Several software reliability growth models （SRGM） have been developed to monitor the reliability growth during the testing phase of software development. In most of the existing research available in the literatures...Several software reliability growth models （SRGM） have been developed to monitor the reliability growth during the testing phase of software development. In most of the existing research available in the literatures, it is considered that a similar testing effort is required on each debugging effort. However, in practice, different types of faults may require different amounts of testing efforts for their detection and removal. Consequently, faults are classified into three categories on the basis of severity： simple, hard and complex. This categorization may be extended to r type of faults on the basis of severity. Although some existing research in the literatures has incorporated this concept that fault removal rate （FRR） is different for different types of faults, they assume that the FRR remains constant during the overall testing period. On the contrary, it has been observed that as testing progresses, FRR changes due to changing testing strategy, skill, environment and personnel resources. In this paper, a general discrete SRGM is proposed for errors of different severity in software systems using the change-point concept. Then, the models are formulated for two particular environments. The models were validated on two real-life data sets. The results show better fit and wider applicability of the proposed models as to different types of failure datasets.展开更多
Statisticians are usually concerned with the proposition of new distributions. In this paper we point out that a unified and concise derivation procedure of the distribution of the minimum or maximum of a random numbe...Statisticians are usually concerned with the proposition of new distributions. In this paper we point out that a unified and concise derivation procedure of the distribution of the minimum or maximum of a random number N of indepen-dent and identically distributed continuous random variables Yi,{i = 1,2,…,N} is obtained if one compounds the probability generating function of N with the survival or the distribution func-tion of Yi. Expressions are then derived in closed form for the density, hazard and quantile func-tions of the minimum or maximum. The methodology is illustrated with examples of the distributions proposed by Adamidis and Loukas (1998), Kus (2007), Tahmasbi and Rezaei (2008), Barreto-Souza and Cribari-Neto (2009), Cancho, Louzada, and Barriga (2011) and Louzada, Roman and Cancho (2011).展开更多
In this paper,we develop an M/M/c queueing system in a Markovian environment with waiting servers,balking and reneging,under both synchronous single and multiple working vacation policies.When the system is in operati...In this paper,we develop an M/M/c queueing system in a Markovian environment with waiting servers,balking and reneging,under both synchronous single and multiple working vacation policies.When the system is in operative phase j,j=1,K¯,customers are served one by one.Once the system is empty,the servers have to wait a random period of time before leaving,causing the system to move to vacation phase 0 at which new arrivals can be served at lower rate.Using the method of the probability generating functions,we establish the steady-state analysis of the system.Special cases of the queueing model are presented.Then,explicit expressions of the useful system characteristics are derived.In addition,a cost model is constructed to define the optimal values of service rates,simultaneously,to minimize the total expected cost per unit time via a quadratic fit search method.Numerical examples are provided to display the impact of different system characteristics.展开更多
In this paper,we discuss some important aspects of the bivariate alternative zero inflated log-arithmic series distribution(BAZILSD)of which the marginals are the alternative zero-inflated logarithmic series ditributi...In this paper,we discuss some important aspects of the bivariate alternative zero inflated log-arithmic series distribution(BAZILSD)of which the marginals are the alternative zero-inflated logarithmic series ditributions of Kumar and Riyaz(2015.An alternative version of zero-inflated logarithmic series distribution and some of its applications.Journal of Statistical Computation and Simulation,85(6),1117-1127).We study some important properties of the distribution by deriving expressions for its probability mass function,factorial moments,conditional probabil-ity generating functions,and recursion formulae for its probilities,raw moments and factorial moments.The parameters of the BAZILSD are estimated by the method of maximum likelihood and certain test procedures are also considered.Further certain real-life data applications are cited for ilustrating the usefulness of the model.A simulation study is conducted for assessing the performance of the maximum likelihood estimators of the parameters of the BAZILSD.展开更多
In this paper,we consider the impatient customers in M/M/1 queueing model under variant working vacation policy.The customer’s impatience is due to its arrival during a working vacation period,where the service rate ...In this paper,we consider the impatient customers in M/M/1 queueing model under variant working vacation policy.The customer’s impatience is due to its arrival during a working vacation period,where the service rate of the customer is lower than a normal busy period.If the system is non-empty when the server returns from the working vacation,the server resumes the normal service period.Otherwise,the server will take successive working vacations till it reaches the maximum number of K working vacations and then the server remains idle until the next arrival.Closed-form probabilities are obtained by using the identities involving beta functions and degenerate hypergeometric functions,and the performance measures of the system are derived using generating functions.The stochastic decomposition structures of the mean queue length and mean waiting time are verified.The effects of the system parameters on some performance measures had been numerically illustrated.展开更多
We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtai...We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities,mean and variance of the system size at time t by employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.展开更多
