Consider a period reversible nonstop time Markov chain having tiny change rates q

furthermore, restricting probabilities {P). Allow A to mean a bunch of states for this chain, and consider another

constant time Markov chain with change rates q/given by

CLIij in the event that I A , j

something else

where c is a discretionary positive number. snow that this chain remains time reversible, and discover its

restricting probabilities.

Q58

Consider an arrangement of n segments to such an extent that the functioning occasions of segment

are dramatically dispersed with rate Ar. At the point when a segment fizzles, be that as it may, the maintenance rate ot

part I relies upon the number of different segments are down. In particular, assume that the

immediate fix pace of part j, j = 1

, n, when there are a sum of k fizzled

parts, is

(a) Explain now we can break down the former as a ceaseless time Markov chain. Characterize the

states and give the boundaries ot the chain.

(b) Show that, in consistent express, the chain is time reversible and register the restricting probabilities.

Q59

Allow Y to signify an outstanding arbitrary variable witn rate A that is autonomous ot the nonstop

time Markov chain {X(t)) and let

= jlX(O) = i}

(a) Show that

C/ikPkj +

iJ

where .is 1 when I = j and O when

(b) Show that the arrangement ot the previous set ot conditions is given by

where is the lattice ot components Pij, I is the character framework, and R the network determined in

Segment 6.9.

(c) Suppose now that VI,

{X(t)}. snow that

, Yn are autonomous exponentials with rate A that are free ot

is equivalent to the component in line I, section j ottne grid Fn.

(c) Explain tne relaonsnp 0T me going before to Approximation 2 0T area 6.9.

Q60

(a) Show that Approximation 1 of Section 6.9 is comparable to unitormizing the consistent time

Markov chain with a worth v such thatvt n and afterward approximating Pij(t) by Pij

(b) Explain why the former should make a decent estimate.

Clue: What is the standard deviation of a Poisson arbitrary variable with mean N?