Loss probabilities in a simple circuit-switched network

1994 ◽  
Vol 26 (2) ◽  
pp. 456-473 ◽  
Author(s):  
J. A. Morrison
Keyword(s):  
Fixed Point ◽  
Parameter Space ◽  
Asynchronous Transfer Mode ◽  
Practical Interest ◽  
Integral Representations ◽  
Cellular Systems ◽  
Holding Period ◽  
Loss Probabilities ◽  
Erlang Loss ◽  
Exponentially Small

In this paper a particular loss network consisting of two links with C1 and C2 circuits, respectively, and two fixed routes, is investigated. A call on route 1 uses a circuit from both links, and a call on route 2 uses a circuit from only the second link. Calls requesting routes 1 and 2 arrive as independent Poisson streams. A call requesting route 1 is blocked and lost if there are no free circuits on either link, and a call requesting route 2 is blocked and lost if there is no free circuit on the second link. Otherwise the call is connected and holds a circuit from each link on its route for the holding period of the call.The case in which the capacities C1, and C2, and the traffic intensities v1, and v2, all become large of O(N) where N » 1, but with their ratios fixed, is considered. The loss probabilities L1 and L2 for calls requesting routes 1 and 2, respectively, are investigated. The asymptotic behavior of L1 and L2 as N→ ∞ is determined with the help of double contour integral representations and saddlepoint approximations. The results differ in various regions of the parameter space (C1, C2, v1, v2). In some of these results the loss probabilities are given in terms of the Erlang loss function, with appropriate arguments, to within an exponentially small relative error. The results provide new information when the loss probabilities are exponentially small in N. This situation is of practical interest, e.g. in cellular systems, and in asynchronous transfer mode networks, where very small loss probabilities are desired.The accuracy of the Erlang fixed-point approximations to the loss probabilities is also investigated. In particular, it is shown that the fixed-point approximation E2 to L2 is inaccurate in a certain region of the parameter space, since L2 « E2 there. On the other hand, in some regions of the parameter space the fixed-point approximations to both L1 and L2 are accurate to within an exponentially small relative error.

Loss probabilities in a simple circuit-switched network

1994 ◽  
Vol 26 (02) ◽  
pp. 456-473
Author(s):  
J. A. Morrison
Keyword(s):  
Fixed Point ◽  
Parameter Space ◽  
Asynchronous Transfer Mode ◽  
Practical Interest ◽  
Integral Representations ◽  
Cellular Systems ◽  
Holding Period ◽  
Loss Probabilities ◽  
Erlang Loss ◽  
Exponentially Small

In this paper a particular loss network consisting of two links with C 1 and C 2 circuits, respectively, and two fixed routes, is investigated. A call on route 1 uses a circuit from both links, and a call on route 2 uses a circuit from only the second link. Calls requesting routes 1 and 2 arrive as independent Poisson streams. A call requesting route 1 is blocked and lost if there are no free circuits on either link, and a call requesting route 2 is blocked and lost if there is no free circuit on the second link. Otherwise the call is connected and holds a circuit from each link on its route for the holding period of the call. The case in which the capacities C 1, and C 2, and the traffic intensities v 1, and v 2, all become large of O(N) where N » 1, but with their ratios fixed, is considered. The loss probabilities L 1 and L 2 for calls requesting routes 1 and 2, respectively, are investigated. The asymptotic behavior of L 1 and L 2 as N→ ∞ is determined with the help of double contour integral representations and saddlepoint approximations. The results differ in various regions of the parameter space (C 1, C 2, v 1, v 2). In some of these results the loss probabilities are given in terms of the Erlang loss function, with appropriate arguments, to within an exponentially small relative error. The results provide new information when the loss probabilities are exponentially small in N. This situation is of practical interest, e.g. in cellular systems, and in asynchronous transfer mode networks, where very small loss probabilities are desired. The accuracy of the Erlang fixed-point approximations to the loss probabilities is also investigated. In particular, it is shown that the fixed-point approximation E 2 to L 2 is inaccurate in a certain region of the parameter space, since L 2 « E 2 there. On the other hand, in some regions of the parameter space the fixed-point approximations to both L 1 and L 2 are accurate to within an exponentially small relative error.

TERBIUM MOLYBDATE: GROUP THEORY AND FLUCTUATIONS

1987 ◽  
Vol 01 (05n06) ◽  
pp. 239-244
Author(s):  
SERGE GALAM
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Parameter Space ◽  
Landau Theory ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Theory Analysis ◽  
Continuous Transition ◽  
Dimensional Parameter ◽  
First Order

A new mechanism to explain the first order ferroelastic—ferroelectric transition in Terbium Molybdate (TMO) is presented. From group theory analysis it is shown that in the two-dimensional parameter space ordering along either an axis or a diagonal is forbidden. These symmetry-imposed singularities are found to make the unique stable fixed point not accessible for TMO. A continuous transition even if allowed within Landau theory is thus impossible once fluctuations are included. The TMO transition is therefore always first order. This explanation is supported by experimental results.

