Mindaugas Snipas

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.

Application of Stochastic Automata Networks for Creation of Continuous Time Markov Chain Models of Voltage Gating of Gap Junction Channels

The primary goal of this work was to study advantages of numerical methods used for the creation of continuous time Markov chain models (CTMC) of voltage gating of gap junction (GJ) channels composed of connexin protein. This task was accomplished by describing gating of GJs using the formalism of the stochastic automata networks (SANs), which allowed for very efficient building and storing of infinitesimal generator of the CTMC that allowed to produce matrices of the models containing a distinct block structure. All of that allowed us to develop efficient numerical methods for a steady-state solution of CTMC models. This allowed us to accelerate CPU time, which is necessary to solve CTMC models, ~20 times.

Continuous Time Markov Chain Models of Voltage Gating of Gap Junction Channels

The major goal of this study was to create a continuous time Markov chain (CTMC) models of voltage gating of gap junction (GJ) channels formed of connexin protein. This goal was achieved by using the Piece Linear Aggregate (PLA) formalism to describe the function of GJs and transforming PLA into Markov process. Infinitesimal generator of CTMC was used to automate construction of Markov chain model from description of the system using PLA formalism. Developed Markov chain models were used to simulate gap junctional conductance dependence on transjunctional voltage. The proposed method was implemented to create models of voltage gating of GJ channels containing 4 and 12 gates. CTMC modeling results were compared with the results obtained using a discrete time Markov chain (DTMC) model. It was shown that CTMC modeling requires less CPU time than an analogous DTMC model. DOI: http://dx.doi.org/10.5755/j01.itc.43.2.3198

Functional asymmetry and plasticity of electrical synapses interconnecting neurons through a 36-state model of gap junction channel gating

We combined the Hodgkin–Huxley equations and a 36-state model of gap junction channel gating to simulate electrical signal transfer through electrical synapses. Differently from most previous studies, our model can account for dynamic modulation of junctional conductance during the spread of electrical signal between coupled neurons. The model of electrical synapse is based on electrical properties of the gap junction channel encompassing two fast and two slow gates triggered by the transjunctional voltage. We quantified the influence of a difference in input resistances of electrically coupled neurons and instantaneous conductance–voltage rectification of gap junctions on an asymmetry of cell-to-cell signaling. We demonstrated that such asymmetry strongly depends on junctional conductance and can lead to the unidirectional transfer of action potentials. The simulation results also revealed that voltage spikes, which develop between neighboring cells during the spread of action potentials, can induce a rapid decay of junctional conductance, thus demonstrating spiking activity-dependent short-term plasticity of electrical synapses. This conclusion was supported by experimental data obtained in HeLa cells transfected with connexin45, which is among connexin isoforms expressed in neurons. Moreover, the model allowed us to replicate the kinetics of junctional conductance under different levels of intracellular concentration of free magnesium ([Mg2+]i), which was experimentally recorded in cells expressing connexin36, a major neuronal connexin. We demonstrated that such [Mg2+]i-dependent long-term plasticity of the electrical synapse can be adequately reproduced through the changes of slow gate parameters of the 36-state model. This suggests that some types of chemical modulation of gap junctions can be executed through the underlying mechanisms of voltage gating. Overall, the developed model accounts for direction-dependent asymmetry, as well as for short- and long-term plasticity of electrical synapses. Our modeling results demonstrate that such complex behavior of the electrical synapse is important in shaping the response of coupled neurons.

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

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.

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.