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6PLenary session 27
4. Buizza R., Miller M., Palmer T. Stochastic representation of model uncertainties in the ECMWF Ensemble
Prediction System. ECMWF Tech. Memo. V. 279. 1999.
Alternative designs of high load queuing systems with small queue
G. Sh. Tsitsiashvili
Institute for Applied Mathematics FEB RAS
Email: guram@iam.dvo.ru
DOI 10.24412/cl-35065-2021-1-02-35
In this paper, two alternative designs are constructing for queuing systems with a large load and a small
queue. These modes are convenient from an economic point of view, since the service device is almost fully
loaded. On the other hand, this mode is also convenient for users who will not be idle in the queue for a long
time. The first design is an aggregation of a large number of single-channel systems into a multi-channel system.
The second design is basing on the model of a single-channel system, in which random fluctuations are
defining as the degree of tending to zero difference between the unit and the load factor. The exponent of this
degree has a critical value, above which the stationary waiting time tends to zero, and below which it tends to
infinity. A similar phase transition is founding in the multi-channel queuing system. The methods of the
sources [1-4] are using.
References
1. Borovkov A.A. Asymptotic methods in queueing theory. M.: Nauka, 1980.
2. Borovkov A.A. Stochastic processes in queuing theory. M.: Nauka, 1972.
3. Tsitsiashvili G.Sh., Osipova M.A. Phase Transitions in Multiserver Queuing Systems // Information Technologies
and Mathematical Modelling. Queueing Theory and Applications. 2016. V. 638. P. 341-353.
4. Boxma O.J., Cohen, J.W. Heavy-traffic analysis for the GI/G/l queue with heavy-tailed distributions // Queueing
Systems. 1999. V. 33. P. 177-204.
Taking into account a priori information is the most important stage in solving ill-posed problems
V. V. Vasin, A. L. Ageev
1
Institute of Mathematics and Mechanics UB RAS
Ural Federal University, Ekaterinburg
Email: vasin@imm.uran.ru
DOI 10.24412/cl-35065-2021-1-02-18
What is a problem with a priori information? It is a problem, in which together with the basic statement
there is an additional information on a solution (namely, constraints) that is absent in the original statement.
But this information might contain important data on some properties of a solution. It should be noted that in
the case of a non-uniqueness solution, (when a priori information is not used in the algorithm of the problem
to be solved), the approximate solutions could not satisfy to physical reality. In the case of uniqueness of solution,
attraction of the additional constraints permits to localize the desired solution and to raise its stability
w.r.t. the errors in the input data. Majority of a priori constraints that arise in the applied problems can be
presented in the form of the linear relations or systems of the linear and convex inequalities. We investigate
various methods of taking into account a priori constraints, in particular, the most general and economical
method on the basis of the Fejer mappings. Also, we consider the ill-posed problems, for which the solutions
are found by the high-precision algorithms using a priori information [1�3].
28 Plenary session
This work was supported by the Russian Science, project 18-11-00024-�.
References
1. Vasin V.V., Ageev A.L. Ill-Posed problems with a priori information. Utrecht, The Netherlands: VSP, 1995.
2. Vasin V.V., Skorik G.G., Kuchuk F. A new technique for solving pressure-rate deconvolution problem in. pressure
transient testing // J. Eng. Math. 2016. V. 101, Iss. 1. P. 189-200.
3. Vasin V.V. Foundations of ill-posed problems. Novosibirsk: Siberian Branch RAS, 2020.
Study of algorithms for solving the motion equations in the particle-in-cells method
V. A. Vshivkov1, E. S. Voropaeva2, A. A. Efimova1, G. I. Dudnikova1
1
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2
Novosibirsk State University
Email: vsh@ssd.sscc.ru
DOI 10.24412/cl-35065-2021-1-00-65
The paper is dedecated to the study and creation of algorithms and programs for modeling the motion of
plasma particles. The movement of charged particles in an electromagnetic field in a vacuum occurs under the
action of the Lorentz force. Particle motion algorithms are part of the particle-in-cell method. Since the parti-
cle-in-cell method uses a very large number of model particles, it is important to economically calculate the
velocities and coordinates of the particles at the time step. In 1970, Boris proposed a scheme that was economical
in terms of the number of operations, which had a second order of approximation. In recent years,
works have appeared in which modifications of the Boris method have been proposed. It was argued that new
modifications would allow calculations for long physical times. A new economical and more accurate scheme is
proposed for solving the equations of motion at a time step.
Since in practical calculations the values of the electric and magnetic fields at the location of the particle
are obtained by interpolation from the nodes of the computational grid, the influence of interpolation on the
running time of the algorithm was investigated. It turned out that the interpolation time in three-dimensional
calculations takes an order of magnitude longer than it takes to calculate the motion of a particle. A method
for reducing the cost of field interpolation is proposed.
This work was supported by the Russian Science Foundation, project 19-71-20026.
