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Computer Networking: A Top-Down Approach (7th Edition)

7th Edition

ISBN:9780133594140

Author:James Kurose, Keith Ross

Publisher:James Kurose, Keith Ross

Chapter1: Computer Networks And The Internet

Section: Chapter Questions

Problem R1RQ: What is the difference between a host and an end system? List several different types of end...

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Question

Use python code to solve

Job Mobility The lawyers at a law firm are either associates or partners. At the end of each year, 30% of the associates leave the firm, 20% are promoted to partner, and 50% remain associates. Also, 10% of the partners leave the firm at the end of each year. Assume that a lawyer who leaves the firm does not return.

Draw the transition diagram for this Markov process. Label the states A, P, and L.
Set up a stochastic matrix for the Markov process.
Find the state of the matrix that changes no more.
In the long run, what percent of the lawyers will be associates?

Expert Solution

Step 1: Algorithm:

Define the three states: A (associate), P (partner), and L (left the firm).
Define the stochastic matrix P as a 3x3 matrix with the following transition probabilities:
P[1,1] = 0.5 (probability of remaining an associate)
P[1,2] = 0.2 (probability of becoming a partner)
P[1,3] = 0.3 (probability of leaving the firm)
P[2,1] = 0.1 (probability of demotion to associate)
P[2,2] = 0.9 (probability of remaining a partner)
P[2,3] = 0.0 (probability of leaving the firm)
P[3,1] = 0.0 (no transition from state L)
P[3,2] = 0.0 (no transition from state L)
P[3,3] = 1.0 (absorbing state - probability of staying in state L)
Define the initial state vector x0 as a 3x1 vector with the initial probabilities of being in each state. For example, if initially 80% of lawyers are associates, 10% are partners, and 10% left the firm, then x0 would be [0.8, 0.1, 0.1].
Multiply the initial state vector x0 by the stochastic matrix P to get the state vector after one year: x1 = x0 @ P.
Repeat step 4 for as many years as you want to simulate.
To find the state of the matrix that changes no more, use the matrix_power function in NumPy to raise the matrix P to a large power (e.g., 100) and multiply it by the initial state vector x0: x_stable = np.linalg.matrix_power(P, 100) @ x0.
The stable state vector x_stable represents the long-run probabilities of being in each state. For example, the percentage of lawyers who will be associates in the long run is given by x_stable[0] * 100.
Output the results.