The matrix M/M/∞ system: retrial models and Markov Modulated sources

1993 ◽  
Vol 25 (2) ◽  
pp. 453-471 ◽  
Author(s):  
J. Keilson ◽  
L. D. Servi
Keyword(s):  
Numerical Evaluation ◽  
Stability Conditions ◽  
Kummer Functions ◽  
The Matrix ◽  
Service Rates ◽  
System Properties ◽  
Markov Modulated ◽  
The Stability ◽  
Analytical Expressions

The matrix-geometric work of Neuts could be viewed as a matrix variant of M/M/1. A 2 × 2 matrix counterpart of Neuts for M/M/∞ is introduced, the stability conditions are identified, and the ergodic solution is solved analytically in terms of the ten parameters that define it. For several cases of interest, system properties can be found from simple analytical expressions or after easy numerical evaluation of Kummer functions. When the matrix of service rates is singular, a qualitatively different solution is derived. Applications to telecommunications include some retrial models and an M/M/∞ queue with Markov-modulated input.

The matrix M/M/∞ system: retrial models and Markov Modulated sources

1993 ◽  
Vol 25 (02) ◽  
pp. 453-471
Author(s):  
J. Keilson ◽  
L. D. Servi
Keyword(s):  
Numerical Evaluation ◽  
Stability Conditions ◽  
Kummer Functions ◽  
The Matrix ◽  
Service Rates ◽  
System Properties ◽  
Markov Modulated ◽  
The Stability ◽  
Analytical Expressions

The matrix-geometric work of Neuts could be viewed as a matrix variant of M/M/1. A 2 × 2 matrix counterpart of Neuts for M/M/∞ is introduced, the stability conditions are identified, and the ergodic solution is solved analytically in terms of the ten parameters that define it. For several cases of interest, system properties can be found from simple analytical expressions or after easy numerical evaluation of Kummer functions. When the matrix of service rates is singular, a qualitatively different solution is derived. Applications to telecommunications include some retrial models and an M/M/∞ queue with Markov-modulated input.

A NOTE ON INFINITE-SERVER MARKOV MODULATED AND SINGLE-SERVER RETRIAL QUEUES

2014 ◽  
Vol 31 (02) ◽  
pp. 1440003
Author(s):  
ZHE DUAN ◽  
MELIKE BAYKAL-GÜRSOY
Keyword(s):  
Queueing Systems ◽  
Retrial Queue ◽  
System Size ◽  
Retrial Queues ◽  
Stochastic Decomposition ◽  
Applied Probability ◽  
Single Server ◽  
The Matrix ◽  
Service Rates ◽  
Markov Modulated

We reconsider the M/M/∞ queue with two-state Markov modulated arrival and service processes and the single-server retrial queue analyzed in Keilson and Servi [Keilson, J and L Servi (1993). The matrix M/M/∞ system: Retrial models and Markov modulated sources. Advances in Applied Probability, 25, 453–471]. Fuhrmann and Cooper type stochastic decomposition holds for the stationary occupancy distributions in both queues [Keilson, J and L Servi (1993). The matrix M/M/∞ system: Retrial models and Markov modulated sources. Advances in Applied Probability, 25, 453–471; Baykal-Gürsoy, M and W Xiao (2004). Stochastic decomposition in M/M/∞ queues with Markov-modulated service rates. Queueing Systems, 48, 75–88]. The main contribution of the present paper is the derivation of the explicit form of the stationary system size distributions. Numerical examples are presented visually exhibiting the effect of various parameters on the stationary distributions.

Queues with negative arrivals

1991 ◽  
Vol 28 (1) ◽  
pp. 245-250 ◽  
Author(s):  
Erol Gelenbe ◽  
Peter Glynn ◽  
Karl Sigman
Keyword(s):  
Stability Conditions ◽  
Queueing Models ◽  
Single Server ◽  
New Models ◽  
Service Rates ◽  
Service Times ◽  
The Stability ◽  
Negative Arrivals ◽  
The Way

We study single-server queueing models where in addition to regular arriving customers, there are negative arrivals. A negative arrival has the effect of removing a customer from the queue. The way in which this removal is specified gives rise to several different models. Unlike the standard FIFOGI/GI/1 model, the stability conditions for these new models may depend upon more than just the arrival and service rates; the entire distributions of interarrival and service times may be involved.

Queues with negative arrivals

1991 ◽  
Vol 28 (01) ◽  
pp. 245-250 ◽  
Author(s):  
Erol Gelenbe ◽  
Peter Glynn ◽  
Karl Sigman
Keyword(s):  
Stability Conditions ◽  
Queueing Models ◽  
Single Server ◽  
New Models ◽  
Service Rates ◽  
Service Times ◽  
The Stability ◽  
Negative Arrivals ◽  
The Way

We study single-server queueing models where in addition to regular arriving customers, there are negative arrivals. A negative arrival has the effect of removing a customer from the queue. The way in which this removal is specified gives rise to several different models. Unlike the standard FIFO GI/GI/1 model, the stability conditions for these new models may depend upon more than just the arrival and service rates; the entire distributions of interarrival and service times may be involved.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

scholarly journals Causality and stability conditions of a conformal charged fluid

