QP October 2014 Q2. IV

a) How the transition matrix is constructed? I did not get how they got \(\frac {\lambda}{\lambda + \mu} \) for transition from state 1 to state 2. Similarly, all those on right ends in each row.

I tried using \(\frac{\mu_{ij}}{\lambda_i}\). My logic looks contradictory if substituted. Can someone help me identify the missing link?

b)I did not understand the logic. If someone explains it clearly it would be of great help.

Thank you.

 

It is QP October 2014 Q2. VI not IV.

Sorry for the inconvenience.

 

a)
transition matrix is constructed on the basis of the probabilities for state i to j, placed at i^th row and j^th column of the matrix......
as jump process at the time of jump is itself a markov chain, we calculate the probability as jump to state j, given than it jumped i.e. like \(\frac {\lambda}{\lambda + \mu} \)...........
\(\frac{\mu_{ij}}{\lambda_i}\) is also correct as you know \( \lambda_i = \lambda + \mu \)...........( \( \lambda_i \) is force out of state i ).

b)
they ask,probability of currently there are no passenger has arrived and then all desk will be full before anyone can complete. i.e. go to state N from 0 without rollback.....

Edit: I just see, they mixed lambda and mu mistakenly.