onstant length L-KRS, where K is the transmission rate of the channel and k is a large integer. Suppose there odes, each with an infinite number of frames to send. We also assume that dprop < S, so that all nodes can de ollision before the end of a slot time. The protocol is as follows: If, for a given slot, no node has possession hannel, all nodes contend for the channel; in particular, each node transmits in the slot with probability p. If ex ne node transmits in the slot, that node takes possession of the channel for the subsequent k1 slots and transm ntire frame. If some node has possession of the channel, all other nodes refrain from transmitting until the nod ossesses the channel has finished transmitting its frame. Once this node has transmitted its frame, all nodes co or the channel. Note that the channel alternates between two states: the productive state, which lasts exactly k nd the nonproductive state, which lasts for a random number of slots. Clearly, the channel efficiency is the ra /(k+x), where x is the expected number of consecutive unproductive slots. (a) For fixed N and p, determine the efficiency of this protocol. (b) For fixed N, determine the p that maximizes the efficiency. (c) Using the p (which is a function of N) found in (b), determine the efficiency as N approaches infinity. (d) Show that this efficiency approaches 1 as the frame length becomes large.
onstant length L-KRS, where K is the transmission rate of the channel and k is a large integer. Suppose there odes, each with an infinite number of frames to send. We also assume that dprop < S, so that all nodes can de ollision before the end of a slot time. The protocol is as follows: If, for a given slot, no node has possession hannel, all nodes contend for the channel; in particular, each node transmits in the slot with probability p. If ex ne node transmits in the slot, that node takes possession of the channel for the subsequent k1 slots and transm ntire frame. If some node has possession of the channel, all other nodes refrain from transmitting until the nod ossesses the channel has finished transmitting its frame. Once this node has transmitted its frame, all nodes co or the channel. Note that the channel alternates between two states: the productive state, which lasts exactly k nd the nonproductive state, which lasts for a random number of slots. Clearly, the channel efficiency is the ra /(k+x), where x is the expected number of consecutive unproductive slots. (a) For fixed N and p, determine the efficiency of this protocol. (b) For fixed N, determine the p that maximizes the efficiency. (c) Using the p (which is a function of N) found in (b), determine the efficiency as N approaches infinity. (d) Show that this efficiency approaches 1 as the frame length becomes large.
Chapter9: Computer Networks
Section: Chapter Questions
Problem 3VE
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Transcribed Image Text:constant length L=kRS, where R is the transmission rate of the channel and k is a large integer. Suppose there are N
nodes, each with an infinite number of frames to send. We also assume that dprop < S, so that all nodes can detect a
collision before the end of a slot time. The protocol is as follows: If, for a given slot, no node has possession of the
channel, all nodes contend for the channel; in particular, each node transmits in the slot with probability p. If exactly
one node transmits in the slot, that node takes possession of the channel for the subsequent k1 slots and transmits its
entire frame. If some node has possession of the channel, all other nodes refrain from transmitting until the node that
possesses the channel has finished transmitting its frame. Once this node has transmitted its frame, all nodes contend
for the channel. Note that the channel alternates between two states: the productive state, which lasts exactly k slots,
and the nonproductive state, which lasts for a random number of slots. Clearly, the channel efficiency is the ratio of
k/(k+x), where x is the expected number of consecutive unproductive slots.
(a) For fixed N and p, determine the efficiency of this protocol.
(b) For fixed N, determine the p that maximizes the efficiency.
(c) Using the p (which is a function of N) found in (b), determine the efficiency as N approaches infinity.
(d) Show that this efficiency approaches 1 as the frame length becomes large.
Transcribed Image Text:4. In this problem, you will derive the efficiency of a CSMA/CD-like multiple access protocol. In this protocol, time is
slotted and all adapters are synchronized to the slots. Unlike slotted ALOHA, however, the length of a slot (in seconds)
is much less than a frame time (the time to transmit a frame). Let S be the length of a slot. Suppose all frames are of
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