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Farid Taghinavaz
Keyword(s):  
Chemical Potential ◽  
Second Law Of Thermodynamics ◽  
Dense Medium ◽  
Wave Number ◽  
Second Law ◽  
Stability Conditions ◽  
Charged Fluid ◽  
Large Wave ◽  
The Stability ◽  
Large Wave Number

Abstract In this paper, I study the conditions imposed on a normal charged fluid so that the causality and stability criteria hold for this fluid. I adopt the newly developed General Frame (GF) notion in the relativistic hydrodynamics framework which states that hydrodynamic frames have to be fixed after applying the stability and causality conditions. To do this, I take a charged conformal matter in the flat and 3 + 1 dimension to analyze better these conditions. The causality condition is applied by looking to the asymptotic velocity of sound hydro modes at the large wave number limit and stability conditions are imposed by looking to the imaginary parts of hydro modes as well as the Routh-Hurwitz criteria. By fixing some of the transports, the suitable spaces for other ones are derived. I observe that in a dense medium having a finite U(1) charge with chemical potential μ0, negative values for transports appear and the second law of thermodynamics has not ruled out the existence of such values. Sign of scalar transports are not limited by any constraints and just a combination of vector transports is limited by the second law of thermodynamic. Also numerically it is proved that the most favorable region for transports $$ {\tilde{\upgamma}}_{1,2}, $$ γ ˜ 1 , 2 , coefficients of the dissipative terms of the current, is of negative values.

scholarly journals Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

2010 ◽  
Vol 2010 ◽  
pp. 1-23 ◽  
Author(s):  
Josef Diblík ◽  
Denys Ya. Khusainov ◽  
Irina V. Grytsay ◽  
Zdenĕk Šmarda
Keyword(s):  
Lyapunov Functions ◽  
Critical Case ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Simple Eigenvalue ◽  
Discrete Equations ◽  
The Matrix ◽  
The Stability ◽  
Important Characterization

Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalueλ=1of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

scholarly journals Iterative stability analysis for general polynomial control systems

2021 ◽  
Author(s):  
Bo Xiao ◽  
Hak-Keung Lam ◽  
Zhixiong Zhong
Keyword(s):  
Stability Analysis ◽  
Lyapunov Function ◽  
Control Systems ◽  
Polynomial System ◽  
Stability Conditions ◽  
General Polynomial ◽  
Main Challenge ◽  
Detailed Simulation ◽  
Feasible Solutions ◽  
The Stability

AbstractThe main challenge of the stability analysis for general polynomial control systems is that non-convex terms exist in the stability conditions, which hinders solving the stability conditions numerically. Most approaches in the literature impose constraints on the Lyapunov function candidates or the non-convex related terms to circumvent this problem. Motivated by this difficulty, in this paper, we confront the non-convex problem directly and present an iterative stability analysis to address the long-standing problem in general polynomial control systems. Different from the existing methods, no constraints are imposed on the polynomial Lyapunov function candidates. Therefore, the limitations on the Lyapunov function candidate and non-convex terms are eliminated from the proposed analysis, which makes the proposed method more general than the state-of-the-art. In the proposed approach, the stability for the general polynomial model is analyzed and the original non-convex stability conditions are developed. To solve the non-convex stability conditions through the sum-of-squares programming, the iterative stability analysis is presented. The feasible solutions are verified by the original non-convex stability conditions to guarantee the asymptotic stability of the general polynomial system. The detailed simulation example is provided to verify the effectiveness of the proposed approach. The simulation results show that the proposed approach is more capable to find feasible solutions for the general polynomial control systems when compared with the existing ones.

scholarly journals Coupled THM and Matrix Stability Modeling of Hydroshearing Stimulation in a Coupled Fracture-Matrix Hot Volcanic System

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Yong Xiao ◽  
Jianchun Guo ◽  
Hehua Wang ◽  
Lize Lu ◽  
John McLennan ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Thermal Conduction ◽  
Dominant Role ◽  
Injection Well ◽  
Time Step ◽  
Volcanic System ◽  
Induced Stress ◽  
The Matrix ◽  
Matrix Stability ◽  
The Stability

A coupled thermal-hydraulic-mechanical (THM) model is developed to simulate the combined effect of fracture fluid flow, heat transfer from the matrix to injected fluid, and shearing dilation behaviors in a coupled fracture-matrix hot volcanic reservoir system. Fluid flows in the fracture are calculated based on the cubic law. Heat transfer within the fracture involved is thermal conduction, thermal advection, and thermal dispersion; within the reservoir matrix, thermal conduction is the only mode of heat transfer. In view of the expansion of the fracture network, deformation and thermal-induced stress model are added to the matrix node’s in situ stress environment in each time step to analyze the stability of the matrix. A series of results from the coupled THM model, induced stress, and matrix stability indicate that thermal-induced aperture plays a dominant role near the injection well to enhance the conductivity of the fracture. Away from the injection well, the conductivity of the fracture is contributed by shear dilation. The induced stress has the maximum value at the injection point; the deformation-induced stress has large value with smaller affected range; on the contrary, thermal-induced stress has small value with larger affected range. Matrix stability simulation results indicate that the stability of the matrix nodes may be destroyed; this mechanism is helpful to create complex fracture networks.

Sign in / Sign up

Export Citation Format

Share Document