A novel approach to fault tolerant multichannel networks designing problems

– This work presents solution of a bus interconnection network set designing task on the base of a hypergraph model. In order to do this the interconnection network is presented as a multipartite hypergraph. A system with virtual bus connections functioning in an environment of common physical channel was analyzed, which is characteristic of the networks based on the WDM technology. The mathematical reliability model was proposed for two modes of system functioning: with redundancy of communication subsystem and division of communication load. As solution estimation criteria the expected changes of processing eﬃciency changes were used as also a communication delay change criteria and system reliability criteria. The designing task solution is searched in a Pareto set composed of Pareto optima. The selection procedure of a speciﬁc solution in the case of its equivalency in relation to a vector goal function was presented.

1 Introduction different variants of assignment task are used nowadays [2,6,7,8]. To define connections between the system components, generally the agglomeration [9,10] and solution methods of optimal hierarchy building task [11] are used.
The subject of the presented research is implementation and application of hypergraph theory methods and algorithms in order to solve the above designing task. Its aim is creating of the designing task solution method which is characterized by the polynomial time complexity.
2 Connection architecture and system reliability model Let us consider a distributed system shown in Fig. 1 composed of K N computation nodes, where each of them has K l tunable send-receive elements and K B communication buses. Buses B i (i = 1, . . . , K B ) have a logical character and may be realized on the basis of the wavelength division multiplexing methods in a single physical bus D F . Any node N i logical channels connecting to a physical bus B F is achieved using physical connection channels l and bus channels separator C. The analyzed system connection architecture may be dynamically reconfigured by tuning send-receive elements. In the case of a traffic pattern change the send-receive elements change the wavelength and actually connect to other logical bus. The system may work in two modes: a. with a communication subsystem redundancy; b. with communication load divi-sion. To estimate characteristics of the system working in one of these three modes, a prob-abilistically-combinatorial model was proposed, especially for estimating its reliability R. In this model, the set: send-receive device (≈), physical connection link (l) and physical channel separator (C) is treated as one single connection device. Let p io -be is a functioning probability of a computation node send-receive device, p l -a physical connection channel functioning probability; Pobrane z czasopisma Annales AI-Informatica http://ai.annales.umcs.pl Data: 26/08/2022 00:40:51 U M C S p c -a bus channel separator functioning probability and p f k -a physical bus channel functioning probability. Then, the selected node connection to a logical bus channel probability p ku can be expressed as: p ku = p io p l p c , a probability p uku of selected node connection accomplishment with other computation nodes is equal: p uku = p io p l p c p f k .
Let us consider the bus distributed system reliability with complete connections (each computa-tion node is connected to each logical bus) and processing equal rights computation nodes, which is the most generalized example of the system of this class. The probability p we (k we ) of connection sets functioning which assures connection to a bus channel not less than k min we computation nodes from the general amount of K N is expressed as: Let K B denote the number of logical buses, p sp we -the computation node functioning proba-bility. Then, using (1), according to the proposed condition, for a system in a redundancy mode, the reliability R is expressed as: Let us consider a computation system with equal computation nodes which works in a com-munication load assignment mode. Let W (k we , k m , σ) denote the number of system functioning states consisting of k we computation nodes connected by k m bus channels with occurrence of σ denies of send-receive elements. In order to obtain the number of system functioning states W (k we , k m , σ), in the case of failure elements appearance, the methods of inclusion-and-exclusion was proposed. The number H 1 (σ) of the whole system failure states in the case of deny not less than one minimal system cut with equal computation nodes may be expressed as: Then the value W may be expressed as: is the summary number of system states for σ denies of send-receive elements; i σthe maximal number of minimal cuts included in estimation; H i (σ) -the number of computation system failure states with denies of send-receive elements not less than i minimal cuts. The coherence assurance probability P ku (k we , k m , σ) of k wu nodes with the use of k m buses, in the case of occurrence σ denies can be expressed as: Pobrane z czasopisma Annales AI-Informatica http://ai.annales.umcs.pl Data: 26/08/2022 00:40:51 Using expression (2), the coherence assurance probability P ku (k we , k m ) of K we computa-tion nodes with the use of k m buses, in the case of occurrence σ denies can be expressed as: Using expression (3), let us define coherence assurance probability P uku (k we ) of K we computation nodes: where: K min m is the minimal required number of buses necessary for capacity assuring. Then, the computing system reliability is equal to: For a client-server computing system in the redundancy mode we can define a bus communica-tion channel reliability probability as P ks (K K , K S ) where through correct send-receive elements therer are connected not fewer than k min k clients from the whole client number K K and k min s servers from the whole server quantity K S : Let p sp k and p sp s be the client and server nodes activity probability, respectively. Then, using expression (4), the reliability R may be expressed as: Let us consider the client-server system reliability where the system has complete connections be-tween computing elements and communication subsystem working in a load division mode. Let W (k s , k k , k m , σ) be a number of correct system states composed of k s servers, k k clients, k m buses, with occurrence of σ denies of send-receive elements.
For the client-server system, the number of H 1 (σ) failures states for not fewer than one minimal cut can be defined as: where: C km(k k +ks) -the general number of computing system states with occurrence of σ denies. The coherence assurance probability P ks (k s , k k , k m , σ) of k s servers and k k clients with the use of k m buses in the case of occurrence σ send-receive elements denies can be expressed as: The coherence assurance probability P ku (k s , k k , k m ) of k s servers and k k clients with the use of k m buses in the case of send-receive elements deny appearance is expressed as: and the coherence probability P ku (k s , k m ) of k s servers and k k clients can be defined as: Using expressions (5), (6) and (7) the searched reliability R may be expressed as:

