Submit Manuscript | http://medcraveonline.com Introduction Reliability of a system is an important aspect of any system design since any user of the system would expect some type of guarantee that the system will function to some level of confdence. Failing to meet such guarantee will result in disastrous consequences. On the other hand, overly exceeding such guarantee level may incur additional and unnecessary expense to the developers. Moreover, for any non-trivial software system, an exhaustive testing among the entire input domain can be very expensive. By adopting the partition testing strategy, we attempt to break up the testable input domain of possible test cases into partitions, which must be non-overlapping, such that if test case i belongs to partition j , then no partition other than j will contain i . Sayre and Poore[11] have given several possible mechanics to partition the domain into fnitely many subdomains, { 1,if test taken from partition is processed correctly 0, otherwise ij j i X = , such that : 1 ; , k i i j i D D D D i j = = =∅ ∩ ≠  which allows us to defne the system reliability by a weighted sum of reliabilities of these subdomains, i.e. 1 k i i i R pR = = ∑ Where R denotes the system reliability and i R is the reliability of each subdomain i D ; and i p , parameters of the operational profle is the likelihood of this test case belongs to partition i D , which are assumed to be known. 12 As mentioned above, a complete testing of any software system of non-trivial size is practically impossible, i R are usually unknown parameters to us. So as to gain knowledge about i R , we must distribute the k test cases among these k partitions, and generate reasonable estimates for each . Specifcally, we denote 1 2 , , , k n n n … as sizes of the samples which are taken from sub domain 1 2 , , ,' k D D D … , respectively, where 1 k i i n N = ∑ = . We model the outcome of the th j taken from the th i partition as a Bernoulli random variable ij X such that: { 1,if test taken from partition is processed correctly 0, otherwise ij j i X = and each ij X follows a Bernoulli distribution with parameter i R . Then, the estimate of the overall system reliability R , denoted by ˆ R can thus be defned as: 1 ˆ ˆ k i R p i R i = =〉 where ˆ i R is the estimate of i R after i n test cases have been allocated to partition such that: 1 ˆ n i ij j i i X n R = ∑ = and () ( ) 2 1 1 ˆ i i i i i k p R R R Var n = − = ∑ (1.1) Optimal sampling scheme Ideally, we would like to execute all possible test paths through the software and determine the true overall reliability of the system. In practice though, resources are often limited, sample test cases must be chosen and allocated strategically to attain the best reliability estimate possible given all kinds of constraints. One of the criteria of distributing test cases among the partitions, which proceeds from rewriting (1.1) as follows: Biom Biostat Int J. 2015;2(4):109‒113. 109 ©2015 Rekab et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Second order optimality of sequential designs with application in software reliability estimation Volume 2 Issue 4 - 2015 Kamel Rekab, Xing Song Department of Mathematics and Statistics, University of Missouri-Kansas City, USA Correspondence: Kamel Rekab, Department of Mathematics and Statistics, University of Missouri-Kansas City, PO Box 32464 Kansas City, MO 64171, USA, Tel: 816-269-4432; Email Received: April 11, 2015 | Published: April 29, 2015 Abstract We propose three efficient sequential designs in the software reliability estimation. The fully sequential design the multistage sequential design and the accelerated sequential design. These designs make allocation decisions dynamically throughout the testing process. We then refine these estimated reliabilities in an iterative manner as we sample. Monte Carlo simulation seems to indicate that these sequential designs are second order optimal. Keywords: software reliability, partition testing, fully sequential design, multistage sequential design, accelerated sequential design. Biometrics & Biostatistics International Journal Research Article Open Access () ( ) ( ) ( ) 2 2 1 1 1 1 1 1 1 1 ˆ k k k i i i i j j j j i i i i i j i i j p R R np R R np R R Var R N N nn − = = =+ − − − − = +         ∑ ∑∑