We consider processing networks where multiple dispatchers are connected to single-server queues by a bipartite compatibility graph, modeling constraints that are common in data centers and cloud networks due to geographic reasons or data locality issues. We prove lower bounds for the steady-state occupancy, i.e., the complementary cumulative distribution function of the empirical queue length distribution. The lower bounds are geometric with ratios given by two flexibility metrics: the average degree of the dispatchers and a novel metric that averages the minimum degree over the compatible dispatchers across the servers. Using these lower bounds, we establish that the asymptotic performance of a growing processing network cannot match that of the classic Power-of-$d$ or JSQ policies unless the flexibility metrics approach infinity in the large-scale limit.

翻译：我们考虑处理网络，其中多个调度器通过二部兼容图连接到单服务器队列，这种建模约束由于地理原因或数据局部性问题在数据中心和云网络中很常见。我们证明了稳态占用的下界，即经验队列长度分布的互补累积分布函数。这些下界是几何形式的，其比率由两个灵活性度量给出：调度器的平均度数以及一个新颖的度量，该度量对服务器各兼容调度器的最小度数进行平均。利用这些下界，我们证明了增长的处理网络的渐近性能无法与经典的Power-of-$d$或JSQ策略相匹配，除非在大规模极限下这些灵活性度量趋近无穷大。