Erlang Distribution Research Articles

The Two Component Generalized Finite Erlang Mixture and its Properties and Special Cases

This research proposes the construction of a two-component generalized finite Erlang mixture with four parameters. Three distinct cases of this mixture are presented, differing in their mixing weights and corresponding component probability (Erlang) distributions. Special cases of these mixtures, including the one-, two-, and three-parameter finite Erlang mixtures, have also been derived. The statistical properties examined for these mixtures include the distribution function, survival function, hazard function, moment generating function, raw and central moments, mean, variance, coeffcient of skewness, coeffcient of kurtosis, and order statistics. Parameter estimation for the finite Erlang mixtures was conducted using both the method of moments and maximum likelihood estimation. Furthermore, the 4-parameter mixed Erlang distributions were applied to a real dataset on the relief times of patients receiving an analgesic, to evaluate their goodness of fit. The results demonstrate the potential of these mixtures to provide robust modeling for empirical data, suggesting their applicability in various statistical and practical contexts.

A deterministic fluid model for production and energy mode control of a single machine

Improving machines’ energy efficiency through dynamic energy mode control to meet demand requirements with minimal energy consumption is a promising approach. This study considers a machine operating in working, idle, off, and warmup energy modes with different energy consumption in each mode. A deterministic fluid model is developed to analyze an energy mode control policy that determines when to keep the machine working, off or idle, and switch to other modes based on the inventory/backlog level to minimize the total energy, inventory, and backlog costs. This approach facilitates the derivation of closed-form expressions for the optimal thresholds and the associated costs. This modeling approach allows us to prove that a policy that operates the machine between the working and off modes or the working and idle modes is always better than a hybrid policy that operates the machine in working, off, and idle modes simultaneously. We use the solution of the deterministic fluid model to propose an approximate policy for machines with stochastic production, warmup, and demand processes. We compare the results of the proposed approximation method with the optimal solution of a stochastic system where the production and warmup times are exponential and the demand inter-arrival times have Erlang distribution. The optimal solution for the stochastic system is determined by solving a Markovian Decision Process (MDP). Our numerical experiments show that the proposed approximation method predicts the optimal policy type for the stochastic case with a 89.3% accuracy, and the average error between the optimal cost and the cost obtained with the approximation method is 1.37% for 729 different cases tested. Furthermore, the computational efficiency of the proposed approximation is around 250 times better than the effort to determine the optimal policy using an MDP approach. We propose this approximation method where the control parameters are given in closed form as an easy-to-implement and effective policy to control energy modes to minimize the total energy, inventory, and backlog costs. Furthermore, we present the deterministic fluid modeling approach as a versatile approach to analyze energy mode control problems.

ANALYSIS OF THE RESULTS OF SIMULATIVE MODELLING OF THE INFORMATION SECURITY SYSTEM IN THE CORPORATE NETWORKS OF HIGHER EDUCATION INSTITUTIONS

The analysis shows that the insufficient level of information security in service networks is the main cause of huge losses for enterprises. Despite the appearance of a number of works to solve this problem, there is currently no unified system for assessing information security. This shows that this problem has not yet been sufficiently studied and relevant. This work is one of the steps towards creating a system for assessing information security in service networks. The purpose of the work is to develop an algorithm and simulation model, analyze the results of the simulation model to determine the main characteristics of the information security system (ISS), providing the ability to completely close all possible channels of threats by controlling all unauthorized access (UA) requests through the protection mechanism (PM). To solve the problem, a simulation method was applied using the principles of queuing systems (QS). This method makes it possible to obtain the main characteristics of the ISS from the UA with an unlimited amount of buffer memory (BM). Models, an algorithm and a methodology for the development of ISS from UA are proposed, which is considered as a single-phase multi-channel QS with an unlimited volume of BM. The process of obtaining simulation results was implemented in the GPSS World modeling system and comparative analyzes of the main characteristics of the ISS were carried out for various laws of distribution of output parameters. At the same time, UA requests were the simplest flows, and the service time was subject to exponential, constant and Erlang distribution laws. Conducted experiments based on the proposed models and algorithm for analyzing the characteristics of the ISS from the UA as a single-phase multi-channel QS with unlimited waiting time for requests in the queue confirmed the expected results. The results obtained can be used to build new or modify existing ISS in corporate networks for servicing objects for various purposes. This work is one of the approaches to generalizing the problems under consideration for systems with an unlimited volume of BM. Prospects for further research include research and development of the principles of hardware and software implementation of ISS in service networks.

