278 Multiaccess Communication Chap.4 of new arrivals transmitted in a slot is a Poisson random variable with parameter A.If the retransmissions from backlogged nodes are sufficiently randomized.it is plausible to approximate the total number of retransmissions and new transmissions in a given slot as a Poisson random variable with some parameter G>A.With this approximation. the probability of a successful transmission in a slot is Ge-C.Finally,in equilibrium. the arrival rate.A.to the system should be the same as the departure rate.Ge-C.This relationship is illustrated in Fig.4.2. We see that the maximum possible departure rate (according to the argument above) occurs at G =I and is 1/e 0.368.We also note,somewhat suspiciously.that for any arrival rate less than l/e,there are two values of G for which the arrival rate equals the departure rate.The problem with this elementary approach is that it provides no insight into the dynamics of the system.As the number of backlogged packets changes. the parameter G will change:this leads to a feedback effect,generating further changes in the number of backlogged packets.To understand these dynamics.we will have to analyze the system somewhat more carefully.The simple picture below,however, correctly identifies the maximum throughput rate of slotted Aloha as 1/e and also shows that G.the mean number of attempted transmissions per slot,should be on the order of 1 to achieve a throughput close to 1/e.If G<1,too many idle slots are generated,and if G>1.too many collisions are generated. To construct a more precise model.assume that each backlogged node retransmits with some fixed probability gr in each successive slot until a successful transmission occurs.In other words.the number of slots from a collision until a given node involved in the collision retransmits is a geometric random variable having value iI with probability (I-g).The original version of slotted Aloha employed a uniform distribution for retransmission,but this is more difficult to analyze and has no identifiable Departure rate Ge-G Arrival rate Equilibrium G=0 G=1 Figure 4.2 Departure rate as a function of attempted transmission rate G for slotted Aloha.Ignoring the dynamic behavior of G.departures (successful transmissions)occur at a rate G-,and arrivals occur at a rate A.leading to a hypothesized equilibrium point as shown.278 Multiaccess Communication Chap. 4 of new arrivals transmitted in a slot is a Poisson random variable with parameter A. If the retransmissions from backlogged nodes are sufficiently randomized, it is plausible to approximate the total number of retransmissions and new transmissions in a given slot as a Poisson random variable with some parameter G > A. With this approximation. the probability of a successful transmission in a slot is Gc- G . Finally. in equilibrium, the arrival rate. A. to the system should be the same as the departure rate. Gc- G . This relationship is illustrated in Fig. 4.2. We see that the maximum possible departure rate (according to the argument above) occurs at G = I and is 1/e 0.368. We also note, somewhat suspiciously, that for any arrival rate less than lie. there are two values of G for which the arrival rate equals the departure rate. The problem with this elementary approach is that it provides no insight into the dynamics of the system. As the number of backlogged packets changes. the parameter G will change; this leads to a feedback effect, generating further changes in the number of backlogged packets. To understand these dynamics. we will have to analyze the system somewhat more carefully. The simple picture below, however. correctly identifies the maximum throughput rate of slotted Aloha as IIe and also shows that G. the mean number of attempted transmissions per slot. should be on the order of I to achieve a throughput close to IIe. If G < I. too many idle slots are generated, and if G > I, too many collisions are generated. To construct a more precise model, assume that each backlogged node retransmits with some fixed probability (jr in each successive slot until a successful transmission occurs. In other words, the number of slots from a collision until a given node involved in the collision retransmits is a geometric random variable having value i 2> I with probability (jr(1 - (jr )i-I. The original version of slotted Aloha employed a uniform distribution for retransmission, but this is more difficult to analyze and has no identifiable : Arrival rate I I I I I I I G = 1 Figure 4.2 Departure rate as a function of attempted transmission rate G for slotted Aloha. Ignoring the dynamic behavior of G. departures (successful transmissions) occur at a rate Gr- c;. and arrivals occur at a rate A. leading to a hypothesized equilibrium point as shown