On the Transient Behavior of the Maximum Level Length in Structured Markov Chains

2011 ◽  
pp. 379-390
Author(s):  
Jesús R. Artalejo
Keyword(s):  
Markov Chains ◽  
Maximum Level ◽  
Transient Behavior

Bounds and Approximations for the Transient Behavior of Continuous-Time Markov Chains

1989 ◽  
Vol 3 (2) ◽  
pp. 175-198 ◽  
Author(s):  
Bok Sik Yoon ◽  
J. George Shanthikumar
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
First Passage Time ◽  
Transient Behavior ◽  
Passage Time ◽  
Lower And Upper Bounds ◽  
Simple Method ◽  
Continuous Time Markov Chains ◽  
Recent Addition ◽  
Dependent Probability

Discretization is a simple, yet powerful tool in obtaining time-dependent probability distribution of continuous-time Markov chains. One of the most commonly used approaches is uniformization. A recent addition to such approaches is an external uniformization technique. In this paper, we briefly review these different approaches, propose some new approaches, and discuss their performances based on theoretical bounds and empirical computational results. A simple method to get lower and upper bounds for first passage time distribution is also proposed.

scholarly journals Quantum fast-forwarding: Markov chains and graph property testing

2019 ◽  
Vol 19 (3&4) ◽  
pp. 181-213 ◽  
Author(s):  
Simon Apers ◽  
Alain Scarlet
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Quantum State ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Walk ◽  
Transient Behavior ◽  
Quantum Walks ◽  
Learning Context

We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with P the Markov chain transition matrix and D = \sqrt{P\circ P^T} its discriminant matrix (D=P if P is symmetric), we construct a quantum walk algorithm that for any quantum state |v> and integer t returns a quantum state \epsilon-close to the state D^t|v>/\|D^t|v>. The algorithm uses O(|D^t|v>|^{-1}\sqrt{t\log(\epsilon\|D^t|v>})^{-1}}) expected quantum walk steps and O(\|D^t|v>|^{-1}) expected reflections around |v>. This shows that quantum walks can accelerate the transient dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We show that this tool leads to speedups on random walk algorithms in a very natural way. Specifically we consider random walk algorithms for testing the graph expansion and clusterability, and show that we can quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, our quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, we use QFF for determining whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of s-t connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks.

Explicit transient probabilities of various Markov models

2021 ◽  
pp. 97-151
Author(s):  
Alan Krinik ◽  
Hubertus von Bremen ◽  
Ivan Ventura ◽  
Uyen Nguyen ◽  
Jeremy Lin ◽  
...  
Keyword(s):  
Markov Chains ◽  
Transition Probability ◽  
Scale Up ◽  
Transient Behavior ◽  
Transition Probability Matrix ◽  
Circulant Matrices ◽  
Transition Matrices ◽  
Content Type ◽  
Sample Paths ◽  
Birth Death

In analyzing finite-state Markov chains knowing the exact eigenvalues of the transition probability matrix P P is important information for predicting the explicit transient behavior of the system. Once the eigenvalues of P P are known, linear algebra and duality theory are used to find P k P^{k} where k = 2 , 3 , 4 , … k= 2,3,4,\ldots . This article is about finding explicit eigenvalue formulas, that scale up with the dimension of P P for various Markov chains. Eigenvalue formulas and expressions of P k P^{k} are first presented when P P is tridiagonal and Toeplitz. These results are generalized to tridiagonal matrices with alternating birth-death probabilities. More general eigenvalue formulas and expression of P k P^{k} are obtained for non-tridiagonal transition matrices P P that have both catastrophe-like and birth-death transitions. Similar results for circulant matrices are also explored. Applications include finding probabilities of sample paths restricted to a strip and generalized ballot box problems. These results generalize to Markov processes with P k P^{k} being replaced by e Q t e^{Qt} where Q Q is a transition rate matrix.

Markov models—Markov chains

2019 ◽  
Vol 16 (8) ◽  
pp. 663-664 ◽  
Author(s):  
Jasleen K. Grewal ◽  
Martin Krzywinski ◽  
Naomi Altman
Keyword(s):  
Markov Chains ◽  
Markov Models

An Analytical Transient Tire Model

2001 ◽  
Vol 29 (2) ◽  
pp. 108-132 ◽  
Author(s):  
A. Ghazi Zadeh ◽  
A. Fahim
Keyword(s):  
Transient Behavior ◽  
Stability Control ◽  
Vehicle Stability ◽  
Tire Model ◽  
Steering Angle ◽  
First Order ◽  
Vehicle Stability Control ◽  
Proposed Model ◽  
Depth Study ◽  
And Performance

Abstract The dynamics of a vehicle's tires is a major contributor to the vehicle stability, control, and performance. A better understanding of the handling performance and lateral stability of the vehicle can be achieved by an in-depth study of the transient behavior of the tire. In this article, the transient response of the tire to a steering angle input is examined and an analytical second order tire model is proposed. This model provides a means for a better understanding of the transient behavior of the tire. The proposed model is also applied to a vehicle model and its performance is compared with a first order tire model.

A Modular Race Tire Model Concerning Thermal and Transient Behavior using a Simple Contact Patch Description

2013 ◽  
Vol 41 (4) ◽  
pp. 232-246
Author(s):  
Timo Völkl ◽  
Robert Lukesch ◽  
Martin Mühlmeier ◽  
Michael Graf ◽  
Hermann Winner
Keyword(s):  
Load Distribution ◽  
Thermal Condition ◽  
Transient Behavior ◽  
Wheel Load ◽  
Contact Patch ◽  
Fundamental Parameters ◽  
Dry Conditions ◽  
Simple Contact ◽  
Tire Forces ◽  
Descriptive Set

ABSTRACT The potential of a race tire strongly depends on its thermal condition, the load distribution in its contact patch, and the variation of wheel load. The approach described in this paper uses a modular structure consisting of elementary blocks for thermodynamics, transient excitation, and load distribution in the contact patch. The model provides conclusive tire characteristics by adopting the fundamental parameters of a simple mathematical force description. This then allows an isolated parameterization and examination of each block in order to subsequently analyze particular influences on the full model. For the characterization of the load distribution in the contact patch depending on inflation pressure, camber, and the present force state, a mathematical description of measured pressure distribution is used. This affects the tire's grip as well as the heat input to its surface and its casing. In order to determine the thermal condition, one-dimensional partial differential equations at discrete rings over the tire width solve the balance of energy. The resulting surface and rubber temperatures are used to determine the friction coefficient and stiffness of the rubber. The tire's transient behavior is modeled by a state selective filtering, which distinguishes between the dynamics of wheel load and slip. Simulation results for the range of occurring states at dry conditions show a sufficient correlation between the tire model's output and measured tire forces while requiring only a simplified and descriptive set of parameters.

Sign in / Sign up

Export Citation Format

Share Document