Main definitions
Let us consider a hypergraph H = (V, E) with a node set V = {v} and an edge set E = {e} which represents subsets of the V set e.g. e ⊆ V . Hypergraph H is called k -regular, if its each edge e ∈ E consists of k nodes. Then the hypergraph H is called l-partite if its node set is divided into l subsets. In a l -partite hypergraph a node set V is divided into l subsets V 1 , V 2 , . . . , V l , in such a manner that all edge nodes e = (v 1 , v 2 , . . . , v l ) ∈ E are parts of other graph regions, e.g. v i ∈ V i , where: i=1,. . . ,l. To denote l-partite hypergraphs we use a notation H = (V 1 , V 2 , . . . , V l ).
Let us consider the l-partite hypergraph denoted as H = (V 1 , V 2 , . . . , V l ). In this graph, its part a = V A 1 , . . . , V A i , . . . , V A l , E A is called a star, for i = 1, . . . , l and V A l ⊆ V l , in which any two edges e 1 , e 2 ∈ E A cover in one and the same node v ∈ V A 1 and don not cover in any other v ∈ V A l . It means that the set cardinality V A 1 is equal to 1, and the node v ∈ V A 1 is called a star center. We distinguish simple and complex stars. If any edge pair e 1 , e 2 ∈ E A covers only in one node v ∈ V A 1 , then such a star is called a simple star. In another case a star is called a complex star. A number of star edges is called a star degree.
For a hypergraph H = (V, E) its subhypergraph H 1 = (W, U ) is called a hypergraph in which the node set W is a subset of a node set V of the hypergraph H e.g. W ⊆ V and an edge set U is a subset of an edge set E of the hypergraph H, whereas, if (x, y) ∈ E and x, y ∈ W , then (x, y) ∈ U . The hypergraph nodes set is called a hypergraph coherence component, in which for its any two elements there exists a path from a node belonging to this set to any other node apart from this node set. If in a subhyprgraph H 1 = (V 1 , E 1 ) of the hypergraph H each coherence component is a star with a center in a node v ∈ V 1 , than H 1 is called a hypergraph H star cover.

