Conditional risk measures and their associated risk contribution measures are commonly employed in finance and actuarial science for evaluating systemic risk and quantifying the effects of risk interactions. This paper introduces various types of contribution ratio measures based on the multivariate conditional value-at-risk (MCoVaR), multivariate conditional expected shortfall (MCoES), and multivariate marginal mean excess (MMME) studied in [34] (Ortega-Jiménez, P., Sordo, M., & Suárez-Llorens, A. (2021). Stochastic orders and multivariate measures of risk contagion. Insurance: Mathematics and Economics, vol. 96, 199–207) and [11] (Das, B., & Fasen-Hartmann, V. (2018). Risk contagion under regular variation and asymptotic tail independence. Journal of Multivariate Analysis 165(1), 194–215) to assess the relative effects of a single risk when other risks in a group are in distress. The properties of these contribution risk measures are examined, and sufficient conditions for comparing these measures between two sets of random vectors are established using univariate and multivariate stochastic orders and statistically dependent notions. Numerical examples are presented to validate these conditions. Finally, a real dataset from the cryptocurrency market is used to analyze the spillover effects through our proposed contribution measures.

Decentralized consensus protocols have a variety of parameters to be set during their deployment for practical applications in blockchains. The analysis given in most research papers proves the security state of the blockchain, at the same time usually providing a range of acceptable values, thus allowing further tuning of the protocol parameters. In this paper, we investigate Ouroboros Praos, the proof-of-stake consensus protocol deployed in Cardano and other blockchains. In contrast to its predecessor, Praos allows multiple honest slot leaders that lead to fork creation and resolution, consequently decreasing the block rate per time unit. In our analysis of dependence on protocol parameters such as active slot coefficient and p2p network block propagation time, we obtain new theoretical results and explicit formulas for the expectation of the length of the longest chain created during the Praos epoch, the length of the longest unintentional fork created by honest slot leaders, the efficiency of block generation procedure (the ratio of blocks included in the final longest chain vs the total number of created blocks), and other characteristics of the blockchain throughput.

We study these parameters as stochastic characteristics of the block generation process. The model is described in terms of the two-parametric family ξij of independent Bernoulli random variables which generate deformation of the binomial distribution by a positive integer parameter—the delay (deterministic or random). An essential part of our paper is a study of this deformation in terms of denumerable Markov chains and generating functions.

We consider the moments and the distribution of hitting times on the lollipop graph which is the graph exhibiting the maximum expected hitting time among all the graphs having the same number of nodes. We obtain recurrence relations for the moments of all order and we use these relations to analyze the asymptotic behavior of the hitting time distribution when the number of nodes tends to infinity.

The cumulative residual extropy has been proposed recently as an alternative measure of extropy to the cumulative distribution function of a random variable. In this paper, the concept of cumulative residual extropy has been extended to cumulative residual extropy inaccuracy (CREI) and dynamic cumulative residual extropy inaccuracy (DCREI). Some lower and upper bounds for these measures are provided. A characterization problem for the DCREI measure under the proportional hazard rate model is studied. Nonparametric estimators for CREI and DCREI measures based on kernel and empirical methods are suggested. Also, a simulation study is presented to evaluate the performance of the suggested measures. Simulation results show that the kernel-based estimator performs better than the empirical-based estimator. Finally, applications of the DCREI measure for model selection are provided using two real data sets.

We propose Rényi information generating function (RIGF) and discuss its properties. A connection between the RIGF and the diversity index is proposed for discrete-type random variables. The relation between the RIGF and Shannon entropy of order q > 0 is established and several bounds are obtained. The RIGF of escort distribution is derived. Furthermore, we introduce the Rényi divergence information generating function (RDIGF) and discuss its effect under monotone transformations. We present nonparametric and parametric estimators of the RIGF. A simulation study is carried out and a real data relating to the failure times of electronic components is analyzed. A comparison study between the nonparametric and parametric estimators is made in terms of the standard deviation, absolute bias, and mean square error. We have observed superior performance for the newly proposed estimators. Some applications of the proposed RIGF and RDIGF are provided. For three coherent systems, we calculate the values of the RIGF and other well-established uncertainty measures, and similar behavior of the RIGF is observed. Further, a study regarding the usefulness of the RDIGF and RIGF as model selection criteria is conducted. Finally, three chaotic maps are considered and then used to establish a validation of the proposed information generating function.

