Random Access Networks Random Access networks are characterised by - PDF document

Chapter 3 Random Access Networks Random Access networks are characterised by the absence of a channel- controlled access mechanism. To some extent, a station is free to broadcast its packets on the network at a time determined by the station itself, with never any certainty that another station is not simultaneously attempting a transmission. 3.1 The ALOHA Procedure The network is constructed such that each station is coupled to a single passive channel, for example a coaxial cable. In normal operation packets are transmitted without error from one station to another and each packet is acknowledged. If two stations try to transmit at the same time, then the signals interfere and an error is detected by the receiving station and no acknowledgement is sent. Station�1 Station�2 Station M channel channel interface After an appropriate ’timeout’, which is at least as long as the maximum

EE414 Notes - The ALOHA Procedure 25 two-way propagation time of the cable, the original transmitting stations conclude that a collision has occurred and schedule retransmission of the packets at later times. To avoid repeated collisions of packets, the retransmission times are usually chosen at random by the stations involved. 3.1.1 Throughput Analysis of ALOHA • Let P seconds be the transmission time for a packet (propagation time + transmission time). • Let Pr[ k arrivals in t seconds] = (Λ t ) k e − Λ t k ! be the Poisson distribution describing all transmissions over the chan- nel where Λ = arrival rate in Pkts/Sec . This expression assumes an infinite population of infrequent users. • Let S be the average number of successful transmissions per packet transmission time P . • Let G be the average number of attempted packet transmissions per packet transmission time P . Packet�transmission�time P time start�of�packet conclusion�of�packet transmission transmission P P time start�of�packet start�of�packet 2 P Vunerable�interval�for�start�of�packets�that�colide with�reference�station�packet Figure 3.1: Vulnerable Period for ALOHA

EE414 Notes - The ALOHA Procedure 26 Figure 9.1 (top) shows some packet being transmitted. The lower diagram shows the positioning of other packets that just avoid colliding with the original packet. The ’vunerable’ period can be seen to be of length 2 P . A relationship between S and G is: S = G Pr[Good Transmission] The probability of a good transmission is the probability of no additional transmissions in the vulnerable period of length 2 P about the transmission of the original packet. Pr[0 arrivals in an interval of length 2 P ] = e − 2 G as Λ = G/P therefore S = Ge − 2 G

EE414 Notes - The ALOHA Procedure 27 0.20 1/2 e Throughput, S 0.15 ( G'��S' , ) 0.10 0 0.10 0.5 Offered�traffic, G Figure 3.2: S vs G for ALOHA A plot of this relationship is shown in Figure 9.2. Note that S has a maxi- 1 mum value of 2 e ≈ 0 . 184 for G = 0 . 5. In producing this graph it is assumed that the system is in statistical equi- librium for the average quantities S and G . This is not always the case. Consider the point ( G ′ , S ′ ). If G ′ increases slightly due to statistical fluctu- ations in the offered load, the throughput S decreases. This further increases the backlog and hence load G . Therefore the operating point moves further to the right. This continues and the system becomes saturated. Thus the system is in an unstable operating region for G > 0 . 5.

EE414 Notes - The ALOHA Procedure 28 3.1.2 Average Packet Transfer Delay of ALOHA The average number of attempts per successfully transmitted packet is given by: G/S = e 2 G Therefore, the average number of unsuccessful attempts per successfully transmitted packet is given by: G/S − 1 = e 2 G − 1 If a collision occurs, the station reschedules the colliding packet for some randomly chosen later time. The average backoff delay is denoted by B . The average transfer delay is given by: T = P + ( e 2 G − 1)( P + B ) The value of B depends on the backoff strategy. The lower bound on the delay is at B = 0: T min = Pe 2 G

EE414 Notes - Slotted ALOHA 29 3.2 Slotted ALOHA The Slotted ALOHA procedure segments the time into slots of a fixed length P . Every packet transmitted must fit into one of these slots. A packet arriving to be transmitted at any given station must be delayed until the beginning of the next slot. Slotted ALOHA requires additional overhead to provide the synchronisation. The station can be in either of two states: • Transmitting • Backed-off To avoid repeated collisions, collided packets are rescheduled at later times chosen independently and at random.

