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In this paper we present the mathematical background of the four most used numerical methods of solving equations and few examples of Python applications that find the approximations of the roots of the given equations. We also compare the exact and approximate solutions of polynomial equations of third degree. Exact solutions are obtained with usage of Cardano formulae by the help of Mathematica...
In this article we want to present some simple method of doing time calculations which is not often used by teachers and show that the discussed way does not have to be difficult for students and is based on strict mathematical rules.
Elements of theory of divisibility are present in many various interesting mathematical tasks, especially in tasks that are addressed to talented pupils taking part in mathematical competitions. Good understanding of it lets solve very interesting and difficult (at first glance) issues. On the other hand, there are a lot of problems with understanding such terms as: multiple, divisor, divisibility,...
The current core curriculum in mathematics for lower secondary school (3-rd educational level in Poland) omits formal definitions of concepts related to geometric transformations in the plane and is based on their intuitive sense. Practice shows that the current approach makes teaching very difficult and the students solve the typical tasks, not understanding the meaning of geometrical concepts. The...
We investigate multi-server queueing systems with Poisson arrivals, non-identical servers and customers of random volume, under assumption that customer’s service time having an exponential distribution doesn’t depend on his volume, but service time parameters can be different for different servers. We also assume that the total volume of customers present in the system at arbitrary time instant is...
In many various practical problems we often deal with computing distribution functions of sums of independent non-negative random variables. In applied mathematics (ex. queueing theory) we can find many formulas with Stieltjes convolutions of distribution functions of random variables of the same type. Finding convolutions on the base of definition is not easy and convenient, because there are some...
M/M/n/m queueing systems with identical servers are well known in queueing theory and its applications. The analysis of these systems is very simple thanks to the fact that the number of customers ɳ(t) in the system at arbitrary time instant t forms a Markov chain. The main purpose of this paper is to analyse the M/M/n/m system under assumption that its servers are different, i.e. they have different...
For single-server queueing systems with non-homogeneous customers having some random space requirements we compare processor-sharing and FIFO disciplines and investigate their influence on the total sum of space requirements characteristics (when this sum is not limited, i.e. V = ∞) and customers loss probability (when this sum is limeted, i.e. V < ∞), using analytical modeling and simulation.
In this paper we present generalization of probability density of random variables. It is obvious that probability density is definite only for absolute continuous variables. However, in many practical applications we need to define the analogous concept also for variables of other types. It can be easily shown that we are able to generalize the concept of density using distributions, especially Dirac’s...
A non-classical single-server queueing system with non-homogeneous customers having some random space requirement (capacity, volume) can be used as a model of a wide class of computer and communicating systems. We assume that the total customers capacity in the systems is limited by some constant value V > O that is called the value of memory capacity of the system. Service time of a customer generally...
In the present paper we investigate queueing systems of different types with customers having some random space requirements connected via common memory space. For such systems combinations we determine the stationary loss probability and the distribution of customers present in each system.
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