We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the proce...We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.展开更多
In this paper we study a Geo/T-IPH/1 queue model,where T-IPH denotes the discrete time phase type distribution defined on a birth-and-death process with countably many states.The queue model can be described by a quas...In this paper we study a Geo/T-IPH/1 queue model,where T-IPH denotes the discrete time phase type distribution defined on a birth-and-death process with countably many states.The queue model can be described by a quasi-birth-anddeath(QBD)process with countably phases.Using the operator-geometric solution method,we first give the expression of the operator and the joint stationary distribution.Then we obtain the probability generating function(PGF)for stationary queue length distribution and sojourn time distribution,respectively.展开更多
Technology of cognitive radio networks has emerged as an effective method to enhance the utilization of the radio spectrum where the primary users have priority to use the spectrum, and the secondary users try to expl...Technology of cognitive radio networks has emerged as an effective method to enhance the utilization of the radio spectrum where the primary users have priority to use the spectrum, and the secondary users try to exploit the spectrum unoccupied by the primary users. In this paper, considering the non-saturated condition, the performance analysis for the IEEE 802.11-based cognitive radio networks is presented with single-channel and multi-channel, respectively. For the single-channel case, an absorbing Markov chain model describing the system transitions is constructed, and one-step transition probability matrix of the Markov chain is given. By using the method of probability generating function, the non-saturated throughput of the secondary users is obtained. For the multi-channel case, taking into account the negotiation-based sensing policy, the mean number of unused channels perceived by the second users is given, and then the non-saturated aggregate throughput of the secondary users is derived. Finally, numerical examples are provided to show the influences of the non-saturated degree, the number of the secondary users and the channel utilization of the primary users on the performance measures for the non-saturated throughput with single-channel and the non-saturated aggregate throughput with multi-channel.展开更多
In this paper, we present a random graph model with spatial reuse for a mobile ad hoc network （MANET） based on the dynamic source routing protocol. Many important performance parameters of the MANET are obtained, su...In this paper, we present a random graph model with spatial reuse for a mobile ad hoc network （MANET） based on the dynamic source routing protocol. Many important performance parameters of the MANET are obtained, such as the average flooding distance （AFD）, the probability generating function of the flooding distance, and the probability of a flooding route to be symmetric. Compared with the random graph model without spatial reuse, this model is much more effective because it has a smaller value of AFD and a larger probability for finding a symmetric valid route.展开更多
文摘Several software reliability growth models （SRGM） have been developed to monitor the reliability growth during the testing phase of software development. In most of the existing research available in the literatures, it is considered that a similar testing effort is required on each debugging effort. However, in practice, different types of faults may require different amounts of testing efforts for their detection and removal. Consequently, faults are classified into three categories on the basis of severity： simple, hard and complex. This categorization may be extended to r type of faults on the basis of severity. Although some existing research in the literatures has incorporated this concept that fault removal rate （FRR） is different for different types of faults, they assume that the FRR remains constant during the overall testing period. On the contrary, it has been observed that as testing progresses, FRR changes due to changing testing strategy, skill, environment and personnel resources. In this paper, a general discrete SRGM is proposed for errors of different severity in software systems using the change-point concept. Then, the models are formulated for two particular environments. The models were validated on two real-life data sets. The results show better fit and wider applicability of the proposed models as to different types of failure datasets.
文摘Statisticians are usually concerned with the proposition of new distributions. In this paper we point out that a unified and concise derivation procedure of the distribution of the minimum or maximum of a random number N of indepen-dent and identically distributed continuous random variables Yi,{i = 1,2,…,N} is obtained if one compounds the probability generating function of N with the survival or the distribution func-tion of Yi. Expressions are then derived in closed form for the density, hazard and quantile func-tions of the minimum or maximum. The methodology is illustrated with examples of the distributions proposed by Adamidis and Loukas (1998), Kus (2007), Tahmasbi and Rezaei (2008), Barreto-Souza and Cribari-Neto (2009), Cancho, Louzada, and Barriga (2011) and Louzada, Roman and Cancho (2011).
文摘In this paper,we develop an M/M/c queueing system in a Markovian environment with waiting servers,balking and reneging,under both synchronous single and multiple working vacation policies.When the system is in operative phase j,j=1,K¯,customers are served one by one.Once the system is empty,the servers have to wait a random period of time before leaving,causing the system to move to vacation phase 0 at which new arrivals can be served at lower rate.Using the method of the probability generating functions,we establish the steady-state analysis of the system.Special cases of the queueing model are presented.Then,explicit expressions of the useful system characteristics are derived.In addition,a cost model is constructed to define the optimal values of service rates,simultaneously,to minimize the total expected cost per unit time via a quadratic fit search method.Numerical examples are provided to display the impact of different system characteristics.
