Sensitivity analysis is an integral part of virtually every study of system reliability. This paper describes a Monte Carlo sampling plan for estimating this sensitivity in system reliability to changes in component reliabilities. The unique feature of the approach is that sample data collected on K independent replications using a specified component reliability vector p are transformed by an importance function into unbiased estimates of system reliability for each component reliability vector q in a set of vectors Q. Moreover, this importance function together with available prior information about the given system enables one to produce estimates that require considerably less computing time to achieve a specified accuracy for all |Q| reliability estimates than a set of |Q| crude Monte Carlo sampling experiments would require to estimate each of the |Q| system reliabilities separately. As the number of components in the system grows, the relative efficiency continues to favor the proposed method.

The paper shows the intimate relationship between the proposal and the method of control variates. Next, it relates the proposal to the estimation of coefficients in a reliability polynomial and indicates how this concept can be used to improve computing efficiency in certain cases. It also describes a procedure that determines the p vector, to be used in the sampling experiment, that minimizes a bound on the worst-case variance. The paper also derives individual and simultaneous confidence intervals that hold for every fixed sample size K. An example illustrates how the proposal works in an s-t connectedness problem.

In this paper, the problem of goodness of fit test for the inverse Rayleigh distribution based on progressively Type-II censored samples is studied. We develop two test statistics via entropy and propose one new non-entropy test statistic via a pivotal method. We also study the properties of these test statistics. Critical values are obtained by simulations. Then, we do power analysis of these test statistics against various alternatives under different censoring schemes. We conclude that the tests we proposed perform well against various alternatives, especially for non-monotone hazard alternatives. Finally, one real data set is analyzed.

Digital-to-analog converters (DAC) transform signals from the abstract digital domain to the real analog world. In many applications, DACs play a crucial role. Due to variability in the production, various errors arise that influence the performance of the DAC. We focus on the current errors, which describe the fluctuations in the currents of the various unit current elements in the DAC. A key performance measure of the DAC is the Integrated Nonlinearity (INL), which we study in this article. There are several DAC architectures. The most widely used architectures are the thermometer and the binary and the segmented architectures. We study the two extreme architectures, namely the thermometer and the binary architectures. We assume that the current errors are independent and identically normally distributed and reformulate the INL as a functional of a Brownian bridge. We then proceed by investigating these functionals. For the thermometer case, the functional is the maximal absolute value of the Brownian bridge, which has been investigated in the literature. For the binary case, we investigate properties of the functional, such as its mean, variance, and density.

The embedded Markov chain approach is widely used in queuing theory, in particular in M/G/1 and GI/M/c queues. In these cases, one has to relate the embedded equilibrium probablities to the corresponding random-time probabilities. The classical method to do this is based on Markov renewal theory, a rather complex approach, especially if the population is finite or if there is balking. In this article we present a much simpler method to derive the random-time probabilities from the embedded Markov chain probabilities. The method is based on conditional probability. Our approach might also be applicable in such situations.

Consider a continuous nonnegative random variable X with mean μ and hazard function h. Assume further that a≤h(t)≤b for all t≥0. Under these constraints, we obtain sharp two-sided bounds for . An application to birth and death processes is discussed.

Considering the Gaussian noise channel, Costa [4] investigated the concavity of the entropy power when the input signal and noise components are independent. His argument was connected to the first-order derivative of the Fisher information. In real situations, however, the noise can be highly dependent on the main signal. In this paper, we suppose that the input signal and noise variables are dependent. Then, some well-known copula functions are used to define their dependence structure. The first- and second-order derivatives of Fisher information of the model are obtained. Then, by using these derivatives, we will generalize two inequalities based on the Fisher information and a functional that is closely associated to Fisher information for the case when the input signal and noise variables are dependent. We will also show that the previous results for the independent case are recovered as special cases of our result. Several applications are provided to support the usefulness of our results. Finally, the channel capacity of the Gaussian noise channel model with dependent signal and noise is studied.

