We consider a cellular network equipped with a retransmission mechanism where the signal transmissions in a single cell depends on interference with simultaneous traffic in surrounding cells. In this framework where failed signals are either retransmitted or lost we study the channel capacity performance of a single-tier model with downlink or uplink traffic. For this purpose, a tractable model that allows for a precise theoretical analysis of coverage probability and coverage rate is developed further. Specifically, we obtain the Shannon capacity in the model and introduce relevant performance measures to guide in the identification of those systems which in a precise sense are able to process all incoming work. To emphasize the generic patterns that arise we extend and simplify the results under a scaling regime of balanced densification, which highlights that performance essentially falls into three categories, for pure-loss, buffered, and no-loss systems.
For some repairable systems, the “initial” failures or defects, if repaired within the specified time period, do not result in the final/functional failures. This can also be interpreted as some time redundancy in various applications. However, with each consecutive repair, this specified time period is usually decreasing, showing a certain deterioration in the quality of repair that can also be interpreted as some imperfectness of repair. On the other hand, the threshold can be fixed, whereas the repair times can be stochastically increasing with each repair. To obtain the corresponding survival probabilities for systems under the described criteria of failures, we employ integral equations of the Volterra type that are solved via the Laplace transforms. The specific cases of exponential distributions of times to defect and repair, and of the geometrically decreasing redundancy time with each repair are considered and analyzed. Provided numerical examples illustrate our findings.
This paper proposes a Palm-based framework for component importance measures conditioned on system failure events, uniformly applicable to nonrepairable, repairable, and phase-mission systems. The formulation does not require renewal, stationary, or Markovian assumptions. We classify measures into transition- and state-based importance. The former captures component failure transitions, recovering Birnbaum and Barlow-Proschan importance, while the latter provides an event-based interpretation of Fussell-Vesely importance. This framework yields well-defined finite-time and phase-conditioned measures for nonstationary systems, establishing a rigorous probabilistic foundation for failure-event-based analysis.
The paper investigates a non-zero-sum differential investment and reinsurance problem between two alpha-robust, risk-averse competitive insurers under a time-consistent mean-variance criterion. The claim arrival processes for both insurers follow the classical Cram'er-Lundberg model, and the reinsurance premium is calculated using the variance premium principle. Each insurer can invest its surplus in one risk-free asset, one risky asset, and a defaultable corporate bond. The paper also considers the effect of bounded memory, which is characterized by the wealth process with delay. Using the game-theoretic approach, we derive the non-zero-sum differential alpha-robust equilibrium strategy and the corresponding value function by solving the extended Hamilton-Jacobi-Bellman equation system. In the numerical simulation section, we observe a phenomenon where the optimal strategy for investing in defaultable bonds decreases as the competitor's risk aversion coefficient increases, provided that one's own risk aversion coefficient remains constant. However, when there is a change in one's own risk aversion coefficient, even if the competitor's risk aversion changes in the opposite direction, the optimal investment strategy in defaultable bonds still decreases as one's own risk aversion coefficient increases.
In this work, we consider two coherent systems with shared components in the case when the component lifetimes are independent and identically distributed with a discrete-time distribution instead of a continuous-time distribution. Then, a discrete-time joint signature is proposed for the two systems by generalizing the traditional joint signature for systems with continuous lifetimes. Some stochastic properties of the proposed joint signature are studied in detail, including joint distribution, stochastic ordering, and transformation formula for comparison of pairs of systems of different sizes. Some illustrative examples are also presented.
We introduce the concept of arbitrable stochastic games, which appears to be new. To do so, we consider a reward criterion different from the standard gamma-weighted criterion. This allows us to define the fair price to play a non-competitive stochastic game. We then illustrate the concept through three variations of the classical coin-toss game with chips, providing proofs via Doob's theorem for supermartingales and practical algorithms. These examples deepen our understanding of the Bitcoin protocol.
The distribution of a random initial age (age composition) of an item is crucial for obtaining its remaining lifetime. A random initial age naturally arises when, for example, an item is drawn at random from a population of continuously manufactured and incepted into operation items. We consider heterogeneous populations of items with lifetime distributions indexed by a frailty parameter. We study different stochastic comparisons for the random age and the remaining (residual) lifetime for items from these populations. The ageing properties for the age composition and remaining lifetime are also discussed. Some examples are provided.
This paper pays attention to the frequency polygon, which is constructed by connecting with straight lines the mid-bin values of a histogram. As a density estimator based on the histogram technique, the frequency polygon has the advantage of computational simplicity and has been widely used in many fields. The purpose of this article is to investigate the weak consistency, the uniformly weak consistency, and the rate of the uniformly weak consistency for frequency polygon estimation of the density function under $\alpha$-mixing samples, which improve and extend the corresponding ones in the literature. In addition, the simulation study and real data analysis are also presented to verify the validity of the theoretical results based on finite samples.
