Erlang Distribution Research Articles

Model parameter estimation and residual life prediction for a partially observable failing system

We consider a partially observable degrading system subject to condition monitoring and random failure. The system's condition is categorized into one of three states: a healthy state, a warning state, and a failure state. Only the failure state is observable. While the system is operational, vector data that is stochastically related to the system state is obtained through condition monitoring at regular sampling epochs. The state process evolution follows a hidden semi-Markov model (HSMM) and Erlang distribution is used for modeling the system's sojourn time in each of its operational states. The Expectation-maximization (EM) algorithm is applied to estimate the state and observation parameters of the HSMM. Explicit formulas for several important quantities for the system residual life estimation such as the conditional reliability function and the mean residual life are derived in terms of the posterior probability that the system is in the warning state. Numerical examples are presented to demonstrate the applicability of the estimation procedure and failure prediction method. A comparison results with hidden Markov modeling are provided to illustrate the effectiveness of the proposed model. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 190–205, 2015

ON A GENERALIZATION OF THE STATIONARY EXCESS OPERATOR

We show that overshoots over Erlang random variables give rise to a natural generalization of the stationary excess operator and its iterates. The new operators can be used to derive expansions for the expectation Eg(X) of a non-negative random variable, similar to Taylor-like expansions encountered when studying stationary excess operators.

On Two Modifications of Er/Es/1/m Queuing System Subject to Disasters

The paper deals with modelling a finite single-server queuing system with the server subject to disasters. Inter-arrival times and service times are assumed to follow the Erlang distribution defined by the shape parameter r or s and the scale parameter rλ or sμ respectively. We consider two modifications of the model − server failures are supposed to be operate-independent or operate-dependent. Server failures which have the character of so-called disasters cause interruption of customer service, emptying the system and balking incoming customers when the server is down. We assume that random variables relevant to server failures and repairs are exponentially distributed. The constructed mathematical model is solved using Matlab to obtain steady-state probabilities which we need to compute the performance measures. At the conclusion of the paper some results of executed experiments are shown.

On the uniform random upper bound family of first significant digit distributions

Abstract The first significant digit patterns arising from a mixture of uniform distributions with a random upper bound are revisited. A closed-form formula for its first significant digit distribution (FSD) is obtained. The one-parameter model of Rodriguez is recovered for an extended truncated Pareto mixing distribution. Considering additionally the truncated Erlang, gamma and Burr mixing distributions, and the generalized Benford law, for which another probabilistic derivation is offered, we study the fitting capabilities of the FSD's for various Benford like data sets from scientific research. Based on the results, we propose the general use of a fine structure index for Benford's law in case the data is well fitted by the truncated Erlang member of the uniform random upper bound family of FSD's.

Fractal analysis of rainfall event duration for microwave and millimetre networks: rain queueing theory approach

Rain fade in radio networks is generated from random fluctuations of rainfall rates, within rain events of spatiotemporal dimensions. These events can be represented as a catenation of single rain spikes occurring as a possible three-stage process – birth, overlap and death. Using the queueing theory approach, the birth–death characteristics of single spikes are investigated as inter-arrival and service time distributions. A total of 548 spike samples from rainfall events in Durban (29°52'S, 30°58'E), South Africa are examined based on distrometer measurements. Rainfall regime analysis of drizzle, widespread, shower and thunderstorm bounds is applied to determine the queue pattern. It is found that the queue patterns in Durban exhibit an Erlang-k distribution (Ek) for both the service and overlap times, while exponential distribution (M) is suitable for inter-arrival time. The mean error statistics for the regimes give root-mean-square errors of 0.64, 1.3 and 2.02% for the service, inter-arrival and overlap distribution, respectively, with acceptable Chi-Squared (χ2) statistics. The M/Ek/s/∞ steady-state analysis is later undertaken to investigate the performance of the proposed queue system. Based on the overall data, a power–law relationship is found to exist between the service time and peak rain rate per spike.

Modeling correlated frequencies with application in operational risk management

In this paper, we propose a copula-free approach for modeling correlated frequency distributions using an Erlang-based multivariate mixed Poisson distribution. We investigate some of the properties possessed by this class of distributions and derive a tailormade expectation-maximization algorithm for fitting purposes. The applicability of the proposed distribution is illustrated in an operational risk management context, where this class is used to model the operational loss frequencies and their complex dependence structure in a high-dimensional setting. Furthermore, by assuming that operational loss severities follow the mixture of Erlang distributions, our approach leads to a closed-form expression for the total aggregate loss distribution and its value-at-risk can be calculated easily by any numerical method. The efficiency and accuracy of the proposed approach are analyzed using a modified real operational loss data set.

