By Qi-Ming He
Fundamentals of Matrix-Analytic Methods pursuits advanced-level scholars in arithmetic, engineering and machine technology. It makes a speciality of the basic components of Matrix-Analytic tools, Phase-Type Distributions, Markovian arrival procedures and dependent Markov chains and matrix geometric strategies.
New fabrics and strategies are provided for the 1st time in examine and engineering layout. This booklet emphasizes stochastic modeling through providing probabilistic interpretation and confident proofs for Matrix-Analytic equipment. Such an technique is principally worthy for engineering research and layout. workouts and examples are supplied during the book.
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Example text
The probabilistic approach used in the above proof will become increasingly useful not only in the development of matrix-analytic methods in this and the next two chapters, but also in stochastic modeling of queueing and inventory models in Chaps. 4 and 5. 1, explain intuitively that the 2-tuple (β α, S I + I T) is also a PH-representation of X ¼ min{X1, X2}. 1 to X ¼ min{X1, . , Xn}) Assume that {X1, . , Xn} are independent and have PH-distributions. Show that X ¼ min{X1, . , Xn} has a PH-distribution.
0. Denote by Xλ the random variable corresponding to (α, T À λI). Show that Xλ is stochastically larger than Xμ, if λ < μ. Show that E[Xλ] ! E[Xμ]. (Hint: Use exp((T À λI)t) ¼ eÀλtexp(Tt). ) For n random variables {X1, X2, . , Xn}, their k-th order statistic is defined as the k-th smallest and denoted as X[k], for k ¼ 1, . , n. It is easy to see X[1] ¼ min {X1, X2, . , Xn} and X[n] ¼ max{X1, X2, . , Xn}. 26* Show that all the order statistics of n independent phase-type random variables have phase-type distributions.
0} and set J(t) ¼ 2. If I2(t) ¼ 2, set J(t) ¼ 3. The process {J(t), t ! 0} stays in state 3, once it enters state 3. It is clear that {J(t), t ! 0} is a continuous time Markov chain. Then it is easy to see that X1 + X2 is the time until absorption into state 3 for the Markov chain {J(t), t ! 0}. 3). 1 can be generalized as follows. 32) 24 1 From the Exponential Distribution to Phase-Type Distributions respectively. Denote by {I1(t), t ! 0} and {I2(t), t ! 0} the underlying Markov chains associated with the two PH-representations.