Designing task and its solution
Designing task solution consists in searching a connection architecture which ensures its exploitation parameters maximization or minimization such as: communication delay, communication medium access errors being a consequence of its business, computation effi-ciency of processing nodes and others. Such a task may be solved by searching a star cover of at least three-partite graph. Let us consider such a task.
Input data. As an input data of a designing task we use: 1) B = {b} -a set of logical communication buses selected in a physical communication channel with the use of a wavelength multiplexing method WDM [12,13]; 2) F = {f } -a set of access protocols, according to which a logical communication bus can function [12]; Designing task definition. Each of processing nodes n ∈ N should have an assigned set M of b buses, which is a subset of the set B, e.g. M ⊆ B, each of those will be functioning on the basis of one access protocol f ∈ F .
Mathematical model. The designing task is solved iterationally. In each step for each of the processing nodes one bus b ∈ B is searched serving for a node communication services with the use of an access protocol f ∈ F . In order to avoid multiple nod joins to the same bus, each step from the set of buses available for clients each of those to which client is connected is excluded.
The mathematical model is based on the three-partite hypergraph H = (V, E) = (X, Y, Z, E). Bus-es from the set B = {b} have equivalent nodes x of its first part (x ∈ X). Each of them (contem-porary each of logical buses) a label η(x) is assigned defining bus transmission characteristics, in the simplest case, the number of clients which this bus may serve.
Pobrane z czasopisma Annales AI-Informatica http://ai.annales.umcs.pl Data: 26/08/2022 00:40:51 The elements f from the set F of bus access protocols correspond to nodes y of the second part of the hypergraph H (y ∈ Y ), and for the n elements of N processing nodes nodes z of the third part of the hypergraph (z ∈ Z) corresponds. The edge set E = {e} has all node trios (x, y, z) such as x ∈ X, y ∈ Y , z ∈ Z. Only such edges are allowed for which the selected bus b i ∈ X may serve client n j ∈ Z 3 with the use of communication protocol f l ∈ Y 2 . The set E = {e} of all edges is defined as a set of all allowed node trios e = (x, y, z). Including the value of parameter n(x), for x ∈ X in the hypergraph H = (V, E) = (X, Y, Z, E) the allowed step solution of the designing task is any of its subhypergraphs β = (V β , E β ), for V β ⊆ V and E β ⊆ E, which each coherence component represents a simple star with the degree η(x) and the center in a node x ∈ X. As S = S(H) = {s} let us denote a set of all allowed solu-tions of the hypergraph H star covering task.
Each edge e ∈ E of the hypergraph H = (V, E) has assigned three weights, defining the following solution characteristics: 1. ω(e) = ϕ(x, y, z) -the expected change of client processing efficiency in a system in which a client z is communicationally served by a bus x, in which communication protocol y is used. To estimate processing efficiency we may use characteristics proposed in [14,15,16]. The characteristics presented there were modified in order to reflect the efficiency change which is an effect of connection architecture changes. Taking into consideration a bus character of the interconnection network the basic efficiency characteristic is a node number for which a communication network is accessible [17]. 2. ξ(e) = ϕ(x, y, z) -the expected change of communication delay on a client demand for the above conditions. The level of delay changes may be defined on the base of stochastic model which uses the method presented in [18]; 3. ψ(e) = ϕ(x, y, z) -the expected change of the error level while realizing client communication demand for a saved condition in step 1. To estimate error level changes the method presented in Section 2 was used.
Designing task solution. The solution estimation will have a multicriteria character. The Proposed criteria set has obviously sample character and its selection depends on the designer needs especially related to future exploitation of the interconnection network. For a considered case these are three functions presented below .
Let us consider a set A = {a} of acceptable solutions of designing task. For each of them we define the following characteristics which estimate the solution quality: The possibilities of the above method do not limit applying summarizing criteria in a form of min or max. To estimate a solution quality it is allowed to use freely parameter convolution methods, including those involving weights of individual subparameters.
The above partial criteria are connected by the goal function expressed as Φ(a) = (Φ 1 (a), Φ 2 (a), Φ 3 (a)). The multicriteria goal function Φ(a) is used which defines in a set A of accepted solutions a Pareto set A p composed of Pareto optima a p . Let us assume that two solutions a 1 , a 2 ∈ A of a vector goal function Φ(a) are equivalent. Then from the Pareto set A p a full set of alternatives A A is selected which is generally a maximal system of different vectored Pareto optima.

Research, results and further work
The approach presented in this work was used to create a designing methodology of multichan-nel bus connections in order to communicationally serve client-server computing systems.
Existing methodologies are concentrated on a defined capacity level assurance of the whole computing system, skipping its reliability parameters. In this work the presented methodology searches an optimal solution in relation to the multicriteria goal function where one criterion is reliability. Depending on a partial criterion relationship manner and constraints set the proposed methodology allows also to specify a connection architecture characterizing: a. maximal reliability with defined: minimal efficiency and maximal communication delay in an interconnection network; b. minimal communication delay with a constraint in minimal reliability and efficiency; c. maximal computing efficiency with a defined acceptable level of reliability and delay. Also there was testet a solution in which a designing goal is communication buses load balancing. For each of searched solutions maximal building costs were limited. We analyzed networks with complete and partial connections as also flat and hierarchical.
Simulation research of the obtained architectures for a client-server model was based on the methodology presented in [19] and presented in the Fig. 2. In both cases the optimal structure in relation to a vector criterion was searched. The following notation was used: K s , K k , -the server and client number respectively; ω -the request submitting probability; ω p -the probability of a request submitted to a preferred server; ω k -the probability of a request submitted to a server of the same group; ω othe probability of a request submitted to any server of other groups. Further research will concentrate on searching the effective methods of multipartite graph coverage searched with a simple star which allowing using graphs of any partitions in a design-ing process, thus allows to introduce further designing criteria.