Erlang Distribution Research Articles

Optimizing Erlangization-based approximations for finite discrete distributions and discrete phase-type distributions

In He et al.[8], continuous phase-type (PH) distributions are constructed to approximate finite discrete probability distributions and discrete PH-distributions. The approximations are based on Erlangization with a fixed number of phases. In this article, we first introduce continuous PH approximations with Erlang distributions of different orders. Then we develop an algorithm to find the continuous PH approximation with the minimum variance, among all such PH approximations with the same total number of phases. Thus, the proposed continuous PH approximations lead to a smaller gap between the variances of the Erlangization-based approximations and the original discrete random variables, which is achieved without adding more phases. The new approximations are useful to mitigate the burden in computation caused by the large number of phases needed in Erlangization approximation. Stochastic dominance is shown between the original (discrete) distributions and the approximations, which leads to bounds on the quantities of original distributions and/or stochastic models (e.g., reliability models). The approximation method is applied to analyze reliability models and a COVID-19 isolation program.

Features of Analytical Modeling of QS with Hypererlang and Erlang Distributions

The article is devoted to the construction of a mathematical model for delaying claims in a queue in the form of a queuing system (QS) described by right-shifted hyper-Erlang and second-order Erlang distributions. This article is a logical continuation of the previous works of the author. 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, we used the method of spectral solution of the Lindley integral equation.

On The Product and Ratio of Pareto and Erlang Random Variables

The distributions of products and ratios of random variables are of interest in many areas of the sciences.In this paper, we find analytically the probability distributions of the product XY and the ratio X/Y ,when X and Y are two independent random variables following Pareto and Erlang distributions, respectively

DEVELOPMENT OF A MODIFIED METHOD OF NETWORK TRAFFIC FORMING

Based on the analysis of the statistical characteristics of heterogeneous network traffic of "Quadruple Play" mobile subscribers, it is shown that it cannot be represented by Poisson or Erlang distributions. It is shown that for such similar traffic, the rate of growth of the required buffer volume increases as the Hurst parameter increases. A method of adaptive traffic formation using the control of the length of intervals of the intensity of the arrival of data packets has been developed. Congestion management is carried out by changing the frequency of the marker generator (GM) on the basis of the results of forecasting the required bandwidth of the system and the required buffer size. The shaper adapts to changes in the length and instantaneous intensity of packet input in real-time.

On Characterization of the Exponential Distribution Via Hypoexponential Distributions

The sum of independent, but not necessary identically distributed, exponential random variables follows a hypoexponential distribution. We focus on a particular case when all but one rate parameters of the exponential variables are identical. This is known as exponentially modified Erlang distribution in molecular biology. We prove a characterization of the exponential distribution, which complements previous characterizations via hypoexponential distribution with all rates different from each other.

Multiple cohort study of hospitalized SARS-CoV-2 in-host infection dynamics: Parameter estimates, identifiability, sensitivity and the eclipse phase profile

Within-host SARS-CoV-2 modelling studies have been published throughout the COVID-19 pandemic. These studies contain highly variable numbers of individuals and capture varying timescales of pathogen dynamics; some studies capture the time of disease onset, the peak viral load and subsequent heterogeneity in clearance dynamics across individuals, while others capture late-time post-peak dynamics. In this study, we curate multiple previously published SARS-CoV-2 viral load data sets, fit these data with a consistent modelling approach, and estimate the variability of in-host parameters including the basic reproduction number, R0, as well as the best-fit eclipse phase profile. We find that fitted dynamics can be highly variable across data sets, and highly variable within data sets, particularly when key components of the dynamic trajectories (e.g. peak viral load) are not represented in the data. Further, we investigated the role of the eclipse phase time distribution in fitting SARS-CoV-2 viral load data. By varying the shape parameter of an Erlang distribution, we demonstrate that models with either no eclipse phase, or with an exponentially-distributed eclipse phase, offer significantly worse fits to these data, whereas models with less dispersion around the mean eclipse time (shape parameter two or more) offered the best fits to the available data across all data sets used in this work. This manuscript was submitted as part of a theme issue on “Modelling COVID-19 and Preparedness for Future Pandemics”.