This work studies the reliability function of K-out-of-N systems with a general repair time distribution and a single repair facility. It introduces a new repair mechanism using an effort function, described by a nonlinear ordinary differential equation. Three theoretical results are obtained: regularity properties preventing simultaneous failures and repairs, derivation of a Kolmogorov forward system for micro-state and macro-state probabilities, and comparison of reliability functions of two K-out-of-N systems. An additional hypothesis on the model’s parameters allows us to obtain an ordering relation between the reliability functions. A numerical example demonstrates the model’s practical application and confirms the theoretical results.

In this paper, we investigate the number of customers that overlap or coincide with a virtual customer in an Erlang-A queue. Our analysis starts with the fluid and diffusion limit differential equations to obtain the mean and variance of the queue length. We then develop precise approximations for waiting times using fluid limits and the polygamma function. Building on this, we introduce a novel approximation scheme to calculate the mean and variance of the number of overlapping customers. This method facilitates the assessment of transient overlap risks in complex service systems, offering a useful tool for service providers to mitigate significant overlaps during pandemic seasons.

We consider the count of subgraphs with an arbitrary configuration of endpoints in the random-connection model based on a Poisson point process on {\mathord{\mathbb R}}^d${\mathord{\mathbb R}}^d$. We present combinatorial expressions for the computation of the cumulants and moments of all orders of such subgraph counts, which allow us to estimate the growth of cumulants as the intensity of the underlying Poisson point process goes to infinity. As a consequence, we obtain a central limit theorem with explicit convergence rates under the Kolmogorov distance and connectivity bounds. Numerical examples are presented using a computer code in SageMath for the closed-form computation of cumulants of any order, for any type of connected subgraph, and for any configuration of endpoints in any dimension d{\geq} 1$d{\geq} 1$. In particular, graph connectivity estimates, Gram–Charlier expansions for density estimation, and correlation estimates for joint subgraph counting are obtained.

We consider a nonnegative random variable T representing the lifetime of a system. We discuss the residual lifetime T_X=(T-X|T \gt X)$T_X=(T-X|T \gt X)$, where X denotes the random age of the system. We also discuss the mean residual life (MRL) of T at the random time X. It is shown that the MRL at random age (MRLR) is closely related to some well-known variability measures. In particular, we show that the MRLR can be considered a generalization of Gini’s mean difference (GMD). Under the proportional hazards model, we show that the MRLR gives the extended GMD and the extended cumulative residual entropy as special cases. Then, we provide a decomposition result indicating that the MRLR has a covariance representation. Some comparison results are also established for the MRLs of two systems at random ages.

In the reliability analysis of multicomponent stress-strength models, it is typically assumed that strengths are either independent or dependent on a common stress factor. However, this assumption may not hold true in certain scenarios. Therefore, accurately estimating the reliability of the stress-strength model becomes a significant concern when strengths exhibit interdependence with both each other and the common stress factor. To address this issue, we propose an Archimedean copula (AC)-based hierarchical dependence approach to effectively model these interdependencies. We employ four distinct semi-parametric methods to comprehensively estimate the reliability of the multicomponent stress-strength model and determine associated dependence parameters. Furthermore, we derive asymptotic properties of our estimator and demonstrate its effectiveness through both Monte Carlo simulations and real-life datasets. The main original contribution of this study is the first attempt to evaluate the reliability problem under dependent strengths and stress using a hierarchical AC approach.

Reliability analysis of stress–strength models usually assumes that the stress and strength variables are independent. However, in numerous real-world scenarios, stress and strength variables exhibit dependence. This paper investigates the reliability estimation in a multicomponent stress–strength model for parallel-series system assuming that the dependence between stress and strength is based on the Clayton copula. The estimators for the unknown parameters and system reliability are derived using the two-step maximum likelihood estimation and the maximum product spacing methods. Additionally, confidence intervals are constructed by utilizing asymptotically normal distribution theory and bootstrap method. Furthermore, Monte Carlo simulations are conducted to compare the effectiveness of the proposed inference methods. Finally, a real dataset is analyzed for illustrative purposes.

In this paper, the model of bisexual branching processes affected by viral infectivity and with random control functions in independent and identically distributed (i.i.d.) random environments is established and the Markov property is given firstly. Then the relations of the probability generating functions of this model are studied, and some sufficient conditions for process extinction under common mating functions are presented. Finally, the limiting behaviors of the considered model after proper normalization, such as the sufficient conditions for the convergence in L1 and L2 and almost everywhere convergence, are investigated under the condition that the random control functions are super additive.

