State Probability Equations Research Articles

Location and Complex Status Update Strategy Optimization in UAV-Assisted IoT

Complex status updates have attracted widespread attention in real-time monitoring services (e.g., real-time fire gas monitoring and wildfire spread prediction). In complex status updates, the status information needs to be obtained by processing the perceived original data. However, as lightweight terminals, temporarily deployed Internet of things (IoT) devices have no computing modules. Unmanned aerial vehicle (UAV) can act as a edge server to help IoT devices complete computing tasks by mobile edge computing (MEC). To this end, this paper considers a complex status update in UAV-assisted IoT, where an UAV moves in hovering-flight-hovering mode to ensure that it can serve IoT devices in different areas. When the UAV hovers, it obtains the status information based on the original data transmitted by the IoT device and sends it to the control center. During the complex status update, the short packet communication and time-varying channel are considered. To realize the trade-off optimization of the average age of information (AoI) and average power consumption of both IoT device and UAV within a long time, we formulate a location and dynamic status update strategy optimization problem for UAV hovering-flight-hovering mode. In order to solve the problem with Markov properties, we derive the state probability equations and further establish the linear programming problem with fixed UAV location. Then, we propose a probability-based algorithm to obtain UAV location and dynamic status update strategy. To adapt to more urgent scenarios, we propose a AoI threshold based strategy to reduce the complexity of the problem. State probability equations are derived under the strategy and a linear programming problem with a fixed AoI threshold are established. Next, we propose a low-complexity algorithm to obtain the optimal AoI threshold. Simulation results show that the proposed algorithms can optimize the three performance metrics in a balanced way and we need to select the appropriate transmit power of the IoT device and the computing capacity of the UAV to achieve better performance.

On the use of Kolmogorov equations for determining ion channels characteristics

The paper proposes a method for determination of the dynamic properties of conduction within voltage-gated ion channels with several sequentially located energy barriers named gate particles using a theory of Markov random processes in continuous time and with a discrete state space. The number of states was taken equal to the number of energy barriers in the channel plus one. If the hypothesis that random flows switching gate particles between an open and a closed are Poisson process, is true, then mathematical description of the system’s state of similar channels can be a system of Kolmogorov linear differential equations for state probabilities. Using this model and based on Volt-Clamp test results, published in open access journals, it is possible to find out how fixed values of membrane potential depend on the intensities of Poisson process for different types of potassium channels. Function parameters, that describe the intensity of ion transition across the channel, in the Kolmogorov equation can be restored using the generalized least squares method. This paper contains the examples of determining the intensities of transitions for two types of voltage-dependent potassium channels known as «delayed rectifiier potassium channels» (IKdr, two identical activation gate particles) and channels with fast activation and inactivation processes (IKa, three identical activation and one inactivation gate particles). It is shown that channel activation and deactivation is described by solving the general Kolmogorov equation.

Construction of Schemes, Models and Algorithm for Detection Network Attacks in Computer Networks

This article is devoted to develop network attack detection schemes to search for vulnerable servers and the likelihood of determining the type of attacks by the contents of network packets, a network attack recognition scheme that allows to filter external network traffic by processing incoming requests is proposedas well a network detection model attacks as signs of detecting the position of a security policy is offered. Based on the analysis of time series, a network attack detection model that allows identifying network attacks by a threshold value is developed and a mathematical model for real-time recognition of network attacks is proposed.Model of the behavior of the information flows are shown that the linear model does not provide an adequate assessment of the current process to the critical states. Within the framework of developing models for detecting network attacks, an algorithm for detecting and identifying network attacks is proposed, which allows one to perform not only an exhaustive search for the classification features of network attacks, but to limit itself to a shortened search.The behavior of the queue of half-open compounds are described with an absorbing state, and a system of differential equations for state probabilities are obtained. Also new requests to belong to a particular cluster are analyzed.

