Search Results (27)

An Erlang loss system, which is an $M/G/s/s$ queue, is a model used in various applications. In this paper, a controlled version of the process is defined. The objective is to maximize the expected time until the system is full when the service time is exponentially distributed. The control variable is the service rate. The dynamic programming equation satisfied by the value function F, from which the optimal control follows at once, is derived, and F is found explicitly when $s=2$ and $s=3$. The problem of minimising the probability of the system being saturated is also considered. Full article

In this paper, we study nonlinear differential equations with Caputo fractional derivatives with respect to other functions and impulses. The main characteristic of the impulses is that the time between two consecutive impulsive moments is defined by random variables. These random variables are independent. As the distribution of these random variables is very important, we consider the Erlang distribution. It generalizes the exponential distribution, which is very appropriate for describing the time between the appearance of two consecutive events. We describe a detailed solution to the studied problem, which is a stochastic process. We define the p-exponential stability of the solutions and obtain sufficient conditions. The study is based on the application of appropriate Lyapunov functions. The obtained sufficient conditions depend not only on the nonlinear function and impulsive functions, but also on the function used in the fractional derivative. The obtained results are illustrated using some examples. Full article

Retrial queues are used extensively to model many practical problems in service systems, call centers, data centers, and computer network systems. The average waiting time is the main observable characteristic of the retrial queues. Long queues may cause negative impacts such as waste of manpower and unnecessary crowding leading to suffocation, and can even cause trouble for customers and institutions. Applying control chart technology can help managers analyze customers’ waiting times to improve the effective performance of service and attention. This paper pioneers the developing and detailed study of a waiting time control chart for a retrial queue with general service times. Two waiting time control charts, the Shewhart control chart, and a control chart using the weighted variance method are constructed in this paper. We present three cases for the Shewhart control chart in which the service time obeys special distributions, such as exponential, Erlang, and hyper-exponential distributions. The case of an exponentially distributed service time is also presented for the control chart using the weighted variance method. Based on the numerical simulations conducted herein, managers can better monitor and analyze the customers’ waiting times for their service systems and take preventive measures. Full article

This study evaluates the reliability measures for a system consisting of a single unit that is subject to random shocks at random times with mismatches between demand and production. The system may sometimes be subject to random shocks leading to a system failure with probability p. The system may also fail completely for various reasons other than shocks. The unit is serviced by a technician if it is affected by a shock. However, the system may fail during operation for various other reasons, such as a programming error, human error, operational continuity and stress, and weather conditions. A single technician immediately repairs or maintains the failed unit. The system is in an operational state when there is demand or in an idle state when there is no demand. The distributions of all times in this system follow a negative exponential, while the time to repair follows a two-stage Erlang distribution. The expressions for the reliability metrics are obtained as functions of time using the Laplace transform and the supplementary variables technique. Sensitivity and relative sensitivity analyses were also performed for the system parameters. Finally, the efficiency of the system is illustrated using numerical examples. Full article

This paper analyses the disruptions occurring in a production system determining the operating states of a single machine. A system with a convergent production character, in which both single flows (streams) and multi-stream flows occur, was considered. In this paper, a two-level formalisation of the production system (PS) was made according to complex systems theory. The continuity analysis was performed at the operational level (manufacturing machine level). The definition of the kth survival value and the quasi-coherence property defined on chains of synchronous relations were used to determine the impact of interruption of the processed material flow on uninterrupted machine operation. The developed methodology is presented in terms of shaping the energy efficiency of technical objects with the highest power demand (the furnace of an automatic paint shop and the furnace of a glass tempering line were taken into consideration). The proposed methodology is used to optimise energy consumption in complex production structures. The model presented is utilitarian in nature—it can be applied to any technical system where there is randomness of task execution times and randomness of unplanned events. This paper considers the case in which two mutually independent random variables determining the duration of correct operation $T_P$ and the duration of breakdown $T_B$ are determined by a given distribution: Gaussian and Gamma family distributions (including combinations of exponential and Erlang distributions). A formalised methodology is also developed to determine the stability of system operation, as well as to assess the potential risk for arbitrary system evaluation parameters. Full article

