We consider scheduling of colored packets with transition costs which form a general metric space. We design a @math competitive algorithm. Our main result is a hardness result of @math which matches the competitive ratio of the algorithm for each metric space separately. In particular, we improve the hardness result of Azar at el. 2009 for uniform metric spaces. We also extend our result to weighted directed graphs which obey the triangular inequality and show a @math competitive algorithm and a nearly-matching hardness result. In proving our hardness results we use some interesting non-standard embedding.
In the bounded delay model @cite_14 we are given a sequence of incoming packets. Each packet is characterized by a value and a deadline. The goal is to find a schedule that maximizes the number of packets that were transmitted before the deadline. The earliest deadline first (EDF) strategy is known to achieve an optimal throughput when all the packets have the same value. For arbitrary values the following results are known. A deterministic competitive algorithm of about @math @cite_27 @cite_31 and a hardness result of @math @cite_5 @cite_30 @cite_20 . A randomized competitive algorithm of @math @cite_18 @cite_19 and a hardness result of @math @cite_30 . For additional papers related to the bounded delay model see @cite_16 @cite_24 @cite_8 .
@cite_30 @cite_18 @cite_14 @cite_8 @cite_24 @cite_19 @cite_27 @cite_5 @cite_31 @cite_16 @cite_20
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We study the online problem of multiplexing packets with arbitrary deadlines in bounded multi-buffer switch. In this model, a switch consists of m input buffers each with bounded capacity B and one output port. Each arriving packet is associated with a value and a deadline that specifies the time limit till the packet can be transmitted. At each time step the switch can select any non-empty buffer and transmit one packet from that buffer. In the preemptive model, stored packets may be preempted from their buffers due to lack of buffer space or discarded due to the violation of the deadline constraints. If preemption is not allowed, every packet accepted and stored in the buffer must be transmitted before its deadline has expired. The goal is to maximize the benefit of the packets transmitted by their deadlines. To date, most models for packets with deadlines assumed a single buffer. To the best of our knowledge this is the first time a bounded multi-buffer switch is used with arbitrary deadline constraints Our main result is a 9.82-competitive deterministic algorithm for packets with arbitrary values and deadlines. Note that the greedy algorithm is not competitive. For the non-preemptive model we present a 2-competitive deterministic algorithm for the unit value packets. For arbitrary values we present a randomized algorithm whose competitiveness is logarithmic in the ratio between the largest and the smallest value of the packets in the sequence We consider packet scheduling in a network providing differentiated services, where each packet is assigned a value. We study various queueing models for supporting QoS (Quality of Service). In the nonpreemptive model, packets accepted to the queue will be transmitted eventually and cannot be dropped. The FIFO preemptive model allows packets accepted to the queue to be preempted (dropped) prior to their departure, while ensuring that transmitted packets are sent in the order of arrival. In the bounded delay model, packets must be transmitted before a certain deadline, otherwise it is lost (while transmission ordering is allowed to be arbitrary). In all models the goal of the buffer policy is to maximize the total value of the accepted packets.Let α be the ratio between the maximal and minimal value. For the non-preemptive model we derive a Θ(log &alpha) competitive ratio, both exhibiting a buffer policy and a general lower bound. For the interesting case of two distinct values, we give an 2α--1 α competitive buffer policy, which exactly matches the lower bound. We also analyze a RED-like policy and derive its competitive ratio, which is approximately 2α--0.5 α for two values and Θ(log α) for multiple values. In addition we improve the previous known lower and upper bounds of the Fixed Partition and Flexible Partition policies.For the FIFO preemptive model, we improve the general lower bound and show a tight bound for the special case of queue size 2. We prove that the bounded delay model with uniform delay 2 is equivalent to a modified FIFO preemptive model with queue size 2. We then give improved upper and lower bounds on the 2-uniform bounded delay model. We also give lower bound for the 2-variable bounded delay model, which matches the previously known upper bound. We consider buffer management of unit packets with deadlines for a multi-port device with reconfiguration overhead. The goal is to maximize the throughput of the device, i.e., the number of packets delivered by their deadline. For a single port or with free reconfiguration, the problem reduces to the well-known packets scheduling problem, where the celebrated earliest-deadline-first (EDF) strategy is optimal 1-competitive. However, EDF is not 1-competitive when there is a reconfiguration overhead. We design an online algorithm that achieves a competitive ratio of 1 - o(1) when the ratio between the minimum laxity of the packets and the number of ports tends to infinity. This is one of the rare cases where one can design an almost 1-competitive algorithm. One ingredient of our analysis, which may be interesting on its own right, is a perturbation theorem on EDF for the classical packets scheduling problem. Specifically, we show that a small perturbation in the release and deadline times cannot significantly degrade the optimal throughput. This implies that EDF is robust in the sense that its throughput is close to the optimum even when the deadlines are not precisely known. Motivated by the quality-of-service (QoS) buffer management problem, we consider online scheduling of packets with hard deadlines in a finite capacity queue. At any time, a queue can store at most b isin Z + packets. Packets arrive over time. Each packet is associated with a non-negative value and an integer deadline. In each time step, only one packet is allowed to be sent. Our objective is to maximize the total value gained by the packets sent by their deadlines in an online manner. Due to the Internet traffic's chaotic characteristics, no stochastic assumptions are made on the packet input sequences. This model is called a finite-queue model. We use competitive analysis to measure an online algorithm's performance versus an unrealizable optimal offline algorithm who constructs the worst possible input based on the knowledge of the online algorithm. For the finite-queue model, we first present a deterministic 3-competitive memoryless online algorithm. Then, we give a randomized (Phi 2 = (1+radic(5) 2) 2 ap 2.618)-competitive memoryless online algorithm. The algorithmic framework and its theoretical analysis include several interesting features. First, our algorithms use (possibly) modified characteristics of packets; these characteristics may not be same as those specified in the input sequence. Second, our analysis method is different from the classical potential function approach. We use a simple charging scheme, which depends on a clever modification (during the course of the algorithm) on the packets in the queue of the optimal offline algorithm. We then prove that a set of invariants holds at the end of each time step. Finally, we analyze the two proposed algorithm in a relaxed model, in which packets have no hard deadlines but an order. We conclude that both algorithms have the same competitive ratios in the relaxed model.