Analytically Computing the Moments of a Conic Combination of Independent Noncentral Chi-Square Random Variables and Its Application for the Extended Cox–Ingersoll–Ross Process with Time-Varying Dimension

This paper focuses mainly on the problem of computing the γth, γ>0, moment of a random variable Yn:=∑i=1nαiXi in which the αi’s are positive real numbers and the Xi’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 Yn, especially when not all αi’s are equal. We analytically solve this problem by showing that the γth moment of Yn can be expressed in terms of generalized hypergeometric functions. Additionally, we extend our result to computing the γth moment of Yn when Xi is a combination of statistically independent Zi2 and Gi in which the Zi’s are distributed according to normal or Maxwell–Boltzmann distributions and the Gi’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 γth conditional moment. Finally, we conduct Monte Carlo simulations to demonstrate the accuracy and efficiency of our explicit formulas through several numerical tests.

Minimum capital requirement and portfolio allocation for non-life insurance: a semiparametric model with Conditional Value-at-Risk (CVaR) constraint

We present an optimization problem to determine the minimum capital requirement for a non-life insurance company. The optimization problem imposes a non-positive Conditional Value-at-Risk (CVaR) of the insurer’s net loss and a portfolio performance constraint. When expressing the optimization problem in a semiparametric form, we demonstrate its convexity for any integrable random variable representing the insurer’s liability. Furthermore, we prove that the function defining the CVaR constraint in the semiparametric formulation is continuously differentiable when the insurer’s liability has a continuous distribution. We use the Kelley-Cheney-Goldstein algorithm to solve the optimization problem in the semiparametric form and show its convergence. An empirical analysis is carried out by assuming three different liability distributions: a lognormal distribution, a gamma distribution, and a mixture of Erlang distributions with a common scale parameter. The numerical experiments show that the choice of the liability distribution plays a crucial role since marked differences emerge when comparing the mixture distribution with the other two distributions. In particular, the mixture distribution describes better the right tail of the empirical distribution of liabilities with respect to the other two distributions and implies higher capital requirements and different assets in the optimal portfolios.

Randomization and the valuation of guaranteed minimum death benefits

In this article, we focus on death-linked contingent claims (GMDBs) paying a random financial return at a random time of death in the general case where financial returns follow a regime switching model with two-sided phase-type jumps. We approximate the distribution of the remaining lifetime by either a series of Erlang distributions or a Laguerre series expansion, whose capability to fit the tail of the observed mortality data turns out to be much better than the commonly used series of exponential distributions.More precisely, we develop a Laurent series expansion of the discounted Laplace transform of the subordinated process at an Erlang distributed time, which leads to explicit formulae for European-type GMDB as well as related risk measures such as the Value-at-Risk (VaR) and the Conditional-Tail-Expectation (CTE). We further concentrate upon path-dependent GMDBs with lookback features like dynamic fund protection or dynamic withdrawal benefits, by relying on a Sylvester equation approach. The advantage of our approaches is that our results are of semi-closed form, avoiding numerical Fourier inversion or Monte-Carlo simulation, leading to fast evaluation. This is necessary in risk-management, in particularly for nested simulation in the framework of Solvency II. Several numerical experiments are included.Our results have implications beyond life-insurance and GMDBs, namely in all situations where randomization or Erlangization replaces known quantities, like, for example, model parameters, by random variables. In Finance, it is for example well-known that a random maturity time leads to much more convenient valuation formulas that well approximate its non-random counterpart.

Penaksiran Parameter Pada Distribusi Erlang Berdasarkan Metode Maksimum Likelihood Dengan Menggunakan Algoritma Newton Raphson Dan Fisher Scoring

The Erlang distribution is a special case of the Gamma distribution with the k shape parameter and the λ rate parameter. In this study, the parameter estimation of the Erlang distribution was carried out using the Maximum Likelihood method. In maximizing the function, an implicit and non-linear form is obtained, then it is solved using the Newton Raphson algorithm. Apart from Newton Raphon, the estimation of parameters was also carried out using the Fisher Scoring algorithm. The Fisher Scoring algorithm is similar to the Newton Raphson algorithm, the difference is that Fisher Scoring uses an matrix information. The result of parameter estimation in Erlang distribution using Newton Raphson algorithm which is applied to outgoing telephone call data that generated by Matlab R2010a software cannot be done simultaneously. Therefore, the parameter assessment is carried out on the k parameter first, then followed by the λ parameter estimation and the parameter and = 0.6886812 are obtained. Meanwhile, the parameter estimation using the Fisher Scoring algorithm produces an equation that is not different from the Newton Raphson algorithm

Classical and Bayesian Methods in Estimation of a Scale Parameter of an Erlang Distribution