EE414 Notes - Slotted ALOHA 30 Packet No Ready? Yes Delay�to Beginning�of Slot Transmit Delay kP Wait�2-Way Prop�Delay Quantised�to Slots Compute No Positive Random Ack? Backoff�Integer ( ) k Yes Backoff Algorithm Exit Figure 3.3: Flow Diagram for Slotted ALOHA Procedure

EE414 Notes - Slotted ALOHA 31 3.2.1 Throughput Analysis of Slotted ALOHA An expression for S in terms of G is required. This will be derived for an infinite population and a finite population of stations. • Throughput S • Offered Load G (per interval P ) As in the Pure ALOHA derivation, both new arrivals and retransmissions due to collisions, form a Poisson process with mean arrival rate Λ pack- ets/sec. (Note that Λ = G/P ). P P Reference�Packet time Vunerable�period�for slotted ALOHA Figure 3.4: Vulnerable Period for slotted ALOHA The vulnerable period for Slotted ALOHA is reduced from 2 P to P seconds. Throughput and offered load are related by the same equation as before: S = G Pr[Successful Transmission] Thus we have for Slotted ALOHA: Pr[no arrival in an interval of length P ] = e − Λ P = e − G P P ⇒ S = Ge − G Compare this with the Pure Aloha case: S = Ge − 2 G The maximum throughput for Slotted ALOHA is 1 e ≈ 0 . 368 and the maxi- mum occurs for G = 1.

EE414 Notes - Slotted ALOHA 32 0.40 1/ e Slotted 0.30 ALOHA Throughput, S 0.20 0.10 ALOHA 0 0.01 0.05 0.1 0.5 1.0 Offered�traffic, G Figure 3.5: S vs G for ALOHA and Slotted ALOHA Finite Population Case In the above we have assumed the number of users is infinite. Here we will revise the model for a finite population and determine under what conditions the infinite population model gives a reasonable approximation. Consider a network with M independent stations. The input from each station is modelled as a sequence of independent Bernoulli Trials . Let S i be the probability of station i successfully transmitting a packet in any slot. 1 − S i is the probability of not successfully transmitting a packet. Let G i and 1 − G i be the probabilities for attempting and not attempting transmissions respectively. The probability that station j has a successful transmission is the probability that station j is the only station to attempt a transmission in a particular slot. Therefore: M � S j = G j (1 − G i ) i =1 i � = j Assuming all stations share the load equally then S = MS j and G = MG j , therefore: S = G (1 − G/M ) M − 1

EE414 Notes - Slotted ALOHA 33 Note that: � n � 1 + X = e X lim as M → ∞ n n →∞ S = Ge − G ∴ as M → ∞ (as before) For M stations the maximum throughput S max is calculated by differenti- ating and equating to zero: � 1 − G � � M − 1 � 0 = − G M M which results in G = 1 (as for the infinite case) � M − 1 � 1 − 1 S max = M This approaches the infinite result (0.368) as M → ∞ and is reasonably close for M ≥ 20. Thus it is reasonable to assume an infinite population in the analysis when we are modelling any reasonably large network. 3.2.2 Analysis of Slotted ALOHA Taking Backoff into Ac- count The analyses given earlier account for retransmissions only by increasing the offered traffic. These analyses assumed that both new and collided packets come from the same process and did not account for the delays that collided packets experience. The following analysis accounts for newly arrived and retransmitted packets as separate variables to give greater accuracy in the analysis. Here we wish to derive: 1. Average Transfer Delay - Throughput relation 2. Throughput - Offered Load expression which is more accurate The following assumptions and definitions are made: • The population of users is infinite. • New arrivals to the network come from a Poisson process with mean arrival rate S packets/slot or Λ packets/sec where S = Λ P (where P = ¯ X/R ).