CONSTRUCTION OF SYSTEM FOR ORGANIZATION OF PRODUCTION IN CONDITIONS OF TECHNICAL AND ECONOMIC RISKS OF INDUSTRIAL ENTERPRISES

One of the key components of effective production organization at the enterprises of the radio electronic industry is the need to take into account the risk environment, and take measures to minimize the likelihood of risk occurrence. The main problem is the probabilistic nature of the manifestation of risk. The probability is manifested in the realization of the risk and its duration, which, in the final analysis, will lead to financial losses. The article proposes an approach to building the core of the system of technical and economic risks and their interrelationship in the organization of production at the enterprises of the radio electronic industry. In the article it is proposed to use basic modules for estimating the limiting probabilities of various states of technical and economic risks for which special systems of linear algebraic equations are developed. In this case, to determine (calculate) the estimate, we can use the results of solving a system of linear algebraic equations. In addition, these results allow for the assessment of the stationary state of the basic modules of the technical and risks system in the organization of production at the enterprises of the radio electronic industry. When solving a system of algebraic equations, it is suggested to use the standard approach, namely, the limit transformation of the system of Kolmogorov's differential equations for state probabilities. The article concludes that to assess the impact of technical and economic risks on the effectiveness of the organization of production of enterprises of the electronic industry, it is necessary to assess the likelihood of finding a system of technical and economic risks in the state of their implementation. The results of such an assessment will allow to determine two main components - the intensity of the events' flows associated with the emergence (implementation) of technical and economic risks and the time (duration) of their finding in the state of realization (negative state).

FINDING NON-STATIONARY STATE PROBABILITIES OF G-NETWORK WITH SIGNALS AND CUSTOMERS BATCH REMOVAL

This paper is devoted to the research of an open Markov queueing network with positive customers and signals, and positive customers batch removal. A way of finding in a non-stationary regime time-dependent state probabilities has been proposed. The Kolmogorov system of difference-differential equations for state probabilities of such network was derived. The technique of its building, based on the use of the modified method of successive approximations combined with a series method, has been proposed. It is proved that the successive approximations converge over time to the stationary state probabilities, and the sequence of approximations converges to the unique solution of the Kolmogorov equations. Any successive approximation can be represented as a convergent power series with infinite radius of convergence, the coefficients of which satisfy the recurrence relations; that is useful for estimations. Model example illustrating the finding of time-dependent state probabilities of the network has been provided.

Mathematical analysis of machining system with warm spares and variable service rate

This paper studies the repairable system considering warm spares and two heterogeneous repairmen. First repairman repairs the failed machine with variable service rate. When number of failed machine reached to N, the service rate of this repairman becomes constant. This repairman is always available and never goes for vacation. When number of failed machine reached to N, the second repairman becomes active, coming after random vacation. In this paper partial sever vacation policy is used. Steady state probability equations are developed using Markov process theory and we solve these equations by recursive approach. A cost model is developed in order to obtain optimum number of failed machines. A numerical example is provided, in which performance measures of the system are obtained under optimum operating conditions and we have also performed sensitivity analysis.

Real Time Systems with Nonpreemptive Priorities and Ample Maintenance Facilities

We consider a real time data acquisition and processing multiserver system with identical servers (such as unmanned aerial vehicles, machine controllers, overhearing devices, medical monitoring devices, etc.) which can be maintained/programmed for different kinds of activities (e.g. passive or active). This system provides a service for real time tasks arriving via several channels (such as surveillance regions, assembly lines, communication channels, etc.) and involves maintenance. We focus on the worst case analysis of the system with ample maintenance facilities exponentially distributed time to failure and maintenance times. We consider two kinds of models (with and without nonpreemptive priorities) and provide balance equations for steady state probabilities and various performance measures, when both operation and maintenance times are exponentially distributed.

Influence of order acceptance policies on optimal capacity investment with stochastic customer required lead times

Abstract The influence of applying queue state dependent order acceptance policies, where either decision is with customer or with manufacturer, on optimal capacity investment is discussed. Therefore, three order acceptance policies are developed where either the customer has a certain service level threshold for each order or the manufacturer has an overall service level threshold. The third policy, modeling queue state independent order acceptance, is used to identify performance gains of including queue state knowledge into this decision. Equations for state probabilities, order acceptance rate, work-in-process, finished-goods-inventory, backorders and service level are developed for a system with stochastic customer-required lead times applying queuing methodology. An optimization problem minimizing capacity, work-in-process, finished-goods-inventory, backorder and lost sales cost (for rejected orders) in a single stage MTO production system is presented. The system is modeled as an M/M/1 queue with input rates depending on queue length and random customer required lead time. For the optimization problem, which cannot be solved explicitly, a solution heuristic is developed and a broad numerical study is conducted. The numerical study shows that allowing the customer to know the expected production lead time and—based on this knowledge—decide whether or not to place an order can have positive or negative influences on the overall costs, depending on the customer's service level target. Furthermore, the study shows that a high cost reduction potential exists for simultaneously optimizing capacity investment and order acceptance policy if the production system can decide whether or not to accept an order.