Classical and Bayesian Methods in Estimation of a Scale Parameter of an Erlang Distribution

Optimal Isolation Strategies in an SIR Model with Erlang–Distributed Infectious Period

In classical epidemic models, a major simplification consists in assuming that the infectious period is exponentially distributed. Here, we first attempt to investigate the consequences of relaxing this assumption on the performances of time–variant disease control strategies by using optimal control theory. In the framework of a basic susceptible–infected– removed (SIR) model, an Erlang distribution of the infectious period is considered and optimal isolation strategies are searched for. The objective functional takes into account the cost of the isolation efforts and the sanitary costs due to the incidence of the epidemic outbreak. Applying the Pontryagin's minimum principle, we prove that the problem admits only bang–bang solutions with at most two switches. Finally, by means of numerical simulations, we show how the shape of the optimal solutions is affected by the different distributions of the infectious period, and by the relative weight of the two cost components.

Ruin probabilities as functions of the roots of a polynomial

A new formula for the ultimate ruin probability in the Cramér–Lundberg risk process is provided when the claims are assumed to follow a finite mixture of m Erlang distributions. Using the theory of recurrence sequences, the method proposed here shifts the problem of finding the ruin probability to the study of an associated characteristic polynomial and its roots. The found formula is given by a finite sum of terms, one for each root of the polynomial, and allows for yet another approximation of the ruin probability. No constraints are assumed on the multiplicity of the roots and that is illustrated via a couple of numerical examples.

The effect of screening on the health burden of chlamydia: An evaluation of compartmental models based on person-days of infection.

Sexually transmitted diseases (STDs) are detrimental to the health and economic well-being of society. Consequently, predicting outbreaks and identifying effective disease interventions through epidemiological tools, such as compartmental models, is of the utmost importance. Unfortunately, the ordinary differential equation compartmental models attributed to the work of Kermack and McKendrick require a duration of infection that follows the exponential or Erlang distribution, despite the biological invalidity of such assumptions. As these assumptions negatively impact the quality of predictions, alternative approaches are required that capture how the variability in the duration of infection affects the trajectory of disease and the evaluation of disease interventions. So, we apply a new family of ordinary differential equation compartmental models based on the quantity person-days of infection to predict the trajectory of disease. Importantly, this new family of models features non-exponential and non-Erlang duration of infection distributions without requiring more complex integral and integrodifferential equation compartmental model formulations. As proof of concept, we calibrate our model to recent trends of chlamydia incidence in the U.S. and utilize a novel duration of infection distribution that features periodic hazard rates. We then evaluate how increasing STD screening rates alter predictions of incidence and disability adjusted life-years over a five-year horizon. Our findings illustrate that our family of compartmental models provides a better fit to chlamydia incidence trends than traditional compartmental models, based on Akaike information criterion. They also show new asymptomatic and symptomatic infections of chlamydia peak over drastically different time frames and that increasing the annual STD screening rates from 35% to 40%-70% would annually avert 6.1-40.3 incidence while saving 1.68-11.14 disability adjusted life-years per 1000 people. This suggests increasing the STD screening rate in the U.S. would greatly aid in ongoing public health efforts to curtail the rising trends in preventable STDs.

On a queueing inventory with common life time and reduction sale consequent to increase in age

Consider an inventoried item for which reduction in sales price is declared as the age of the item increases. Decision to maintain sales price at the same level/reduce, is taken at stages 2, · · · , k − 1, k. On the items attaining CLT, they are sold at scrap value, provided items are still left in stock. Customer arrival forms a non-homogenous Poisson process, with rate increasing with each sales price reduction. Service time follows exponential distribution.The items are replenished according to (S, s) policy with positive lead time. Each stage of CLT is iid which follows a Phase type distribution with representation(α, S) of orderm. The k-fold convolution of this distribution is the CLT of the inventoried items. The problem is modelled as a queueing-inventory problem which is a continuous time Markov chain (CTMC). The stationary distribution of this CTMC is computed and various performance measures are discussed. A cost function is constructed to compute the optimal order quantity and reorder level.The model is compared with queueing inventory model in which the CLT follows Erlang Distribution of order k.

