This paper uses three objects -- the queue matrix Q, the matching matrix P, and the Lyapunov energy function V = ||Q||^2 -- as a shared mathematical language to explain, within a single framework, the scheduling objective of maximum weight matching (MWM), queue stability under admissible traffic (per-port loads strictly below 1), and the mechanics of iSLIP's Grant-Accept row-column decoupling together with the long-run average service matrix P-bar. The setting throughout is an N-by-N SoC crossbar, where each clock cycle permits at most one cell transfer per input-output port pair. For the experimental comparison, we built a C++ discrete-event simulator and used exact MWM (solved by the Hungarian algorithm) as the performance reference. All three approximate algorithms are given a fixed iteration budget: r = 3 rounds per cycle for iSLIP and for spectral scheduling, and r_sink = 10 Sinkhorn normalization rounds for entropy-regularized optimal transport (OT). Throughput and average cell delay are measured across four traffic patterns. Spectral scheduling and entropy-regularized OT track MWM closely in both throughput and delay across most tested conditions. iSLIP, by contrast, hits a throughput ceiling of roughly 80% under non-uniform admissible traffic at high load (unbalanced pattern w = 0.5, rho_load >= 0.9), with bottleneck queues growing without bound and delays reaching two orders of magnitude above MWM. Under uniform traffic this breakdown does not occur: at rho_load = 0.99 iSLIP delay is about 3.7x that of MWM. The performance gains of spectral scheduling and OT come at an additional per-cycle compute cost on the order of O(r*N^2) multiply-accumulate or exponential operations; whether this overhead is feasible in real hardware -- in terms of die area, power, and timing closure -- remains to be evaluated.

翻译：本文以三个对象——队列矩阵Q、匹配矩阵P和李雅普诺夫能量函数V=||Q||²——作为共享数学语言，在统一框架内阐释了以下内容：最大权重匹配（MWM）的调度目标、可接受流量下的队列稳定性（每端口负载严格小于1），以及iSLIP的Grant-Accept行列解耦机制及其长期平均服务矩阵P-bar。整个场景设定在N×N片上系统交叉开关中，每个时钟周期每个输入输出端口对最多允许一个信元传输。实验对比方面，我们构建了C++离散事件模拟器，并以精确MWM（由匈牙利算法求解）作为性能基准。三种近似算法均分配固定迭代预算：iSLIP与频谱调度每周期r=3轮，熵正则化最优传输（OT）的Sinkhorn归一化轮数r_sink=10。吞吐量和平均信元延迟在四种流量模式下测量。在大多数测试条件下，频谱调度和熵正则化OT在吞吐量和延迟方面均紧密跟随MWM。相比之下，iSLIP在高负载非均匀可接受流量下（非平衡模式w=0.5，rho_load>=0.9）遭遇约80%的吞吐量上限，瓶颈队列无限增长，延迟达到MWM的两个数量级。在均匀流量下未出现此崩溃：当rho_load=0.99时，iSLIP延迟约为MWM的3.7倍。频谱调度和OT的性能提升以每周期额外O(r*N²)的乘累加或指数运算计算开销为代价；该开销在真实硬件中是否可行——涉及芯片面积、功耗和时序收敛——仍有待评估。