failure number of software system or its sub system in ‘t’ units of time. In this section we evaluate the decision rules based on the considered mean value function for Five different data sets of the above form, borrowed from (Xie, 2002), (Pham, 2006) and (LYU,1996). The procedure adopted in estimating the parameters is a two step approach (Satyaprasad et al.,2013). Based on the estimates of the parameter ‘b’ in each mean value function, we have chosen the specifications of b0 = b −δ , b1 = b +δ equidistant on either side of estimate of b obtained through a Data Set to apply SPRT such that b0 < b < b1. Assuming the value of δ = 0.025 , the choices are given in the following table. Table 5.1: Estimates of a, b & Specifications of b0, b1 for Time domain Data Set Estimate of ‘a’ Estimate of ‘b’ b0 b1 31.524466 0.029012 0.004012 0.054012 XIE 28.767254 0.075071 0.050071 0.100071 NTDS 18.610725 0.105451 0.080451 0.130451 IBM 23.655141 0.026521 0.001521 0.051521 AT&T 27.139750 0.470899 0.445899 0.495899 LYU Using the selected b0 , b1 and subsequently the m0 (t ), m1 (t ) for the model, we calculated the decision rules given by Equations 4.4 and 4.5, sequentially at each ‘t’ of the data sets taking the strength ( α, β ) as (0.05, 0.2). These are presented for the model in Table 5.2. The following consolidated table reveals the iterations required to come to a decision about the software of each Data Set. 282
  7. 7. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 5, September - October (2013), © IAEME Table 5.2: SPRT analysis for 5 data sets of Time domain data Data Set T N(t) Acceptance region (≤) Rejection Region (≥) Decision 2.096161 3.892803 Accept Xie 30.02 1 3.289091 10.007634 Accept NTDS 9 1 9.537308E-01 1.161761E+01 10 1 -1.043887E+00 1.484484E+01 19 2 -9.708590E+00 2.581256E+01 32 3 -2.609443E+01 5.180791E+01 43 4 -8.665526E+01 1.568921E+02 58 5 -2.236631E+02 3.995060E+02 70 6 -9.364992E+02 1.667050E+03 88 7 -3.110639E+03 5.535421E+03 Continue IBM 103 8 -1.819048E+04 3.236855E+04 125 9 -1.357475E+05 2.415514E+05 150 10 -6.257971E+05 1.113554E+06 169 11 -6.991078E+06 1.244004E+07 199 12 -9.174644E+07 1.632552E+08 231 13 -6.856016E+08 1.219971E+09 256 14 -1.712571E+10 3.047378E+10 296 15 0.346755 1.578489 5.5 1 AT&T Reject 0.549381 1.782911 7.33 2 -1.197242E+01 2.915316E+01 0.5 1 -9.029384E+00 3.620727E+01 1.7 2 -1.761479E+01 6.114396E+01 4.5 3 -5.297372E+01 1.243779E+02 7.2 4 -1.608839E+02 3.145575E+02 10 5 -5.273155E+02 9.641456E+02 13 6 -1.085730E+03 1.956316E+03 14.8 7 -1.563321E+03 2.805418E+03 15.7 8 -2.768828E+03 4.949402E+03 17.1 9 -1.181216E+04 2.103861E+04 20.6 10 -4.952239E+04 8.813843E+04 24 11 -8.249745E+04 1.468140E+05 25.2 12 LYU Continue -1.211256E+05 2.155489E+05 26.1 13 -2.508898E+05 4.464526E+05 27.8 14 -4.581086E+05 8.151803E+05 29.2 15 -1.470789E+06 2.617159E+06 31.9 16 -5.904594E+06 1.050674E+07 35.1 17 -1.757032E+07 3.126493E+07 37.6 18 -4.214090E+07 7.498623E+07 39.6 19 -3.036268E+08 5.402786E+08 44.1 20 -1.417638E+09 2.522570E+09 47.6 21 -1.407760E+10 2.504991E+10 52.8 22 -3.410741E+11 6.069131E+11 60 23 -3.942281E+13 7.014960E+13 70.7 24 From the Table 5.2, a decision of either to accept, reject the system or continue is reached much in advance of the last time instant of the data. 283
  8. 8. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 5, September - October (2013), © IAEME VI. CONCLUSION The above consolidated table of HLSRGM as exemplified for five Data Sets indicates that the model is performing well in arriving at a decision. The model has given a decision of acceptance for 2 Data Sets i.e Xie and NTDS at 1st and 1st instances respectively, a decision of rejection for 1 Data Sets i.e AT&T at 2nd instance and a decision of continue for 2 Data Sets i.e IBM and LYU. Therefore, we may conclude that, applying SPRT on data sets we can come to an early conclusion of reliability / unreliability of software. VII. REFERENCES 1. Lyu, M.R. (1996). “The hand book of software reliability engineering”, McGrawHill & IEEE Computer Society press. 2. Pham. H., (2006). “System software reliability”, Springer. 3. Satya Prasad (2007).”Half logistic Software reliability growth model “Ph.D Thesis of ANU, India. 4. Satyaprasad. R, Shaheen and Krishna Mohan. G., (July-August, 2013). “Two Step approach for Software Process control:HLSRGM”, International Journal of Emerging Trends & Technology in Computer Science(IJETTCS), Volume 2, Issue 4. 5. Stieber, H.A. (1997). “Statistical Quality Control: How To Detect Unreliable Software Components”, Proceedings the 8th International Symposium on Software Reliability Engineering, 8-12. 6. Wald. A., 1947. “Sequential Analysis”, John Wiley and Son, Inc, New York. 7. Xie, M., Goh. T.N., Ranjan.P. (2002). “Some effective control chart procedures for reliability monitoring” -Reliability engineering and System Safety 77 143 -150. 8. Sandeep P. Chavan, Dr. S. H. Patil and Amol K. Kadam, “Developing Software Analyzers Tool using Software Reliability Growth Model for Improving Software Quality”, International Journal of Computer Engineering & Technology (IJCET), Volume 4, Issue 2, 2013, pp. 448 - 453, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375. AUTHOR’S PROFILE Dr. R. Satya Prasad Received Ph.D. degree in Computer Science in the faculty of Engineering in 2007 from Acharya Nagarjuna University, Andhra Pradesh. He received gold medal from Acharya Nagarjuna University for his outstanding performance in a first rank in Masters Degree. He is currently working as Associative Professor in the Department of Computer Science & Engineering, Acharya Nagarjuna University. His current research is focused on Software Engineering. He published 70 research papers in National & International Journals. Miss. Shaheen working as Assistant Professor, Institute of Public Enterprise (IPE), Osmania Univeristy Campus. She did her Master of Computer Science. She was associated with INFOSYS, Gacchibowli, Hyderabad and Genpact, Uppal as an Academic Consultant training fresh recruits on technical subjects. She is currently pursuing Ph. D. in Computer Science from Acharya Nagarjuna University. Mr. G. Krishna Mohan, working as a Reader in the Department of Computer Science, P.B.Siddhartha College, Vijayawada. He obtained his M.C.A degree from Acharya Nagarjuna University, M.Tech from JNTU, Kakinada, M.Phil from Madurai Kamaraj University and pursuing Ph.D from Acharya Nagarjuna University. He qualified, AP State Level Eligibility Test. His research interests lies in Data Mining and Software Engineering. He published 19 research papers in various National and International journals. 284

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