Breaking the Barrier Of 2 for the Competitiveness of Longest Queue Drop

We consider the problem of managing the buffer of a shared-memory switch that transmits packets of unit value. A shared-memory switch consists of an input port, a number of output ports, and a buffer with a specific capacity. In each time step, an arbitrary number of packets arrive at the input port, each packet designated for one output port. Each packet is added to the queue of the respective output port. If the total number of packets exceeds the capacity of the buffer, some packets have to be irrevocably rejected. At the end of each time step, each output port transmits a packet in its queue and the goal is to maximize the number of transmitted packets. The Longest Queue Drop (LQD) online algorithm accepts any arriving packet to the buffer. However, if this results in the buffer exceeding its memory capacity, then LQD drops a packet from the back of whichever queue is currently the longest, breaking ties arbitrarily. The LQD algorithm was first introduced in 1991, and is known to be 2-competitive since 2001. Although LQD remains the best known online algorithm for the problem and is of practical interest, determining its true competitiveness is a long-standing open problem. We show that LQD is 1.707-competitive, establishing the first (2-?) upper bound for the competitive ratio of LQD, for a constant ? > 0

Online packet scheduling for CIOQ and buffered crossbar switches

We consider the problem of online packet scheduling in Combined Input and Output Queued (CIOQ) and buffered crossbar switches. In the widely used CIOQ switches, packet buffers (queues) are placed at both input and output ports. An N×N CIOQ switch has N input ports and N output ports, where each input port is equipped with N queues, each of which corresponds to an output port, and each output port is equipped with only one queue. In each time slot, arbitrarily many packets may arrive at each input port, and only one packet can be transmitted from each output port. Packets are transferred from the queues of input ports to the queues of output ports through the internal fabric. Buffered crossbar switches follow a similar design, but are equipped with additional buffers in their internal fabric. In either model, our goal is to maximize the number or, in case the packets have weights, the total weight of transmitted packets. Our main objective is to devise online algorithms that are both competitive and efficient. We improve the previously known results for both switch models, both for unweighted and weighted packets. For unweighted packets, Kesselman and Rosén (J. Algorithms 60(1):60–83, 2006) give an online algorithm that is 3-competitive for CIOQ switches. We give a faster, more practical algorithm achieving the same competitive ratio. In the buffered crossbar model, we also show 3-competitiveness, improving the previously known ratio of 4. For weighted packets, we give 5.83- and 14.83-competitive algorithms with an elegant analysis for CIOQ and buffered crossbar switches, respectively. This improves upon the previously known ratios of 6 and 16.24

Breaking the barrier of 2 for the competitiveness of longest queue drop

We consider the problem of managing the buffer of a shared-memory switch that transmits packets of unit value. A shared-memory switch consists of an input port, a number of output ports, and a buffer with a specific capacity. In each time step, an arbitrary number of packets arrive at the input port, each packet designated for one output port. Each packet is added to the queue of the respective output port. If the total number of packets exceeds the capacity of the buffer, some packets have to be irrevocably rejected. At the end of each time step, each output port transmits a packet in its queue and the goal is to maximize the number of transmitted packets. The Longest Queue Drop (LQD) online algorithm accepts any arriving packet to the buffer. However, if this results in the buffer exceeding its memory capacity, then LQD drops a packet from the back of whichever queue is currently the longest, breaking ties arbitrarily. The LQD algorithm was first introduced in 1991, and is known to be 2-competitive since 2001. Although LQD remains the best known online algorithm for the problem and is of practical interest, determining its true competitiveness is a long-standing open problem. We show that LQD is 1.707-competitive, establishing the first (2-ε) upper bound for the competitive ratio of LQD, for a constant ε > 0