Preface to the Special Issue on Probability and Stochastic Processes with Applications to Communications, Systems and Networks, 2nd Edition

Probability theory and stochastic processes are fundamental mathematical tools that play a critical role in understanding and modeling randomness and uncertainty in various fields, including communications, systems engineering, and network design. These concepts provide the foundation for analyzing and designing systems that operate in unpredictable environments, such as wireless communication networks, internet traffic, and signal processing systems.

The Special Issue is the continuation of the work initiated in [1], representing a reprint of the second edition of the Special Issue “Probability and Stochastic Processes with Applications to Communications, Systems and Networks, 2nd Edition”. The authors’ geographical distribution is shown in

Table 1. Geographic distribution of authors by country.

Country	Number of Authors
Republic of Korea	5
India	6
Russia	4
France	2
Italy	1
Iran	1
Saudi Arabia	3
Tunisia	1
Canada	1
Egypt	1

; the 25 authors are from ten different countries. Note that it is usual for a paper to be written by more than one author and for authors to collaborate with authors with different or multiple affiliations.

In Contribution 1, the authors explore diversity selection in three-user interference-aligned (IA) MIMO channels, emphasizing reliability enhancement through diversity order (DO) analysis. While degrees of freedom (DoF) have traditionally been prioritized for bandwidth optimization in IA, diversity order has received less attention for improving error performance. This paper introduces the concept of conditional diversity order and proposes a beamforming vector selection method that achieves a conditional diversity order of M²/2, where M is the number of antennas per transceiver. The proposed two-stage decoding approach integrates zero-forcing for interference suppression with maximum likelihood (ML) decoding and orthogonalization techniques to recover the desired signal. The authors demonstrate that their scheme outperforms traditional IA methods in both error probability and conditional DO, particularly in scenarios with a large number of antennas. These findings offer valuable insights into optimizing IA for enhanced reliability and lay the groundwork for future research on advanced decoding and beamforming strategies.

In Contribution 2, a new linear transceiver optimization problem is addressed for correlated MIMO interference channels in the presence of channel state information (CSI) errors. This scenario is more realistic and practical compared to previous studies focusing on uncorrelated MIMO interference channels. The optimization problem is formulated as minimizing the mean square error (MSE) under general power constraints. Since the objective function for precoders and receive filters is not jointly convex, the problem is decomposed into two convex subproblems, which are solved iteratively to obtain linear precoders and receive filters. The proposed algorithm is guaranteed to converge to a local minimum. Numerical results show that the algorithm significantly reduces sensitivity to CSI errors compared to existing robust schemes in correlated MIMO interference channels.

In Contribution 3, the authors extend the k-out-of-n: G reliability system to a multi-server queueing model with N repair policies. The system consists of n servers with identically and exponentially distributed service times. Servers fail at an exponential rate, and repairs follow N distinct policies. Although servers operate independently, maintenance is only performed if at least k servers are operational. The steady-state model is analyzed using a matrix-analytic method, and an optimization problem is formulated. Performance measures are evaluated and the results are presented, providing insights into the system’s behavior under various repair policies.

In Contribution 4, a two-server queueing system is analyzed, where the arrival and service processes are interdependent. The system’s evolution is governed by transitions on the product space of three Markov chains, which describe the arrival and service processes. Transitions follow a semi-Markov rule, with dwell times exponentially distributed. The authors derive the stability condition for the system and compute the stationary distribution at equilibrium. Several performance measures are evaluated, and numerical illustrations are provided to validate the model’s effectiveness.

In Contribution 5, positive recurrence is established for a single-server queueing system under generalized service intensity conditions. Unlike previous studies, the authors do not assume the existence of a service distribution density function but instead use a lower bound of integral type as a sufficient condition. Positive recurrence ensures the existence of an invariant distribution and guarantees slow convergence to it in the full variation metric.

In Contribution 6, weak convergence results are presented for a family of parameter-dependent Gibbs measures. The authors show that the limiting distribution is centered on the set of global minima of the limiting Gibbs potential. An explicit calculation of the limit distribution is provided, and the method is applied as an alternative to Lyapunov function or direct convergence approaches for stationary probabilities. The theoretical findings are illustrated with a numerical example, including an application to the repairman problem.

In Contribution 7, the redundancy allocation problem (RAP) is studied under the assumption that component failure times follow an Erlang distribution with a time-dependent rate parameter. Redundancy choices for each subsystem include none, active, standby, or mixed. A genetic algorithm is employed to solve the optimal allocation problem. Numerical examples are analyzed to investigate the impact of time dependence, and a case study from the literature is examined. The results highlight that the time dependence of the mean time between failures (MTBF) distribution parameters can significantly influence optimal redundancy allocation.