Waiting Time Control Chart for M/G/1 Retrial Queue

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.

NUMERICAL-ANALYTICAL DELAY MODEL BASED ON QS WITHOPERATIONAL SHIFT OF DISTRIBUTION PRINCIPLES

This article demonstrates results for a queuing system formed by right-shifting an Erlang distribution and a second-order hyperexponential distribution. As is known, the first distribution provides a coefficient of variation less than one, and the second one – more than one. From the general queuing theory, it is known that the average delay of requests in the queue in the QS G/G/1 is related to the coefficients of variation of arrival intervals and service times by a quadratic dependence, and the system we are considering belongs precisely to this type. An operational shift in the distribution laws leads to a multiple reduction in delay compared to a conventional system, and this value depends on the value of the shift parameter. To construct a mathematical model of the delay, the method of spectral solution of the Lindley integral equation for the system under consideration was applied. For the practical application of the obtained results, the standard method of oprobability theory moments is used. The obtained results of numerical and analytical modeling in Mathcad clearly confirm the adequacy of the proposed mathematical delay model.

Application of Queue Theory in Cafe Services with Erlang Distribution

As urban lifestyles evolve, culinary businesses, particularly cafes, have experienced rapid growth. This surge in popularity has led to an increase in customers and, consequently, longer queues. These extended wait times can frustrate customers and pose challenges to cafe management. To address this issue, we conducted a comprehensive eval___uation and optimization of the service system at a Samarinda cafe using the Erlang distribution queuing system. Primary data was meticulously collected over six days, amounting to a total of 12 hours of observation. Kolmogorov-Smirnov distribution fitting tests were employed, revealing that customer service times adhered to an exponential distribution. The average customer arrival rate was determined to be 0.351 per minute, while the average service time was calculated at 5.546 minutes per customer. Our analysis confirmed that the system operates in a steady state with a utility value of 0.06, indicating sufficient service capacity to handle the current customer load. Therefore, the study concludes that the cafe's service system is currently optimal.

A size-dependent division strategy accounts for leukemia cell size heterogeneity

Heterogeneity in the size distribution of cancer cell populations is linked to drug resistance and invasiveness. However, understanding how such heterogeneity arises is still damped by the difficulties of monitoring the proliferation at the typical timescales of mammalian cells. Here, we show how to infer the growth regime and division strategy of leukemia cell populations using live cell fluorescence labeling and flow cytometry in combination with an analytical model where cell growth and division rates depend on powers of the size. We found that the dynamics of the size distribution of Jurkat T-cells is reproduced by (i) a sizer-like division strategy, with (ii) division times following an Erlang distribution and (iii) fluctuations up to ten percent of the inherited fraction of size at division. Overall, our apparatus can be extended to other cell types and environmental conditions allowing for a comprehensive characterization of the growth and division model different cells adopt.

The Reliability and Availability Analysis of a Single-Unit System under the Influence of Random Shocks and the Variation in Demand from Production with Erlang Distribution

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.

An Efficient Neutrosophic Method for Generating Random Variates from Erlang Distribution

The simulation process depends on generating a series of random numbers subject to a regular probability distribution in the field [0, 1]. The generation of these numbers is based on the cumulative distribution function of the regular distribution, but we encounter many systems that do not, by nature, follow the regular distribution adopted in the simulation process. Therefore, it is necessary to Convert these random numbers into random variables that follow the probability distribution in which the system to be simulated operates. In classical logic, many techniques can be used in the conversion process, which results in random variables that follow irregular probability distributions. However, the results we obtain are specific results that do not take into account the changes that may occur in the system’s operating environment. To obtain more accurate results, we presented in previous research a study to generate neutrosophic random numbers that follow a regular distribution with no specificity that can be enjoyed by both ends of the field [0, 1]. One or both of them together, and for systems that operate according to probability distributions other than the regular distribution defined in the field [0, 1], we have presented some techniques through which we can obtain neutrosophic random variables based on the neutrosophic random numbers that were generated, in this research and using Previous information: We present a neutrosophic study to generate random variables that follow the Erlang distribution, which is one of the most important and widely used distributions in scientific fields.

