Time-dependent State Probabilities Research Articles

Analysis of a Single Server Queueing System Controlled by a Random Switch

In this paper, a single server queueing system operating in a doubly stochastic environment is analysed. The random environment makes transitions among N levels controlled by a random switch which performs the task of assigning a job to the server. The queueing system resides in level k of the environment till the completion of the service of the last customer in the system and immediately reports to the random switch to get a new service job. The random switch generates a set up time and waits till a customer arrives and assigns a job to the server in any one of the N levels with a positive probability governed by a binomial distribution. In each level r of the environment, the queueing system behaves like M(λr)/M(μr)/1 queue subject to the condition that the server reports to the random switch immediately after performing exhaustive service for getting a new assignment. For this model, time-dependent state probabilities are explicitly found and the corresponding steady-state probabilities are deduced. Some key performance measures are also obtained. A numerical study is also made.

Analyzing Reliability of CGS Station by Continuous Time Markov Chains (CTMC)

Improving the system's reliability is one way to achieve a secure system. City Gas Station (CGS) has a key role in the timely and safe supply of Natural gas (NG) to residential, commercial, and industrial customers. With complexities inherent in systems, having a proper and all-embracing model of the entirety of a system is not readily possible. The continuous-time Markov chain (CTMC) model is regarded as a great help in communicating, comparing, and integrating partial system models. In this study, we have exploited CTMC for reliability analysis in CGS stations. The CTMC model can solve both time-dependent and stationary state probabilities. Therefore, it can potentially develop the state enumeration method into a sequential one. Implementing this procedure leads to identifying critical components and failure probability, eventually enhancing the station's reliability. Additionally, some suggestions are presented for optimizing the performance of the station.

Application of Decomposable Semi-Regenerative Processes to the Study of k-out-of-n Systems

This paper aimed to demonstrate the capabilities of decomposable semi-regenerative processes for the investigation of the k-out-of-n system. Proposed in 1955 by W. Smith, the regeneration idea has come a long way in terms of development and has found widespread applications. First, we briefly recall the history of the development of the regeneration idea and the main results of the theory of regenerative, semi-regenerative, and decomposable semi-regenerative processes. Then, the methods of the theory of decomposable semi-regenerative processes are used for the study of a k-out-of-n renewable system with exponentially distributed life and generally distributed repair times of its components. This system is very important for practice and its special cases have previously been considered (including by the authors); however, only special cases and using other methods are considered herein. In the current paper, two scenarios of system repair after its failure are considered for the first time: the partial and the full system repair scenarios. For both scenarios, the time-dependent system state probabilities are calculated in terms of their Laplace transforms. The closed form representation of the stationary probabilities for both scenarios are also presented. These latest results represent a new contribution to the study of this system.

On Reliability of a Double Redundant Renewable System with a Generally Distributed Life and Repair Times

The paper provides reliability analysis of a cold double redundant renewable system assuming that both life-time and repair time distributions are arbitrary. The proposed approach is based on the theory of decomposable semi-regenerative processes. We derive the Laplace–Stieltjes transform of two main reliability measures like the distribution of the time between failures and the time to the first failure. The transforms are used to calculate corresponding mean times. It is further derived in closed form the time-dependent and time stationary state probabilities in terms of the Laplace transforms. Numerical results illustrate the effect of the type of distributions as well as their parameters on the derived reliability and probabilistic measures.

Industrial equipment reliability estimation: A Bayesian Weibull regression model with covariate selection

A three-state continuous-time semi-Markov process is used to model the degradation of an industrial equipment. The transition times are assumed Weibull-distributed and influenced by a set of covariates. A Weibull Regression Model is developed within the Bayesian probability framework, to account for the influence of these covariates and estimate the model parameters with the related uncertainty, on the basis of few data and expert judgment. The number of covariates is reduced by a two-step selection procedure derived from the condition monitoring engineering practice. The developed model enables estimating reliability and time-dependent state probabilities for a component degrading in given operational and ambient conditions, represented by a vector of covariates. The model is illustrated by way of a real case study concerning the degradation process affecting diaphragm valves used in the biopharmaceutical industry.