Optimal System for Warm Standby Components in the Presence of Standby Switching Failures, Two Types of Failures and General Repair Time

different system configurations with warm standby components, standby switching failures, two types of failures common cause and hardware failure and general repair are compared based on the availability. The time-to- for each of the primary and warm standby components are assumed to follow the exponential distribution. Laplace transforms of state probability equations are developed by using the supplementary variable technique. We develop the explicit expressions for the steady-state availability, Av, for two configurations. For all configurations, comparisons are made for specific values of distribution parameters and of the cost of the components. The configurations are ranked based on Av and cost/benefit, for three various repair time distributions: Gamma (G), Weibull (W) and Lognormal (L), where benefit is Av. KeywordsStandby switching failures; Common cause CCF; Supplementary variable; General repair times.

A tutorial on hypercube queueing models and some practical applications in Emergency Service Systems

This paper presents some extensions and applications of hypercube queueing models todescribe server-to-customer type Emergency Service Systems. The classical hypercube is a well-known spatially distributed queueing model effective in analyzing these systems, based on Markovian analysis approximations. Experience has shown that each real life Emergency Service System may have its own unique characteristics so that each system may require a particular hypercube queueing model incorporating those characteristics. Some of these distinctive characteristics are considered in the extensions presented in this tutorial such as dispatch policy based on random selection of the server to take an incoming call; partial cooperation among servers whereby depending on where the call is coming from, some servers cannot take the call; servers with additional workload coming from walk-in nonemergency customers such as in customer-to-server systems; existence of calls requiring the dispatch of more than one server; existence of more than one type of servers in the system, for instance paramedical and medical units in emergency medical systems. In this study, we present a set of these models based on the smallest non-trivial service systems. For each model, the construction of the system of equations for equilibrium hypercube state probabilities and the evaluation of particular operational characteristics are described.

Kinetics of proton pumping in cytochrome c oxidase

We propose a simple model of cytochrome c oxidase, including four redox centers and four protonable sites, to study the time evolution of electrostatically coupled electron and proton transfers initiated by the injection of a single electron into the enzyme. We derive a system of master equations for electron and proton state probabilities and show that an efficient pumping of protons across the membrane can be obtained for a reasonable set of parameters. All four experimentally observed kinetic phases appear naturally from our model. We also calculate the dependence of the pumping efficiency on the transmembrane voltage at different temperatures and discuss a possible mechanism of the redox-driven proton translocation.

A continuous-time semi-markov bayesian belief network model for availability measure estimation of fault tolerant systems

In this work it is proposed a model for the assessment of availability measure of fault tolerant systems based on the integration of continuous time semi-Markov processes and Bayesian belief networks. This integration results in a hybrid stochastic model that is able to represent the dynamic characteristics of a system as well as to deal with cause-effect relationships among external factors such as environmental and operational conditions. The hybrid model also allows for uncertainty propagation on the system availability. It is also proposed a numerical procedure for the solution of the state probability equations of semi-Markov processes described in terms of transition rates. The numerical procedure is based on the application of Laplace transforms that are inverted by the Gauss quadrature method known as Gauss Legendre. The hybrid model and numerical procedure are illustrated by means of an example of application in the context of fault tolerant systems.

Probabilistic safety analysis of maintainable systems

This paper presents two mathematical models to perform probabilistic safety analysis in maintenance environment. Both models assume system operating in safe and unsafe environments. Model I represents a repairable system with three states: working normally, working in unsafe mode, and failed safely. Model II represents a repairable system with four states: working normally, working in unsafe mode due to human error, working in unsafe mode due to hardware failure, and failed safely. Equations for state probabilities, system availabilities, reliability, and mean time to failure are developed.

