[n this problem you will derive the efficiency of a CSMNCD-like multiple access protocol. [n 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 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 intinite number of Ii'ames to send. We also assume that d prup < S, so that all nodes can detect a collision before the end of a slot time. TI1e protocol is a'; 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 k - I slots and transmits its entire frame .
[1' 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 Nand 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 I as the frame length becomes large.