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Give an intuitive explanation of why the maximum throughput, for small beta, is approximately the same...

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Programming

Give an intuitive explanation of why the maximum throughput, forsmall beta, is approximately the same for CSMA slotted Aloha andFCFS splitting with CSMA. Show the optimal expected number ofpackets transmitted after a state transmission in an Aloha is thesame at the beginning of a CRP for FCFS splitting. Note that aftera collision, the expected numbers are slightly different in twosystems, but the difference is unimportant since collisions arerare.

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Carrier Sensing We assume that a node can hear whether other nodes are transmitting after some small propagation and detection delay We allow nodes to initiate transmission after detecting an idle period no need to wait for slot boundary This strategy is called Carrier Sense Multiple Access CSMA even though it doesnt necessarily imply using a carrier but only some possibility to detect idle periods quickly Information Networks p142 Carrier Sensing Let denote the propagation and detection delay measured in expected packet transmission time units thus with this time in second C the raw channel bit rate in bitssecond and L the expected number of bits in a packet CL The performance of CSMA degrades with increasing thus with increasing channel rate and with decreasing packet size A simple model for CSMA is to model it as a slotted system where idle slots terminates after time units we thus no longer assume equalduration time slots Information Networks p242 CSMA assumptions Slotted system but not with equalduration time slots We no longer assume data packets of equal length but normalize time so that expected packet transmission is 1 time unit 0 1 efeedback with a maximum delay For simplicity we assume infinite set of nodes Poisson arrivals with overall intensity Information Networks p342 CSMA Slotted Aloha Major difference to slotted Aloha is that idle slots have duration Another difference is that newly arriving packets when channel is busy are regarded as backlogged and will transmit with probability qr after each subsequent idle slot packets arriving during an idle slot will be transmitted in next slot as usual This is called nonpersistent CSMA to distinguish from two slight variations Information Networks p442 CSMA Slotted Aloha variants Persistent CSMA arrivals during busy slot transmit at end of that slot thus causing collision with relatively high probability Ppersistent CSMA collided packets and newly arrived packets waiting for the end of a busy period use different probabilities for transmission in next slot We will focus on nonpersistent CSMA Information Networks p542 Nonpersistent CSMA Slotted Aloha W e again use Mar k o v chain with number of backlogged pac kets n as state and end of idle slots as state transition times Each busy slot success or collision must be follo wed b y an idle slot since this is nonpersistent CSMA For simplicity assume all data pac kets ha v e unit length Time between state transitions are either idle slot or 1 busy slot follo wed b y idle slot Information Networks p642 Nonpersistent CSMA Slotted Aloha Probability of idle slot is probability of no arrivals in previous idle slot and no retransmissions b y backlogged nodes thus e 1 q r n Expected time between state transitions in state n is 1 e 1 q r n Expected number of arrivals between state transitions is 1 e 1 q r n Expected number of departures between state transitions in state n is probability of successful transmission q r n 1 q r e 1 q r n Information Networks p742 Nonpersistent CSMA Slotted Aloha The drift in state n is as before the expected number of arrivals less expected numbers of departures D n 1 e 1 q r n q r n 1 q r e 1 q r n For small q r w e mak e the approximation 1 q r n 1 1 q r n e q r n and get D n 1 e gn g n e gn where g n q r n is expected number of attempted transmissions Information Networks p842 Nonpersistent CSMA Slotted Aloha The drift is negativ e if g n e gn 1 e gn where the numerator is the expected number of departures per state transition and the denominator is the expected duration of a state transition so it can be interpreted as the departure rate in state n W e can plot departure rate as function of attempted rate as before for small this function has a maximum of approximately 1 1 2 for g 2 Information Networks p942 Nonpersistent CSMA Slotted Aloha 0 02 04 06 08 1 12 14 16 18 2 0 01 02 03 04 05 06 07 08 09 Information Networks p1042    See Answer
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