Birth and death process

4. Vehicles arrive in College Station for football weekends at varying rates. Cars arrive according to a Poisson process with rate A = 3 per minute, trucks arrive according to a Poisson process with rate A = 1.5 per minute, motorcycles arrive according to a Poisson process with rate A = 0.75 per minute, and RVs arrive according to a Poisson process with rate A = 0.25 per minute. Assume that arrivals of each vehicle type are independent of arrivals of all other types. (a) What is the expected time between arrivals of vehicles regardless of their type? (b) What is the expected time of the arrival of the 20th vehicle with more than 2 wheels? (1 mark each) For each of the given matrices, select all decompositions that can be applied to it. You do not have to show your work for this question. 2 ) A = [61 O Diagonalization (i.e., A = PDP-1 with D diagonal) O Schur triangularization O Spectral decomposition O Singular value decomposition b ) B = 0 1.001 Diagonalization O Schur triangularization O Spectral decomposition Singular value decomposition c) C is the 75 x 75 matrix with every entry equal to 1. Diagonalization Schur triangularization O Spectral decomposition Singular value decomposition d) D = 0 1 1 Diagonalization OSchur triangularization Spectral decomposition Singular value decomposition\f2. [5 points each] The birth and death process {X(t), t 2 0} on non-negative integers with parameters An = 0, / = (2 0), n >0 and / = 0 is called a pure death process. (a) Sketch the sample paths of this pure death process with X(0) = 1. (b) Explain briefly how to simulate this pure death process. (c) When ( = 1, find E[X(() X(0) = 1].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