Discrete and Continuous Models and Applied Computational Science

We continue to describe tools for simplifying the typesetting of mathematics in accordance with ISO standards. In this article, we briefly describe the mismath package for the LaTeX system.

Background: In fifth-generation (5G) New Radio (NR) network slicing, a mobile network operator leases part of its spectrum to a mobile virtual network operator under service-level agreements. Under equal-rate (Round Robin) scheduling, the base station assigns time-frame shares so that every user in a slice receives the same bitrate regardless of location. Existing analytical queueing frameworks for dynamic spectrum slicing, however, cover only the fixed-power case, and no such framework yet exists for equal-rate scheduling. Purpose: To develop a resource queueing model for dynamic spectrum slicing under equal-rate scheduling, extending the authors' prior fixed-power framework to this regime. Method: We describe each user's channel quality by an inverse spectral efficiency (ISE) and derive its cumulative distribution function from the path-loss model. Under equal-rate scheduling, all users in a slice share a common bitrate equal to the allocated bandwidth divided by the sum of independent, identically distributed ISEs; we characterize this sum exactly through multi-fold convolution and, for computational tractability, approximate it through the central limit theorem. These distributions feed a continuous-time Markov chain (CTMC) that models the joint spectrum partition and per-slice user counts, with state-dependent admission control and stochastic frequency reallocation that preserves isolation guarantees between slices. Results: The framework yields closed-form expressions for the blocking probability, the average per-user bitrate, the average number of active users in each slice, and overall resource utilization. Conclusions: The proposed model provides an analytical tool for spectrum dimensioning and admission control in multi-tenant 5G NR networks under equal-rate scheduling.

Purpose. To study a two-queue system with a single server operating under the Join-the-Shortest-Queue (JSQ) routing discipline and the Serve-the-Longest-Queue (SLQ) service discipline, where the destination queue for an arriving customer and the queue to be served are selected according to probabilistic tie-breaking rules when the queue lengths are equal, with absolute preemptive priority and non-zero server switch-over times. Methods. The system is studied using a Markov description, stationary balance equations, conditional probability generating functions, and numerical analysis. Results. The system dynamics are described by a two-dimensional Markov process whose components are the length of the first queue and a phase variable representing the difference between the queue lengths together with the server state. For this process, the stationary balance equations are derived. Conditional probability generating functions are introduced and used to obtain a matrix form of the balance equations and to consider the condition for the existence of a stationary regime. Based on the obtained generating functions, expressions are derived for the main stationary performance measures, including the mean queue lengths, variances, correlation coefficient, effective arrival rates, the probability that the server is in a switch-over state, and the probability that the server is idle. Numerical experiments are performed to evaluate the effect of the arrival rate, service rates, switch-over rates, and probabilistic tie-breaking parameters on the stationary characteristics of the system and on the phase distribution of the underlying Markov process. Conclusions. The results show that the combined use of the JSQ and SLQ disciplines contributes to balancing the queue lengths. However, the system behavior and its stationary characteristics significantly depend on the combination of the arrival rate, service rates, server switch-over rates, and probabilistic tie-breaking parameters.

Background: The extension of NB-IoT to LEO satellite systems enables mMTC in remote areas beyond terrestrial coverage. However, LEO NB-IoT faces significant physical layer challenges, including extreme path loss, rapid orbital dynamics, and blockage by buildings, forests, and hills. Existing Markov chain propagation models fail to capture the fine-grained elevation-dependent variability required for transmission optimization. Purpose: This study develops an accurate BLER model for LEO NB-IoT that accounts for elevation angle, deployment-specific shadowing, and scintillation, enabling dynamic scheduling to improve reliability and energy efficiency. Methods: We propose a fine-grained model incorporating ITU-R P.681-11 Loo parameterization with PCHIP interpolation for arbitrary elevations, integrating free-space loss, elevation-dependent scintillation, atmospheric absorption, and two-state Markov shadowing with urban/village/rural parameters. The BLER is estimated via Monte Carlo simulations with $10^5$ samples and link-budget analysis for repetitions $N_{\text{rep}}\in\{1,2,4,8,16,32,64,128\}$. Results: Elevation angle is the dominant factor: at EIRP=42 dBm and $N_{\text{rep}}=1$, BLER falls from 1 at $20^\circ$ to <0.1 at $70^\circ$. Deployment conditions have secondary but measurable effects; urban scenarios show up to 0.4 higher BLER than villages at the same elevation. Increasing the number of repetitions to 128 shifts the BLER curves by 8-10 dBm, whereas $N_{\text{rep}}=4$ yields a near-zero BLER at $50^\circ$ in all scenarios. Conclusions: Fixed-time transmission is suboptimal; UEs can achieve near-zero BLER by waiting for elevations above $50^\circ$ with moderate repetitions. Our model enables elevation-aware scheduling from ephemeris data without requiring real-time sounding, offering a practical framework for reliable and energy-efficient LEO NB-IoT communications under diverse environmental conditions.

