Search Results (65)

In this paper, we introduce two analytic deformations of probability measures that unify and extend two classical deformations from free probability theory, namely the $\mathbb{T}=(s,t)$-deformation $U^{\mathbb{T}}$ and the $T_a$-deformation, where $a,t\in \mathbb{R}$ and $s>0$. The corresponding operators, denoted by ${\mathbb{Y}}_{(a,s,t)}^{\prime }$ and ${\mathbb{Y}}_{(a,s,t)}^{\prime \prime }$, are defined via a functional equation involving the Cauchy–Stieltjes transform (CST). This framework recovers the classical cases as particular instances, specifically ${\mathbb{Y}}_{(0,s,t)}^{\prime }={\mathbb{Y}}_{(0,s,t)}^{\prime \prime }=U^{\mathbb{T}}$ and ${\mathbb{Y}}_{(a,1,1)}^{\prime }={\mathbb{Y}}_{(a,1,1)}^{\prime \prime }=T_a$. We analyze the analytic and structural properties of the operators ${\mathbb{Y}}_{(a,s,t)}^{\prime }$ and ${\mathbb{Y}}_{(a,s,t)}^{\prime \prime }$ within the concept of Cauchy–Stieltjes kernel (CSK) families, with particular emphasis on their action on variance functions (VFs). In particular, we derive explicit formulas for the VFs associated with measures deformed by ${\mathbb{Y}}_{(a,s,t)}^{\prime }$ and ${\mathbb{Y}}_{(a,s,t)}^{\prime \prime }$. As an application, we establish an invariance property showing that the class of free Meixner family (FMF) is stable under both deformations. Furthermore, by restricting the parameters to ${\mathbb{Y}}_{(a,1,t)}^{\prime }$ and ${\mathbb{Y}}_{(a,1,t)}^{\prime \prime }$, we obtain two new characterizations of the semicircle law. These results highlight the role of symmetry in the analytic deformation and in the stability properties of fundamental distributions in free probability. Full article

In this paper, a cold standby repairable system comprising two heterogeneous components, each characterized by multiple types of mutually independent failure modes, is investigated. The operational lifetimes of the components follow exponential distributions, while their repair times after failure are governed by general distributions. By applying the theory of the Markov renewal process together with the Laplace and the Laplace–Stieltjes transform techniques, we derive analytical expressions for the time to the first system failure, system availability, and the rate of occurrence of system failures. Some results for these reliability measures under several special cases are also presented. Finally, numerical examples are provided under different repair time distributions to analyze the influence of model parameters on the system’s reliability performance. Full article

The resummation of Stieltjes series remains a key challenge in mathematical physics, especially when Padé approximants fail, as in the case of superfactorially divergent series. Weniger’s $\delta$-transformation, which incorporates a priori structural information on Stieltjes series, offers a superior framework with respect to Padé. In the present work, the following fundamental question is addressed: Is the $\delta$-transformation, once it is applied to a typical Stieltjes series, capable of correctly simulating the branch cut structure of the corresponding Stieltjes function? Here, it is proved that the intrinsic log-convexity of the Stieltjes moment sequence (guaranteed via the positivity of Hankel’s determinants) allows the necessary condition for $\delta$ to have all real poles to be satisfied. The same condition, however, is not sufficient to guarantee this. In attempting to bridge such a gap, we propose a mechanism rooted in the iterative action of a specific linear differential operator acting on a class of suitable auxiliary log-concave polynomials. To this end, we show that the denominator of the $\delta$-approximants can always be recast as a high-order derivative of a log-concave polynomial. Then, on invoking the Gauss–Lucas theorem, a consistent geometrical justification of the $\delta$ pole positioning is proposed. Through such an approach, the pole alignment along the negative real axis can be viewed as the result of the progressive restriction of the convex hull under differentiation. Since a fully rigorous proof of this conjecture remains an open challenge, in order to substantiate it, a comprehensive numerical investigation across an extensive catalog of Stieltjes series is proposed. Our results provide systematic evidence of the potential $\delta$-transformation ability to mimic the singularity structure of several target functions, including those involving superfactorial divergences. Full article

The Peak Age of Information (PAoI) quantifies the freshness of updates used in cyber-physical systems (CPSs), realized within the Internet of Things (IoT) paradigm, encompassing devices, networks, and control algorithms. Consequently, PAoI is a critical metric for real-time applications enabled by Ultra-Reliable Low Latency Communication (URLLC). While highly useful for system evaluation, the direct analysis of this metric is complicated by the correlation between the random variables constituting the PAoI. Thus, it is often evaluated using only the mean value rather than the full distribution. Furthermore, since CPS communication technologies like Wi-Fi or DECT-2020 involve multiple processing stages, modeling them as tandem queueing systems is essential for accurate PAoI analysis. In this paper, we develop an analytical model for a DECT-2020 network segment represented as a two-phase tandem queueing system, enabling detailed PAoI analysis via Laplace–Stieltjes transforms (LST). We circumvent the dependence between generation and sojourn times by classifying updates into four mutually exclusive groups. This approach allows us to derive the LST of the PAoI and determine the exact Probability Density Function (PDF) for $M\left|M\right|1\to M\left|M\right|1$ system. We also calculate the mean and variance of the PAoIs and validate our results through numerical experiments. Additionally, we evaluate the impact of different service time distributions on PAoI variability. These findings contribute to the theoretical understanding of PAoI in tandem queueing systems and provide practical insights for optimizing DECT-2020-based communication systems. Full article