M/E<sub>n</sub>/1 대기모형에서 얼랑분포의 성질을 이용한 오버슛의 분포에 대한 근사

본 논문은 <TEX>$M/E_n/1$</TEX> 대기모형에서 작업부하량과정의 오버슛의 분포함수에 대한 근사식을 제안한다. 오버슛이란 작업부하량과정이 미리 정해진 한계점을 처음으로 초과할 때 초과하는 양을 말하는데 정확한 분포함수는 수학적인 표현으로만 얻어졌을 뿐 분포함수를 실제로 계산하는 것은 거의 불가능하다. 그래서 기존 연구에서는 오버슛에 관한 몇가지 성질을 이용하여 오버슛의 분포함수에 대한 근사식이 구해졌다. 본 논문은 고객의 서비스시간의 분포가 얼랑분포라는 점을 활용하여 기존에 얻어진 근사식보다 더 정확한 근사식을 제안한다. 그리고 제안한 근사식이 얼마나 참값에 가까운지 판단하기 위하여 시뮬레이션을 통하여 얻어진 오버슛의 분포함수와 비교한다. We consider an <TEX>$M/E_n/1$</TEX> queueing model where customers arrive at a facility with a single server according to a Poisson process with customer service times assumed to be independent and identically distributed with Erlang distribution. We concentrate on the overshoot of the workload process in the queue. The overshoot means the excess over a threshold at the moment where the workload process exceeds the threshold. The approximation of the distribution of the overshoot was proposed by Bae et al. (2011); however, but the accuracy of the approximation was unsatisfactory. We derive an advanced approximation using the property of the Erlang distribution. Finally the newly proposed approximation is compared with the results of the previous study.

Development of a Hybrid Prediction Mechanism using SMA and EXS Methods for GSM Logical Channel Load Variables

GSM logical channel load are stochastic (random), distinct in time (Erlang) distribution data; and as such it requires robust means of its prediction. The method employed in this work for the predictions is a hybrid of Simple Moving Average (SMA) and Exponential Smoothing (ExS), which can fit in to predict logical channel load variables with it peculiarities. A three (3) month Data were used in determining the number of observations for the prediction (n) for SMA and smoothing constant (α) for ExS. The determinant values obtained are n = 28, and α = 0.077. These values are used to predict the logical control and traffic channels load variables that characterizes its utilization.

Multivariate Mixtures of Erlangs for Density Estimation Under Censoring and Truncation

Multivariate mixtures of Erlang distributions form a versatile, yet analytically tractable, class of distributions making them suitable for multivariate density estimation. We present a flexible and effective fitting procedure for multivariate mixtures of Erlangs, which iteratively uses the EM algorithm, by introducing a computationally efficient initialization and adjustment strategy for the shape parameter vectors. We furthermore extend the EM algorithm for multivariate mixtures of Erlangs to be able to deal with censored and truncated data. The effectiveness of the proposed algorithm, which has been implemented in R, is demonstrated on simulated as well as real data sets.The Addendum for this paper are available at the following URL: http://ssrn.com/abstract=2593016

A Study of Transverse Momentum Distributions of Jets Produced inp-p,p-p-,d-Au, Au-Au, and Pb-Pb Collisions at High Energies

The transverse momentum distributions of jets produced inp-p,p-p-,d-Au, Au-Au, and Pb-Pb collisions at high energies with different selected conditions are analyzed by using a multisource thermal model. The multicomponent (mostly two-component) Erlang distribution used in our description is in good agreement with the experimental data measured by the STAR, D0, CDF II, ALICE, ATLAS, and CMS Collaborations. Related parameters are extracted from the transverse momentum distributions and some information on different interacting systems is obtained. In the two-component Erlang distribution, the first component has usually two or more sources which are contributed by strong scattering interactions between two quarks or more quarks and gluons, while the second component has mostly two sources which are contributed by harder head-on scattering between two quarks.

Continuous-time Random Walk Model of Relaxation of Two-state Systems

Using the continuous-time random walk (CTRW) approach, we study the phenomenon of relaxation of two-state systems whose elements evolve according to a dichotomous process. Two characteristics of relaxation, the probability density function of the waiting times difference and the relaxation law, are of our particular interest. For systems characterized by Erlang distributions of waiting times, we consider different regimes of relaxation and show that, under certain conditions, the relaxation process can be non-monotonic. By studying the asymptotic behavior of the relaxation process, we demonstrate that heavy and superheavy tails of waiting time distributions correspond to slow and superslow relaxation, respectively.