Reliability analysis for shock systems based on damage evolutions via Markov processes

AbstractReliability evaluation under shock models has attracted considerable attention in the literature. Existing research on shock models has focused primarily on damage effects of shocks treated as constants after the arrivals of shocks. Due to the presence of self‐healing, deterioration, and variation over time, the damage effects caused by the shocks may have some evolutions. In this work, we focus on arrivals of shocks that follow a continuous‐time renewal process, and a Markov process that describes the evolution of shock damage. In this regard, we develop here two kinds of shock models and then derive system reliabilities under these two kinds of shock models. The asymptotic behavior of the damage evolution process of each shock is then discussed, and the probabilities of each shock eventually disappearing and destroying the system are derived through the use of aggregated stochastic processes. Finally, the results and numerical examples are presented for three special cases, namely: (1) when the inter‐arrival times of the shocks follow exponential distribution, (2) when the inter‐arrival times of the shocks follow Erlang distribution, and (3) when the inter‐arrival times of the shocks follow uniform distribution.

Numerical methods and hypoexponential approximations for gamma distributed delay differential equations.

Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic gamma distributed DDEs have not previously been implemented. Modellers have therefore resorted to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations (ODEs). In this work, we address the lack of appropriate numerical tools for gamma distributed DDEs in two ways. First, we develop a functional continuous Runge-Kutta (FCRK) method to numerically integrate the gamma distributed DDE without resorting to Erlang approximation. We prove the fourth-order convergence of the FCRK method and perform numerical tests to demonstrate the accuracy of the new numerical method. Nevertheless, FCRK methods for infinite delay DDEs are not widely available in existing scientific software packages. As an alternative approach to solving gamma distributed DDEs, we also derive a hypoexponential approximation of the gamma distributed DDE. This hypoexponential approach is a more accurate approximation of the true gamma distributed DDE than the common Erlang approximation but, like the Erlang approximation, can be formulated as a system of ODEs and solved numerically using standard ODE software. Using our FCRK method to provide reference solutions, we show that the common Erlang approximation may produce solutions that are qualitatively different from the underlying gamma distributed DDE. However, the proposed hypoexponential approximations do not have this limitation. Finally, we apply our hypoexponential approximations to perform statistical inference on synthetic epidemiological data to illustrate the utility of the hypoexponential approximation.

Teletraffic analysis of a mobile crowdsensing system: The pedestrian-to-vehicle scenario

Crowdsensing systems are developed in order to use the computational and communication capabilities of registered users to monitor specific variables and phenomena in an opportunistic manner. As such, the Quality of Experience is not easily attained since these systems heavily rely on the user’s behavior and willingness to cooperate whenever an event with certain interest needs to be monitored. In this work, we analyze the data acquisition phase, where pedestrians opportunistically transmit to vehicles to further disseminate it in the city according to their trajectory. This highly dynamic environment (sensors and data sinks are mobile, and the number of users varies according to the region and time) poses many challenges for properly operating a crowdsensing system. We first study the statistical properties of vehicular traffic in different regions of Luxembourg City where pedestrians share their computational resources and send data to passing cars. Then we propose an Erlang distribution to model the vehicles’ dwelling times and develop a Markov chain accordingly. We model the system using two different queues: we use a single server queue to model the vehicle traffic, while we use an infinite server queue system to model the pedestrian traffic.

Ergodicity of Two Class Priority Queues with Preemptive Priority

Well-known results on the ergodicity of queues with preemptive priority were obtained under the assumption that jobs arrive according to the Poisson process. This assumption, however, does not always hold true in practice. In our work we find sufficient ergodicity conditions for queues with two priority classes with a single server, where interarrival times of high-priority jobs have either Erlang or hyperexponential distribution and interarrival times of low-priority jobs and service times of jobs of both classes have arbitrary continuous distributions. We present results in disciplines with loss of an interrupted job and with retrial of an interrupted job.

Spectral decomposition for a QS based delay model with Erlang and hyperexponential distributions

This article is devoted to the study and obtaining a closed-form solution for the average delay of claims in a queue for a QS formed by two flows with Erlang and hyperexponential distributions of the second order for time intervals. As is known, the Erlang distribution ensures the coefficient of variation of the arrival intervals is less than one, and the hyperexponential distribution is greater than one. It is also known that the main characteristic of the QS, the average delay, is related to these coefficients of variations by a quadratic dependence. Studies of G/G/1 systems in queuing theory are topical due to the fact that they are used in modeling data transmission systems for teletraffic analysis. To solve the problem, the method of spectral decomposition of the solution of the Lindley integral equation was used. The spectral decomposition for the system under consideration made it possible to obtain a closed-form solution for the average delay of requests in the queue. For the practical application of the results obtained, the method of moments is used.