On-line analysis of the TCP acknowledgment delay problem

We study an on-line problem that is motivated by the networking problem of dynamically adjusting delays of acknowledgments in the Transmission Control Protocol (TCP). We provide a theoretical model for this problem in which the goal is to send acks at times that minimize a linear combination of the cost for the number of acknowledgments sent and the cost for the additional latency introduced by delaying acknowledgments. To study the usefulness of applying packet arrival time prediction to this problem, we assume there is an oracle that provides the algorithm with the times of the next L arrivals, for some L greater than or equal to 0. We give two different objective functions for measuring the cost of a solution, each with its own measure of latency cost. For each objective function we first give an 0(n(2))-time dynamic programming algorithm for optimally solving the off-line problem. Then we describe an on-line algorithm that greedily acknowledges exactly when the cost for an acknowledgment is less than the latency cost incurred by not acknowledging. We show that for this algorithm there is a sequence of n packet arrivals for which it is Omega (rootn)-competitive for the first objective function, 2-competitive for the second function for L = 0, and 1-competitive for the second function for L = 1. Next we present a second on-line algorithm which is a slight modification of the first, and we prove that it is 2-competitive for both objective functions for all L. We also give lower bounds on the competitive ratio for any deterministic on-line algorithm. These results show that for each objective function, at least one of our algorithms is optimal. Finally, we give some initial empirical results using arrival sequences from real network traffic where we compare the two methods used in TCP for acknowledgment delay with our two on-line algorithms. In all cases we examine performance with L = 0 and L = 1.