FINDING NONSTATIONARY STATE PROBABILITIES OF OPEN MARKOV NETWORKS WITH MULTIPLE CLASSES OF CUSTOMERS AND VARIOUS FEATURES

This paper discusses a system of difference-differential equations (DDE) that is satisfied by the time-dependent state probabilities of open Markov queueing networks with various features. The number of network states in this case and the number of equations in this system is infinite. Flows of customers arriving at the network are a simple and independent, the time of customer services is exponentially distributed. The intensities of transitions between the network states are deterministic functions depending on its states.To solve the system of DDE, we propose a modified method of successive approximations, combined with the method of series. The convergence of successive approximations with time to a stationary probability distribution, the form of which is indicated in the paper has been proved. The sequence of approximations converges to a unique solution of the system of equations. Any successive approximation can be represented as a convergent power series with an infinite radius of convergence, the coefficients of which satisfy recurrence relations, which is convenient for calculations on a computer. Examples of the analysis of Markov G-networks with various features have been presented.

Transient analysis of an M/M/c queuing system with retention of reneging customers

In this paper, we study the transient behaviour of an M/M/c queuing system with reneging and retention of reneging customers. The probability generating function technique along with Bessel function properties is used to derive the time-dependent state probabilities explicitly. The transient behaviour of system size probabilities, the expected system size, the average reneging rate, and the average retention rate is studied with the help of a numerical example.

Transient analysis of an M/M/c queuing system with retention of reneging customers

In this paper, we study the transient behaviour of an M/M/c queuing system with reneging and retention of reneging customers. The probability generating function technique along with Bessel function properties is used to derive the time-dependent state probabilities explicitly. The transient behaviour of system size probabilities, the expected system size, the average reneging rate, and the average retention rate is studied with the help of a numerical example.

Transient analysis of an M/M/c queuing system with balking and retention of reneging customers

ABSTRACTIn this paper, we study an infinite capacity multi-server Markovian queuing system with balking and retention of reneging customers. The transient analysis of the model is performed. The probability generating function technique along with Bessel function properties is used to derive the time-dependent state probabilities explicitly.

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.

Time dependent system state probabilities of single server queuing system with infinite queue

Analytical expression for time dependent system state probabilities of single server queuing system with infinite queue capacity M/M/1 is derived. Expression is derived by finding the limit value of expression for time dependent system state probabilities of single server queuing system with finite queue capacity M/M/1/K, when number of places in the queue tens to infinity, in the case that system is empty at the beginning. Only elementary mathematical operations are used.

Transient analysis of a single server discrete-time queue with system disaster

A discrete-time Geo/Geo/1 queue with system disaster is considered in this paper. The time-dependent and steady state probabilities of number of customers present in the system are obtained in terms of ballot numbers by solving the underlying system of difference equations using the generating function and continued fractions. Further, the busy period distribution is derived in terms of Catalan numbers. For special cases, time-dependent system size probabilities and busy period distribution are verified with the existing results in the literature. Numerical illustrations are provided for different parameter values to see their effect on performance measures and to get more insight of the model behavior.

Development of a Bayesian multi-state degradation model for up-to-date reliability estimations of working industrial components

We consider a three-state continuous-time semi-Markov process with Weibull-distributed transition times to model the degradation mechanism of an industrial equipment. To build this model, an original combination of techniques is proposed for building a semi-Markov degradation model based on expert knowledge and few field data within the Bayesian statistical framework. The issues addressed are: i) the prior elicitation of the model parameters values from experts, avoiding possible information commitment; ii) the development of a Markov-Chain Monte Carlo algorithm for sampling from the posterior distribution; iii) the posterior inference of the model parameters values and, on this basis, the estimation of the time-dependent state probabilities and the prediction of the equipment remaining useful life. The developed Bayesian model offers the possibility of updating the system reliability estimation every time a new evidence is gathered. The application of the modeling framework is illustrated by way of a real industrial case study concerning the degradation of diaphragms installed in a production line of a biopharmaceutical industry.

An Optimal Age Replacement Policy for Multi-State Systems

We consider an age replacement policy (ARP) for a system which consists of several multi-state elements. These elements can be in different states with performance levels ranging from perfectly functioning (the highest state) to total failure (the zero state). The multi-state system (MSS) is considered to be in a failure or unacceptable state if its performance level, determined by the multiple elements and the configuration, falls below the user demand level, and is considered as in a working or acceptable state if its performance level is greater than or equal to the user demand level. Under an ARP, a multi-state system is replaced at a failure, or at age T, whichever comes first. The deterioration of the multi-state element is assumed to follow the non-homogenous continuous-time Markov chain (NHCTMC). We use the recursion to solve the Chapman-Kolmogorov's (C-K) forward equation to obtain the time-dependent state probabilities of each element of the system. Then we compute the state probabilities of the entire system by using the Lz-transform method. Finally, we derive the expected cost and profit functions, and determine the cost minimization or profit maximization ARPs. The multi-state model under the ARP is a generalization of the classic two-state maintenance model, and can be applied to analyze more complex aging systems. Numerical examples are presented to demonstrate our results.

