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 doesn’t necessarily imply using a carrier but only some possibility to detect idle periods quickly Information Networks – p.1/42
Carrier Sensing Let β denote the propagation and detection delay measured in expected packet transmission time units, thus with τ this time in second, C the raw channel bit rate in bits/second and L the expected number of bits in a packet β = τC L The performance of CSMA degrades with increasing β , 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 β time units, we thus no longer assume equal-duration time slots Information Networks – p.2/42
CSMA, assumptions Slotted system but not with equal-duration 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 , e ) -feedback with a maximum delay β For simplicity we assume infinite set of nodes Poisson arrivals with overall intensity λ Information Networks – p.3/42
CSMA Slotted Aloha Major difference to slotted Aloha is that idle slots have duration β Another difference is that newly arriving packets when channel is busy are regarded as backlogged and will transmit with probability q r 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 – p.4/42
CSMA Slotted Aloha, variants Persistent CSMA: arrivals during busy slot transmit at end of that slot, thus causing collision with relatively high probability P-persistent 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 – p.5/42
Nonpersistent CSMA Slotted Aloha We again use Markov chain with number of backlogged packets, n , as state and end of idle slots as state transition times Each busy slot (success or collision) must be followed by an idle slot (since this is nonpersistent CSMA) For simplicity assume all data packets have unit length Time between state transitions are either β (idle slot) or 1 + β (busy slot followed by idle slot) Information Networks – p.6/42
Nonpersistent CSMA Slotted Aloha Probability of idle slot is probability of no arrivals in previous idle slot and no retransmissions by backlogged nodes, thus e − λβ (1 − q r ) n Expected time between state transitions in state n is β + (1 − e − λβ (1 − q r ) n ) Expected number of arrivals between state transitions is λ ( β + 1 − e − λβ (1 − q r ) n ) Expected number of departures between state transitions in state n is probability of successful transmission � q r n � e − λβ (1 − q r ) n λβ + 1 − q r Information Networks – p.7/42
Nonpersistent CSMA Slotted Aloha The drift in state n is as before the expected number of arrivals less expected numbers of departures � � q r n D n = λ ( β +1 − e − λβ (1 − q r ) n ) − e − λβ (1 − q r ) n λβ + 1 − q r For small q r we make the approximation (1 − q r ) n − 1 ≈ (1 − q r ) n ≈ e − q r n and get D n ≈ λ ( β + 1 − e − g ( n ) ) − g ( n ) e − g ( n ) where g ( n ) = λβ + q r n is expected number of attempted transmissions Information Networks – p.8/42
Nonpersistent CSMA Slotted Aloha The drift is negative if g ( n ) e − g ( n ) λ < β + 1 − e − g ( n ) 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 We can plot departure rate as function of attempted rate as before, for small β this function has a maximum of approximately 1 / (1 + √ 2 β ) for g = √ 2 β Information Networks – p.9/42
Nonpersistent CSMA Slotted Aloha 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Information Networks – p.10/42
Nonpersistent CSMA Slotted Aloha We have the same stability problem as in ordinary slotted Aloha For fixed q r , g ( n ) grows with n and when n becomes too large, departure rate is less than arrival rate, leading to yet larger backlogs Expected idle time that a backlogged node must wait before attempting retransmission is β ( q r + 2 q r (1 − q r ) + 3 q r (1 − q r ) 2 + . . . ) = β/q r , for small β and modest λ , q r can be quite small without causing appreciable delay, this means that backlog must be very large before instability sets in and the problem is less serious than for ordinary Aloha Information Networks – p.11/42
CSMA Slotted Aloha P-persistent CSMA, in which packets are transmitted after idle slots with probability p if they are new arrivals and with some much smaller probability q r if they have had collisions will give a little extra protection against instability A more fundamental way to achieve stability is to do a pseudo-Bayesian stabilization as for the ordinary slotted Aloha All packets are considered backlogged immediately after entering the system At end of each idle slot, each backlogged packet is transmitted with probability q r which will vary with the estimated channel backlog ˆ n Information Networks – p.12/42
Pseudo-Bayesian stabilization In state n , expected number of packets transmitted at end of idle slot is g ( n ) = nq r , packet departure rate is maximized (for small β and q r ) when g ( n ) = √ 2 β so we choose � √ 2 β � � q r (ˆ n ) = min n , 2 β ˆ Backlog estimate is updated according to n k )) + λβ , for idle n k (1 − q r (ˆ ˆ n k +1 = ˆ n k (1 − q r (ˆ ˆ n k )) + λ (1 + β ) , for success n k + 2 + λ (1 + β ) , ˆ for collision Information Networks – p.13/42
Pseudo-Bayesian stabilization Again the update rule for this Pseudo-Bayesian stabilization can be motivated by showing that for an a priori Poisson distribution of n k with mean ˆ n k , the a posteriori distribution of n k is Poisson with mean ˆ n k )) given an idle slot n k (1 − q r (ˆ Poisson with mean 1 + ˆ n k )) given a n k (1 − q r (ˆ successful transmission approximately Poisson with mean ˆ n k + 2 given a collision Adding the expected arrivals in the three cases yields the suggested update rule Information Networks – p.14/42
Pseudo-Bayesian stabilization When n k and ˆ n k are small then q r is relatively large and new arrivals are scarcely delayed at all When ˆ n k ≈ n k and n k is large, the departure rate is approximately 1 / (1 + √ 2 β ) , so for λ < 1 / (1 + √ 2 β ) the backlog decreases on average If | n k − ˆ n k | is large the expected change in backlog can be positive, but the expected change in | n k − ˆ n k | is negative so eventually ˆ n k will be close to n k and backlog will decrease; similar to pseudo-Bayesian stabilization of ordinary slotted Aloha Information Networks – p.15/42
Delay for Pseudo-Bayesian stabilization We can do a similar analysis of the expected queueing delay as for pseudo-Bayesian stabilization of ordinary slotted Aloha Let W i be the delay from arrival of i th packet until beginning of i th successful transmission Average of W i over all i is the expected queueing delay W Let n i be the number of backlogged packets at the instant before packet i ’s arrival, not counting any packet currently in successful transmission Information Networks – p.16/42
Delay for Pseudo-Bayesian stabilization n i � W i = R i + t j + y i j =1 where R i is residual time until next state transition, t j is the sequence of subsequent intervals until each of the next n i successful transmissions are completed, and y i is the remaining interval until the i th successful transmission starts The backlog is at least 1 in all of the state transition intervals and we make the simplifying approximation that the number of attempted transmissions in each of these intervals are Poisson with parameter g Information Networks – p.17/42
Delay for Pseudo-Bayesian stabilization The difference from analysis of ordinary slotted Aloha is that there we assumed a successful transmission always occurred, this is motivated by our new q r is kept small The expected value for each t j is given by E [ t ] = e − g ( β + E [ t ])+ ge − g (1+ β )+[1 − (1+ g ) e − g ](1+ β + E [ t ]) The first term corresponds to an idle transmission in first state transmission interval, second term for a success, and third term for collision Information Networks – p.18/42
Delay for Pseudo-Bayesian stabilization Solving for E [ t ] gives E [ t ] = 1 + β − e − g ge − g This is the reciprocal of expected departure rate and thus is approximately minimized by g = √ 2 β Averaging over i and using Little’s theorem we get W (1 − λE [ t ]) = E [ R ] + E [ y ] Information Networks – p.19/42
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