This paper proposes an options pricing model that incorporates stochastic volatility, stochastic interest rates, and stochastic jump intensity. Market shocks are modeled using a jump process, with each jump governed by an asymmetric double-exponential distribution. The model also integrates a Markov regime-switching framework for volatility and the risk-free rate, allowing the market to alternate between a finite number of distinct economic states. A closed-form solution for European option pricing is derived. To demonstrate the significance of the proposed model, a comparison with various other models is performed, and the sensitivity of the various model parameters is illustrated.

In this paper, we analyze a polling system on a circle. Random batches of customers arrive at a circle, where each customer, independently, obtains a location that is uniformly distributed on the circle. A single server cyclically traverses the circle to serve all customers. Using mean value analysis, we derive the expected number of waiting customers within a given distance of the server. We exploit this to obtain closed-form expressions for both the mean batch sojourn time and the mean time to delivery.

In this work, we consider extensions of the dual risk model with proportional gains by introducing dependence structures among gain sizes and gain interarrival times. Among others, we further consider the case where the proportionality parameter is randomly chosen, the case where it is a uniformly random variable, as well as the case where we may have upward as well as downward jumps. Moreover, we consider the case with causal dependence structure, as well as the case where the dependence is based on the generalized Farlie–Gumbel–Morgenstern copula. The ruin probability and the distribution of the time to ruin are investigated.

We consider the task completion time of a repairable server system in which a server experiences randomly occurring service interruptions during which the server works slowly. Every service-state change preempts the task that is being processed. The server may then resume the interrupted task, it may replace the task with a different one, or it may restart the same task from the beginning, under the new service-state. The total time that the server takes to complete a task of random size including interruptions is called completion time. We study the completion time of a task under the last two cases as a function of the task size distribution, the service interruption frequency/severity, and the repair frequency. We derive closed form expressions for the completion time distribution in Laplace domain under replace and restart recovery disciplines and present their asymptotic behavior. In general, the heavy tailed behavior of completion times arises due to the heavy tailedness of the task time. However, in the preempt-restart service discipline, even in the case that the server still serves during interruptions albeit at a slower rate, completion times may demonstrate power tail behavior for exponential tail task time distributions. Furthermore, we present an M/G/\infty$M/G/\infty$ queue with exponential service time and Markovian service interruptions. Our results reveal that the stationary first order moments, that is, expected system time and expected number in the system are insensitive to the way the service modulation affects the servers; system-wide modulation affecting every server simultaneously vs identical modulation affecting each server independently.

For coherent systems with components and active redundancies having heterogeneous and dependent lifetimes, we prove that the lifetime of system with redundancy at component level is stochastically larger than that with redundancy at system level. In particular, in the setting of homogeneous components and redundancy lifetimes linked by an Archimedean survival copula, we develop sufficient conditions for the reversed hazard rate order, the hazard rate order and the likelihood ratio order between two system lifetimes, respectively. The present results substantially generalize some related results in the literature. Several numerical examples are presented to illustrate the findings as well.

In this paper, we use an information theoretic approach called cumulative residual extropy (CRJ) to compare mixed used systems. We establish mixture representations for the CRJ of mixed used systems and then explore the measure and comparison results among these systems. We compare the mixed used systems based on stochastic orders and stochastically ordered conditional coefficients vectors. Additionally, we derive bounds for the CRJ of mixed used systems with independent and identically distributed components. We also propose the Jensen-cumulative residual extropy (JCRJ) divergence to calculate the complexity of systems. To demonstrate the utility of these results, we calculate and compare the CRJ and JCRJ divergence of mixed used systems in the Exponential model. Furthermore, we determine the optimal system configuration based on signature under a criterion function derived from JCRJ in the exponential model.

This paper investigates the precise large deviations of the net loss process in a two-dimensional risk model with consistently varying tails and dependence structures, and gives some asymptotic formulas which hold uniformly for all x varying in t-intervals. The study is among the initial efforts to analyze potential risk via large deviation results for the net loss process of the two-dimensional risk model, and can provide a novel insight to assess the operation risk in a long run by fully considering the premium income factors of the insurance company.

In this paper, a new multivariate counting process model (called Multivariate Poisson Generalized Gamma Process) is developed and its main properties are studied. Some basic stochastic properties of the number of events in the new multivariate counting process are initially derived. It is shown that this new multivariate counting process model includes the multivariate generalized Pólya process as a special case. The dependence structure of the multivariate counting process model is discussed. Some results on multivariate stochastic comparisons are also obtained.