In this paper, when the errors in the semi-parametric errors-in-variables model are asymptotic negatively associated (or rho-, for short) random variables, the estimators of parameter, non-parameter, and error variances in the model are $\widehat{\beta}_{n}$, $\widehat{g}_{n}(t)$, and $ \widehat{\sigma}_{n}<^>{2}$, respectively, by using wavelet smoothing and least square method. Under some general assumptions, we also establish some results on the strong consistency of the estimators. Furthermore, simulations are conducted to assess the finite sample behavior of the estimators and confirm the validity of the theoretical results.
Fluid queues governed by birth-death processes have been used to analyze buffer dynamics and stability behavior of the fluid flow systems. However, most existing studies primarily focus on classical single-ended queues, often ignore double-ended queue flow dynamics, or rely heavily on simulation-based approaches. Specifically, the study of the fluid flow systems modulated by double-ended queues subject to the catastrophic failure and subsequent repair processes is challenging and interesting and has not yet received attention in the literature. Even when such systems are considered, explicit closed-form analytic expressions for equilibrium buffer content distributions and related performance measures are rarely available. To overcome these limitations, this article investigates a fluid flow system regulated by a double-ended queue and exposed to catastrophic failures with subsequent repair processes. Such a driven queue can be equivalently represented as a one-dimensional bilateral birth-death process, namely a continuous-time randomized random walk on the integers with catastrophic failures and repairs. The stability condition for the fluid occupancy in the credit buffer is rigorously established, and explicit closed-form analytical expressions for both the probability density function and the cumulative distribution function of the buffer content in the equilibrium regime are determined. These analytic results provide deeper insight into the steady-state behavior of the system and enable the derivation of several vital performance measures of practical interest. Furthermore, graphical illustrations are presented to highlight the influence of the system parameters on the performance descriptors of the fluid content, thereby enhancing the interpretability and applicability of the proposed fluid queueing system.
Failure extropy, introduced by Nair and Sathar Nair [(2020). On dynamic failure extropy. J. Indian Soc. Probab. Stat. 21: 287--313], provides a complementary perspective to entropy for quantifying uncertainty in lifetime distributions. However, it becomes mathematically invalid for distributions with unbounded support. To overcome this limitation, Tahmasebi and Toomaj [(2022). On negative cumulative extropy with applications. Commun. Stat. Theory Methods 51(15): 5025--5047] proposed the concept of negative cumulative extropy (NCEx), offering a bounded and interpretable alternative. In this paper, we extend the notion of NCEx to the bivariate dynamic setting, where uncertainty is assessed for systems whose components have failed at specified times. The proposed formulation effectively captures the uncertainty associated with past lifetimes under dependence, which the existing NCEx cannot address. The measure is further generalized to a vector-valued form, and its fundamental properties are established, including monotonicity, invariance, bounds expressed in terms of the expected inactivity time, and key characterizations. A new stochastic ordering based on the proposed measure is also established. To facilitate practical implementation, a nonparametric estimator is developed and its performance evaluated through extensive Monte Carlo simulations. The practical relevance of the proposed measure is demonstrated using a real dataset, and its superiority over existing entropy-based approaches is shown on an additional dataset.
This paper investigates the complexity of residual lifetimes of live components in coherent systems through the lens of cumulative residual extropy and its divergence-based extension, Jensen-cumulative residual extropy. Unlike classical reliability metrics that focus on system inactivity or mean residual life, our framework quantifies the hidden informational structure of components that remain alive at the system failure time. We derive closed-form expressions for the cumulative residual extropy of conditional residual lifetimes using system signatures and establish stochastic bounds and comparisons that highlight the impact of structural configuration. A novel divergence measure, the Jensen-cumulative residual extropy, is introduced to capture discrepancies between coherent systems and benchmark $k$-out-of- $n$ structures. Numerical illustrations with gamma-distributed lifetimes demonstrate the sensitivity of cumulative residual extropy and Jensen-cumulative residual extropy to redundancy patterns and dependence structures. Furthermore, by integrating cost considerations into the divergence framework, we provide a rigorous optimization scheme for selecting system signatures that jointly minimize informational complexity and economic expenditure. The proposed approach enriches the theoretical foundation of reliability analysis and offers practical guidelines for designing resilient, cost-effective, and information-efficient engineering systems.
