The mechanism of the Counter Scheme (CS) has been shown to be an effective statistical approach for the reorganization of linear lists, where the records in the list are referenced independently with a time homogeneous multinomial distribution. In this paper we show that derivative schemes can be used effectively in other contexts as well. Specifically, we consider (a) linear lists that are doubly linked, so that they may be accessed at both ends, (b) multilists, which result from dissecting a linear list into several pieces that are accessed independently and reside in WORM (write-once-read-many) store, and (c) reorganizing a disk, by copying its contents to another disk, so as to minimize the expected seek time required to access a record.

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.

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.

Motivated by stochastic allocation problems, we study sufficient conditions that imply a general class of stochastic arrangement inequalities. This class includes many known and useful arrangement inequalities in the literature. Some illustrative examples in reliability theory are also given.

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.

Various large-scale suppliers frequently use web-based spot markets, along with discount stores and foreign distributors, for inventory liquidation. Recognizing the potential benefits of such practices, we consider a multi-period, integrated replenishment, and liquidation problem for a capacitated supplier facing stochastic demand from a spot market along with its primary market (with higher priority contractual customers). In each period, the supplier must decide: (i) how much to produce, and (ii) if there are excess units left after sales to the primary market, how many of these to liquidate. We show that the optimal policy is characterized by two quantities: the critical produce-up-to level and the critical retain-up-to level. We establish bounds for these two quantities. We identify two practical benchmark policies and establish thresholds on the unit revenue earned from the spot market such that one of the two benchmark policies is optimal. We provide closed form expressions to determine these thresholds for the infinite horizon problem under specific conditions on the available production capacity. In general, it is difficult, if not impossible, to theoretically determine these thresholds in closed form for the finite horizon problem. Hence, we report results of a computational study to gain insights regarding the behavior of the optimal policy with respect to the spot market revenue. Our computational results also quantify the benefits of the optimal policy relative to the benchmark policies and examine the effects of demand correlation.

The double-mean-reverting model, introduced by Gatheral [(2008). Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society London, July 18], is known to be a successful three-factor model that can be calibrated to both CBOE Volatility Index (VIX) and S&P 500 Index (SPX) options. However, the calibration of this model may be slow because there is no closed-form solution formula for European options. In this paper, we use a rescaled version of the model developed by Huh et al. [(2018). A scaled version of the double-mean-reverting model for VIX derivatives. Mathematics and Financial Economics 12: 495–515] and obtain explicitly a closed-form pricing formula for European option prices. Our formulas for the first and second-order approximations do not require any complicated calculation of integral. We demonstrate that a faster calibration result of the double-mean revering model is available and yet the practical implied volatility surface of SPX options can be produced. In particular, not only the usual convex behavior of the implied volatility surface but also the unusual concave down behavior as shown in the COVID-19 market can be captured by our formula.

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.

This paper considers a two-person zero-sum stochastic differential game. The dynamics of the game are given by a one-dimensional stochastic differential equation whose diffusion coefficient may be controlled by the players. The drift coefficient is held constant and cannot be controlled. Player l's objective is to maximize the probability that the state at final time, T, is positive, while Player 2's objective is to maximize the probability that the state is negative.

We introduce a Markov queueing system with Poisson arrivals, exponential services, and jockeying between two parallel and equivalent servers. An arriving customer admits to the shortest line. Every transition, of only the last customer in line, from the longer line to the shorter line may accompanied by a certain fixed cost. Thus, a transition from the longer queue to the shorter queue occurs whenever the difference between the lines reaches a certain discrete threshold (d = 2, 3, …). In this study, we focus on the stochastic analysis of the number of transitions of an arbitrary customer.

We introduce a level-crossing analysis of the finite time-t probability distributions of the excess life, age, total life, and related quantities of renewal processes. The technique embeds the renewal process as one cycle of a regenerative process with a barrier at level t, whose limiting probability density function leads directly to the time-t quantities. The new method connects the analysis of renewal processes with the analysis of a large class of stochastic models of Operations Research. Examples are given.

