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Studying various models in queueing theory is essential for improving the efficiency of queueing systems. In this paper, from the family of models {E_r/M/c; r,c in N}, the E_r/M/3 model is introduced, and quantities such as the distribution of the number of customers in the system, the average number of customers in the queue and in the system, and the average waiting time in the queue and in the system for a single customer are obtained. Given the crucial role of the traffic intensity parameter in performance evaluation criteria of queueing systems, this parameter is estimated using Bayesian, E‑Bayesian, and hierarchical Bayesian methods under the general entropy loss function and based on the system’s stopping time. Furthermore, based on the E‑Bayesian estimator, a new estimator for the traffic intensity parameter is proposed, referred to in this paper as the E^2‑Bayesian estimator. Accordingly, among the Bayesian, E‑Bayesian, hierarchical Bayesian, and the new estimator, the one that minimizes the average waiting time in the customer queue is considered the optimal estimator for the traffic intensity parameter in this paper. Finally, through Monte Carlo simulation and using a real dataset, the superiority of the proposed estimator over the other mentioned estimators is demonstrated.

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In this paper, a first-order integer-valued autoregressive process with non-negative integer values is introduced, based on the binomial thinning operator and driven by Poisson-Komal distributed noise. To estimate the parameters of the proposed model, two estimation methods are investigated: Conditional Maximum Likelihood Estimation and the Yule–Walker Method. Furthermore, the performance of these estimation techniques is evaluated through a simulation study. In addition, the practical applicability of the proposed model is demonstrated using two real-world datasets from the field of veterinary sciences.

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This study investigates the wavelet energy distribution in high-frequency fractal systems and analyzes its characteristics using information-theoretic measures. The main innovation of this paper lies in modeling the wavelet energy distribution ($p_j$) using a truncated geometric distribution and incorporating the concept of extropy to quantify system complexity. It is demonstrated that this distribution is strongly influenced by the fractal parameter $alpha$ and the number of decomposition levels $M$. By computing wavelet entropy and extropy as measures of disorder and information, respectively—the study provides a quantitative analysis of the complexity of these systems. The paper further examines key properties of this distribution, including its convergence to geometric, uniform, and degenerate distributions under limiting conditions (e.g., $M to infty$ or $alpha to 0$). Results indicate that entropy and extropy serve as complementary tools for a comprehensive description of system behavior: while entropy measures disorder, extropy reflects the degree of information and certainty. This approach establishes a novel framework for analyzing real-world signals with varying parameters and holds potential applications in the analysis of fractal signals and modeling of complex systems in fields such as finance and biology.

To validate the theoretical findings, synthetic fractal signals (fractional Brownian motion) with varying fractal parameters ($alpha$) and decomposition levels ($M$) were simulated. Numerical results show that wavelet entropy increases significantly with the number of decomposition levels ($M$), whereas extropy exhibits slower growth and saturates at higher decomposition levels. These findings underscore the importance of selecting an appropriate decomposition level. The proposed combined framework offers a powerful tool for analyzing and modeling complex, non-stationary systems in domains such as finance and biology.

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In this paper, a novel index entitled the Jensen cumulative residual extropy divergence is investigated for the analysis and measurement of the behavioral complexity of conditional mixed systems. First, using the vector of conditional coefficients obtained from the signature vector, the behavior of this measure is analytically examined for a class of coherent systems as well as their dual systems, in the case where the components follow gamma distributions. Then, simulations are performed to evaluate the obtained results. The results of this paper show that the minimum complexity is achieved by coherent $k$-out-of-$n$ systems with the Jensen cumulative residual extropy divergence equal to zero. Moreover, the results indicate that duality of systems does not necessarily lead to equality of the Jensen cumulative residual extropy divergence in conditional mixed systems; rather, this index is sensitive to component weighting, order statistics, and the structural interaction among the components of the system.

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This paper presents a nonparametric Bayesian method for estimating nonstationary covariance structures in big spatial datasets. The approach extends the Vecchia approximation and assumes conditional independence among ordered data points, leading to a sparse precision matrix and sparse Cholesky decomposition. This enables modeling an $n$-dimensional Gaussian process as a sequence of Bayesian linear regressions. Data ordering via maximum minimum distance improves model performance. Applying the grouping algorithm to ordered data removes weak dependencies and defines a block-sparse covariance structure, significantly reducing computational burden and enhancing accuracy. Simulations and real data analysis show that posterior samples from the proposed method yield narrower uncertainty intervals than those from ungrouped approaches.

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In this paper, we present a comprehensive review and comparative analysis of estimation methods for periodic autoregressive (PAR) models driven by scale mixture of skew-normal (SMSN) innovations, a flexible class suitable for modeling both symmetric and asymmetric data. Expectation-conditional maximization algorithms are employed to develop maximum likelihood, maximum a posteriori, and Bayesian estimation procedures. A thorough evaluation of these methods is conducted using simulation studies, with particular attention to asymptotic properties and robustness against outliers, high peaks, and heavy tails. To demonstrate their practical utility, these methods are applied to monthly Google stock price data.