Aspects of Lagrangian probability distributions

1994 ◽  
Vol 31 (A) ◽  
pp. 185-197 ◽  
Author(s):  
Masaaki Sibuya ◽  
Norihiko Miyawaki ◽  
Ushio Sumita
Keyword(s):  
Random Walks ◽  
Random Forests ◽  
Busy Period ◽  
Probability Distributions ◽  
Probabilistic Method ◽  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Queuing Systems ◽  
Transit System ◽  
Watson Process

Lagrangian distributions are reviewed from the viewpoint of the Galton-Watson process. They are related to the busy period in queuing systems and to the first visit in random walks.A property of the distributions is remarked for the application to vacant vehicles in a new transit system. Combinatorial identities of multinomial and binomial coefficients and related recurrences are shown by a probabilistic method. Based on the identities and recurrences, random forests generated by the Poisson and geometric Galton-Watson processes are characterized.

Aspects of Lagrangian probability distributions

1994 ◽  
Vol 31 (A) ◽  
pp. 185-197 ◽  
Author(s):  
Masaaki Sibuya ◽  
Norihiko Miyawaki ◽  
Ushio Sumita
Keyword(s):  
Random Walks ◽  
Random Forests ◽  
Busy Period ◽  
Probability Distributions ◽  
Probabilistic Method ◽  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Queuing Systems ◽  
Transit System ◽  
Watson Process

Lagrangian distributions are reviewed from the viewpoint of the Galton-Watson process. They are related to the busy period in queuing systems and to the first visit in random walks. A property of the distributions is remarked for the application to vacant vehicles in a new transit system. Combinatorial identities of multinomial and binomial coefficients and related recurrences are shown by a probabilistic method. Based on the identities and recurrences, random forests generated by the Poisson and geometric Galton-Watson processes are characterized.

q-SERIES IN MARKOV CHAINS WITH BINOMIAL TRANSITIONS

2008 ◽  
Vol 23 (1) ◽  
pp. 75-99 ◽  
Author(s):  
Antonis Economou ◽  
Stella Kapodistria
Keyword(s):  
Busy Period ◽  
Hypergeometric Series ◽  
Binomial Coefficients ◽  
Single Server ◽  
Transition Rates ◽  
Extensive Analysis ◽  
Service Completion ◽  
Markovian Queue ◽  
Number Of Customers ◽  
Sojourn Time Distributions

We consider a single-server Markovian queue with synchronized services and setup times. The customers arrive according to a Poisson process and are served simultaneously. The service times are independent and exponentially distributed. At a service completion epoch, every customer remains satisfied with probability p (independently of the others) and departs from the system; otherwise, he stays for a new service. Moreover, the server takes multiple vacations whenever the system is empty.Some of the transition rates of the underlying two-dimensional Markov chain involve binomial coefficients dependent on the number of customers. Indeed, at each service completion epoch, the number of customers n is reduced according to a binomial (n, p) distribution. We show that the model can be efficiently studied using the framework of q-hypergeometric series and we carry out an extensive analysis including the stationary, the busy period, and the sojourn time distributions. Exact formulas and numerical results show the effect of the level of synchronization to the performance of such systems.

scholarly journals Comparison of Multiple Random Walks Strategies for Searching Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Zhongtuan Zheng ◽  
Hanxing Wang ◽  
Shengguo Gao ◽  
Guoqiang Wang
Keyword(s):  
Random Walks ◽  
Random Network ◽  
Probability Distributions ◽  
First Passage Time ◽  
Shortest Paths ◽  
Passage Time ◽  
Practical Significance ◽  
First Passage ◽  
The Mean ◽  
High Degree

We investigate diverse random-walk strategies for searching networks, especially multiple random walks (MRW). We use random walks on weighted networks to establish various models of single random walks and take the order statistics approach to study corresponding MRW, which can be a general framework for understanding random walks on networks. Multiple preferential random walks (MPRW) and multiple simple random walks (MSRW) are two special types of MRW. As search strategies, MPRW prefers high-degree nodes while MSRW searches for low-degree nodes more efficiently. We analyze the first passage time (FPT) of wandering walkers of MRW and give the corresponding formulas of probability distributions and moments, and the mean first passage time (MFPT) is included. We show the convergence of the MFPT of the first arriving walker and find the MFPT of the last arriving walker closely related with the mean cover time. Simulations confirm analytical predictions and deepen discussions. We use a small random network to test the FPT properties from different aspects. We also explore some practical search-related issues by MRW, such as detecting unknown shortest paths and avoiding poor routings on networks. Our results are of practical significance for realizing optimal routing and performing efficient search on complex networks.

