Eimutis Valakevičius

Modeling reliability of power systems substations by using stochastic automata networks

In this paper, stochastic automata networks (SANs) formalism to model reliability of power systems substations is applied. The proposed strategy allows reducing the size of state space of Markov chain model and simplifying system specification. Two case studies of standard configurations of substations are considered in detail. SAN models with different assumptions were created. SAN approach is compared with exact reliability calculation by using a minimal path set method. Modeling results showed that total independence of automata can be assumed for relatively small power systems substations with reliable equipment. In this case, the implementation of Markov chain model by a using SAN method is a relatively easy task.

Continuous Time Markov Chain Model of Asset Prices Distribution

The aim of this paper is to introduce a new model of a financial asset prices distribution. It is known that the probability distribution of an asset prices or returns is unknown in reality. The general model of asset prices based on continuous time Markov chains is proposed. For this reason the interarrivals between two price states are approximated by mixture of exponential distributions. Numerical-analytic approach is used to obtain the probability distribution of asset prices. The developed software allows creating the space of an asset prices, the matrix of transition rates among states, a system of equations to find the steady state probabilities of price states and solves the system of equations by method of imbedded Markov chains.

COMPLEXITY OF EMBEDDED CHAIN ALGORITHM FOR COMPUTING STEADY STATE PROBABILITIES OF MARKOV CHAIN

The paper presents the theoretical evaluation of the complexity of an algorithm, based on embedded Markov chains, for computing steady state probabilities. Experimental research with different infinitesimal generator matrices was performed to support theoretical evaluations. Results showed that modified algorithm can be more effective for sparse matrices. An example of a queuing system is presented to demonstrate the automatic creation of the model of the system based on the proposed modelling method. http://dx.doi.org/10.5755/j01.itc.40.2.425

On Structured Initial Solution Generation for Phase Type Fitting with EM Method

The choice of initial solutions when fitting with a phase-type distribution (PH), using the expectation maximization method (EM) is investigated. It is known that the EM method can converge to a local solution, especially when fitting with a general structure. The problem of how to choose an initial solution for which the EM method would converge to a global solution is still open. We contribute to the research of this problem by studying the use of structures for initial solution generation. The proposed approach is tested by fitting with four state phase-type distributions (PH(4)). Numerical results show that the EM method converges faster from initial solutions of various structures. DOI: http://dx.doi.org/10.5755/j01.itc.47.2.18169

Numerical solution of reliability models described by stochastic automata networks

Abstract This paper presents the solution of Markov chain reliability models with a large state-space. To specify a system reliability model, we use our previously proposed methodology, which is based on the Stochastic Automata Networks formalism. We model parts of the system by arrowhead matrices with functional transition rates. As a result, the infinitesimal generator matrix of the reliability model has a distinctive structure. In this paper, we demonstrate that a block Gauss–Seidel method can be applied very efficiently to such a structure. The application of the proposed methodology is illustrated by an example of a standard 3/2 substation configuration. Even though its Markov chain reliability model has almost two million states, its steady-state probabilities can be estimated in just a few seconds of CPU time.

The risk–return profile of Lithuanian private pension funds

Abstract The introduction of a private pension funds in conjunction with the public social security system is the essence of pension system reform that was implemented in Lithuania. The performance of private funds is mainly presented by fund’s net asset value and few classical risk estimates. Such evaluation shows the management company’s ability to profitably invest funds, but does not give the evidential risk–return evaluation. This paper refers to the overall statistical analysis of 26 private pension funds over a certain time period. The objective of the research is to determine the risk–return profile of pension funds and to answer the question whether the categories specified based on investment strategy in equities reflect fund’s empirical behaviour. Research methodology includes the statistical analysis, risk measuring, performance ratio estimation, and K-means clustering. The conclusions obtained by the research allow determining whether the distinct pension funds have beaten a low risk reference and are adequately assigned to a certain risk category.

On Numerical Approach to Stochastic Systems Modelling

The paper considers the problem of representing non-Markovian systems that evolve stochastically over time. It is often necessary to use approximations in the case the system is non-Markovian. Phase type distribution is by now indispensable tool in creation of stochastic system models. In the paper is suggested a method and software for evaluating stochastic systems approximations by Markov chains with continuous time and countable state space. The performance of a system is described in the event language is used for generating the set of states and transition matrix between them. The example of a numerical model is presented.

Numerical-Analytic Model of Multi-Class, Multi-Server Queue with Nonpreemptive Priorities

We consider a multi-class, multi-server queuing system with preemptive priorities. We distinguish three groups of priority classes that consist of multiple customer types, each having their own arrival and service rate. We assume Poisson arrival processes and exponentially distributed service times. The performance of the system is described in event language. The created software automatically constructs and solves system of equilibrium equations to find steady state probabilities. We suggest a numerical-analytic method to estimate the probabilities. Based on these probabilities, we can compute a wide range of relevant performance characteristics, such as average number of customers of a certain type in the system and expected postponement time for each customer class.

A Decomposition Method in Modeling Queuing Systems

The analysis of stochastic systems is notoriously hard, especially when their performance is described by multidimensional Markov chains. Approximation of general distribution functions by phase-type distributions rise to large Markov chains. If the model is too large to analyze in its entirety, it is divided into subsystems. Each subsystem is analyzed separately and global solution is constructed from the partial solutions. Numerical algorithm to solve for the steady-state probabilities of these Markov chains from a system of linear equations by decomposition method is presented in the paper. Based on these probabilities, we can compute a wide range of relevant performance characteristics, such as average number of customers of a certain type in the system and expected postponement time for each customer class.

Stochastic volatility models and option prices

It is an observed fact in the market that the implied volatility of traded options vary from day to day. An alternative and straightforward explanation is that the instantaneous volatility of a stock is a stochastic quantity itself. The assumptions of the Black and Scholes model no longer hold. This is, therefore, one reason why Black and Scholes prices can differ from market prices of options. Having decided to make the instantaneous volatility stochastic, it is necessary to decide what sort of process it follows. The article analyzes three stochastic volatility models and considers how stochastic volatility can be incorporated into model prices of options. The investigation of stochastic volatility influence for pricing options traded in the SEB Vilnius Bank is done.