Background Modeling processes in thermonuclear targets requires reliable data on the cross sections and rates of thermonuclear reactions. The most reliable source of such data is the regression of experimental measurements. However, existing experiments are conducted for colliding particle energies of 2-3 keV and higher, while target ignition calculations require data at energies starting from 0.1 keV. Therefore, the approximating formula must allow extrapolation to lower energies. The physical basis for such extrapolation is Gamow's formula, according to which the reaction S-factor should tend to a constant. However, the formulas for S-factors given in the literature are applicable within the corresponding experimental intervals. The question of the possibility of their downward extrapolation in energy remains open. Purpose In this regard, the aim of this work is to develop a new method for approximating and extrapolating experimental data on thermonuclear reaction cross sections. Method The method proposed in this paper is based on constructing a special type of global spline. Results Four reactions involving hydrogen and helium isotopes traditionally used in thermonuclear target calculations were processed. The resulting approximations are applicable in the energy range of 0.01 keV to 3.3 MeV. Their accuracy is 0.2-1%. Conclusions A comparison of the obtained results with the formulas most commonly used in the literature shows that the accuracy of the proposed method is 2-4 times higher than the world standard.

Background. Active Queue Management (AQM) algorithms of the RED family remain widely deployed in network equipment, yet a systematic three-way comparison of RED, Adaptive RED (ARED), and Gentle RED under strictly identical conditions is lacking in the literature. Purpose. The aim of this study is to quantify the differences in buffer management dynamics, throughput stability, and congestion signaling among the three algorithms and to formulate practical selection guidelines. Method. A dumbbell-topology network model was implemented in the ns-3 discrete-event simulator (version 3.41) with ten TCP bulk-send sources creating sustained congestion at a 2 Mbit/s bottleneck link. All three algorithms were evaluated with identical queue thresholds, load profile, and topology over a 60-second simulation. An extended metric set including EWMA zone distribution and temporal drop structure was applied. Results. All algorithms achieved approximately 90% link utilization (mean throughput 1.80 Mbit/s). RED provided the most predictable throughput (coefficient of variation 18.7%) at the cost of the highest drop count (1240 packets). ARED minimized drops (1003 packets, 19% fewer than RED) but produced the largest queue oscillations (p99 of 49 packets). Gentle RED achieved the lowest mean queue length (6.40 packets) with the highest throughput variability (coefficient of variation 24.7%). Conclusions. No single algorithm is universally optimal; the choice depends on scenario priorities: Gentle RED for minimum delay, ARED for minimum packet loss, and RED for maximum throughput predictability.

Background. Metal foils heated by an electric current are commonly used as evaporation elements in ion sources. Because the evaporation rate is highly sensitive to temperature, predicting the temperature distribution accurately and efficiently is essential for choosing the foil geometry and operating conditions. Purpose. This study develops a two-dimensional electric-thermal model of a wing-shaped thulium foil and evaluates numerical strategies for repeated simulations. Methods. Owing to symmetry, only one quarter of the foil is included in the computational domain. The current paths are determined by solving a Laplace problem with constant electrical conductivity. The resulting field is then scaled to the prescribed total current and used to calculate the spatially varying Joule heat source. The transient heat equation accounts for the temperature dependence of density, heat capacity, thermal conductivity, and electrical resistivity, as well as radiative and evaporative heat losses from both faces of the foil. The equation is discretized on a masked Cartesian grid and integrated using an alternating-direction implicit scheme, with Picard iterations applied at each half-step. Results. At a current of 2.5 A and a contact temperature of 78 K, the stationary maximum temperature is 829.803 K. All tested tridiagonal solvers produce the same converged temperature field. The fastest partition solver, combined with a CUDA task graph, completes the simulation in 36.3 s, giving a 24.1-fold speedup over the serial central-processing-unit implementation. Conclusions. The model accounts for nonuniform electric heating and nonlinear heat losses in the foil. Its graphics-processing-unit implementation is sufficiently efficient for repeated optimization of the foil geometry and operating current.