Performance Proof for Slotted AlohaT. Charles ClancyElectrical EngineeringUniversity of MarylandENEE 426, Spring 2008February 14, 2008We consider n users sharing a s lotted channel, with probability p of trans-mitting in each time slot. The first goal is to determine the optimal probabilityp.The probability of success PSfor a single time slot is the probability thatonly one of n users transmits.PS= (n1) p(1 − p)n−1= np(1 − p)n−1(1)This equation is due to the fact that there are (n1) different users that cantransmit, and the probability of that user transmitting is p while the probabilityof the other n − 1 users not transmitting is (1 − p)n−1.Our goal is to find a value of p that maximizes PS. To accomplish this wetake the derivative and solve it equal to zero.ddpPS= −np(1 − n)(1 − p)n−2+ n(1 − p)n−1(2)And then solving P0S= 0np(n − 1)(1 − p)n−2= n(1 − p)n−1p(n − 1) = 1 − pnp − p = 1 − pnp = 1p = 1/n(3)Substituting this back into our original equation we get the optimal PS:PS= np(1 − p)n−1=1 −1nn−1(4)1This gives us the probability of a successful transmission in any time slot.If we multiply PSby the number of bits per slot divided by the time per slot,we can compute the theoretical maximal rate in bits per second, w hich is thenetwork capacity. The per-user capacity is this value divided by n.If we are interested in the asymptotic case, we can consider the limit asn → ∞. First, let’s manipulate the expression a bit:PS=1 −1nn−1=(n − 1)n−1nn−1= n−(n−1)n−1Xk=0n−1knn−1−k(−1)k=n−1Xk=0n−1kn−k(−1)k=n−1Xk=0(−1)kk!·(n − 1)!nk(n − 1 − k)!(5)If we take the limit as n → ∞, we c an see thatlimn→∞(n − 1)!nk(n − 1 − k)!= 1 (6)because the numerator’s largest degree term is nn−1and the denominator’slargest degree term is nk· nn−1−k= nn−1. Since both have coefficient 1, thelimit is 1. Thuslimn→∞PS= limn→∞n−1Xk=0(−1)kk!=∞Xk=0(−1)kk!= e−1(7)The last step is due to the definition of the Taylor series of ex. Thus we haveproven a rate of