This paper establishes a comprehensive analytical framework for a new transformation of probability measures, denoted by ${\mathcal{T}}_{(a,t)}$, which unifies the classical t- and $T_a$-transformations in free probability. We derive the functional equation characterizing ${\mathcal{T}}_{(a,t)}$ through the Cauchy–Stieltjes transform and explicitly show how it specializes to known deformations when $a=0$ or $t=1$. Within the setting of Cauchy-Stieltjes kernel families, we prove structural symmetry and invariance properties of the transformation, demonstrating in particular that both the free Meixner family and the free analog of the Letac-Mora class remain invariant under ${\mathcal{T}}_{(a,t)}$. Furthermore, we obtain several new limiting theorems that uncover symmetric relationships among fundamental free distributions, including the semicircular, Marchenko–Pastur, and free binomial laws. Full article

This paper proves a new lemma that characterizes controllability for linear Volterra control systems and shows that the usual controllability assumption for the variational linearized system near an optimal pair is superfluous. Building on this, it establishes a Pontryagin-type maximum principle for Volterra optimal control with general control and state constraints (fixed terminal constraints and time-dependent state bounds), where the cost combines a terminal term with a state-dependent and integral term. Using the Dubovitskii–Milyutin framework, we construct conic approximations for the cost, dynamics, and constraints and derive necessary optimality conditions under mild regularity: (i) a classical adjoint system when only terminal constraints are present and (ii) a Stieltjes-type adjoint with a non-negative Borel measure when pathwise state constraints are active. Furthermore, under convexity of the cost functional and linear Volterra dynamics, the maximum principle becomes a sufficient criterion for global optimality (recovering the classical sufficiency in the differential case). The differential case recovers the classical PMP, and an SIR example illustrates the results. A key theme is symmetry/duality: the adjoint differentiates in the state while the maximum condition differentiates in the control, reflecting operator transposition and the primal–dual geometry of Dubovitskii–Milyutin cones. Full article

The paper is devoted to the study of a class of singular high-order fractional integro-differential equations with p-Laplacian operator, involving both the Riemann–Liouville fractional derivative and the Caputo fractional derivative. First, we investigate the problem by the method of reducing the order of fractional derivative. Then, by using the Schauder fixed point theorem, the existence of solutions is proved. The upper and lower bounds for the unique solution of the problem are established under various conditions by employing the Banach contraction mapping principle. Furthermore, four numerical examples are presented to illustrate the applications of our main results. Full article

This study is devoted to the detailed examination of the concept of $(a,b)$-deformation, defined for parameters $a\in \mathbb{R}$ and $b>0$. The analysis is conducted within the framework of Cauchy–Stieltjes kernel (CSK) families of probability measures, with particular attention given to the role of their variance functions (VFs). Using the VF as the main analytical tool, it is shown that the $(a,b)$-deformation of any measure belonging to the free Meixner family (FMF) remains within the same family. Moreover, the VF framework provides a powerful and flexible means for establishing new limit theorems associated with $(a,b)$-deformed free convolution. In particular, several novel limiting behaviors are derived, which naturally encompass both free and Boolean additive convolutions as special cases. Full article

This paper analyzes a finite-capacity $GI/M/2/N$ queue with two heterogeneous servers operating under a multiple working-vacation policy, Bernoulli feedback, and customer impatience. Using the supplementary-variable technique in tandem with a tailored recursive scheme, we derive the stationary distributions of the system size as observed at pre-arrival instants and at arbitrary epochs. From these, we obtain explicit expressions for key performance metrics, including blocking probability, average reneging rate, mean queue length, mean sojourn time, throughput, and server utilizations. We then embed these metrics in an economic cost function and determine service-rate settings that minimize the total expected cost via the Bat Algorithm. Numerical experiments implemented in R validate the analysis and quantify the managerial impact of the vacation, feedback, and impatience parameters through sensitivity studies. The framework accommodates general renewal arrivals ($GI$), thereby extending classical ($M/M/2/N$) results to more realistic input processes while preserving computational tractability. Beyond methodological interest, the results yield actionable design guidance: (i) they separate Palm and time-stationary viewpoints cleanly under non-Poisson input, (ii) they retain heterogeneity throughout all formulas, and (iii) they provide a cost–optimization pipeline that can be deployed with routine numerical effort. Methodologically, we (i) characterize the generator of the augmented piecewise–deterministic Markov process and prove the existence/uniqueness of the stationary law on the finite state space, (ii) derive an explicit Palm–time conversion formula valid for non-Poisson input, (iii) show that the boundary-value recursion for the Laplace–Stieltjes transforms runs in linear time $\mathcal{O}\left(N\right)$ and is numerically stable, and (iv) provide influence-function (IPA) sensitivities of performance metrics with respect to $({\mu }_1,{\mu }_2,\nu ,\alpha ,\varphi ,\beta )$. Full article