In this paper, we are going to study the strong limit theorem for the relative entropy density rates between two finite asymptotically circular Markov chains. Firstly, we prove some lammas on which the main result based. Then, we establish two strong limit theorem for non-homogeneous Markov chains. Finally, we obtain the main result of this paper. As corollaries, we get the strong limit theorem for the relative entropy density rates between two finite non-homogeneous Markov chains. We also prove that the relative entropy density rates between two finite non-homogeneous Markov chains are uniformly integrable under some conditions.

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.

An external uniformization technique was developed by Ross [4] to obtain approximations of transition probabilities of finite Markov chains in continuous time. Yoon and Shanthikumar [7] then reported through extensive numerical experiments that this technique performs quite well compared to other existing methods. In this paper, we show that external uniformization results from the strong law of large numbers whose underlying distributions are exponential. Based on this observation, some remarks regarding properties of the approximation are given.

We consider preemptive scheduling on parallel machines where processing times of jobs are i.i.d. but jobs may already have received distinct amounts of service. We show that when processing times are increasing in likelihood ratio, SEPT (shortest expected [remaining] processing time first) stochastically minimizes any increasing and Schur-concave function of the job completion times. The same result holds when processing times are exponential with possibly different means.

We consider a two-player game in which the first player (the Guesser) tries to guess, edge-by-edge, the path that second player (the Chooser) takes through a directed graph. At each step, the Guesser makes a wager as to the correctness of her guess and receives a payoff proportional to her wager if she is correct. We derive optimal strategies for both players for various classes of graphs, and we describe the Markov-chain dynamics of the game under optimal play. These results are applied to the infinite-duration Lying Oracle Game, in which the Guesser must use information provided by an unreliable Oracle to predict the outcome of a coin toss.

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.

We analyze service systems where Nservers (N > 2) move one at a time along an array of stations to satisfy a known number of requests for service. Processing a request consists of determining the server to satisfy the request. The cost of processing a request is determined by the distance the server that performs the request has to move. We determine qualitative properties of sequencing strategies that minimize the expected cost incurred by the service of all requests when the array of stations is on a circle or an interval.

We revisit the random m-ary search tree and study a finer profile of its node outdegrees with the purpose of exploring possibilities of data structure compression. The analysis is done via Pólya urns. The analysis shows that the number of nodes of each individual node outdegree has a phase transition: Up to m = 26, the number of nodes of outdegree k, for k = 0, 1, …, m, is asymptotically normal; that behavior changes at m = 27. Based on the analysis, we propose a compact m-ary tree that offers significant space saving.

By using the martingale method, we show that thinning of an arbitrary point process produces independent thinned processes if and only if the original point process is (nonhomogeneous) Poisson. Thinning is a classical problem for point processes. It is well-known that independent homogeneous Poisson processes result from constant Bernoulli thinnings of homogeneous Poisson processes. In fact, the conclusion remains true for nonconstant Bernoulli thinnings of nonhomogeneous Poisson processes. processes. In fact, the conclusion remains true for nonconstant Bernoulli thinnings of nonhomogeneous Poisson processes. It is easy to see that the converse is also true, i.e., if the thinned processes are independent nonhomogeneous Poisson processes, so are the original processes. But if we only suppose the thinned processes are independent, nothing is concerned with their distribution law, the problem of whether or not the original processes are (nonhomogeneous) Poisson becomes interesting and challenging, which is the objective of this paper. It is considerably surprising for us to arrive at the affirmative answer. So far as this problem is concerned, the most work was done under the renewal assumption. For example, Bremaud [1] showed that for arbitrary delayed renewal processes, the existence of a pair of uncorrelated thinned processes is sufficient to guarantee that the original process is Poisson. It is natural that the mathematical tools to solve the problem in this case be typical ones for renewal theory, such as renewal equations and Laplace-Stieltjes transformations. Obviously, they are not available for nonrenewal processes. We find out that the martingale method is the most efficient one to solve this problem in the general case. More precisely, we use mainly the dual predictable projections of point processes. In fact, the distribution law of a point process is determined uniquely by its dual predictable projection (see [5]), and Ma [6] offered us a very useful criterion of independence of jump processes having no common jump time through their dual predictable projections. Based on these results, it is not a long way to arrive at the destination.