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Unformatted text preview: Final Exam, Spring Semester, 2008 – Data Networks Professor: Sunghyun Choi 2008-06-13 (Fri), 11am-1pm 1. (20 points) Consider a slotted Aloha with an infinite set of nodes and with “perfect capture.” That is, if more than one packet are transmitted in a slot, the receiver “locks onto” one of the transmissions and receives it correctly; feedback immediately informs each transmitting node about which node was successful and the unsuccessful packets are retransmitted in the next slot. Find the expected system delay assuming Poisson arrivals with overall rate ( 1) λ λ < . Sol1: 1 When there are backlogged nodes upon arrival of i-th packet, (5points) :delay from the arriv i i n i i j i j i n W R t y W = = + + al of the ith packet until the beginning of the ith successful transmission, : residual time to the beginning of the next slot, : interval from the end of the (j-1)th subsequent success to the end i j R t of the jth subsequent success, : remaining interval until the beginning of the next successful transmission, Due to the perfect capture, E[ ] 1, E[ ] 0. (5po ints) E[ ] i j i i y t y R = = 1 , E[ ] (due to Little's theorem) 2 1 1 , 2 2(1 ) Expected syst (3points) (5points em dela ) i n W W W W λ λ λ = = ∴ = + =- 1 3-2 y T = W + 1 slottime = 1 2(1 ) (2points) 2-2 λ λ λ + =- Sol2: 2 2 2 2 Using P-K formula for M/G/1 queue with vacations [ ] [ ] , (5points) 2(1 ) 2 [ ] [ ] [ ] [ ] 1 (due to perfect capture) E X E V W E V E X E V E V λ ρ = +- = = = /1 1 2(1 ) 2 3 3- 2 Expected system delay T = W + 1= 2(1 ) 2 (8points) (5points) (2poi 2 - 2 nts) W ρ λ λ λ λ λ λ λ λ = = ∴ = +- + =- 2. (25 points) Consider the system of the figure below. A computer CPU is connected to two I/O devices. Jobs enter the system according to a Poisson process with rate λ , use the CPU, and with probability ( 1,2) i P i = , are routed to the i-th I/O device while with probability P they exit the system. The service time of a job at the CPU (or the i-th I/O device, i =1,2) is exponentially distributed with mean 1/ (or 1/ , 1,2, respectively) i i μ μ = . We assume that all ....
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- Spring '08
- Poisson Distribution, Probability theory, CSMA