Convergence to Uniformity in a Cellular Automaton via Local Coevolution

1997 ◽  
Vol 08 (05) ◽  
pp. 1013-1024 ◽  
Author(s):  
Moshe Sipper ◽  
Marco Tomassini
Keyword(s):  
Parallel Machines ◽  
Parallel Implementation ◽  
A Priori ◽  
Practical Interest ◽  
Optimal Solution ◽  
Cellular Systems ◽  
Evolving System ◽  
Uniform Solution ◽  
Specialized Hardware ◽  
Density Problem

Cellular programming is a coevolutionary algorithm by which parallel cellular systems evolve to solve computational tasks. The evolving system is a massively parallel, locally interconnected grid of cells, where each cell operates according to a local interaction rule. If this rule is identical for all cells, the system is referred to as uniform, otherwise, it is non-uniform. This paper describes an experiment that addresses the following question: Employing a local coevolutionary process to solve a hard problem, known as density classification, can an optimal uniform solution be found? Since our approach involves the evolution of non-uniform CAs, where cellular rules are initially assigned at random, such convergence to uniformity cannot be a priori expected to easily emerge. The question is of both theoretical and practical interest. As for the latter, one major advantage of local evolutionary processes is their amenability to parallel implementation, using commercially available parallel machines or specialized hardware. Our experiment shows that when such local evolution is applied to the density problem, the optimal solution can be found.

scholarly journals On Locating and Counting Satellite Components Born along the Stability Circle in the Parameter Space for a Family of Jarratt-Like Iterative Methods

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 839 ◽  
Author(s):  
Young Hee Geum ◽  
Young Ik Kim
Keyword(s):  
Fixed Point ◽  
Iterative Methods ◽  
Parameter Space ◽  
Control Parameter ◽  
Elementary Theory ◽  
Linear Stability Theory ◽  
Bifurcation Points ◽  
Local Bifurcations ◽  
The Stability ◽  
Bifurcation Behavior

This paper is devoted to an analysis on locating and counting satellite components born along the stability circle in the parameter space for a family of Jarratt-like iterative methods. An elementary theory of plane geometric curves is pursued to locate bifurcation points of such satellite components. In addition, the theory of Farey sequence is adopted to count the number of the satellite components as well as to characterize relationships between the bifurcation points. A linear stability theory on local bifurcations is developed based upon a small perturbation about the fixed point of the iterative map with a control parameter. Some properties of fixed and critical points under the Möbius conjugacy map are investigated. Theories and examples on locating and counting bifurcation points of satellite components in the parameter space are presented to analyze the bifurcation behavior underlying the dynamics behind the iterative map.

Dynamic Stability of Wave-Convection Transport

1982 ◽  
Vol 24 (4) ◽  
pp. 225-227
Author(s):  
M. D. Greenberg ◽  
C. Y. Harrell
Keyword(s):  
Dynamic Stability ◽  
Nonlinear Equations ◽  
Parameter Space ◽  
Stability Criterion ◽  
Practical Interest ◽  
Single Equation ◽  
Sine Wave ◽  
Second Order ◽  
Spherical Object

A flexible inextensible horizontal belt is assumed to be formed, by closely spaced vertical push rods, into a traveling sine wave. A spherical object resting at the bottom of a trough will tend to be convected with the trough as the wave travels. The dynamic stability of such wave-convection transport is considered. Assuming the wave to be shallow, the governing nonlinear equations are expanded (through second order) in the ‘shallowness parameter’, and thus reduced to a single equation, essentially of forced Duffing type, which is integrated numerically, over the parameter space of practical interest, to yield a stability criterion.

scholarly journals Analytic moduli for unfoldings of germs of generic analytic diffeomorphisms with a codimension parabolic point

2013 ◽  
Vol 35 (1) ◽  
pp. 274-292 ◽  
Author(s):  
C. ROUSSEAU
Keyword(s):  
Fixed Point ◽  
Normal Form ◽  
Parameter Space ◽  
Parabolic Point ◽  
Codimension One ◽  
Analytic Classification ◽  
Natural Domains ◽  
Generic Family ◽  
Parabolic Fixed Point

AbstractIn this paper we provide a complete modulus of analytic classification for germs of generic analytic families of diffeomorphisms which unfold a parabolic fixed point of codimension$k$. We start by showing that a generic family can be ‘prepared’, i.e. brought to a prenormal form${f}_{\epsilon } (z)$in which the multi-parameter$\epsilon $is almost canonical (up to an action of$ \mathbb{Z} / k \mathbb{Z} $). As in the codimension one case treated in P. Mardešić, R. Roussarie and C. Rousseau [Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms.Mosc. Math. J. 4(2004), 455–498], we show that the Ecalle–Voronin modulus can be unfolded to give a complete modulus for such germs. For this purpose, we define unfolded sectors in$z$-space that constitute natural domains on which the map${f}_{\epsilon } $can be brought to normal form in an almost unique way. The comparison of these normalizing changes of coordinates on the different sectors forms the analytic part of the modulus. This construction is performed on sectors in the multi-parameter space$\epsilon $such that the closure of their union provides a neighborhood of the origin in parameter space.

