We propose a matrix-free comparison method for renewal processes with dependent, heterogeneous, and mixed inter-renewal laws. After its domination, non-degeneracy, and moment conditions have been verified, Stage I yields a rigorous bound sup _t≥ 0𝔼W_t≤ C_GL on the expected forward recurrence time. A generalised Lorden-type inequality expresses C_GL through the first two moments of two comparison distributions. The primary route forms these distributions from the pointwise envelopes of the conditional survival functions. A fallback route uses a generalised intensity measure, extending the classical hazard rate to laws with atoms, when conditional laws are unavailable but per-clock intensity bounds are known. Evaluating C_GL requires four one-dimensional tail integrals, with linear work per quadrature node in the number of regimes and no state-space matrices. Stage II uses this constant in a coupling-to-stationarity argument and yields a total-variation rate whenever the required model-specific coupling inputs are available. We study two reliability models. For an i.i.d. atom-plus-exponential model, both stages are explicit: a rare-trial coupling with a closed-form minorisation coefficient gives a rigorous total-variation bound. For a Markov-modulated mixed model, a quantile coupling verifies the pathwise domination required for an unconditional Stage I bound; the distinct coupling of the joint age–environment process needed for Stage II remains open. Phase-type, larger-state-space, and sensitivity experiments assess conservatism, scaling, and route-specific robustness.
Let (X_n)_n be a sequence of i.i.d. ℝ^d -valued random variables uniformly distributed on a compact subset 𝕄 of ℝ^d . In this work, we study the asymptotic behavior of maximal packings of this compact subset 𝕄 . Under some assumptions on 𝕄 (in particular, that it is a smooth submanifold of ℝ^d without boundary), we establish a precise extreme value law (of Gumbel type) for the minimal distances from the sample (X_1,⋯ ,X_n) to the centers of maximal packings. This result leads to explicit asymptotic confidence bounds for this maximal packing and thus to the support 𝕄 . A distinctive feature of our contribution is the explicit derivation of the scaling sequences in the Gumbel convergence, depending only on the geometry of the support 𝕄 . These formulas provide interpretable confidence bounds for 𝕄 , which represent a novel complement to previous approaches such as those by Fasy et al. (2014). Our approach bridges geometric and combinatorial probability arguments and relies on analytic tools such as the Lambert W function. In addition, we provide examples of main sample spaces in directional and circular statistics (circle, sphere, torus), along with simulations that illustrate and support the theoretical findings.
We develop nonparametric inference for reliability indicators of discrete-time semi-Markov systems from independent trajectories observed over a common fixed horizon. Augmenting the physical state by the backward recurrence time yields a finite coupled Markov representation on the observed age range. We distinguish the resulting age-restricted failure-or-exit time from calendar truncation, since these two finite-horizon quantities coincide only in special cases. The framework covers restricted factorial moments and moment characteristics, calendar-truncated failure-time summaries, and the discrete-time intensity of the hitting time. Under explicit row-exposure conditions, strong consistency and joint asymptotic normality are established for the empirical initial law, the required transition rows and the corresponding plug-in functionals. The Gaussian random-matrix representation gives pointwise and joint covariance formulas, simultaneous confidence envelopes, Wald procedures for linear summaries, and curvature-adjusted Gaussian approximations. Restriction diagnostics and a target-specific horizon-selection rule based on exposure, boundary interaction and nested-horizon stability are developed separately. Numerical experiments assess the inferential formulas and the diagnostics, while a complete-case illustration from the European Group for Blood and Marrow Transplantation (EBMT) reports calendar-truncated failure-time summaries and finite-dimensional hitting intensities.
We develop a Dickman approximation to the small jumps of a large class of multivariate stochastic integral processes. We then apply these results to the simulation of multivariate continuous-time autoregressive moving average (MCARMA) processes driven by general multivariate gamma distributions (GMGD). These include GMGD driven Ornstein-Uhlenbeck processes as an important special case. A small-scale simulation study suggests that the method works well.
This article introduces the Turing–Entropic Tail Classifier (TENT), a nonparametric information-theoretic framework for tail inference in discrete and continuous distributions. Central to the approach is the tail profile, a collection of information-theoretic quantities motivated by Turing’s formula and domain-of-attraction theory on countable alphabets. TENT classifies the qualitative form of tail decay by comparing empirical tail profiles with theoretical benchmarks corresponding to exponential-type, near-exponential, sub-exponential, and power-law regimes, including Zipf and Pareto-type behavior. For heavier-than-exponential tails, the framework further yields point and interval estimates for selected tail parameters. Although the method is primarily designed to support preliminary model selection for discrete parametric families, a supporting result shows that, under suitable regularity conditions, discretizing continuous observations by common-width binning preserves the relevant tail decay rate. This provides a principled route for extending the classifier to continuous data. Simulation studies illustrate the finite-sample performance of TENT across multiple tail classes and sampling regimes.