scholarly journals Efficient circuits for quantum walks

2010 ◽  
Vol 10 (5&6) ◽  
pp. 420-434
Author(s):  
C.-F. Chiang ◽  
D. Nagaj ◽  
P. Wocjan
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Problem Size ◽  
Integrable Probability ◽  
The Difference ◽  
General Method

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions \cite{GroverRudolph}. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk \cite{Szegedy}, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents \cite{Vigoda, Vazirani} and sampling from binary contingency tables \cite{Stefankovi}. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

scholarly journals Effect of Database Generation on Damage Consequences’ Assessment Based on Random Forests

2021 ◽  
Vol 9 (11) ◽  
pp. 1303
Author(s):  
Luca Braidotti ◽  
Jasna Prpić-Oršić ◽  
Marko Valčić
Keyword(s):  
Machine Learning ◽  
Monte Carlo ◽  
Decision Support ◽  
Decision Support Systems ◽  
Random Forests ◽  
Support Systems ◽  
Probability Distributions ◽  
New Generation ◽  
Main Effects ◽  
Further Development

Recently, the application of machine learning has been explored to assess the main damage consequences without employing flooding sensors. This method can be the base of a new generation of onboard decision support systems to help the master during the progressive flooding of the ship. In particular, the application of random forests has been found suitable to assess the final fate of the ship and the damaged compartments’ set and estimate the time-to-flood. Random forests have to be trained using a database of precalculated progressive flooding simulations. In the present work, multiple options for database generation were tested and compared: three based on Monte Carlo (MC) sampling based on different probability distributions of the damage parameters and a parametric one. The methods were tested on a barge geometry to highlight the main effects on the damage consequences’ assessment in order to ease the further development of flooding-sensor-agnostic decision support systems for flooding emergencies.

Effect of Uncertainty in the Balancing Weights on the Vibration Response of a High-Speed Rotor

2021 ◽  
Vol 143 (6) ◽  
Author(s):  
Janina Datz ◽  
Mahmoud Karimi ◽  
Steffen Marburg
Keyword(s):  
High Speed ◽  
Probability Distributions ◽  
Probabilistic Method ◽  
Turbine Rotor ◽  
Vibration Response ◽  
Galerkin Projection ◽  
System Response ◽  
Influence Coefficient ◽  
Uncertain Parameters ◽  
Influence Coefficient Method

Abstract This work investigates how uncertainties in the balancing weights are propagating into the vibration response of a high-speed rotor. Balancing data are obtained from a 166-MW gas turbine rotor in a vacuum balancing tunnel. The influence coefficient method is then implemented to characterize the rotor system by a deterministic multi-speed and multi-plane matrix. To model the uncertainties, a non-sampling probabilistic method based on the generalized polynomial chaos expansion (gPCE) is employed. The uncertain parameters including the mass and angular positions of the balancing weights are then expressed by gPCE with deterministic coefficients. Assuming predefined probability distributions of the uncertain parameters, the stochastic Galerkin projection is applied to calculate the coefficients for the input parameters. Furthermore, the vibration amplitudes of the rotor response are represented by appropriate gPCE with unknown deterministic coefficients. These unknown coefficients are determined using the stochastic collocation method by evaluating the gPCE for the system response at a set of collocation points. The effects of individual and combined uncertain parameters from a single and multiple balancing planes on the rotor vibration response are examined. Results are compared with the Monte Carlo simulations, showing excellent agreement.

The Probabilistic Method of Failure Analysis to Transmission Facilities under Ice Storms

2010 ◽  
Vol 37-38 ◽  
pp. 1525-1528
Author(s):  
Wen Jun Xu ◽  
Hong Ming Yang ◽  
Ming Yong Lai ◽  
Shuang Wang
Keyword(s):  
Transmission Lines ◽  
Power Transmission ◽  
Probability Distributions ◽  
Probabilistic Method ◽  
Joint Probability ◽  
Copula Function ◽  
Value Theory ◽  
Joint Probability Distribution ◽  
Ice Storms ◽  
Power Transmission Systems

Based on Extreme Value Theory (EVT), the Generalized Pareto Distributions (GPDs) of meteorological variables wind speed and freezing precipitation is simulated. Considering the dependence of them, a joint probability distribution is calculated by the Copula function. Further more, the probability distributions of ice loads and wind loads on transmission lines are analyzed, and the failure probability of broken lines and collapsed towers under ice storms is calculated. The accuracy and validity of this analytical method is demonstrated with comparison between numerical results and the historical datas of Chen Zhou power transmission systems.

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