In this study, we introduce a novel transformation of probability measures that unifies two significant transformations in free probability theory: the t-transformation and the $V_a$-transformation. Our unified transformation, denoted ${\mathcal{U}}_{(a,t)}$, is defined analytically via a modified functional equation involving the Cauchy transform, and reduces to the t-transformation when $a=0$, and to the $V_a$-transformation when $t=1$. We investigate some properties of this new transformation from the lens of Cauchy–Stieltjes kernel (CSK) families and the corresponding variance functions (VFs). We derive a general expression for the VF resulting from the ${\mathcal{U}}_{(a,t)}$-transformation. This new expression is applied to prove a central result: the free Meixner family (FMF) of measures is invariant under this transformation. Furthermore, novel limiting theorems involving ${\mathcal{U}}_{(a,t)}$-transformation are proved providing new insights into the relations between some important measures in free probability such as the semicircle, Marchenko–Pastur, and free binomial measures. Full article

We explore the existence of positive solutions to a $\psi$–Riemann–Liouville fractional differential equation with a parameter and a sign-changing singular nonlinearity, supplemented with nonlocal boundary conditions which contain Riemann–Stieltjes integrals and various fractional derivatives. To establish our main results, we use the Guo–Krasnosel’skii fixed-point theorem. Full article

Padé approximants are computational tools customarily employed for resumming divergent Stieltjes series. However, they become ineffective or even fail when applied to Stieltjes series whose moments do not satisfy the Carleman condition. Differently from Padé, Levin-type transformations incorporate important structural information on the converging factors of a typical Stieltjes series. For example, the computational superiority of Weniger’s $\delta$-transformation over Wynn’s epsilon algorithm is ultimately based on the fact that Stieltjes series converging factors can always be represented as inverse factorial series. In the present paper, the converging factors of an important class of superfactorially divergent Stieltjes series are investigated via an algorithm developed one year ago from the first-order difference equation satisfied by the Stieltjes series converging factors. Our analysis includes the analytical derivation of the inverse factorial representation of the moment ratio sequence of the series under investigation, and demonstrates the numerical effectiveness of our algorithm, together with its implementation ease. Moreover, a new perspective on the converging factor representation problem is also proposed by reducing the recurrence relation to a linear Cauchy problem whose explicit solution is provided via Faà di Bruno’s formula and Bell’s polynomials. Full article

This paper addresses the existence of solutions to a Hadamard fractional differential equation of arbitrary order on an infinite interval, subject to integral boundary conditions that incorporate both Riemann–Stieltjes integrals and Hadamard fractional derivatives. Due to the presence of nontrivial solutions in the associated homogeneous boundary value problem, the problem is classified as resonant. The Mawhin continuation theorem is utilized to derive the main findings. Full article

In the present work, a Cauchy (initial) problem for a fractional linear system with distributed delays and Caputo-type derivatives of incommensurate order is considered. As the main result, a new straightforward approach to study the considered initial problem via an equivalent Volterra–Stieltjes integral system is introduced. This approach is based on the existence and uniqueness of a global fundamental matrix for the corresponding homogeneous system, which allows us to prove that the corresponding resolvent system possesses a unique measure resolvent kernel. As a consequence, an integral representation of the solutions of the studied system is obtained. Then, using the obtained results, relations between the stability of the zero solution of the homogeneous system and different kinds of boundedness of its other solutions are established. Full article

In this study, the linear renewal equation is obtained by using the integral equation, the renewal function and the Fourier–Stieltjes transform. It is proven that the linear renewal equation can be obtained by taking the derivative of the integral equation. Analytical methods for the solution of the obtained linear renewal equation are discussed. It is shown that the linear renewal equation is a powerful tool that can model the direct relationship between stochastic processes and density functions. It is shown that the Fourier–Stieltjes transform allows the equation to be simplified in the frequency domain and analytical solutions to be obtained, and the Laplace transform provides an effective analytical solution method, especially for uniform distribution and exponential density functions. The integral equation-based linear renewal density equation derived in this study preserves the temporal and structural symmetries of the system, allowing for the analytical derivation of symmetric forms in the solution space. In the light of the findings, predictions were made about what kind of studies would be done in the future. Full article