Cascading failures are encountered commonly in various practical systems. As one of the most well-known models in reliability, consecutive k-type systems, especially two/three-dimensional ones, have not been studied under the assumption of cascading component failures. In this paper, we consider cascading component failures based on generalized urn model for some linear one/two/three-dimensional consecutive k-type systems. These systems include linear consecutive-k-out-of-n: F systems, linear l-consecutive-k-out-of-n: F systems without/with overlapping, and their two/three-dimensional versions. By applying finite Markov chain imbedding approach (FMCIA) via an original manner, reliability functions of these systems are derived together with several illustrative examples.
Since decades, the problem of finding the moment generating function and the distribution function of compound renewal sums with discounted claims has been studied from various perspectives but not yet within of a fractional calculus framework. In this first approach, our aim is therefore to fill this gap by examining these specific questions when the counting process is a fractional renewal process.
We propose the matched filter subset scan (MFSS), a new approach for flexibly incorporating prior information into subset scanning while maintaining computational efficiency. MFSS encodes structured information about which subsets of the data are a priori more likely by (i) defining a set of subsets (the “filter set” Θ ) that collectively describe what we expect the true signal to look like; and (ii) optimizing a penalized log-likelihood ratio statistic, where the penalty is proportional to the Hamming distance to the nearest filter in Θ . We demonstrate that this approach maintains computational efficiency and exact detection of the highest-scoring subset for small-sized filter sets, and present initial results for scaling to larger-sized filter sets. Finally, we prove several theoretical results, including asymptotic identification of the correct filter and asymptotic improvement in detection accuracy, under reasonable simplifying assumptions.
The matching probability between two sequences X_t _t = 1^n and Y_t _t = 1^n plays an important role in genomic studies. The length of the longest matching L_n^0 between two aligned sequences is known as a good indicator for measuring the closeness of the two sequences. The distribution of L_n^0 is often computed under the assumption that the two nucleic acid sequences are independent and identically distributed with known parameters. In many real situations, the above basic assumption does not hold. In this article, three extensions are considered: (i) the two sequences are independent Markov dependent random variables, (ii) allowing d ( d = 0, 1, 2 , … ) mutations, and (iii) with unknown parameters. The finite Markov chain imbedding technique will be used for studying the above three extensions. Examples and numerical results will be provided to illustrate the theoretical results.
This paper addresses a fundamental and practically significant problem in call centre operations, determining optimal call allocation policies and evaluating their performance so as to meet client service targets while minimising staffing costs. Motivated by a problem presented by an industry partner, we examine a real-world setting involving a relatively small call centre with hierarchical structure among agents. It is natural to model the operation of such a centre as a continuous-time Markov chain. To gain insight into the structure of optimal policies, we first (i) apply backward induction based on Bellman’s equation to a finite-horizon discrete-time model, and (ii) derive stationary policies for an infinite-horizon continuous-time model with discounting. Subsequently, we evaluate the performance of these policies in the original finite-horizon continuous time setting by computing the expected number of abandonments and the waiting time distributions of customers. This is achieved using first-step analysis combined with Laplace transform methods. The effectiveness of the proposed approach is illustrated through numerical examples.
The performance and lifetime of energy-harvesting (EH) wireless sensor networks (WSNs) and other low-power autonomous devices are fundamentally constrained by the stochastic nature of harvested energy and the finite capacity of their energy storage units. Fluctuations in renewable energy inflow often lead to energy scarcity, service interruptions, increased delay, and reduced reliability, highlighting the need for rigorous analytical modelling. Accurately capturing these dynamics is essential for designing efficient storage systems and determining cost-optimal battery capacities. This paper develops a Markov Fluid Vacation Queue (MFVQ) framework to jointly characterise the evolution of stored energy and the discrete workload in EH-powered devices. The model couples a finite-capacity fluid queue describing the battery with a discrete request queue modelling data or service arrivals, while periods of inactivity caused by battery depletion are represented through phase-type (PH) distributed vacations. This formulation enables realistic modelling of intermittently powered operation under stochastic charging and consumption dynamics. Using the Matrix-Analytic Method (MAM), we derive closed-form expressions for the steady-state energy distribution, mean delay, blocking probability, and depletion likelihood. Numerical experiments further quantify the impact of battery capacity, harvesting variability, and data arrival rates on system performance. The results provide practical guidelines for cost-optimal battery sizing and reliable storage utilisation in EH-WSNs. Beyond sensor networks, the proposed framework applies broadly to renewable-powered and intermittently operated systems—including off-grid IoT devices and solar- or wind-powered EV charging stations—offering a unified analytical tool for energy storage modeling and design optimisation.