An unmanned aerial vehicle (UAV) observation platform obtained the first vertical profiles of particle number size distribution (PNSD) from 7 to 16 July 2022 on the eastern slope of the Tibetan Plateau (ESTP). The results were from two flanks at the Chuni (CN) and Tianquan (TQ) sites, which are alongside a mountain (Mt. Erlang). The observations revealed a significant negative correlation between the planetary boundary layer height (PBLH) and the particle number concentration (PNC), and the correlation coefficient was −0.19. During the morning, the rise in the PBLH at the CN and TQ sites caused decreases of 16.43% and 58.76%, respectively, in the PNC. Three distinct profile characteristics were classified: Type I, the explosive growth of fine particles with a size range of 130–272 nm under conditions of low humidity, strong wind shear, and northerly winds; Type II, the process of particles with a size range of 130–272 nm showing hygroscopic growth into larger particles (e.g., 226–272 nm) under high humidity conditions (RH > 85%), with a maximum vertical change rate of about −1653 # cm⁻³ km⁻¹ for N_130–272 and about 3098 # cm⁻³ km⁻¹ for N_272–570; and Type III, in which during the occurrence of a surface low-pressure center and an 850 hPa low-vortex circulation in the Sichuan Basin, polluting air masses originating from urban agglomeration were transported to the ESTP region, resulting in an observed increase in the PNC below 600 nm. Overall, this study sheds light on the various factors affecting the vertical profiles of PNSD in the ESTP region, including regional transport, meteorological conditions, and particle growth processes, helping us to further understand the various features of the aerosol and atmospheric physical character in this key region. Full article

We study classification of random sequences of characters selected from a given alphabet into two classes characterized by distinct character selection probabilities and length distributions. The classification is based on the sign of the log-likelihood score (LLS) consisting of a random sum and a random term depending on the length distributions for the two classes. For long sequences selected from a large alphabet, computing misclassification error rates is not feasible either theoretically or computationally. To mitigate this problem, we computed limiting distributions for two versions of the normalized LLS applicable to long sequences whose class-specific length follows a translated negative binomial distribution (TNBD). The two limiting distributions turned out to be plain or transformed Erlang distributions. This allowed us to establish the asymptotic accuracy of the likelihood-based classification of random sequences with TNBD length distributions. Our limit theorem generalizes a classic theorem on geometric random sums due to Rényi and is closely related to the published results of V. Korolev and coworkers on negative binomial random sums. As an illustration, we applied our limit theorem to the classification of DNA sequences contained in the genome of the bacterium Bacillus subtilis into two classes: protein-coding genes and standard noncoding open reading frames. We found that TNBDs provide an excellent fit to the length distributions for both classes and that the limiting distributions capture essential features of the normalized empirical LLS fairly well. Full article

The Redundancy Allocation Problem (RAP) is well-known in the field of reliability optimization. In this paper, RAP is investigated assuming that the distribution of the time to failure of the components has the form of an Erlang distribution with a time-dependent rate parameter and considering that the choice of redundancy for each subsystem can be none, active, standby or mixed. A genetic algorithm is used to solve the problem of optimal allocation. To analyze the effect of the time dependence, some numerical examples are worked out. Then, a case study of RAP from the literature is analyzed. The obtained results show that time dependence of the failure time distribution parameters can lead to significant differences in the optimal redundancy allocation. Full article

The squared momentum transfer spectra of $\eta$ and ${\eta }^0$, produced in high-energy photon–proton ($\gamma p$) $\to \eta \left({\eta }^0\right)+p$ processes in electron–proton ($ep$) collisions performed at CEBAF, NINA, CEA, SLAC, DESY, and WLS are analyzed. The Monte Carlo calculations are used in the analysis of the squared momentum transfer spectra, where the transfer undergoes from the incident $\gamma$ to emitted $\eta \left({\eta }^0\right)$ or equivalently from the target proton to emitted proton. In the calculations, the Erlang distribution and Tsallis–Levy function are used to describe the transverse momentum ($p_T$) spectra of emitted particles. Our results show that the average transverse momentum ($⟨p_T⟩$), the initial-state temperature ($T_i$), and the final-state temperature ($T_0$) roughly decrease from the lower center-of-mass energy (W) to the higher one in the concerned energy range of a few GeV, which is different from the excitation function from heavy-ion collisions in the similar energy range. Full article

We explore the effects of cross-diffusion dynamics in epidemiological models. Using reaction–diffusion models of infectious disease, we explicitly consider situations where an individual in a category will move according to the concentration of individuals in other categories. Namely, we model susceptible individuals moving away from infected and infectious individuals. Here, we show that including these cross-diffusion dynamics results in a delay in the onset of an epidemic and an increase in the total number of infectious individuals. This representation provides more realistic spatiotemporal dynamics of the disease classes in an Erlang $SEIR$ model and allows us to study how spatial mobility, due to social behavior, can affect the spread of an epidemic. We found that tailored control measures, such as targeted testing, contact tracing, and isolation of infected individuals, can be more effective in mitigating the spread of infectious diseases while minimizing the negative impact on society and the economy. Full article

Traditionally, Parisian ruin is said to occur when the insurer’s surplus process has stayed below level zero continuously for a certain grace period. Inspired by this concept, in this paper we propose a modification by assuming that once a grace period has been granted when the surplus becomes negative, the surplus level will not be monitored continuously in the interim, but instead it will be checked at the end of the grace period to see whether the business has recovered. Under an Erlang distributed grace period, a computationally tractable formula for the Gerber–Shiu expected discounted penalty function is derived. Numerical examples regarding the modified Parisian ruin probability are also provided. Full article

