In this work, we consider used coherent systems in the case when the component lifetimes are independent and identically distributed (i.i.d.) with a discrete-time distribution instead of a continuous-time distribution. Then, discrete-time dynamic signature and discrete-time residual signature are proposed for single coherent systems, and discrete-time dynamic joint signature and discrete-time residual joint signature are proposed for two coherent systems with shared components. Associated properties of these proposed discrete-time dynamic signatures are discussed in detail. Some illustrative examples are also presented.

We study the problem of scheduling packets on communication channels whose service availability is modeled by independent Gilbert–Elliot (ON,OFF) Markov chains. Classical models assume that channel states are always known to the scheduler and that transmission is perfect for ON channels. We consider the often more realistic problem of non-preemptive scheduling on imperfect and unobservable channels. In particular, in our model, ON channels have a nonzero probability of unsuccessful transmission, the state of the channel is only observed upon successful transmission, and once a channel is activated, it must remain active until successful transmission. We give conditions under which simple policies are optimal. We also investigate the performance impact of not being able to preempt and of not having access to channel information. Combining known and novel results, we characterize the “Preemption Gain” and “Observability Gain.” Numerically, we show that these gains can be significant, highlighting the critical value of channel state information and preemption, and providing insight into the possible benefit of redesigning systems to permit preemption and/or to provide state information in communication channels.

Let $\{X_n\}_{n \geq 1}$ be a Markov chain taking values in $\{1, -1\}$ with transition probabilities

\begin{align*} p = P(X_{n+1} = 1 \,\,|\,\, X_n = 1)\ ,\ q = P(X_{n+1} = -1 \,\,|\,\, X_n = -1). \end{align*}

Define a correlated random walk $(S_n)_{n\ge0} ,S_0=0$ and $S_n=\sum_{i=1}^n X_i,n\ge1$. Let $A$ and $B$ be positive integers, and define a stopping time

\begin{equation*} \tau = \inf \{ n \geq 1 : S_n = A \text{ or } S_n = -B \}. \end{equation*}

Using the martingale approach, Pozdnyakov [4] (Martingale approach to gambler’s ruin problem for correlated random walks. Journal of Applied Probability 62(4): 1594–1606.) derived the expected stopping time $E(\tau)$.

This study investigates the monotonicity of $E(\tau)$ with respect to the parameters $p$ and $q$ under the condition of symmetric boundaries $A = B$. We prove that when $p + q = t$ is fixed, $E(\tau)$ increases as $p$ increases from $0$ to $\frac{t}{2}$, which implies that $E(\tau)$ attains its maximum at $p = q = \frac{t}{2}$.

In this paper, we develop a quantile-based framework for weighted extropy and residual weighted extropy for uncertainty quantification in reliability and lifetime analysis. The proposed measures are formulated in terms of the quantile density and hazard quantile functions, providing a flexible alternative to conventional density-based formulations. Explicit expressions are derived for several important lifetime distributions, and fundamental properties, including monotonicity, stochastic ordering, characterization results, and extensions to order statistics, are established. In particular, the proposed weighted measures provide additional discriminatory ability and can distinguish distributions having the same classical extropy. Nonparametric estimators based on order-statistic spacings are also developed, and their finite-sample performance is investigated through simulation studies. Applications to real lifetime datasets further illustrate the practical utility of the proposed methodology. The results demonstrate that quantile-based weighted extropy provides a flexible and useful tool for uncertainty quantification, with potential applications in reliability, survival analysis, and risk assessment.

Reliability assessment under scheduled inspections is fundamentally an inference problem, because remaining lifetime must be evaluated conditionally on partial evidence revealed at inspection times. This paper develops a signature-based information-theoretic framework for coherent systems observed under two periodic inspections and studies the conditional residual lifetime beyond the second inspection. Uncertainty is quantified using cumulative residual extropy, yielding explicit representations and sharp comparison results that link inspection timing and system architecture to uncertainty reduction. To capture structural heterogeneity beyond uncertainty magnitude, we introduce divergence measures built from cumulative residual extropy that quantify complexity induced by coherent design under a common inspection policy. Stochastic-ordering results provide rigorous ranking tools for both inspection-conditioned lifetimes and the induced mixing structure. For statistical inference, we propose nonparametric plug-in and kernel-smoothed estimators, prove their large-sample behavior, and assess finite-sample accuracy by Monte Carlo experiments. An application to medical image intensities shows that the same two-inspection protocol produces distinct uncertainty and complexity profiles that discriminate between clinically relevant photometric transformations.

This paper develops a reliability framework for series–parallel systems subject to run-dependent Marshall–Olkin (MO) shocks, in which each parallel block may contain an arbitrary number of components sharing a block-common shock source. Within a parallel block, the component lifetimes are dependent, since they share the block-common run-shock time. This dependence prevents a direct application of the phase-type maximum closure. We resolve this obstacle through a structural decomposition lemma that expresses each block lifetime in terms of mutually independent random variables. This lemma allows exact phase-type representations at all system levels through nested Kronecker product constructions. From these representations, we obtain survival functions, mean time to failure, variance, hazard rate, and mean residual life in closed matrix-exponential form. We also show that the conditional mean residual life of any component, given the survival of all other components in its block, depends on their observation times only through their maximum. This is a structural invariance property of the MO parallel-block model. Closed-form dimension formulas describe how the state-space size grows with the system size, the phase dimensions, and the run-length threshold $k$. The methodology is illustrated on a three-block series–parallel system with Erlang-2 interarrival times.

We consider a cellular network equipped with a retransmission mechanism, where signal transmissions in a single cell depend 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 traffic. For this purpose, a tractable model that allows for a precise theoretical analysis of coverage probability and coverage rate is developed further in multiple new directions. Specifically, we obtain the Shannon capacity in the model and introduce relevant performance measures to guide the identification of those systems that, 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 balanced densification scaling regime, which highlights that performance essentially falls into three categories: 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.

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.

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.

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.

In this paper, when the errors in the semi-parametric errors-in-variables model are asymptotic negatively associated (or ρ−, 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.

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.

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.

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.

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.

We study the multiserver-job setting in the load-focused multilevel scaling limit, where system load approaches capacity much faster than the growth of the number of servers $n$. We consider the “1 and $n$” system, where each job requires either one server or all $n$. Within the multilevel scaling limit, we examine three regimes: load dominated by $n$-server jobs, 1-server jobs, or balanced. In each regime, we characterize the asymptotic growth rate of the boundary of the stability region and the scaled mean queue length. We demonstrate that mean queue length peaks near balanced load via theory, numerics, and simulation.

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.