We study waiting time problems for first-order Markov dependent trials via conditional probability generating functions. Our models involve α frequency cells and β run cells with prescribed quotas and an additional ...We study waiting time problems for first-order Markov dependent trials via conditional probability generating functions. Our models involve α frequency cells and β run cells with prescribed quotas and an additional γ slack cells without quotas. For any given and , in our Model I we determine the waiting time until at least frequency cells and at least run cells reach their quotas. For any given τ ≤ α + β, in our Model II we determine the waiting time until τ cells reach their quotas. Computer algorithms are developed to calculate the distributions, expectations and standard deviations of the waiting time random variables of the two models. Numerical results demonstrate the efficiency of the algorithms.展开更多
In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1) i.e. we have a sequence {(Xn/Yn), n>=0} of S={(0/0),(0/1),(1/0),(1/1)...In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1) i.e. we have a sequence {(Xn/Yn), n>=0} of S={(0/0),(0/1),(1/0),(1/1)}-valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of (X0n,k10,X1n,k11;Y0n,k20,Y1n,k21) where for i=0,1,Xin,k1i denotes the number of occurrences of i-runs of length k1i in the first component and Yin,k2i denotes the number of occurrences of i-runs of length k2i in the second component of Markov dependent bivariate trials. Further we consider two patterns Λ1 and Λ2 of lengths k1 and k2 respectively and obtain the pgf of joint distribution of (Xn,Λ 1,Yn,Λ2 ) using method of conditional probability generating functions where Xn,Λ1(Yn,Λ2) denotes the number of occurrences of pattern Λ1(Λ2 ) of length k1 (k2) in the first (second) n components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distributions are studied using the joint distribution of runs.展开更多
文摘We study waiting time problems for first-order Markov dependent trials via conditional probability generating functions. Our models involve α frequency cells and β run cells with prescribed quotas and an additional γ slack cells without quotas. For any given and , in our Model I we determine the waiting time until at least frequency cells and at least run cells reach their quotas. For any given τ ≤ α + β, in our Model II we determine the waiting time until τ cells reach their quotas. Computer algorithms are developed to calculate the distributions, expectations and standard deviations of the waiting time random variables of the two models. Numerical results demonstrate the efficiency of the algorithms.
文摘In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1) i.e. we have a sequence {(Xn/Yn), n>=0} of S={(0/0),(0/1),(1/0),(1/1)}-valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of (X0n,k10,X1n,k11;Y0n,k20,Y1n,k21) where for i=0,1,Xin,k1i denotes the number of occurrences of i-runs of length k1i in the first component and Yin,k2i denotes the number of occurrences of i-runs of length k2i in the second component of Markov dependent bivariate trials. Further we consider two patterns Λ1 and Λ2 of lengths k1 and k2 respectively and obtain the pgf of joint distribution of (Xn,Λ 1,Yn,Λ2 ) using method of conditional probability generating functions where Xn,Λ1(Yn,Λ2) denotes the number of occurrences of pattern Λ1(Λ2 ) of length k1 (k2) in the first (second) n components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distributions are studied using the joint distribution of runs.
