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#
Authors:
James F. Kurose, Keith W. Ross
Chapter:
The Link Layer: Links,access Networks, And Lans
Exercise:
Problems
Question:8 | ISBN:9780132856201 | Edition: 6

Question

In Section 5.3, we provided an outline of the derivation of the efficiency of slotted ALOHA. In this problem we’ll complete the derivation.

a. Recall that when there are N active nodes, the efficiency of slotted ALOHA

is Np(1 – p) N–1 . Find the value of p that maximizes this expression.

b. Using the value of p found in (a), find the efficiency of slotted ALOHA by

letting N approach infinity. Hint: (1 – 1/N) N approaches 1/e as N approaches infinity.

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Answer

Solution:

a. The efficiency of slotted ALOHA is  

              E(p) = 𝑁𝑝(1 − 𝑝)𝑁−1

  By derivation, we have

              E ′ (p) = N(1 − p)N−1 − 𝑁𝑝(𝑁 − 1)(1 − 𝑝)𝑁−2 = 𝑁(1 − 𝑝)𝑁−2 ((1 − 𝑝) − 𝑝(𝑁 − 1))

Let

              E ′ (p) = 0

Thus, when

              p = 1/𝑁

We have the maximum efficiency.

b. When p equals to 1/N , the efficiency of slotted ALOHA is

              E(p) = 𝑁*1/𝑁(1 − 1/𝑁 )𝑁−1  =  (1 − 1/𝑁 )𝑁−1  =  (1 − 1 𝑁 )𝑁 /  1 − 1/N

Since

              lim N→∞   (1 − 1/𝑁 ) = 1

then       lim N→∞   (1 − 1/𝑁 )𝑁 = 1/𝑒

Thus when n approaching infinity, the efficiency is

              lim N→∞   𝐸(𝑝) = 1/𝑒

 

CIte Reference:

http://course.duruofei.com/wp-content/uploads/2015/05/Networking_HW4.pdf

 

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