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# hw2 - EE 547 Homework 2 Due Tue Sept 23 2008 1 Suppose you...

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EE 547: Homework 2 Due: Tue. Sept. 23, 2008 1. Suppose you walk to MacDonald at the Northwestern Ave for a lunch and then back to your office. Identify which portions of time that you spend correspond to the propagation delay, the transmission delay, the processing delay and the queueing delay, respectively. 2. Assume that N ( t ) , t 0 is a Poisson process. Prove or disprove the following: the random variables N (1) and N (2) are independent. 3. Carefully show that definitions I and II given in the notes for a Poisson process are equivalent. Hint: In deriving II from I, set up a differential equation in the following manner. First define P 0 ( t + h ) := P ( { N ( t + h ) = 0 } ). Then P 0 ( t + h ) = P ( { N ( t ) = 0 , N ( t + h ) - N ( t ) = 0 } ) = P ( { N ( t ) = 0 } ) P ( { N ( t + h ) - N ( t ) = 0 } ) = P 0 ( t )(1 - λh + o ( h )) , hence P 0 ( t + h ) - P 0 ( t ) h = - λP 0 ( t ) + o ( h ) h . Now let h 0 and set up and solve the appropriate differential equation. After solving for P 0 ( t ) proceed to solve for P n ( t ) := P ( { N ( t ) = n } ) in much the same manner. Alternatively, you can use the method on Page 26-27 of Schwartz.
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