construct the rate diagram

1. Consider a birth and death process with 4 attainable states: (0, 1, 2, 3). The steady state probabilities of these states are PO, PI, P2, and P3. The birth and death rates are summarized in the below table: State Birth Rate Death Rate 0 NNN 0 a) Construct the rate diagram for this birth and death process. b) Develop the balance equations c) Solve these equations to find PO, PI, P2, P3 (steady state probabilities). d) Calculate L.Consider a birth and death process X(t), t 2 0, such as the branching process, that has state space {0, 1, 2, ...} and birth and death rates of the form Ax = x1 and Hx = XH, x 2 0, where 1 and u are nonnegative constants. Set my(t) = E.(X(t)) = > yP x,(t). )=0 (a) Write the forward equation for the process. (b) Use the forward equation to show that my(t) = (2 - u)m.(t). (c) Conclude that my(t) = xe(2-4)t7. (12 points) Consider a birth and death process with birth rate i + 1 and death rate i when there are i individuals. (a) Starting at state 2, find the expected time until the process first leaves state 2. (b) Find the expected time to go from state 2 to state 4.7. (12 points) Consider a birth and death process with birth rate i + 1 and death rate i when there are i individuals. (a) Starting at state 2, find the expected time until the process first leaves state 2. (b) Find the expected time to go from state 2 to state 4.Consider a birth-and-death process with just three attainable states (0, 1, and 2), for which the steady-state probabilities are PO, P1, and P2, respectively. The birth-and-death rates are summarized in the following table: state Birth Death rate rate 0 1 1 1 2 2 2 a)Construct the rate diagram for this birth-and-death process. b) Develop the balance equation. c) Solve these equations to find PO, P1, and P2 d) Use the general formulas for the birth-and-death process to calculate PO, P1 and P2.Also calculate L, Lq, W, and Wq.Q13 (1 mark). For liner system , the following expression is WRONG. (a) If is strictly diagonally dominant, then Jacobi method converged. (b) If is strictly diagonally dominant, then Gauss- Seidel method converged. (c) If Jacobi method converged, then Gauss-Seidel method converged. (d) If Jacobi method converged, then Gauss-Seidel method may not converged