文摘In this paper,we discuss some important aspects of the bivariate alternative zero inflated log-arithmic series distribution(BAZILSD)of which the marginals are the alternative zero-inflated logarithmic series ditributions of Kumar and Riyaz(2015.An alternative version of zero-inflated logarithmic series distribution and some of its applications.Journal of Statistical Computation and Simulation,85(6),1117-1127).We study some important properties of the distribution by deriving expressions for its probability mass function,factorial moments,conditional probabil-ity generating functions,and recursion formulae for its probilities,raw moments and factorial moments.The parameters of the BAZILSD are estimated by the method of maximum likelihood and certain test procedures are also considered.Further certain real-life data applications are cited for ilustrating the usefulness of the model.A simulation study is conducted for assessing the performance of the maximum likelihood estimators of the parameters of the BAZILSD.
文摘In this paper,we consider the impatient customers in M/M/1 queueing model under variant working vacation policy.The customer’s impatience is due to its arrival during a working vacation period,where the service rate of the customer is lower than a normal busy period.If the system is non-empty when the server returns from the working vacation,the server resumes the normal service period.Otherwise,the server will take successive working vacations till it reaches the maximum number of K working vacations and then the server remains idle until the next arrival.Closed-form probabilities are obtained by using the identities involving beta functions and degenerate hypergeometric functions,and the performance measures of the system are derived using generating functions.The stochastic decomposition structures of the mean queue length and mean waiting time are verified.The effects of the system parameters on some performance measures had been numerically illustrated.
基金Supported by the National Natural Science Foundation of China(11671204)
文摘We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities,mean and variance of the system size at time t by employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.
基金the National Natural Sciente Foundation of China (Grant Nos. 10771021, 10471012)Scientific Research Foundation for Returned Scholars, Ministry of Education of China (Grant No. [2005]564)
文摘We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.
基金This work was supported by the National Natural Science Foundation of China(No.61174160)Post-Doctoral Science Foundation of Central South University(No.125011)The authors are grateful to the three anonymous referees and the editor for their careful reading and invaluable comments and suggestions,which are helpful to improve the paper.This research completed when the first author was a post-doctoral fellow in School of Mathematics,Central South University。
文摘In this paper we study a Geo/T-IPH/1 queue model,where T-IPH denotes the discrete time phase type distribution defined on a birth-and-death process with countably many states.The queue model can be described by a quasi-birth-anddeath(QBD)process with countably phases.Using the operator-geometric solution method,we first give the expression of the operator and the joint stationary distribution.Then we obtain the probability generating function(PGF)for stationary queue length distribution and sojourn time distribution,respectively.
基金supported by the National Natural Science Foundation of China(71101124, 11201408)the Natural Science Foundation of Heibei Province(2012203093)
文摘Technology of cognitive radio networks has emerged as an effective method to enhance the utilization of the radio spectrum where the primary users have priority to use the spectrum, and the secondary users try to exploit the spectrum unoccupied by the primary users. In this paper, considering the non-saturated condition, the performance analysis for the IEEE 802.11-based cognitive radio networks is presented with single-channel and multi-channel, respectively. For the single-channel case, an absorbing Markov chain model describing the system transitions is constructed, and one-step transition probability matrix of the Markov chain is given. By using the method of probability generating function, the non-saturated throughput of the secondary users is obtained. For the multi-channel case, taking into account the negotiation-based sensing policy, the mean number of unused channels perceived by the second users is given, and then the non-saturated aggregate throughput of the secondary users is derived. Finally, numerical examples are provided to show the influences of the non-saturated degree, the number of the secondary users and the channel utilization of the primary users on the performance measures for the non-saturated throughput with single-channel and the non-saturated aggregate throughput with multi-channel.
基金the National Natural Science Foundation of China(No.10471088No.60572126)the Key Foundation for research of Shanghai Municipal Education Commission(No.06ZZ84)
文摘In this paper, we present a random graph model with spatial reuse for a mobile ad hoc network （MANET） based on the dynamic source routing protocol. Many important performance parameters of the MANET are obtained, such as the average flooding distance （AFD）, the probability generating function of the flooding distance, and the probability of a flooding route to be symmetric. Compared with the random graph model without spatial reuse, this model is much more effective because it has a smaller value of AFD and a larger probability for finding a symmetric valid route.
