Question

( Markov jump process and Kolmogorov forward equation ) In a particular company,the salary scale has only two different levels. On average, an employee spends 2 years at level 1 before moving on to the higher level, or leaving the company. An employee at the maximum level spends an average of 5 years before leaving. Nobody is demoted, and promotion can occur at any time. Upon leaving level 1, the probability that an employee moves to level 2 is 50%. (a) Explain how you could model this as a Markov jump process, commenting on any assumptions that you make. Draw the state diagram of the Markov process. (b) From the average waiting time, find the diagonal element of the intensity matrix,and the transition probability matrix of the jump Markov process. c) Show that the intensity matrix of the Markov jump process is given by Q=\left(\begin{array}{ccc} -0.50 & 0.25 & 0.25 \\ 0 & -0.20 & 0.20 \\ 0 & 0 & 0 \end{array}\right) (d) Use the Kolmogorov's forward equation to show that the probability of moving from level 1 to level 2 in t periods of time is given by P_{12}(t)=\frac{5}{6}\left(e^{-0.2 t}-e^{-0.5 t}\right)

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