Suppose Ethernet physical addresses are chosen at random (using true random bits). (a) What is the probability that on a 1024-host network, two addresses will be the same? (b) What is the probability that the above event will occur on one or more of 220 networks? (c) What is the probability that, of the 230 hosts in all the networks of (b), some pair has the same address? 1 Hint: The calculation for (a) and (c) is a variant of that used in solving the so-called Birthday Problem: Given N people, what is the probability that two of their birthdays (addresses) will be the same? The second person has probability 1 - 35 of having a different birthday from the first, the third has probability 1 - 365 of having a different birthday from the first two, and so on. The probability that all birthdays are different is thus 2 2 N (1-365) × (1-385) ××(1-¹) 365 which for smallish N is about 1 - 1+2+...+(N − 1) 365

Suppose Ethernet physical addresses are chosen at random (using true random bits). (a) What is the probability that on a 1024-host network, two addresses will be the same? (b) What is the probability that the above event will occur on one or more of 220 networks? (c) What is the probability that, of the 230 hosts in all the networks of (b), some pair has the same address? 1 Hint: The calculation for (a) and (c) is a variant of that used in solving the so-called Birthday Problem: Given N people, what is the probability that two of their birthdays (addresses) will be the same? The second person has probability 1 - 35 of having a different birthday from the first, the third has probability 1 - 365 of having a different birthday from the first two, and so on. The probability that all birthdays are different is thus 2 2 N (1-365) × (1-385) ××(1-¹) 365 which for smallish N is about 1 - 1+2+...+(N − 1) 365

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter20: Queuing Theory

Section20.4: The M/m/1/gd/∞/∞ Queuing System And The Queuing Formula L = Λw

Problem 7P

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