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 Ն 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 O(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 ⍀ ( ͌ n)-competitive for the first objective function, 2-competitive for the second function for L ϭ 0, and 1-competitive for the second function for L ϭ 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 ϭ 0 and L ϭ 1.

AbotractWQ study an on-line problem that is motivated by the networking problem of dynamically adjusting delays of acknowlcdgmcnts in the Transmission Control Protocol (TCP). The thcoroticnl problem we study is the following. There is a scqucnco of n packet arrival times .4 = (or,. . . , an) and a look-nhoad coefficient L, The goal is to partition A into 1; subsequences orroe,. . . , a~: (where a subsequence end is deilncd by an acknowledgment) that minimizes a linear combination of the cost for the number of acknowledgments sent and the cost for the additional latency introduced by delaylng acknowledgments. At each arrival, an oracle provides the algorithm with the times of the next L arrivals.First we give an 0(n*) dynamic programming algorithm for optimally solving the off-line problem. Then we deacribc an on-line algorithm that greedily acknowledges exactly when the cost for an acknowledgment is less than the latency cost obtained by not acknowledging. We show that for this algorithm there ls a sequence of n packet arrivals for which it is S-l (fi) -competitive. Next we present a second on-line algorithm which is a slight modification of the first that we prove is 2-competitive. Let Copt be the cost of the optimal solution and let CA be the cost of the solution produced by algorithm A. We then show that for any on-line algorithm A with any constant look-ahead L, CA > SC+-c where c is a factor that can be made arbitrarily small with rcnpcct to Capt. Thus, in the worst case, our result for L = 0 IQ the hcst possible even for on-line algorithms that can use nny constant look-ahead.We then 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 mitli L = 0 and L = 1, Finally, we consider an alternate dcilnition for the latency cost in our objective function, *Supported in part by NSF NY1 Grant CC%9357707 with mntching funds provfdcd by WUTA.

The continuing need for high-throughput Automated Storage and Retrieval Systems (AS/RS) has lead to the introduction of storage/retrieval machines that can carry more than one unit-load. However, this technology involves a large capital investment so careful operating methods are desired to make the most of its capabilities. In this paper, we study a shift-based sequencing problem for twin-shuttle AS/RS, where depletion (retrieval operations) and replenishment (storage operations) of items occur over different shifts. For example, certain warehouses or distribution depots deplete their items in stock during morning shifts and replenish during later shifts. We show that this problem can be transformed into the minimum-cost perfect matching problem and present an efficient polynomial-time optimum method that can achieve a large throughput gain over other methods. We also provide average-case and lower bound analyses for this problem.

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