This paper studies an optimal deterministic investment problem for a DC pension plan member with inflation risk. We describe the price processes of the inflation-indexed bond and the stock by a continuous diffusion process and a jump diffusion process with random parameters, respectively. The contribution rate linked to the income of the DC plan member is assumed to be a non-Markovian adapted process. Under the mean-variance criterion, we use Malliavin calculus to derive a characterization for the optimal deterministic investment strategy. In some special cases, we obtain the explicit expressions for the optimal deterministic strategies.

Motivated by many applications in production planning, system reliability, queueing networks, and wireless communication, this work is devoted to singularly perturbed Markov chains with finite states. Focusing on nonstationary processes with the inclusion of transient states, asymptotic error bounds of a sequence of suitably scaled occupation measures are derived. The main tools used include martingales and differential equations. The results are useful for analyzing structural properties of the underlying Markov chains and for designing nearly optimal and hierarchical controls of large-scale and complex systems.

Alternating renewal processes have been widely used to model social and scientific phenomenal where independent “on” and “off” states alternate. In this paper, we study a model where the value of a process cumulates and declines according to two modes of compound Poisson processes with respect to an underlying alternating renewal process. The model discussed in the present paper can be used as a revenue management model applied to inventory or to finance. The exact distribution of the process is derived as well as the double Laplace transform with respect to the level and time of the process.

The paper deals with prognosis estimation for industrial systems in a series configuration, modeled by superimposed renewal processes (SRP), when the cause of failures is not available. In the presence of missing information, an SRP is commonly approximated by a Poisson process or a virtual age model. The performance of the approximations was assessed in the ideal configuration where all parameters of the models are known. The current article adopts a practitioner's perspective by assuming that the parameters of the models are unknown and must be estimated. In addition to inference procedures, the assessment of the prognosis indicators, such as the remaining useful life, is discussed. Finally, we investigate a fleet of infrastructure components of the Norwegian railway network operated by Bane NOR.

We study the optimal buffer capacity K for the M/M/1/K queue under some standard cost and reward structures by comparing various Markov reward processes. Using explicit expressions for the deviation matrix of the underlying Markov chains, we find the bias optimal value for K in the case of a tie between two consecutive optimal gain policies. We show that the bias optimal value depends both on whether the reward is granted upon arrival or departure of the customers, and on the initial queue size. Moreover, we demonstrate that in some specific cases the optimal policy is threshold-based with respect to the initial queue size.

The stability properties of the bandwidth allocation algorithm First Fit are analyzed for some distributions on the sizes of the requests. Fluid limits are used to get the ergodicity results. When there are two possible sizes, the description of the transient behavior involves a finite Markov chain on the exit states of a transient Markov chain on a countable state space. The explicit expression of this exit matrix is given.

Motivated by Ahmadi-Javid (Journal of Optimization Theory Applications, 155(3), 2012, 1105–1123) and Ahmadi-Javid and Pichler (Mathematics and Financial Economics, 11, 2017, 527–550), the concept of Tsallis Value-at-Risk (TsVaR) based on Tsallis entropy is introduced in this paper. TsVaR corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the Value-at-Risk. The main properties and analogous dual representation of TsVaR are investigated. These results partially generalize the Entropic Value-at-Risk by involving Tsallis entropies. Three spaces, called the primal, dual, and bidual Tsallis spaces, corresponding to TsVaR are fully studied. It is shown that these spaces equipped with the norm induced by TsVaR are Banach spaces. The Tsallis spaces are related to the $L^p$ spaces, as well as specific Orlicz hearts and Orlicz spaces. Finally, we derive explicit formula for the dual TsVaR norm.

Network mutual aid platforms is one of the popular risk-sharing models in recent years, and they have almost 200 million members in China. However, current mutual aid platforms does not satisfy the actuarial rules in either the apportionment method or the pricing principle. Hence, a variety of mutual aid models which enable mutual aid members with different risks to exchange their risks in a transparent and actuarial fair way have been proposed in this paper. Besides, the decision-making frameworks for participants choosing between the mutual aid platform and similar insurance products, or choosing no risk sharing are constructed, respectively. Decisions are made based on the principle of maximizing expected utility. Moreover, the optimization problems of maximizing profit and minimizing risk are constructed, respectively. Through the principle of individual fairness and relative fairness, the problem of adverse selection of the platform can also be reduced. Finally, the actual mutual aid plan is compared with similar insurance products to discuss the advantages of the optimized plan.

