An Optimal Lower Bound for Buffer Management in Multi-Queue Switches

Abstract

In the online packet buffering problem (also known as the unweighted FIFO variant of buffer management), we focus on a single network packet switching device with several input ports and one output port. This device forwards unit-size, unit-value packets from input ports to the output port. Buffers attached to input ports may accumulate incoming packets for later transmission; if they cannot accommodate all incoming packets, their excess is lost. A packet buffering algorithm has to choose from which buffers to transmit packets in order to minimize the number of lost packets and thus maximize the throughput.We present a tight lower bound of e/(e − 1) ≈ 1.582 on the competitive ratio of the throughput maximization, which holds even for fractional or randomized algorithms. This improves the previously best known lower bound of 1.4659 and matches the performance of the algorithm RANDOM SCHEDULE. Our result contradicts the claimed performance of the algorithm RANDOM PERMUTATION; we point out a flaw in its original analysis.

Competitive buffer management for multi-queue switches in QoS networks using packet buffering algorithms

“…Moreover, they showed that for B = 2, the same algorithm achieves the competitive ratio of 13/7(≈ 1.858) and gave a matching lower bound for large enough m. Schmidt [37] also proposed a deterministic algorithm whose competitive ratio is at most 17/9 for any even B ≥ 4 and at most 17/9 + 2/9(B + 1) for any odd B ≥ 3. Furthermore, he showed a lower bound e e−1 (≈ 1.581) of deterministic online algorithms for any B and large enough m, and a lower bound 1.465 of randomized online algorithms for any B and large enough m. Azar and Litichevskey [8] showed a e e−1 (≈ 1.58)-competitive deterministic algorithm for large enough B > log m. Schmidt [36] claimed he showed a 3/2-competitive randomized algorithm, whose flaw was pointed out in [13]. Also, in the case of m = 2, Schmidt [36] showed a lower bound of 16/13 ≈ 1.230 for any online algorithm for large enough B. Bienkowski and Madry [12] and Kobayashi et al [32] proved 16/13-competitive algorithms for the randomized and deterministic cases respectively.…”

“…Also, in the case of m = 2, Schmidt [36] showed a lower bound of 16/13 ≈ 1.230 for any online algorithm for large enough B. Bienkowski and Madry [12] and Kobayashi et al [32] proved 16/13-competitive algorithms for the randomized and deterministic cases respectively. Bienkowski [13] showed a lower bound of e e−1 for any online algorithm for any B and large enough m. As for the single-queue models, the current upper and lower bounds on competitive ratios are summarized in Table 2. There are several other models, such as shared-memory switches [19,24,31], CIOQ switches [26,11,27,30,3] and crossbar switches [28,29,3], are also extensively studied.…”

“…On the other hand, the k ′ OP T r -events can be f -events. Hence, the number of f -events for (OP T r , σ r ) is at most T OS (g OS (σ r )) + k ′ − k. Therefore, (13) and 14)…”

“…Randomized Algorithm e e−1 ≈ 1.581 [13] e e−1 ≈ 1.581 [13] 2.5 * [9] 2.270 * (C5.4) * large enough B, † any B, § B ≥ 3 Bold values indicate our results, where (C5.x) after each value shows the corresponding Corollary number, and α is chosen to maximize each value. if there is a c ′ -competitive online algorithm for the 2-value single-queue switch model.…”

Buffer management of multi-queue QoS switches with class segregation

“…If s = j, then r j (e ℓ |g, o) = r j (e ℓ−1 |g, o), r j (e ℓ |g, o) = r j (e ℓ−1 |g, o), and α j (e ℓ ) = α j (e ℓ−1 ) hold. Thus from Equation (8), it follows that Equation (7) holds for h = ℓ. So we assume that s = j and let us consider the following cases: (c-1) both greedy and opt accept the v j -packet; (c-2) both greedy and opt reject the v j -packet; (c-3) greedy rejects and opt accepts the v j -packet; (c-4) greedy accepts and opt rejects the v j -packet.…”