Analyzing Nonblocking Switching Networks using Linear Programming (Duality)

The main task in analyzing a switching network design (including circuit-, multirate-, and photonic-switching) is to determine the minimum number of some switching components so that the design is non-blocking in some sense (e.g., strict- or wide-sense). We show that, in many cases, this task can be accomplished with a simple two-step strategy: (1) formulate a linear program whose optimum value is a bound for the minimum number we are seeking, and (2) specify a solution to the dual program, whose objective value by weak duality immediately yields a sufficient condition for the design to be non-blocking. We illustrate this technique through a variety of examples, ranging from circuit to multirate to photonic switching, from unicast to $f$-cast and multicast, and from strict- to wide-sense non-blocking. The switching architectures in the examples are of Clos-type and Banyan-type, which are the two most popular architectural choices for designing non-blocking switching networks. To prove the result in the multirate Clos network case, we formulate a new problem called {\sc dynamic weighted edge coloring} which generalizes the {\sc dynamic bin packing} problem. We then design an algorithm with competitive ratio 5.6355 for the problem. The algorithm is analyzed using the linear programming technique. A new upper-bound for multirate wide-sense non-blocking Clos networks follow, improving upon a decade-old bound on the same problem

Blocking and Nonblocking Multirate Clos Switching Networks

This paper investigates in detail the blocking and nonblocking behavior of multirate Clos switching networks at the connection/virtual connection level. The results are applicable to multirate circuit and fast-packet switching systems. Necessary and sufficient nonblocking conditions are derived analytically. Based on the results, an optimal bandwidth partitioning scheme is proposed to reduce switch complexity while maintaining the nonblocking property. The blocking behavior of blocking switches supporting multicast connections is investigated by means of simulation. We propose a novel simulation model that filters out external blocking events without distorting the bandwidth and fanout (for multicasting) distributions of connection requests. In this way, the internal blocking statistics that truly reflect the switch performance can be gathered and studied. Among many simulation results, we have shown that for point-to-multipoint connections, a heuristic routing policy that attempts to build a narrow multicast tree can have relatively low blocking probabilities compared with other routing policies. In addition, when small blocking probability can be tolerated, our results indicate that situations with many large-fanout connection requests do not necessarily require a switch architecture of higher complexity compared to that with only point-to-point requests