Real-time battery system aided by Unmanned Aerial Vehicles for attacking multiple moving targets

The timely repositioning of a long-range weapon, such as missiles, is crucial to attack moving targets. This paper proposes a stochastic model where Unmanned Aerial Vehicles (UAVs) are utilised to aid a battery of long-range weapons in acquiring moving targets, assessing their battle damage, and incorporating such information into the repositioning. A closed-form expression for the time-dependent state probabilities of the system is developed, which is used to estimate several important combat measures: the time-varying mean and variance of the number of surviving UAVs and surviving enemy targets; the mission success and failure rate and the probability of a combat draw; the mean and variance of the combat duration time.

Online short-term reliability evaluation using a fast sorting technique

To consider time varying system operating conditions and operating reserve units, the online short-term reliability of a power system is investigated. Real-time system reliability based on the online operation data provided by the SCADA/EMS/WAMS is assessed using time-dependent state probabilities of components. Considering both speed and accuracy requirements, a fast sorting technique (FST) is proposed to quickly select the required number of system states in descending order probability. The number of computations and comparisons used in the evaluation is minimum. The relative accuracy of a given reliability index is defined as the stopping rule for the evaluation. As only a small number of system states are required to achieve the high accuracy of the short-term reliability indices when using the FST, the evaluation can be performed in an online environment. The proposed evaluation technique is illustrated by the application to the IEEE-RTS and China Southern Power Grid.

A Markov time related to a robot-safety device system

We consider a robot-safety device system attended by two different repairmen. The twin-system is characterized by the natural feature of cold standby and an admissible risky state. Apart from tangible results obtained in the previous Literature, we introduce a Markov time called the recovery time of the system. In order to obtain the corresponding distribution, we employ a stochastic process endowed with time dependent state probabilities related to the point availability of a renewable robot without safety device. Finally, as an example, we consider the case of Weibull repair (for the robot) and deterministic repair (for the safety device). We provide several computer-plotted graphs obtained by advanced numerical methods.

The equivalence of the transient behaviour formulae for the single server queue

It is indeed obvious to expect that the different results obtained for some problem are equal, but it needs to established. For the M/M/1/N queue, using a simple algebraic approach we will prove that the results obtained by Takâcs [17] and Sharma and Gupta [13] are equal. Furthermore, a direct proof to the equivalence between all formulae of the M/M/l/∞ queue is established. At the end of this paper, we will show that the time-dependent state probabilities for M/M/l/N queue can be written in series form; its coefficients satisfy simple recurrence relations which would allow for the rapid and efficient evaluation of the state probabilities. Moreover, a brief comparison of our technique, Sharma and Gupta's formula and Takâcs result is also given, for the CPU time computing the state probabilities.

The infinite server queue and heuristic approximations to the multi-server queue with and without retrials

In this paper, properties of the time-dependent state probabilities of theM t /G/∞ queue, when the queue is assumed to start empty are studied. Those results are compared with corresponding time-dependent results for theM/M/1 queue. Approximation to the time-dependent state probabilities of theM/G/m/m queue by means of the corresponding time-dependent state probabilities of theM/G/∞ queue are discussed. Through a decomposition formula it is shown that the main performance characteristics of the ergodicM/M/m/m+d queue are sums of the corresponding random variables for the ergodicM/M/m/m andM/M/1/1+(d−1) queues, respectively, weighted by the 3-rd Erlang formula (stationary probability of waiting or being lost for theM/M/m/m+d queue). Successful exact and approximation extensions of this kind of decomposition formula to theM/M/m/m+d queue with retrials are presented.

Combat modeling of remotely targeted vehicles and a battery against a moving target

This paper considers a combined system composed of multiple stand-by remotely piloted vehicles (RPVs) and a single battery against a single passive enemy target, where the target, if not killed, is allowed to change its location after each attack. The RPV has the duty to report on target acquisition, to confirm a target kill, and to pass information on any change in target location after each battery attack. The battery has the duty to attack the target on the basis of the target location information provided to it by the RPV. We develop a closed-form expression for the time-dependent state probabilities of the system, which can be used to compute several important combat measures of effectiveness, including (a) the time-varying mean and variance of the number of the RPVs being alive and of the surviving enemy target, (b) the mission success, mission failure, and combat draw probabilities, and (c) the mean and variance of the combat duration time. Illustrative numerical examples are solved for these combat measures, and sensitivity analyses are performed with respect to target acquisition time and target kill probability. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 645–667, 1998