A New Simheuristic Approach for Stochastic Runway Scheduling

We consider a stochastic, dynamic runway scheduling problem involving aircraft landings on a single runway. Sequencing decisions are made with knowledge of the estimated arrival times (ETAs) of all aircraft due to arrive at the airport, and these ETAs vary according to continuous-time stochastic processes. Time separations between consecutive runway landings are modeled via sequence-dependent Erlang distributions and are affected by weather conditions, which also evolve continuously over time. The resulting multistage optimization problem is intractable using exact methods, and we propose a novel simheuristic approach based on the application of methods analogous to variable neighborhood search in a high-dimensional stochastic environment. Our model is calibrated using flight tracking data for over 98,000 arrivals at Heathrow Airport. Results from numerical experiments indicate that our proposed simheuristic algorithm outperforms an alternative based on deterministic forecasts under a wide range of parameter values, with the largest benefits seen when the underlying stochastic processes become more volatile and also when the on-time requirements of individual flights are given greater weight in the objective function. Funding: This work was supported by the Engineering and Physical Sciences Research Council [Grant EP/M020258/1]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/trsc.2022.0400 .

Investigation of stochastic models of packet generation in computer networks

Stochastic packet generation models are models that are used to generate traffic in computer networks with certain characteristics. These models can be used to simulate network activity and test network performance. Standard data transmission on the network is packet generation with delays, in which packets are sent at certain intervals. Various stochastic models can be used to generate delayed packets, including uniform distribution, exponential distribution, and Erlang distribution. In this work, an experimental setup was assembled and a client-server application was developed to conduct research and analyze the performance of the data transmission channel. An algorithm has been proposed that allows to restore the moment characteristics of a random value of the interval between packets for further use of queuing models. The analysis of the distribution laws on the performance of the experimental network sample was performed and estimates of the efficiency of channel use and the average packet generation time in network segments, as well as histograms of delays according to the distribution laws, were obtained. An experimental setup was created, and a client-server application was developed to analyze the performance of the data transmission channel. An algorithm for restoring the moment characteristics of the time intervals between packets is proposed. The analysis of the distribution laws on network performance was carried out, estimates of the efficiency of channel use and the average packet generation time in network segments were obtained, as well as histograms of delays according to the distribution laws. The generation of packets with delays according to stochastic distribution laws (uniform, exponential, Erlang) is of great importance in modeling and analyzing the operation of network systems. Also, the generation of packets with delays according to the above-mentioned distribution laws allows testing and debugging of network applications and devices in conditions close to real ones. This allows to identify possible problems and improve the operation of network systems. As a result of the experiment, an algorithm was proposed that allows to restore the moment characteristics of a random value of the interval between packets for further use of queuing models. Also, the analysis of the influence of distribution laws on the performance of the experimental network sample was performed and estimates of the efficiency of channel use and the average packet generation time in network segments, as well as histograms of delays according to distribution laws, were obtained.

Elaborated Analysis of a Nonreplenishable Queue with Erlang Distribution of the Service Time

Elaborated Analysis of a Nonreplenishable Queue with Erlang Distribution of the Service Time

Simulation Modeling of QS with Hyper-Erlang and Erlang Distributions

The article is devoted to the construction of simulation models of customer delay in the queue in the form of a queuing system (QS) described by both ordinary and right-shifted hyper-Erlang and second-order Erlang distributions. This article is a logical continuation of previous works devoted to the construction of numerical-analytical QS models with shifted distribution laws. In the article, the Erlang distribution is considered as a special case of the more general Gamma distribution law, in contrast to the normalized Erlang distribution. These two forms of the Erlang distribution differ in numerical characteristics, except for the coefficient of variation. To solve the problem, the system of discrete-event modeling GPSS WORLD was used.

