A Commnuications device has 3 modes; standyby, receive or send. It is modelled by a Markov Chain(MC) in continous times with states 1,2 and 3 corresponding to these modes in the above order. The transition probability rate out of any mode is always 2. If it is in standyby mode it is equally likely to switch to receive or send. Otherwise it is twice as likely to switch to receive or send as to switch back to standby. Write down the transition probability rate matrix Q for this MC. Show that this MC has an asymptotic distribution and find it.
Markov Chain Question
- Thread starter mamoth
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pka
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Yes, I saw your posting at another website.
I doubt that you are going to receive any help on those problems.
The difficultly is the matter of definitions.
For example, I have taught ‘Markov Chains’ and I do not understand these questions! That is to say that these questions depend upon the definitions in your text material. We do not know those definitions!
I doubt that you are going to receive any help on those problems.
The difficultly is the matter of definitions.
For example, I have taught ‘Markov Chains’ and I do not understand these questions! That is to say that these questions depend upon the definitions in your text material. We do not know those definitions!
mamoth said:
If I recall correctly, the Q matrix is as follows:
r(i,j) = q(i,j)/lamda_i
where Q(i,j) = q(i,j) if i != j and -lamba_i otherwise.
so, since lamda_i = 2:
Q = [-2 1 1;
2/3 -2 4/3;
2/3 4/3 -2 ];
since:
q(1,2) = 1/2*2
q(1,3) = 1/2*2
and were told that:
p + 2p = 1 or
p(standby) = 1/3,
p(receive) = 2/3
so:
q(2,1) = 2*1/3 = 2/3
q(2,3) = 2*2/3 = 4/3
and
q(3,1) = 2*1/3 = 2/3
q(3,2) = 2*2/3 = 4/3
I'm thinking that these are correct since I just started working on problems like these myself.
PS: I checked this out with the problems that I was working on and my facts seem OK. My interpretation of the Q matrix is that the row entries are "thinned poissons."