辅导案例-ISE 3414
ISE 3414 - Stochastic Modeling and Analysis (P.O.R.) Midterm Exam I Fall 2020 - Prof. M. R. Taaffe) NAME: STUDENT ID NUMBER: SECTION: 10:10 a.m. 11:15 a.m. This exam is open-book and open-notes, and use of MATLAB. You have one week to complete and upload the exam via Canvas (under the “Practice Quiz” Section of the “Quizzes” page. There are a total of 100 points. PLEASE READ ALL QUESTIONS VERY CAREFULLY! You should upload only ONE .pdf file. Write out complete answers, including fully specified algorithms and all numbers filled in the various vectors and matrices, to all questions. Also provide actual numbers for all questions. Use MATLAB for doing your calculations unless you really enjoy doing arithmetic and matrix manipulations by hand. Include (for example, via “Snip and Paste” (Press Winkey + Shift + S)) MATLAB output for numerical results that you obtained via MATLAB. You do NOT have to include any code provided by the Instructor via the Canvas pages. Attach extra pages to your .pdf file if necessary to fit in your MATLAB computations. IMPORTANT For the purposes of this Midterm, you may not state an answer in terms of P(∞), or P∞, or P(n ∗) where n∗ is some sufficiently large number, unless you are specifically asked to do so. In other words you cannot state an answer in a manner that includes just raising the single-step transition matrix to a very large power, unless asked to do so. The Virginia Tech Honor Code Pledge: “I have neither given nor received unauthorized assistance on this take-home test. ” Signature: SCORE: ISE 3414 1. (25 total points): Here are P,P2, and P3. In answering the following questions you do not have to provide a numerical answer; it will be sufficient to show an expression with all of the numbers in the right places, so that only simple arithmetic needs to be completed. P = 0.4 0.1 0.5 0.3 0.6 0.1 0.1 0.2 0.7 , P2 = 0.24 0.2 0.56 0.31 0.41 0.28 0.17 0.27 0.56 , P3 = 0.212 0.256 0.532 0.275 0.333 0.392 0.205 0.291 0.504 (a) (5 points): What is the value of p (4) 2,3? (b) (5 points): If the process starts in State 3, what is the expected number of visits to State 2 in the first 3 epochs? Midterm I, Fall 2020 Page 2 of 14 Taaffe ISE 3414 (c) (5 points): For this part, assume that the initial state is equally likely to be any one of the three states. What is the probability that the process is in State 3 at epoch 3. You need to write an expression with all of the numbers in the right places to show us how you computed your result. (d) (5 points): What is the variance of the sojourn time for State 1? (e) (5 points): What is the State-1 sojourn-time probability-mass function, and its associ- ated parameters? (State this completely and clearly). Midterm I, Fall 2020 Page 3 of 14 Taaffe ISE 3414 2. (10 points): Consider the following transition diagram. Write down a sufficient set of linear equations (preferably in matrix form) to solve for the mean first-passage time from State 2 to State 4. Solve the equations in MATLAB (or by hand if you do want to), but write the equations with specific numerical coefficients. You may use ugly simultaneous linear equations, explicitly, or you may write your answer in matrix-vector form, as long as every quantity or expression is defined. Include “Snip and Paste” MATLAB for MATLAB computations.