Energy criterion for stability of equilibrium plasma given by stationary distribution function

We consider a fluid queue driven by a discouraged arrivals queue and obtain explicit expressions for the stationary distribution function of the buffer content in terms of confluent hypergeometric functions. We compare it with a fluid queue driven by an infinite server queue. Numerical results are presented to compare the behaviour of the buffer content distributions for both these models.

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On the nonlinear damping of a plasma mode

Journal of Plasma Physics ◽

10.1017/s0022377800007066 ◽

1972 ◽

Vol 8 (2) ◽

pp. 175-182 ◽

Cited By ~ 2

Author(s):

R. Nandan ◽

G. Pocobelli

Keyword(s):

Distribution Function ◽

Stationary Distribution ◽

Particle Motion ◽

Nonlinear Damping ◽

Second Derivative ◽

Poisson System ◽

Temperature Ratio ◽

Plasma Mode ◽

Stationary Distribution Function

We present a theory of the nonlinear damping of a plasma mode based on a solution of the Vlasov–Poisson system. The formulation is exempt from the objectionable separation of the particles into ‘resonant’ and ‘nonresonant’, and is valid for ion as well as for electron modes. The effect of the damping of the mode on particle motion is taken in account. In particular, we evaluate numerically the damping of an ion mode for a temperature ratio Te/Ti = 16. We also obtain a number of small new shifts in the damping of a plasma mode in general, including contributions from the second derivative of the stationary distribution function.

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Stationary distribution functions for ohmic Tokamak-plasmas in the weak-collisional transport regime by MaxEnt principle

Journal of Plasma Physics ◽

10.1017/s0022377814000713 ◽

2014 ◽

Vol 81 (1) ◽

Cited By ~ 2

Author(s):

Giorgio Sonnino ◽

Philippe Peeters ◽

Alberto Sonnino ◽

Pasquale Nardone ◽

György Steinbrecher

Keyword(s):

Distribution Function ◽

Entropy Production ◽

Stationary Distribution ◽

Local Equilibrium ◽

Distribution Functions ◽

Tokamak Plasmas ◽

Minimum Entropy ◽

Scale Invariant ◽

Confined Plasmas ◽

Transport Regime

In previous works, we derived stationary density distribution functions (DDF) where the local equilibrium is determined by imposing the maximum entropy (MaxEnt) principle, under the scale invariance restrictions, and the minimum entropy production theorem. In this paper we demonstrate that it is possible to reobtain these DDF solely from the MaxEnt principle subject to suitable scale invariant restrictions in all the variables. For the sake of concreteness, we analyse the example of ohmic, fully ionized, tokamak-plasmas, in the weak-collisional transport regime. In this case we show that it is possible to reinterpret the stationary distribution function in terms of the Prigogine distribution function where the logarithm of the DDF is directly linked to the entropy production of the plasma. This leads to the suggestive idea that also the stationary neoclassical distribution functions, for magnetically confined plasmas in the collisional transport regimes, may be derived solely by the MaxEnt principle.

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Phase Enlargement of Semi-Markov Systems without Determining Stationary Distribution of Embedded Markov Chain

SPIIRAS Proceedings ◽

10.15622/sp.2020.19.3.3 ◽

2020 ◽

Vol 19 (3) ◽

pp. 539-563

Author(s):

Vadim Kopp ◽

Mikhail Zamoryonov ◽

Nikita Chalenkov ◽

Ivan Skatkov

Keyword(s):

Distribution Function ◽

Markov Chain ◽

State Space ◽

Stationary Distribution ◽

Phase State ◽

Random Variables ◽

Embedded Markov Chain ◽

Discrete State ◽

Markov Systems ◽

The Difference

A phase enlargement of semi-Markov systems that does not require determining stationary distribution of the embedded Markov chain is considered. Phase enlargement is an equivalent replacement of a semi-Markov system with a common phase state space by a system with a discrete state space. Finding the stationary distribution of an embedded Markov chain for a system with a continuous phase state space is one of the most time-consuming and not always solvable stage, since in some cases it leads to a solution of integral equations with kernels containing sum and difference of variables. For such equations there is only a particular solution and there are no general solutions to date. For this purpose a lemma on a type of a distribution function of the difference of two random variables, provided that the first variable is greater than the subtracted variable, is used. It is shown that the type of the distribution function of difference of two random variables under the indicated condition depends on one constant, which is determined by a numerical method of solving the equation presented in the lemma. Based on the lemma, a theorem on the difference of a random variable and a complicated recovery flow is built up. The use of this method is demonstrated by the example of modeling a technical system consisting of two series-connected process cells, provided that both cells cannot fail simultaneously. The distribution functions of the system residence times in enlarged states, as well as in a subset of working and non-working states, are determined. The simulation results are compared by the considered and classical method proposed by V. Korolyuk, showed the complete coincidence of the sought quantities.

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An extremal markovian sequence

Journal of Applied Probability ◽

10.1017/s0021900200027236 ◽

1989 ◽

Vol 26 (02) ◽

pp. 219-232 ◽

Cited By ~ 6

Author(s):

M. Teresa Alpuim

Keyword(s):

Distribution Function ◽

Order Statistics ◽

Stationary Distribution ◽

Random Variable ◽

Stationary Sequence ◽

Left Endpoint ◽

Limit Laws ◽

Marginal Distributions ◽

Common Distribution ◽

Common Distribution Function

In this paper we consider an independent and identically distributed sequence {Yn } with common distribution function F(x) and a random variable X 0, independent of the Yi 's, and define a Markovian sequence {Xn } as Xi = X 0, if i = 0, Xi = k max{Xi − 1, Yi }, if i ≧ 1, k ∈ R, 0 < k < 1. For this sequence we evaluate basic distributional formulas and give conditions on F(x) for the sequence to possess a stationary distribution. We prove that for any distribution function H(x) with left endpoint greater than or equal to zero for which log H(ex ) is concave it is possible to construct such a stationary sequence with marginal distributions equal to it. We study the limit laws for extremes and kth order statistics.

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Mathematical Model of Call Center in the Form of Multi-Server Queueing System

Mathematics ◽

10.3390/math9222877 ◽

2021 ◽

Vol 9 (22) ◽

pp. 2877

Author(s):

Anatoly Nazarov ◽

Alexander Moiseev ◽

Svetlana Moiseeva

Keyword(s):

Distribution Function ◽

Stationary Distribution ◽

Service Time ◽

Call Center ◽

Queueing System ◽

Optimization Problems ◽

Arbitrary Distribution ◽

Hyperexponential Distribution ◽

Poisson Arrivals ◽

Multi Server

The paper considers the model of a call center in the form of a multi-server queueing system with Poisson arrivals and an unlimited waiting area. In the model under consideration, incoming calls do not differ in terms of service conditions, requested service, and interarrival periods. It is assumed that an incoming call can use any free server and they are all identical in terms of capabilities and quality. The goal problem is to find the stationary distribution of the number of calls in the system for an arbitrary recurrent service. This will allow us to evaluate the performance measures of such systems and solve various optimization problems for them. Considering models with non-exponential service times provides solutions for a wide class of mathematical models, making the results more adequate for real call centers. The solution is based on the approximation of the given distribution function of the service time by the hyperexponential distribution function. Therefore, first, the problem of studying a system with hyperexponential service is solved using the matrix-geometric method. Further, on the basis of this result, an approximation of the stationary distribution of the number of calls in a multi-server system with an arbitrary distribution function of the service time is constructed. Various issues in the application of this approximation are considered, and its accuracy is analyzed based on comparison with the known analytical result for a particular case, as well as with the results of the simulation.

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