Teaching elementary queueing theory with a computer algebra system

Computer algebra systems such as Maple, Mathematica and MACSYMA are readily available for a wide range of PC's and workstations. Many college campuses have site licenses for these software tools and make them widely available to students through PC labs, across networks and on time sharing systems. Additionally student versions of these software tools are widely available at nominal cost. Computer algebra systems provide sophisticated computational support and are intuitive to use. In this paper we describe how we use a computer algebra system to support an upper division undergraduate course teaching elementary queueing theory. We discuss the significant enhancement added by using a computer algebra system for this course work. We present the traditional approach to this material, i.e., deriving the closed form solutions for a specific queueing model then we show how we approach the material using numerical solutions based on the general equations for steady state probabilities for a Poisson Birth-Death process. We present several examples and discuss the strengths of our approach.

Using a computer algebra system (Maple) to teach elementary queueing theory

AbstractComputer algebra systems such as Maple, Mathematica, and MACSYMA are readily available for a wide range of PCs and workstations. Many college campuses have site licenses for these software tools and make them widely available to students through PC labs, across networks, and on time‐sharing systems. In addition, student versions of these software tools are widely available at nominal cost. Computer algebra systems provide sophisticated computational support and are intuitive to use. In this article we describe how we use Maple to support an upper‐division undergraduate course teaching elementary queueing theory. We discuss the significant enhancement added by using Maple for this coursework. We present the traditional approach to this material (i.e., deriving the closed form solutions for a specific queueing model), then we show how we approach the material using numerical solutions based on the general equations for steady state probabilities for a Poisson Birth‐Death process. We present several examples and discuss the strengths of our approach.

Common-cause failure analysis of a non-identical unit parallel system with arbitrarily distributed repair times

This paper presents common-cause failure analysis of a two non-identical unit parallel system with arbitrarily distributed repair times. Laplace transforms of state probability equations are developed by using the supplementary variable technique. A general formular for the system steady-state availability is developed for gamma distributed repair times. Time-dependent availability analysis is presented for the case when failed system repair times are gamma distributed with β = 2.

R&M through avionics/electronics integrity program: Wilbur W. Bhagat. Proc. a. Reliab. Maintainab. Symp., 216 (1989)

A device is said to have three-states if it operates normally in its operational mode but fails in either opened or shorted (closed) modes. This paper presents a repairable k-out-of-n three-state device network model with common-cause failures. Laplace transforms of the state probability equations are developed.

Steady state solution and convergence rate of time-dependent Markov chains of queuing networks

The differential equations for transient state probabilities for Markovian processes are examined to derive the rate of convergence of transient states to equilibrium states. There is an acute need to solve the balance equations for large states, particularly for handling computer performance modeling with a network of queues that do not satisfy product form solutions or cannot be cast into the forms convenient for mean value analysis. The rate of convergence to equilibrium states is derived for irreducible aperiodic homogeneous Markov chains on the basis of a geometrical interpretation. A numerical integration method with dynamic step-size adjustments is applied and compared against the power method of Wallace and Rosenberg.

Steady state solution and convergence rate of time-dependent Markov chains of queuing networks

The differential equations for transient state probabilities for Markovian processes are examined to derive the rate of convergence of transient states to equilibrium states. There is an acute need to solve the balance equations for large states, particularly for handling computer performance modeling with a network of queues that do not satisfy product form solutions or cannot be cast into the forms convenient for mean value analysis.The rate of convergence to equilibrium states is derived for irreducible aperiodic homogeneous Markov chains on the basis of a geometrical interpretation. A numerical integration method with dynamic step-size adjustments is applied and compared against the power method of Wallace and Rosenberg.

Solution of transient markovian state probabilities and rate of approach to equilibrium states

The differential equations for transient state probabilities for Markovian processes are examined to derive the rate of convergence of transient states to equilibrium states. There is an acute need to solve the balance equations for large states, particularly for handling computer per- formance modeling with a network of queues that do not satisfy product form solutions or can- not be cast into the forms convenient for mean value analysis. The rate of convergence to equilibrium states is derived for irreducible aperiodic homogeneous Markov chains on the basis of a geometrical interpretation. A numerical integration method with dynamic step-size adjustments is applied and compared against the power method of Wallace and Rosenberg.