In this paper we develop a probabilistic framework for quantifying the economic impact of epidemics under prevalence-based control policies. The epidemic dynamics are described by a novel continuous-time Markovian SIR model in which the infection rates and the cost structure switch when the number of infectious cases crosses a critical threshold. On top of the stochastic dynamics, we consider a state-dependent cost framework that distinguishes (i) time-in-state sojourn costs, (ii) per-infection costs and (iii) activation/deactivation expenses associated with implementing and lifting control measures. Within this setting, we define the total epidemic cost and the cost of a tagged infectious individual as random variables and derive recursive expressions for their Laplace–Stieltjes transforms (LSTs) and moments. Using matrix differential calculus and Kronecker products, we further obtain recursive formulas for the sensitivities and elasticities of the most representative moments, with respect to the main cost parameters. Numerical experiments illustrate how different cost components drive the mean and standard deviation of total and individual costs across a range of thresholds and epidemic scenarios. The proposed methodology thus combines stochastic epidemic modeling and perturbation analysis to provide distribution-aware, parameter-sensitive assessments of epidemic costs, with direct relevance for statistical evaluation and operations research methods in policy design.
In this paper, we present a sequential decision-making problem with spatial and temporal components. We consider a sequential decision problem involving both temporal and spatial decisions. The temporal component is formulated as a multiple stopping problem, while the spatial component consists of selecting subsets of locations at each intervention time. This leads to a multivariate sequential decision problem combining optimal stopping and spatial action selection. We propose a solution based on backward induction, which is a baseline tool to overcome optimal stopping problems, in which we encapsulate a method called Cross Entropy Method to tackle the combinatorial complexity of the spatial optimization. To study the proposed sequential procedure, numerical simulations on both simulated and real dataset have been conducted. In particular, we illustrate the applicability of the method for the mitigation of the pollution impact on health by considering 5 States in the United States of America and by focusing on the pollutant PM2.5. The results clearly highlight the benefits of such a method for decision makers.
As a key branch of probability theory and mathematical statistics, probability limit theory focuses on analyzing the convergence properties of random variable sequences and their associated distribution functions. Complete f-moment convergence is much general than complete convergence and complete moment convergence. In this paper, we study the complete f-moment convergence for rowwise m_n -extended negatively dependent random variables, which is a new dependence structure. The results on complete f-moment convergence are obtained under some suitable conditions, which generalize the corresponding ones in the literature. As an application, we establish the complete consistency for the G-M estimator of nonparametric regression models. Moreover, a series of simulations are implemented to show the numerical performance of theoretical results based on finite samples.
The stationary analysis of continuous-time Markov chains (CTMCs) on countably infinite state spaces is a cornerstone of applied probability, but the direct computation of the stationary measure π is often intractable. A standard remedy is the augmented truncation method, which approximates π by the stationary measure π̃^(k) of a finite-state process. While convergence of π̃^(k) to π is known in certain cases, the central challenge for practical applications lies in obtaining computable, a priori error bounds. A key contribution is a simple truncation/augmentation construction that renders the truncated states transient, thereby enabling a particularly tractable finite-row perturbation structure. This paper applies the perturbation theory of Markov chains to the problem of augmented truncation to derive explicit error bounds on the V-norm distance ‖π̃^(k)- π‖ _V . Using the deviation matrix D of the original process, we first establish an exact identity linking the approximation error to the perturbation operator Δ^(k) , π̃^(k)- π= π̃^(k)Δ^(k)D . We then decompose this identity into two interpretable and computable components: a “tail term” that depends on the original process dynamics outside the truncated state space, and an “augmentation term” controlled by the redistribution policy of the lost probability mass. We prove that this decomposition holds for any arbitrary augmentation scheme and show that the error bound can be minimized by an optimal choice of this policy, which we explicitly characterize. The resulting bounds are particularly powerful as they do not require strong structural assumptions like stochastic monotonicity. The methodology is illustrated on a non-trivial M/M/1 retrial queue, demonstrating its applicability and the behavior of the computable components of the bound.
