This paper considers the problem of optimal distribution of computational resources of multiprocessor systems based on the principles of parallelization and conveyorization. The necessary conditions and criteria of efficiency and optimality of systems of identically distributed competing processes are obtained that take into account the overheads required in the context of their run time.Keywords: system of competing processes, distributed processing, optimality and efficiency criteria.Optimal distribution of computational resources of multiprocessor systems (MSs) based on the principles of parallelization and conveyorization is one of the most important problems in creating efficient system and application software [1,2]. An important place in solving it is occupied by problems of optimal organization of competing processes using common program resources. The solution of these problems determines not only the efficiency of using MSs but also their capabilities to solve complicated problems from various areas of knowledge.It is relevant to note that, in solving problems of optimal organization of competing processes, mathematical models based on lumped processing are most developed [2]. At the present time, MSs based on distributed processing have continued to play an increasingly important part [3][4][5]. In this connection, the problems of construction and investigation of mathematical models of optimal organization of competing processes during distributed processing are especially topical.As in [5], a mathematical model of distributed processing of competing processes includes the following parameters: p p , ³ 2 , is the numbers of processors of a multiprocessor system, n n , ³ 2 , is the number of competing processes, s s , ³ 2 , is the number of blocks Q j of a structured program resource (PR), j s = 1, , and [ ], , , , t i n j s ij = = 1 1 , is the matrix of run times of execution of blocks with the help competing processes. It is assumed that all n processes use one copy of the PR structured in the form of blocks. Let us introduce the parameter e > 0 that characterizes the time of additional system expenditures for the organization of parallel use of PR blocks by a set of distributed competing processes.In this article, the necessary conditions and efficiency and optimality criteria of systems of identically distributed competing processes are considered, taking into account the overheads required depending on their run time.

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Distributed processing is one of the important forms of parallelism. Distributed processing principles are apparent virtually in all parallel computer architectures, and in many multiprocessor systems these principles play a dominant role. It suffices to mention the macropipelined computing complex [1, 2], transputer systems [3], and a number of other systems [4, 5].Because of the widespread use of distributed processing principles, it is relevant to consider the design and analysis of mathematical models of this kind of parallelism, to investigate the potential for acceleration and efficiency improvement of computations through distributed processing in multiprocessor systems (MS).This article constructs a mathematical model of distributed processing of competing processes. Mathematical relationships are derived for the least total time to execute a set of competing nonhomogeneous, homogeneous, and identically distributed processes, and the limiting acceleration and efficiency coefficients are estimated. MATHEMATICAL MODEL OF DISTRIBUTED PROCESSING OF COMPETING PROCESSESAs in [6-8], we consider n competing processes (n >__ 2) that access a software resource Pr structured from s linearly ordered blocks Q1, Q2 ..... Qs (s >_ 2). All n processes use only one copy of the software resource. The processes execute in a computing environment with p identical processors (t9 >__ 2), each with its own local memory and access to shared memory.Here a process involves execution of program blocks in the sequence Q1, (22 ..... Qs. A process is called distributed if all the program blocks (or part of them) required for its execution can be processed on different processors. Otherwise, the process is called concentrated.In what follows we assume that the interaction of processes, processors, and program blocks satisfies the following conditions:i) none of the blocks of the software resource Pr can be processed simultaneously on more than one processor; it) none of the processors can process simultaneously more than one program block; iiO each block is processed without preemption.The distribution of program blocks among processors for each process is carried out cyclically by the following rule: block j = kp + i(j = 1, s, i = 1, p, k > 0) is assigned to processor i. We also assume that processors do not remain idle if program blocks are ready for execution, and program blocks do not "lie idle" if free processors are available. NONHOMOGENEOUS PROCESSESPreviously [6-8] it has been assumed that a system of n competing processes is homogeneous, i.e., the execution times of the blocks of the software resource Pr are independent of the volume of data being processed.Here we assume that the execution times of the blocks QI, Q2 ..... Qs essentially depend on the volume of processed data. Such a system of n processes is called nonhomogeneous.

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How do people learn in general and study astronomy in particular? To develop a coherent educational policy we need an appropriate theory. Does learning consist of the incremental addition of individual “bits” of information into the mind? Or is learning an active process that transforms the mind of the learner? Among different theories on how people learn are: Behaviorism, Neuroscience, Right Brain vs. Left Brain, Communities of Practice, Control Theory, Observational Learning (Social learning theory), Vygotsky and Social Cognition, Learning Styles, Piaget's theory, Constructivism, Brain-based Learning, Multiple Intelligences. These theories are described in brief. All of the above mentioned learning theories may be applicable to some extent in the case of astronomy education in a planetarium environment. Especially the Multiple Intelligences theory can be tested perfectly while teaching in Planetarium and thus should be taken into more thorough consideration. It is discussed what a planetarium may offer to the audience with different types of intelligences, according to the Multiple Intelligences approach.