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 rmcf87 March 12th, 2009 12:35 AM

Markov Chain (transition matrix)

Hey can anyone please help me obtain the transition of the markov chain with states being the number of projects the architect has in hand at the beginning of a month.

Quote:
 An architect can handle up to 3 projects at once. During any month there is a probability of 0.5 of acquiring one new project (to start the next month) provided the architect knows that the project can be handled, otherwise no new projects are acquired. That is, the architect decides to take the honourable course of not taking on a new project for the next month unless it is known that it can be handled. Existing projects are equally-likely to finish by the start of the next month or to continue, independently of each other.
Thank you.

 mattpi March 12th, 2009 07:51 AM

Re: Markov Chain (transition matrix)

Try writing everything in terms of what can happen, with what probabilities. For instance, if $X_n=i$ and $X_{n+1}=j,$ then:

$i=0,\ j=0\ \Rightarrow$ no new projects are acquired, prob. 0.5.
$i=0,\ j=1\ \Rightarrow$ a new project is acquired, prob. 0.5
$i=0,\ j>1\ \Rightarrow=$ not possible - only one project can be acquired in any month, prob. 0

$i=1,\ j=0\ \Rightarrow$ a project is finished and no new projects are acquired, prob. 0.5 x 0.5
$i=1,\ j=1\ \Rightarrow$ ?
$i=1,\ j=2\ \Rightarrow$ ?

etc.

Remember that $\sum_{j\in S} p_{ij}=1$ for any $i\in S$ where $S$ is the state space.

 rmcf87 March 12th, 2009 02:18 PM

Re: Markov Chain (transition matrix)

Thank you very much for your reply. And I can see what you're trying to say, I understand it now, thanks again.

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