This paper analyzes a basic queueing model of service platforms in digital sharing economy, in which seeker arrivals follow a Markovian arrival process and are therefore non-Poisson, and the matching rate between a seeker and an owner depends on the number of the idle owners. The queueing model of service platforms can be expressed as a level-independent quasi-birth-and-death process. We apply the matrix-geometric solution to provide a detailed analysis, including the system stability, the average stationary numbers of seekers and of idle owners, the average sojourn time of an arriving seeker, and the expected revenues for both the service platform and each owner. Specifically, we propose a new effective method for computing the average sojourn time of any seeker by means of the first passage times and the phase-type distributions. Finally, numerical examples verify the theoretical results and illustrate the impact of key parameters, arrival correlation, and full distributional information. We believe that the methodology and results developed in this paper not only can be applied to study more queueing models of service platforms, but also will open a series of promising innovative research topics, such as performance evaluation and queueing-game for digital sharing economy.
This article delves into modeling a retrial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{k}$$\end{document}-out-of-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{(\mathfrak {a}+\mathfrak {b}+\mathfrak {c})}$$\end{document}:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{G}$$\end{document} repairable system which incorporates various standby modes using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{MRSPNs}$$\end{document}. Such a system consists of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{(\mathfrak {a}+\mathfrak {b}+\mathfrak {c})}$$\end{document} units and is considered operational provided that at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{k}$$\end{document} units are functioning. The unit population is composed of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\mathfrak {a}}$$\end{document} active components, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\mathfrak {b}}$$\end{document} warm standby components, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\mathfrak {c}}$$\end{document} cold standby components. The proposed model considers corrective and preventive maintenance. When repairing a failing unit, the repairer may encounter breakdowns (active breakdowns). A failed unit is immediately repaired if the repairer is available; otherwise, if the repairer is repairing a failed unit, out of order or under preventive maintenance, the failed unit enters an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{FCFS}$$\end{document} orbit and waits for repair service after a certain duration. The main stationary probabilities for the system are obtained by adopting the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{MRSPN}$$\end{document} model and its underlying Markov regenerative process, followed by deriving the availability and various performance measures in the steady-state. To get the mathematical expressions that describe how reliably a system operates over time and how long it is expected to function before encountering its first failure, we utilize the Markov renewal equation along with the Laplace transform technique. Through a numerical example, we exhibit how system parameters affect the performance measures and the reliability indices.
This paper studies a single-server priority retrial queue with two classes of incoming calls and one class of outgoing calls. Arrivals follow independent Poisson processes, and class-1 incoming calls have preemptive priority over the other two classes. High-priority calls that find the server busy join a finite buffer, while blocked or preempted low-priority incoming calls enter an orbit; preempted outgoing calls leave the system. A preemptive-resume priority discipline is assumed for orbital calls, and service times are class-dependent with general i.i.d. distributions. Using a regenerative process framework, stability conditions and key performance measures are derived. For the Markovian version of the model, matrix-analytic methods provide explicit performance results, which are shown to be consistent with the regenerative analysis. An important finding is the insensitivity of system performance to the service-time distribution of low-priority incoming calls. Simulations are performed to validate the theoretical results obtained. Numerical results illustrate the effect of buffer capacity under different service distributions, and a cost function is formulated to determine the optimal buffer size in the Markovian case. The effect of service time distributions on the system performance is also studied numerically. The proposed model is motivated by communication scenarios in vehicular ad hoc networks (VANETs), where heterogeneous traffic with different priority levels competes for limited processing resources. However, this study emphasizes probabilistic modeling and analytical methodologies rather than application-specific protocol design. Thus the developed framework and results are applicable to a wide range of stochastic service systems exhibiting prioritized and retrial behavior.
In a market economy with inflationary fluctuations and stochastic rate of return, we study robust Nash equilibrium strategies for two competing asset-liability fund managers. The two fund managers are averse to model uncertainty and prone to money illusion, which leads them to partially overlook inflation risks. The financial market consists of a nominal bond, an inflation-linked index bond, and a stock. The inflation index is characterized using a continuous-time Fisher equation, while the expected rate of return on the stock is modeled by a mean-reverting Ornstein-Uhlenbeck process, which captures the market dynamics of bull and bear phases. Each fund manager aims to develop a robust investment strategy to outperform their competitors by maximizing the expected terminal utility of a composite measure of relative nominal and real surpluses under worst-case scenarios. The robust two-player stochastic differential game is addressed using a backward stochastic differential equation approach, and explicit expressions of robust Nash equilibrium investment strategies, density generator processes under well-defined worst-case misspecification, and the corresponding value functions are derived. The financial implications of money illusion, model ambiguity, and strategic competition are analyzed, and several particular results are recovered. Finally, the economic effects of some model parameters on robust investment policies are demonstrated with numerical examples.
