Function tree

Abstract: Usually, a data flow needs to traverse a series of network functions, which is called a service function chain (SFC), before reaching its destination. The emergence of network function virtualization (NFV) makes the embedding solution of the SFC flexible as far as the deployment location is concerned. When providers embed the SFC into a substrate network, they will hope to minimize the setup cost of the SFC and link connection cost toward clients. For unicast, since there is one path connecting the source node to the destination node, all functions of the SFC are just needed to be deployed along the path. However, when embedding the SFC for a multicast task, the topology of the SFC may change because the function deployment locations have impacts on the traffic delivery cost. Thus, a service function tree (SFT) may be a better choice. Given the huge space of SFT embedding solutions, however, it is extremely hard to find the optimal one such that the total traffic delivery cost is minimized. In this paper, we tackle the optimal SFT embedding problem in the NFV enabled multicast task. Specifically, we formally define the problem and formulate it with an integer linear programming (ILP), which turns out to be NP-hard. Then, a two-stage method is proposed to deal with the problem with an approximation ratio of $1+\rho $ , where $\rho $ is the best approximation ratio of Steiner tree and can be as small as 1.39. With extensive experimental evaluations, we demonstrate that by applying our SFT embedding solution, the delivery cost of multicast traffic can be reduced by 22.05% at most against three benchmarks.

Abstract: Understanding product functions is a key aspect of the work undertaken by engineers involved in complex system design. The support offered to these engineers by existing modeling tools such as the function tree and the function structure is limited because they are not intuitive and do not scale well to deal with real-world engineering problems. A research collaboration between two universities and a major power system company in the aerospace domain has allowed the authors to further develop a method for function analysis known as function analysis diagram that was already in use by line engineers. The capability to generate and edit these diagrams was implemented in the Decision Rationale editor, a software tool for capturing design rationale. This article presents the intended benefits of the method and justifies them using an engineering case study. The results of the research have shown that the function analysis diagram method has a simple notation, permits the modeling of product functions together with structure, allows the generation of rich and accurate descriptions of product functionality, is useful to work with variant and adaptive design tasks, and can coexist with other functional modeling methods.

Abstract: We give a new proof showing that it is not possible to define in monadic second-order logic (MSO) a choice function on the infinite binary tree. This result was first obtained by Gurevich and Shelah using set theoretical arguments. Our proof is much simpler and only uses basic tools from automata theory. We discuss some applications of the result concerning unambiguous tree automata and definability of winning strategies in infinite games. In a second part we strengthen the result of the non-existence of an MSO-definable well-founded order on the infinite binary tree by showing that every infinite binary tree with a well-founded order has an undecidable MSO-theory.