Greedy Algorithms for Multi-Queue Buffer Management with Class Segregation

In this paper, we focus on a multi-queue buffer management in which packets of different values are segregated in different queues. Our model consists of m packets values and m queues. Recently, Al-Bawani and Souza (arXiv:1103.6049v2 [cs.DS] 30 Mar 2011) presented an online multi-queue buffer management algorithm Greedy and showed that it is 2-competitive for the general m-valued case, i.e., m packet values are 0 < v_{1} < v_{2} < … < v_{m}, and (1+v_{1}/v_{2})-competitive for the two-valued case, i.e., two packet values are 0 < v_{1} < v_{2}. For the general m-valued case, let c_i = (v_{i} + \sum_{j=1}^{i-1} 2^{j-1} v_{i-j})/(v_{i+1} + \sum_{j=1}^{i-1}2^{j-1}v_{i-j}) for 1 \leq i \leq m-1, and let c_{m}^{} = \max_{i} c_{i}. In this paper, we precisely analyze the competitive ratio of Greedy for the general m-valued case, and show that the algorithm Greedy is (1+c_{m}^{})-competitive.

💡 Research Summary

The paper studies online buffer management for network switches that employ class‑segregated queues: there are m distinct packet values 0 < v₁ < v₂ < … < v_m and exactly m FIFO queues, each dedicated to one value. Each queue Q_i has a finite capacity B_i ≥ 1 and all packets stored in Q_i have the same value v_i, so the order of transmission inside a queue is irrelevant. Time is discretized into unit slots; packets arrive at non‑integral times and transmission events occur at integer times.

The algorithm under investigation, Greedy, works as follows. In an arrival event, if the destination queue still has free slots the packet is accepted (otherwise it is dropped). In a transmission event Greedy selects the non‑empty queue with the highest packet value and transmits exactly one packet from that queue. Because all packets in a queue share the same value, preemption is useless in this model; every accepted packet will eventually be transmitted.

Al‑Bawani and Souza (2011) previously proved that Greedy is 2‑competitive for the general m‑valued case and (1 + v₁/v₂)‑competitive for the two‑value case. Their analysis, however, leaves a gap: the bound 2 is not tight for many value distributions. This paper closes the gap by introducing a refined parameter that captures how much “value loss” can accumulate when Greedy prefers higher‑value queues.

For each i ∈ {1,…,m‑1} define

c_i = ( v_i + Σ_{j=1}^{i‑1} 2^{j‑1} v_{i‑j} ) / ( v_{i+1} + Σ_{j=1}^{i‑1} 2^{j‑1} v_{i‑j} ).

Let c*m = max{1≤i≤m‑1} c_i. The main theorem (Theorem 4.1) shows that Greedy is (1 + c*_m)‑competitive. Since the definition of c_i always yields a value smaller than 1, we obtain a strict improvement over the previous 2‑competitive guarantee.

The technical core proceeds in several steps. First, Lemma 2.1 (borrowed from Al‑Bawani and Souza) establishes that for any value level h, the total excess of packets accepted by the optimal offline algorithm over Greedy, summed over all values ≥ h, never exceeds the total number of packets Greedy has accepted at those levels. This inequality is the foundation for later bounding the difference in transmitted value.

Next the authors introduce ξ_h(e) = Σ_{ℓ=h}^{m} δ_ℓ(e) − δ*_h(e), where δ_ℓ(e) (resp. δ*_ℓ(e)) denotes the number of ℓ‑value packets transmitted by Greedy (resp. Opt) up to event e. Claims 3.1 and 3.2 prove that ξ_h(e) never decreases at a transmission event, regardless of whether the high‑value queues of Greedy or Opt are empty. Lemma 3.1 then uses induction over all events to show ξ_h(e) ≥ 0 for every h and every event e. Intuitively, this means that up to any point in time Greedy has transmitted at least as many high‑value packets as Opt, after accounting for the lower‑value packets that Opt might have already sent.

With ξ_h(e) non‑negative, the authors relate the total value transmitted by Greedy to that of Opt. By expanding the definition of c_i, they bound the possible deficit at each value level i by c_i times the total value transmitted at higher levels. Taking the worst case over i yields the factor c*_m. Consequently, the total value of Opt is at most (1 + c*_m) times the total value of Greedy, establishing the competitive ratio.

The paper also discusses concrete instances. For the simple arithmetic progression v₁ = 1, v₂ = 2, and v_{i+1} = v_i + Σ_{j=1}^{i‑1}2^{j‑1}v_{i‑j} (i ≥ 2), one computes c*_m = ½, so Greedy becomes 1.5‑competitive, a substantial improvement over the generic bound of 2. The authors note that c*_m < 1 holds for any strictly increasing value sequence, guaranteeing a competitive ratio strictly below 2 for all instances.

The paper situates its contribution among a rich literature on buffer management, summarizing known deterministic and randomized bounds for unit‑valued and general‑valued multi‑queue models. It emphasizes that while more sophisticated algorithms (e.g., transmit‑largest‑head) achieve better constants in some settings, Greedy remains attractive because of its extreme simplicity and now provably tight performance guarantee.

In conclusion, the authors provide a precise analysis that refines the competitive ratio of the Greedy algorithm from a coarse 2‑bound to (1 + c*_m), where c*_m depends explicitly on the packet value distribution. This result both advances theoretical understanding of online buffer management with class segregation and offers practical reassurance that a very simple online policy can be near‑optimal across a wide range of QoS‑driven networking scenarios.