scholarly journals A parametric study of the coalescence of liquid drops in a viscous gas

2014 ◽  
Vol 753 ◽  
pp. 279-306 ◽  
Author(s):  
James E. Sprittles ◽  
Yulii D. Shikhmurzaev
Keyword(s):  
Parameter Space ◽  
Gas Dynamics ◽  
Scaling Laws ◽  
Practical Interest ◽  
Conventional Model ◽  
Liquid Drops ◽  
Viscous Gas ◽  
Initial Stage ◽  
Large Parameter Space

AbstractThe coalescence of two liquid drops surrounded by a viscous gas is considered in the framework of the conventional model. The problem is solved numerically with particular attention paid to resolving the very initial stage of the process which only recently has become accessible both experimentally and computationally. A systematic study of the parameter space of practical interest allows the influence of the governing parameters in the system to be identified and the role of viscous gas to be determined. In particular, it is shown that the viscosity of the gas suppresses the formation of toroidal bubbles predicted in some cases by early computations where the gas’ dynamics was neglected. Focusing computations on the very initial stages of coalescence and considering the large parameter space allows us to examine the accuracy and limits of applicability of various ‘scaling laws’ proposed for different ‘regimes’ and, in doing so, reveal certain inconsistencies in recent works. A comparison with experimental data shows that the conventional model is able to reproduce many qualitative features of the initial stages of coalescence, such as a collapse of calculations onto a ‘master curve’ but, quantitatively, overpredicts the observed speed of coalescence and there are no free parameters to improve the fit. Finally, a phase diagram of parameter space, differing from previously published ones, is used to illustrate the key findings.

MUTATIONS OF THE "GAME OF LIFE": A GENERALIZED CELLULAR AUTOMATA PERSPECTIVE OF COMPLEX ADAPTIVE SYSTEMS

2000 ◽  
Vol 10 (08) ◽  
pp. 1821-1866 ◽  
Author(s):  
RADU DOGARU ◽  
LEON O. CHUA
Keyword(s):  
Cellular Automata ◽  
Boolean Function ◽  
Parameter Space ◽  
Cell Parameter ◽  
Piecewise Linear ◽  
Cellular Systems ◽  
Boolean Logic ◽  
Game Of Life ◽  
Local Activity ◽  
Edge Of Chaos

This paper presents a novel approach for studying the relationship between the properties of isolated cells and the emergent behavior that occurs in cellular systems formed by coupling such cells. The novelty of our approach consists of a method for precisely partitioning the cell parameter space into subdomains via the failure boundaries of the piecewise-linear CNN (cellular neural network) cells [Dogaru & Chua, 1999a] of a generalized cellular automata [Chua, 1998]. Instead of exploring the rule space via statistically defined parameters (such as λ in [Langton, 1990]), or by conducting an exhaustive search over the entire set of all possible local Boolean functions, our approach consists of exploring a deterministically structured parameter space built around parameter points corresponding to "interesting" local Boolean logic functions. The well-known "Game of Life" [Berlekamp et al., 1982] cellular automata is reconsidered here to exemplify our approach and its advantages. Starting from a piecewise-linear representation of the classic Conway logic function called the "Game of Life", and by introducing two new cell parameters that are allowed to vary continuously over a specified domain, we are able to draw a "map-like" picture consisting of planar regions which cover the cell parameter space. A total of 148 subdomains and their failure boundaries are precisely identified and represented by colored paving stones in this mosaic picture (see Fig. 1), where each stone corresponds to a specific local Boolean function in cellular automata parlance. Except for the central "paving stone" representing the "Game of Life" Boolean function, all others are mutations uncovered by exploring the entire set of 148 subdomains and determining their dynamic behaviors. Some of these mutations lead to interesting, "artificial life"-like behavior where colonies of identical miniaturized patterns emerge and evolve from random initial conditions. To classify these emergent behaviors, we have introduced a nonhomogeneity measure, called cellular disorder measure, which was inspired by the local activity theory from [Chua, 1998]. Based on its temporal evolution, we are able to partition the cell parameter space into a class U "unstable-like" region, a class E "edge of chaos"-like region, and a class P "passive-like" region. The similarity with the "unstable", "edge of chaos" and "passive" domains defined precisely and applied to various reaction–diffusion CNN systems [Dogaru & Chua, 1998b, 1998c] opens interesting perspectives for extending the theory of local activity [Chua, 1998] to discrete-time cellular systems with nonlinear couplings. To demonstrate the potential of emergent computation in generalized cellular automata with cells designed from mutations of the "Game of Life", we present a nontrivial application of pattern detection and reconstruction from very noisy environments. In particular, our example demonstrates that patterns can be identified and reconstructed with very good accuracy even from images where the noise level is ten times stronger than the uncorrupted image.

Sign in / Sign up

Export Citation Format

Share Document