The q-Weibull distribution, as a generalization of the Weibull distribution, plays an important role in the field of reliability theory, survival analysis, finance, engineering, medical science, etc. In contrast to the Weibull distribution, which is limited to describing monotonic hazard rate functions, the q-Weibull distribution offers the flexibility to model various behaviors of the hazard rate function, including unimodal, bathtub-shaped, monotonic (both increasing and decreasing), and constant. In this article, we investigated the stochastic comparison of extreme order statistics derived from independent, heterogeneous q-Weibull random variables using various stochastic orderings, including the usual stochastic order, hazard rate order, reversed hazard rate order, and likelihood ratio order. Some of these results are further extended to dependent setups by incorporating Archimedean copulas to model the dependence structure. Finally, we explored the behavior of extreme order statistics when the random variables are subjected to random shocks.
Extropy-based divergence measures offer distinct advantages over entropy-based counterparts, owing to their mathematical simplicity and enhanced interpretability. Relative extropy by Lad et al. [5] is a symmetric divergence measure between two probability distributions, and Mohammadi et al. [8] introduced the asymmetric divergence between two distributions based on extropy. We further study these measures, their properties, and interrelationships in this article. To address the divergence between truncated lifetime distributions, we define dynamic relative extropy for residual and past lifetime scenarios. Exploring the interrelationships of dynamic cases of relative extropy, extropy divergence, and extropy inaccuracy, we derive some unique properties and characterizations for the exponential distribution. A nonparametric estimator for relative extropy is developed, and its performance is assessed through numerical simulation studies. The practical applicability of relative extropy is used to analyze the divergence in lifetime patterns of mice under a lifetime feeding experiment and the shopping patterns of customers based on age and income groups. Further, the application of relative extropy is also applied to find the dissimilarity between two images.
Consecutive $k$-type systems have become important in both reliability theory and applications; in spite of a large literature existing on them, three-dimensional consecutive $k$-type systems have not yet been studied for multi-state case. In this paper, we introduce several different types of multi-state linear three-dimensional consecutive $k$-type systems for the first time, with due consideration to possible overlapping of failure blocks. The finite Markov chain imbedding approach is then used for the derivation of their reliability functions with state spaces and transition matrices provided in a novel way, and the involved computational process is illustrated through several numerical examples. Finally, some possible applications of the work and potential extensions are pointed out.
The generalized Gompertz distribution-an extension of the standard Gompertz distribution as well as the exponential distribution and the generalized exponential distribution-offers more flexibility in modeling survival or failure times as it introduces an additional parameter, which can account for different shapes of hazard functions. This enhances its applicability in various fields such as actuarial science, reliability engineering and survival analysis, where more complex survival models are needed to accurately capture the underlying processes. The effect of heterogeneity has generated increased interest in recent times. In this article, multivariate chain majorization methods are exploited to develop stochastic ordering results for extreme-order statistics arising from independent heterogeneous generalized Gompertz random variables with increased degree of heterogeneity.
Lorenz dominance is a classical criterion for comparing income distributions with respect to inequality and social welfare. However, its binary nature, in which one distribution either dominates another or does not, often leads to inconclusive results when empirical Lorenz curves intersect. To overcome this limitation, we introduce the Lorenz dominance index (LDI), a continuous measure that quantifies the extent to which one Lorenz curve lies above another. The LDI provides an interpretable assessment based on the population, allowing for the evaluation of partial or near dominance and improving its usefulness in empirical settings. We derive the asymptotic distribution of the LDI and propose a nonparametric bootstrap procedure to construct confidence intervals and perform inference. Monte Carlo simulations confirm the estimator's strong performance in finite samples and its nominal coverage. An application to household income data from China highlights the practical value of the LDI in distributional analysis.
In this work, by considering coherent systems comprising independent components with discrete lifetimes, we introduce the notion of discrete-time signature and then discuss some of its properties. With the use of the introduced signature, a stochastic ordering result is also established. We then introduce transformation formulas for the discrete-time signature to facilitate the comparison of systems of different sizes. Some examples are also presented to illustrate all the results developed here.
In this paper, we study the joint distribution of the forward and backward recurrence times in a delayed renewal process, as well as their marginal distributions. We obtain several exact results and bounds for these quantities. Some of these bounds are "general," in the sense that the bounds are valid for any arbitrary distributions of the inter-arrival times, and some are based on aging properties of the distributions of the interarrival times of the renewals. Finally, several numerical examples are presented to illustrate the results.
System components usually attain marginal lifetimes with stochastic dependence in the context of load-sharing reliability structures. This study deals with the load-sharing parallel systems of two components. We prove that two marginal lifetimes are positively quadrant dependent when component lifetimes have continuous probability distributions, and such a stochastic dependence is upgraded to the total positive of order 2 in the setting of component lifetimes having an exponential distribution. In addition, we discuss how these findings shed light on related results for the load-sharing Ross model, the conditional residual lifetime, and the conditional inactivity time.
