Tutorial 9

# Tutorial 9 - IERG4020 Tutorial 9 FENG Shen 1 PROBLEM SET 5...

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IERG4020 Tutorial 9 FENG Shen 1

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PROBLEM SET 5 DISCUSSION 2
Problem 5.1 To show the perfect shuffle achieves a cyclic left shift. Label the N = 2 n links from 0 (00…0) to N – 1 (11…1). In perfect shuffle, links are connected in the following manner: 3 0 (000) 1 (001) 2 (010) 3 (011) 4 (100) 5 (101) 6 (110) 7 (111) Perfect Shuffle 0 (000) 1 (001) 2 (010) 3 (011) 4 (100) 5 (101) 6 (110) , if 2 2 21 , i f 2 j N jN j N jj 

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Problem 5.1 For j < N / 2, its label in binary form is . It is connected to which is a cyclic left shift. For j N / 2, its label in binary form is . It is connected to which is also a cyclic left shift. 4 0 (000) 1 (001) 2 (010) 3 (011) 4 (100) 5 (101) 6 (110) 7 (111) Perfect Shuffle 0 (000) 1 (001) 2 (010) 3 (011) 4 (100) 5 (101) 6 (110) 121 0. . . nn x xx  2. . . 0 j x 1 ... x 0' 21 2 1 . . . 1 1 ... 0 100. ..0 1 ... 1 ns jN x N x x  
Problem 5.2 Route by the MSB first, and then 2 nd MSB 00 00 01; 01 11 10; 10 01 11; 11 10 00. There is no contention among packets. 5 00 01 10 11 0 1 0 1 0 1 0 1

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Problem 5.2 There cannot be contention if there is only one packet at each source (if they are destined for different destinations). One example: Two packets at node 00 with source-destination mapping 00 10 and 00 11. Their nodes traversed are 00 01 10 and 00 01 11, respectively. 6 00 01 10 11 0 1 0 1 0 1 0 1
Problem 5.8 For a nonblocking switch, the probability of a packet NOT being deflected is where ρ is the link loading. 7 4 1/ 4 (1 ) nonblocking p  ρ p

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Problem 5.8 How it is derived? ρ : Link loading = Pr{a packet comes}. ρ /4: A packet is destined for one specific output. 1- ρ /4: A packet is NOT destined for the output. (1- ρ /4) 4 : No packet is destined for the output. 1-(1- ρ /4) 4 :At least a packet is destined for the output. This packet is undeflected. By conservation, Pr{Finding an undeflected packet on an output link} = Pr{Finding a packet on an input link and that it will not be deflected} = p ρ . 8 4 1/ 4 (1 ) nonblocking p  ρ p
Problem 5.8 If a 4 × 4 reverse banyan switch is used Consider a packet A at a particular input.

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Tutorial 9 - IERG4020 Tutorial 9 FENG Shen 1 PROBLEM SET 5...

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