In Contribution 8, the authors address the problem of adaptive event filtering for a class of nonlinear discrete-time systems modeled using interval type-2 fuzzy models. The system is subject to Markov switching and susceptible to cheating attacks. To reduce unnecessary signal transmissions over the communication channel, an improved event-triggering mechanism is proposed. Additionally, an extended dissipativity specification is introduced to quantify the transient dynamics of filtering errors. Using the linear matrix inequalities (LMI) approach and leveraging information from upper and lower membership functions, the authors establish sufficient conditions for the existence of a desired filter. This ensures the root mean square (RMS) stability and extended dissipativity of the augmented filtering system. An optimization algorithm based on particle swarm optimization (PSO) is proposed to compute the filter gains with optimal efficiency. The effectiveness of the developed scheme is validated through numerical experiments on a single-link robotic arm and a lower limb system.

In Contribution 9, the authors study the formation of a particle flow that replicates an image and estimate the distance between the copy and the original using a specialized probabilistic metric. The ability of a ball flow to cover a surface during ball grinding is analyzed using stochastic geometry formulas. The recovery of inhomogeneous Poisson flow characteristics from imprecise observations is examined using the point-coloring theorem of Poisson flow. The dependence of the Poisson distribution parameter on the peak load generated by an inhomogeneous input Poisson flow is evaluated in a queueing system with an infinite number of servers and deterministic service time. The models consist of an inhomogeneous Poisson flow of points, each labeled with attributes such as mass, area, volume, observability, and service time. The authors establish an asymptotic step dependence between the objective functions of the model and the parameters of label splitting. The results have potential applications in nanotechnology, powder metallurgy, ecology, and smart city initiatives, and are supported by real-world observations and experiments.

In Contribution 10, the authors explore the Hawkes process, a self-excited point process with a clustering effect, where the jump rate depends on the entire history of the process. While the Hawkes process is generally non-Markovian, it can exhibit Markovian properties under specific conditions, such as when the excitation function is exponential or a sum of exponentials. The intensity of the Hawkes process is determined by the sum of a base intensity and terms dependent on the process’s history. In this study, the authors consider a linear Hawkes model with a randomly determined baseline intensity and investigate large deviation principles for this new model. The random baseline intensity Hawkes processes have broad applications in insurance, finance, queueing theory, and statistics.

In Contribution 11, the authors analyze the waiting time distribution for a single-server queueing system with finite buffer capacity \(N\). Clients arrive according to a Poisson process, and the server operates under a fast service rule. The service times follow a general distribution independent of the arrival process. The authors propose an alternative approach to derive the probability distribution of the queue length at the epoch following batch departures. Using an embedded Markov chain, Markov renewal theory, and semi-Markov processes, they obtain the queue length distribution at a random epoch. The waiting time distribution for a random customer is determined by establishing a functional relationship between the probability-generating function of the queue length distribution and the Laplace–Stieltjes transform (LST) of the waiting time distribution. The authors derive the probability density function of the waiting time and provide numerical results.

In Contribution 12, the authors address a critical issue in power grids arising from the high energy consumption of mining devices in bitcoin-based financial trading systems (BFTSs). This poses a significant risk, as cybercriminals may exploit vulnerabilities to disrupt optimal energy management. The paper introduces a new type of cyber-attack, termed “miner-misuse”, and proposes an online anomaly detection approach based on reinforcement learning (RL) to counteract such attacks. Due to incomplete system information, the authors implement an Observable Markov Decision Process (OMDP) within the RL framework to block miner attacks. The proposed method is shown to perform optimally and accurately when tuning parameters are correctly set during training. A hybrid mechanism combining optimization and learning ensures both optimal solutions and fast convergence.

Additionally, the authors propose an Intelligent Priority Selection (IPS) algorithm integrated with the RL method to enhance the detection of mining attacks. To validate the approach, they model the energy consumption of mining devices as a function of the hashing rate in the BFTS. Uncertain fluctuations in energy consumption are addressed using the Unscented Transform (UT) method, which accurately models uncertain parameters with high probability. The results demonstrate that the proposed anomaly detection method outperforms other approaches in detecting real-time miner attacks, as evidenced by the F-score value and the probability of successful attack detection.

Thus, the collection covers a wide range of advanced topics in communication systems, queueing theory, stochastic processes, and optimization, with applications ranging from wireless networks to power grids and nanotechnology. Taken together, all of the papers emphasize the importance of advanced mathematical and computational tools for solving complex problems in communication systems, queueing theory, and stochastic processes. The integration of optimization algorithms, machine learning, and probabilistic models enables the development of robust and efficient solutions for real-world applications ranging from improving the reliability of wireless communications to securing power grids and optimizing industrial processes. This work not only advances theoretical knowledge, but also provides practical foundations for solving emerging problems in technology and infrastructure.

Probability theory and stochastic processes are powerful tools for understanding and managing randomness in communications, systems, and networks. From modeling network traffic to designing robust communication systems, these concepts enable engineers and researchers to tackle real-world challenges in the presence of uncertainty. As technology continues to evolve, the importance of probability and stochastic processes will only grow, making them essential knowledge for anyone working in these fields.

This Special Issue serves as a starting point for exploring the rich and fascinating world of probability and stochastic processes. Whether you are a student, researcher, or practitioner, mastering these concepts will open doors to innovative solutions and deeper insights into the behavior of complex systems.