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Probability Operations Research - Quiz 4 with Answers | ISE 3414, Quizzes of Operational Research

Material Type: Quiz; Professor: Bish; Class: Prob Operations Research; Subject: Industrial and Systems Enginee; University: Virginia Polytechnic Institute And State University; Term: Fall 2007;

Typology: Quizzes

Pre 2010

Uploaded on 12/10/2007

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Download Probability Operations Research - Quiz 4 with Answers | ISE 3414 and more Quizzes Operational Research in PDF only on Docsity! ISE 3414 - POR, Quiz 4 POR STUDENT NUMBER: Dr. Bish, Fall 07, Total Points: 20 * You are allowed four letter-size pages only (you can use both sides). No calculators, laptops, cellphones, or any other electronic device. © To obtain any credit, you must show all your work. No credit will be given otherwise. Read FIRST all problems. The easiest. problem may not be the first one! 1. (10 pts) Note that this problem has a slight modification (bolded below) over the problem you have seen in class. Consider a communication system consisting of a buffer with finite capacity and multiple transmission channels. Messages arrive at the buffer randomly over time according to a given probability distribution. The messages are first stored in the buffer, which has capacity for N messages. Each message which finds upon arrival that the buffer is full is lost and does not influence the system. Such an arrival is called a buffer overflow. At fixed clock times n = 0, 1,2,---, messages are taken out from the buffer and transmitted. The transmission time is constant and equals one clock time for each message. Each transmission channel has a probability p (0 < p < 1) of being in working condition during each clock time, independently of all the previous clock times and all other transmission channels. The channel can transmit one message during a clock time only if it is in working condition; otherwise, it cannot transmit any messages during that clock time and has to wait until the next clock time, during which it again has probability p of being in working condition. The transmission of a message can only start at a clock time so that a message which finds upon arrival in the buffer that a transmission line is idle has to wait until a subsequent clock time. For k = 0,1,---, a,=probability that k messages arrive during one time unit. Clearly, D729 a4 = 1. Define X;,=the number of messages in the buffer at clock time n just prior to transmission. Let A, denote the number of messages arriving between clock time n and n + 1. {a) (1 pt) Express X,41 im terms of X, and show that {X;,,” > 0} is a Markov Chain. let doping Tr as #0 numbar of wersyes tronsmitea in pariod A Xngus min f (Xa-Ta)" t An, N | Anat = nin d max (Xn-Ta,0) 4 Ay ,NI Xv = Min (Xn-Th 4An, N) Xn >Th min ( An, N) ; Xy, LT Xna onky depends on Xn Ta ond Aa. He $s nckaiadent of post Vadues Ghvon Xn Tr cad An (b) (9 pts) Consider that N = 3 and ¢ = 2 and determine the one-step transition probability matrix (tpm) of this DTMC. To get any credit, you also need to explain all your work. No credit will be given otherwise. The number of tronsatesion chowals In coring conttHon wl be Bont al rv with poroneders Dard po. ket & ba HL pokrloillty Hot | of the frensutosion channels fs in worry condition, Then, Coe (3).9. U0 Cys (4) p'. (l-p)' Cae (2) -p* (tp)? 5 0 Qo i (cyte), on C24, 3 3° 1 2 CoM (0,402), 3 2 Qa. Qe & . 2 2 . Qa: C00, Herc) ay © 2 + crite) 2 ‘ ” aw CoQ 4 CpOe4 C2 Oa Co 598 4 cy 5, + te Sa é=t fe tes s a F105 4 62% G+ ey 5a +a 280 by od if Xn2 Ol Ao messagh to dransavt) if no mossppe arrlues jth He next shale ull bo -O > Bo = tp Onde “” ” a4 vf M M ” atl be 4 => Aye " ” » + Dea By ca Yo tus massages arrive, + \ xt Y Hee or nord messges jorrlue, He 7 Stole ut! ba three aR. | a: ig
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