Different virtual machines can share servers, subject to resource constraints. Incoming jobs whose resource requirements cannot be satisfied are queued and receive service according to a preemptive-resume scheduling policy. The problem is to evaluate a cost function, including holding and server costs, with a view to searching for the optimal number of servers. A model with two job types is analyzed exactly and the results are used to develop accurate approximations, which are then extended to more than two classes. Numerical examples and comparisons with simulations are presented.

We show that several truncation properties of queueing systems are consequences of a simple property of censored stochastic processes. We first consider a discrete-time stochastic process and show that its censored process has a truncated stationary distribution. When the stochastic process has continuous time, we present a similar result under the additional condition that the process is locally balanced. We apply these results to single-server batch arrival batch service queues with finite buffers and queueing networks with finite buffers and batch movements, and extend the well-known results on truncation properties of the MX/G/1/k queues and queueing networks with jump-over blocking.

In the reliability context, the geometric distribution is a natural choice to model the lifetimes of some equipment and components when they are measured by the number of completed cycles of operation or strokes, or in case of periodic monitoring of continuous data. This paper aims at investigating how the heterogeneity among the parameters affects some characteristics such as the distribution and hazard rate functions of spacings arising from independent heterogeneous geometric random variables. First, refined representations of the distribution functions are provided for both the second spacing and sample range from heterogeneous geometric sample. Second, stochastic comparisons are carried out on the second spacings and sample ranges for two sets of independent and heterogeneous geometric random variables in the sense of the usual stochastic and hazard rate orderings. The results established here not only fill the gap on the study of stochastic properties of spacings from heterogeneous geometric samples, but also are expected to be applied in the fields of statistics and reliability.

Expansions for moments of $\overline{X}$, the mean of a random sample of size n, are given for both the univariate and multivariate cases. The coefficients of these expansions are simply Bell polynomials. An application is given for the compound Poisson variable SN, where $S_{n} = n \overline{X}$ and N is a Poisson random variable independent of X1, X2, …, yielding expansions that are computationally more efficient than the Panjer recursion formula and Grubbström and Tang's formula.

This article deals with Markovian models defined on a finite-dimensional discrete state space and possess a stationary state distribution of a product-form. We view the space of such models as a mathematical object and explore its structure. We focus on models on an orthant [script Z]+n, which are homogeneous within subsets of [script Z]+n called walls, and permit only state transitions whose ∥ ∥∞-length is 1. The main finding is that the space of such models exhibits a decoupling principle: In order to produce a given product-form distribution, the transition rates on distinct walls of the same dimension can be selected without mutual interference. This principle holds also for state spaces with multiple corners (e.g., bounded boxes in [script Z]+n).

In addition, we consider models which are homogeneous throughout a finite-dimensional grid [script Z]n, now without a fixed restriction on the length of the transitions. We characterize the collection of product-form measures which are invariant for a model of this kind. For such models with bounded transitions, we prove, using Choquet's theorem, that the only possible invariant measures are product-form measures and their combinations.

The object of research is G-network with positive customers and signals of multiple classes. The present paper describes an analysis of this network at a non-stationary regime, also provided a description of method for finding non-stationary state probabilities.

At the beginning of the article, a description of the network with positive customers and signals is given. A signal when entering the system destroys a positive customer of its type or moves the customer of its type to another system. Streams of positive customers and signals arriving to each of the network systems are independent. Selection of positive customers of all classes for service – randomly. For non-stationary state probabilities of the network, the system of Kolmogorov difference-differential equations (DDE) has been derived. It is solved by a modified method of successive approximations, combined with the method of series. The convergence of successive approximations with time has been proved to the stationary distribution of probabilities, the form of which is indicated in the article, and the sequence of approximations converges to the unique solution of the DDE system. Any successive approximation is representable in the form of a convergent power series with an infinite radius of convergence, the coefficients of which satisfy recurrence relations, which is convenient for computer calculations.

The obtained results can be applied for modeling behavior of computer viruses and attack in computer systems and networks, for example, as model impact of some file viruses on server resources.