Practical Byzantine Fault Tolerance-Enhanced Blockchain-Enabled Data Sharing System: Latency and Age of Data Package Analysis

Data timeliness, privacy, and security are key enablers for data-sharing systems to support time-sensitive and mission-critical systems and applications. While blockchain-enabled data sharing frameworks can offer reliable security and privacy when properly implemented, the timeliness of data and the related latency are important issues that can limit the adoption of blockchain in large-scale mission-critical applications. This paper thus carried out a performance analysis of the blockchain-enabled data-sharing framework from latency and data age perspectives to investigate the suitability of blockchain technology in data sharing systems. To achieve this, the uniqueness of such systems such as transactions validation latency, transaction generation rate, waiting time, blockchain-appending rate, and overall communication latency were jointly studied. The communication latency was characterized following the spatiotemporal modeling approach. We further adopted the practical Byzantine fault tolerance (PBFT) consensus protocol due to its well discussed suitability in large-scale data sharing applications and captured the validation stages of such a PBFT scheme using the Erlang distribution of order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> . Simulations results show that various influential system parameters must be carefully considered when adopting blockchain technology in time-sensitive data sharing applications. This will guide the adoption of blockchain technology in various data sharing applications and systems.

Coexistence in two-species competition with delayed maturation

Inter- and intraspecific competition is most important during the immature life stage for many species of interest, such as multiple coexisting mosquito species that act as vectors of diseases. Mortality caused by competition that occurs during maturation is explicitly modelled in some alternative formulations of the Lotka–Volterra competition model. We generalise this approach by using a distributed delay for maturation time. The kernel of the distributed delay is represented by a truncated Erlang distribution. The shape and rate of the distribution, as well as the position of the truncation, are found to determine the solution at equilibrium. The resulting system of delay differential equations is transformed into a system of ordinary differential equations using the linear chain approximation. Numerical solutions are provided to demonstrate cases where competitive exclusion and coexistence occur. Stability conditions are determined using the nullclines method and local stability analysis. The introduction of a distributed delay promotes coexistence and survival of the species compared to the limiting case of a discrete delay, potentially affecting management of relevant pests and threatened species.

Modelling the Reliability of Logistics Flows in a Complex Production System

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 TP and the duration of breakdown TB 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.

A Rényi-Type Limit Theorem on Random Sums and the Accuracy of Likelihood-Based Classification of Random Sequences with Application to Genomics

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.

Redundancy Allocation of Components with Time-Dependent Failure Rates

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.

Excitation Functions of Related Temperatures of η and η0 Emission Sources from Squared Momentum Transfer Spectra in High-Energy Collisions

The squared momentum transfer spectra of η and η0, produced in high-energy photon–proton (γp) →η(η0)+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 γ to emitted η(η0) 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 (pT) spectra of emitted particles. Our results show that the average transverse momentum (⟨pT⟩), the initial-state temperature (Ti), and the final-state temperature (T0) 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.

Bayesian Approach for estimating the unknown Scale parameter of Erlang Distribution Based on General Entropy Loss Function

We are used Bayes estimators for unknown scale parameter when shape Parameter is known of Erlang distribution. Assuming different informative priors for unknown scale parameter. We derived The posterior density with posterior mean and posterior variance using different informative priors for unknown scale parameter which are the inverse exponential distribution, the inverse chi-square distribution, the inverse Gamma distribution, and the standard Levy distribution as prior. And we derived Bayes estimators based on the general entropy loss function (GELF) is used the Simulation method to obtain the results. we generated different cases for the parameters of the Erlang model, for different sample sizes. The estimates have been compared in terms of their mean-squared error (MSE). We concluded that the best estimators of the scale parameterof the Erlang distribution, based on GELF for the shape parameter (c=1,2,3) under inverse gamma prior with for all samples sizes(n) where the true cases of the Erlang model are and according to the smallest values of MSE