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.
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