Order statistics of Matrix-Geometric distributions Azucena Campillo - PowerPoint PPT Presentation

Motivation Order statistics for independent Matrix-geometric distributions Conclusion Order statistics of Matrix-Geometric distributions Azucena Campillo Navarro 1 , Bo Friis Nielsen 1 , Mogens Bladt 2 . 1 Technical University of Denmark Department of Applied Mathematics and Compute Science. 2 Autonomous National University of Mexico. Budapest, Hungary, June 2016. 1 / 20

Motivation Order statistics for independent Matrix-geometric distributions Conclusion Outline 1. Motivation: Maximum and minimum of two independent phase-type distributions. 2. The Maximum of three independent Matrix-geometric distributions. 3. Generalization: The r-th order statistics of n independent Matrix-geometric distributions. 2 / 20

Motivation Order statistics for independent Matrix-geometric distributions Conclusion Outline 1. Motivation: Maximum and minimum of two independent phase-type distributions. 2. The Maximum of three independent Matrix-geometric distributions. 3. Generalization: The r-th order statistics of n independent Matrix-geometric distributions. 2 / 20

Motivation Order statistics for independent Matrix-geometric distributions Conclusion Outline 1. Motivation: Maximum and minimum of two independent phase-type distributions. 2. The Maximum of three independent Matrix-geometric distributions. 3. Generalization: The r-th order statistics of n independent Matrix-geometric distributions. 2 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion The Maximum and Minimum Let’s consider two Markov chains: � X 1 � � X 2 � and n ∈ N . n n n ∈ N The space of states are given by E 1 and E 2 , respectively. In both of them it is suppose that the states are transient, except one, which is absorbing. Let α 1 and α 2 , be the initial distributions of the corresponding Markov chains. Let � S 1 � S 2 � � s 1 s 2 Λ 1 = , Λ 2 = , 1 1 0 0 be the transition probability matrices of the corresponding Markov chains, where s i = e − S i e , i = 1 , 2 . 3 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion The Maximum and Minimum Let’s consider two Markov chains: � X 1 � � X 2 � and n ∈ N . n n n ∈ N The space of states are given by E 1 and E 2 , respectively. In both of them it is suppose that the states are transient, except one, which is absorbing. Let α 1 and α 2 , be the initial distributions of the corresponding Markov chains. Let � S 1 � S 2 � � s 1 s 2 Λ 1 = , Λ 2 = , 1 1 0 0 be the transition probability matrices of the corresponding Markov chains, where s i = e − S i e , i = 1 , 2 . 3 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion The Maximum and Minimum Let’s consider two Markov chains: � X 1 � � X 2 � and n ∈ N . n n n ∈ N The space of states are given by E 1 and E 2 , respectively. In both of them it is suppose that the states are transient, except one, which is absorbing. Let α 1 and α 2 , be the initial distributions of the corresponding Markov chains. Let � S 1 � S 2 � � s 1 s 2 Λ 1 = , Λ 2 = , 1 1 0 0 be the transition probability matrices of the corresponding Markov chains, where s i = e − S i e , i = 1 , 2 . 3 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion The Maximum and Minimum Let’s consider two Markov chains: � X 1 � � X 2 � and n ∈ N . n n n ∈ N The space of states are given by E 1 and E 2 , respectively. In both of them it is suppose that the states are transient, except one, which is absorbing. Let α 1 and α 2 , be the initial distributions of the corresponding Markov chains. Let � S 1 � S 2 � � s 1 s 2 Λ 1 = , Λ 2 = , 1 1 0 0 be the transition probability matrices of the corresponding Markov chains, where s i = e − S i e , i = 1 , 2 . 3 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Let Y 1 ∼ DPH ( α 1 , S 1 ) and Y 2 ∼ DPH ( α 2 , S 2 ) , which are independent. Denote Y (1) = m´ ın ( Y 1 , Y 2 ) , Y (2) = m´ ax ( Y 1 , Y 2 ) , the first and the second order statistics. 4 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Let Y 1 ∼ DPH ( α 1 , S 1 ) and Y 2 ∼ DPH ( α 2 , S 2 ) , which are independent. Denote Y (1) = m´ ın ( Y 1 , Y 2 ) , Y (2) = m´ ax ( Y 1 , Y 2 ) , the first and the second order statistics. 4 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain Consider the multivariable Markov chain X 1 n , X 2 � � { X n } = , n ∈ N . n Suppose that E 1 = { 1 , 2 , 3 } and E 2 = { 1 , 2 , 3 } are the space of states of the corresponding Markov chains. Consequently, the space of state of the multivariable Markov chain is: { (1 , 1) , (1 , 2) , (2 , 1) , (2 , 2) , (3 , 1) , (3 , 2) , (1 , 3) , (2 , 3) , (3 , 3) } . 5 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain Consider the multivariable Markov chain X 1 n , X 2 � � { X n } = , n ∈ N . n Suppose that E 1 = { 1 , 2 , 3 } and E 2 = { 1 , 2 , 3 } are the space of states of the corresponding Markov chains. Consequently, the space of state of the multivariable Markov chain is: { (1 , 1) , (1 , 2) , (2 , 1) , (2 , 2) , (3 , 1) , (3 , 2) , (1 , 3) , (2 , 3) , (3 , 3) } . 5 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain Consider the multivariable Markov chain X 1 n , X 2 � � { X n } = , n ∈ N . n Suppose that E 1 = { 1 , 2 , 3 } and E 2 = { 1 , 2 , 3 } are the space of states of the corresponding Markov chains. Consequently, the space of state of the multivariable Markov chain is: { (1 , 1) , (1 , 2) , (2 , 1) , (2 , 2) , (3 , 1) , (3 , 2) , (1 , 3) , (2 , 3) , (3 , 3) } . 5 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain X n (3,3) (2,3) (1,3) (3,2) (3,1) (2,2) (2,1) (1,2) (1,1) n Y (1) Y (2) 6 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain X n (3,3) (2,3) (1,3) (3,2) (3,1) (2,2) (2,1) (1,2) (1,1) n Y (1) Y (2) 6 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain X n (3,3) (2,3) (1,3) (3,2) (3,1) (2,2) (2,1) (1,2) (1,1) n Y (1) Y (2) 6 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain X n (3,3) (2,3) (1,3) (3,2) (3,1) (2,2) (2,1) (1,2) (1,1) n Y (1) Y (2) 6 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain X n (3,3) (2,3) (1,3) (3,2) (3,1) (2,2) (2,1) (1,2) (1,1) n Y (1) Y (2) 6 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain X n (3,3) (2,3) (1,3) (3,2) (3,1) (2,2) (2,1) (1,2) (1,1) n Y (1) Y (2) 6 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain X n (3,3) (2,3) (1,3) (3,2) (3,1) (2,2) (2,1) (1,2) (1,1) n Y (1) Y (2) 6 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain X n (3,3) (2,3) (1,3) (3,2) (3,1) (2,2) (2,1) (1,2) (1,1) n Y (1) Y (2) 6 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain X n (3,3) (2,3) (1,3) (3,2) (3,1) (2,2) (2,1) (1,2) (1,1) n Y (1) Y (2) 6 / 20

Motivation Order statistics for independent Matrix-geometric distributions Order statistics of two independent discrete phase-type distributions Conclusion Multivariable Markov chain X n (3,3) (2,3) (1,3) (3,2) (3,1) (2,2) (2,1) (1,2) (1,1) n Y (1) Y (2) 6 / 20