This paper focuses mainly on the problem of computing the ${\gamma }^{\mathrm{th}}$, $\gamma >0$, moment of a random variable $Y_n:={\sum }_{i=1}^n{\alpha }_i{X}_i$ in which the ${\alpha }_i$’s are positive real numbers and the $X_i$’s are independent and distributed according to noncentral chi-square distributions. Finding an analytical approach for solving such a problem has remained a challenge due to the lack of understanding of the probability distribution of $Y_n$, especially when not all ${\alpha }_i$’s are equal. We analytically solve this problem by showing that the ${\gamma }^{\mathrm{th}}$ moment of $Y_n$ can be expressed in terms of generalized hypergeometric functions. Additionally, we extend our result to computing the ${\gamma }^{\mathrm{th}}$ moment of $Y_n$ when $X_i$ is a combination of statistically independent $Z_i^2$ and $G_i$ in which the $Z_i$’s are distributed according to normal or Maxwell–Boltzmann distributions and the $G_i$’s are distributed according to gamma, Erlang, or exponential distributions. Our paper has an immediate application in interest rate modeling, where we can explicitly provide the exact transition probability density function of the extended Cox–Ingersoll–Ross (ECIR) process with time-varying dimension as well as the corresponding ${\gamma }^{\mathrm{th}}$ conditional moment. Finally, we conduct Monte Carlo simulations to demonstrate the accuracy and efficiency of our explicit formulas through several numerical tests. Full article

In this paper, we consider the batch arrival of customers to a transport station. Customers belonging to each category is considered as a single entity according to a BMMAP. An Erlang clock of order m starts ticking when the transport vessel reaches the station. When the Lth stage of the clock is reached, an order for the next vessel is placed. The lead time for arrival of the vessel follows exponential distribution. There are two types of rooms in this system: the waiting rooms and the service rooms for customers in the transport station and in the vessel, respectively. The waiting room capacity for customers of type 1 is infinite whereas those for customer of type $2,\dots ,k$ are of finite capacities. The service room capacity $C_j$ for customer type $j,j=1,2,\dots ,k$ is finite. Upon arrival, customers of category j occupy seats designated for that category in the vessel, provided there is at least one vacancy belonging to that category. The total number of vessels with the operator is $h^{*}$. The service time of each vessel follows exponential distribution with parameter $\mu .$ Each group of customers belong to category j searches independently for customers of this category to mobilize passengers when the Erlang clock reaches $L_1$ where $L_1<L$. The search time for customers of category j follows exponential distribution with parameter ${\lambda }_j.$ The stability condition is derived. Some performance measures are estimated. Full article

We analyze the $p_T$ spectra of ${\pi }^{\pm }$, $K^{\pm }$, p, and $\overline{p}$ produced in different centralities’ Au-Au collisions at different collision energies from 7.7 to 62.4 GeV using a two-component Erlang distribution in the framework of a multi-source thermal model. The fitting results are consistent with the experimental data, and the yield ratios of negative to positive particles are obtained from the normalization constants. Based on the yield ratios, the chemical potentials of light hadrons ($\pi$, K, and p) and quarks (u, d, and s) are extracted. This study shows that only the yield ratios of p decrease with the increase in centrality. The logarithms of these yield ratios in the same centrality show obvious linear dependence on $1/\sqrt{s_{NN}}$. The extracted chemical potentials (the absolute magnitude for $\pi$) of light hadrons and quarks decrease with the increase in energy. The curves of chemical potential vs. energy for all centralities derived from the linear fits of the logarithms of the yield ratio as a function of energy have their maximum (the absolute magnitude for $\pi$) at the same energy of 3.526 GeV, which is possibly the critical energy of phase transition from a liquid-like hadron state to a gas-like quark state in the collision system. Full article

This paper considers a modified Erlang loss system for cognitive wireless networks and related applications. A primary user has pre-emptive priority over secondary users, and the primary customer is lost if upon arrival all the channels are used by other primary users. Secondary users cognitively use idle channels, and they can stay (either in an infinite buffer or in an orbit) in cases where idle channels are not available upon arrival or they are interrupted by primary users. While the infinite buffer model represents the case with zero sensing time, the infinite orbit model represents the case with positive sensing time. We obtain an explicit stability condition for the cases where arrival processes of primary users and secondary users follow Poisson processes, and their service times follow two distinct arbitrary distributions. The stability condition is insensitive to the service time distributions and implies the maximal throughout of secondary users. Moreover, we extend the stability analysis to the system with outgoing calls. For a special case of exponential service time distributions, we analyze the buffered system in depth to show the effect of parameters on the delay performance and the mean number of interruptions of secondary users. Our simulations for distributions rather than exponential reveal that the mean number of terminations for secondary users is less sensitive to the service time distribution of primary users. Full article