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CS 455 - Introduction to Computer Networks
Homework 5

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Due date: see class webpage

• Question 1 (20 pts)
  • Two nodes A and B use slotted Aloha to access a channel

    We observed the interactions of the two nodes for a long time and found that:

    Node A transmits in 40% of the slots Node B transmits in 30% of the slots

    Based on these observations, we will assume that:

    Node A transmits in a slot with probability 0.4 Node B transmits in a slot with probability 0.3

    Questions:

    What is the probability that node A completes a transmission successfully in a slot ? Let PA be the probability that node A completes a transmission successfully in a slot (= the answer in the previous question) What is the probability that node A succeeds for the first time in slot 3 ? Express your answer using PA only: What is the probability that node B completes a transmission successfully in a slot ? What fraction of slots will contain successful transmission ?

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• Question 2 (20 pts)
  • In an Aloha network, the arrival rate is 20 messages per sec

    Messages have the same length and it takes 0.1 sec to transmit a message.

    What is the throughput if the Aloha network uses unslotted transmissions (10 pts) What is the throughput if the Aloha network uses slotted transmissions (10 pts)

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• Question 3 (20 pts)
  • The maximum network diameter in Ethernet is 2500 m (= meter).

    I.e.:

    the distance between any two nodes on an Ethernet is at most 2500 m.

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  • The speed of electrical signals in copper is 2×10⁸ m/sec.

    Question:

    What is the maximum end-to-end propagation delay in Ethernet ? (10 pts) (What is the maximum propagation delay from one end of the Ethernet to the other end)

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    At time t = 0, an Ethernet host senses that the Ether channel is idle and transmits.

    As you know, the Ethernet uses CSMS/CD and Ethernet hosts will sense the channel (while it is transmitting) for possible collision.

    Question:

    After how many seconds of sensing can this Ethernet host be 100% certain that there will not be any collisions ? I.e., when can it stop sensing (because it is sure there won't be a collision). (10 pts)

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• Question 4 (40 pts)
  • (This question is adapted from Question 37 in the text book on page 164. --- from an old edition :))

    Let A and B be two Ethernet hosts attempting to transmit on an Ethernet network. Each host has a steady queue of frames ready to send.

    A's frames will be numbered as A₁, A₂ and so on, and B's frames will be numbered as B₁, B₂ and so on.

    The backoff time unit T = 50 micro seconds.

    Suppose A and B simultaneously (= at the same time) attempt to send frame 1 and the transmissions collide. Then A chooses backoff time 0×T (i.e., A picked the random number 0) and B chooses backoff time 1×T. So A wins the race and transmits its first frame A₁ successfully (in the mean time, when B retries at time 1×T, B will sense A's transmission; so B will wait for A to finish its transmission.)

    Fact:

    At the end of A's transmission, B will attempt to transmit frame B1 and A will attempt to transmit frame A2. A and B's transmissions will collide and both nodes will backoff. A will back off picking a backoff time from: 0×T or 1×T B will back off picking a backoff time from: 0×T, 1×T, 2×T or 3×T Each backoff time period is selected with same probability (= equally likely) each outcome is equally likely. In other words: A selects 0×T and 1×T each with probability 0.5 B selects 0×T, 1×T, 2×T and 3×T each with probability 0.25

    Questions:

    Find the probability that A will win the second backoff race (10 pts) Find the probability that here is a collision in the second backoff race (10 pts)

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    Suppose A wins the second backoff race.

    A will now attempt to transmit frame A₃ while B is still attempting to transmit frame B₁.

    B will now pick a back-off time from 0×T, 1×T, 2×T, 3×T 4×T, 5×T, 6×T, and 7×T - each outcome is equally likely.

    Question:

    Find the probability than A wins the third backoff race (10 pts) Find the probability that there is a collision (10 pts)