A finite-buffer queue with a single vacation policy: An analytical study with evolutionary positioning

Abstract In this paper, application of an evolutionary strategy to positioning a GI/M/1/N-type finite-buffer queueing system with exhaustive service and a single vacation policy is presented. The examined object is modeled by a conditional joint transform of the first busy period, the first idle time and the number of packets completely served during the first busy period. A mathematical model is defined recursively by means of input distributions. In the paper, an analytical study and numerical experiments are presented. A cost optimization problem is solved using an evolutionary strategy for a class of queueing systems described by exponential and Erlang distributions.

Minimum–Maximum regret redundancy allocation with the choice of redundancy strategy and multiple choice of component type under uncertainty

This paper deals with a redundancy allocation problem in series–parallel systems with the choice of redundancy strategy, including active and cold standby strategies in which component’s time to failure follows an Erlang distribution. The scale parameter of Erlang distribution and consequently the reliability of each component are imprecise in terms of interval data, and only the lower and upper bounds are known. This problem, for the first time, is formulated through Min–Max regret criterion, which is commonly used to define robust solutions. The resulting problem formulation contains an unlimited number of constraints, and a Benders’ decomposition method is implemented to deal with the given problem. This method is compared with an enumeration method to show its effectiveness. The performance of the proposed model using the Benders’ decomposition method is examined over different problem sizes, and the associated results are analyzed.

Multi-Objective Lead-Time Control Problem with Stochastic Constraints

This research intends to find out any development of a robust multi-objective for lead time optimal control problem in amulti-stage assembly system model. Assembly system modeling is possible by the help of the open queue network. Aworking station includes one or infinite servers and just manufacturing or assembly operations are performed therein.Each part has a separate entry process and independent of each other. It is completely based upon Poissonprocess.Serving Lead Time of Stations are also independent of each other and therefore exponential distribution ofeach parameter is controllable. All stations have bounded uncertain unrecyclable wastes which are completelyindependent in compliance with Erlang distribution. Uncertainty in the problem parameters has been suggested asrobust multi-objective optimal control model in which we have three incompatible target functions including cyclicoperation cost minimization, average lead time minimization and lead time variance. Finally, target progress methodhas been applied in order to achieve serving optimal speeds and solve discrete time of the main problemapproximately. The proposed model could present a suitable solution even for the same problem as mentioned inother related papers along with some considerable results in parameter uncertainty conditions.

ON SOME PROPERTIES OF A CLASS OF MULTIVARIATE ERLANG MIXTURES WITH INSURANCE APPLICATIONS

AbstractWe discuss some properties of a class of multivariate mixed Erlang distributions with different scale parameters and describes various distributional properties related to applications in insurance risk theory. Some representations involving scale mixtures, generalized Esscher transformations, higher-order equilibrium distributions, and residual lifetime distributions are derived. These results allows for the study of stop-loss moments, premium calculation, and the risk allocation problem. Finally, some results concerning minimum and maximum variables are derived and applied to pricing joint life and last survivor policies.

Pharmacokinetic tools for the dose adjustment of ciclosporin in haematopoietic stem cell transplant patients

Ciclosporin A (CsA) is used in the prophylaxis and treatment of acute and chronic graft vs. host disease after haematopoietic stem cell (HSCT) transplantation. Our objective was to build and compare three independent Bayesian estimators of CsA area under the curve (AUC) using a limited sampling strategy (LSS), to assist in dose adjustment. The Bayesian estimators were developed using in parallel: two independent parametric modelling approaches (nonmem® and iterative two stage (ITS) Bayesian modelling) and the non-parametric adaptive grid method (Pmetrics®). Seventy-two full pharmacokinetic profiles (at pre-dose and 0.33, 0.66, 1, 2, 3, 4, 6, 8 and 12h after dosing) collected from 40 HSCT patients given CsA were used to build the pharmacokinetic models, while 15 other profiles (n = 7) were kept for validation. For each Bayesian estimator, AUCs estimated using the full profiles were compared with AUCs estimated using three samples. The pharmacokinetic profiles were well fitted using a two compartment model with first order elimination, combined with a gamma function for the absorption phase with ITS and Pmetrics or an Erlang distribution with nonmem. The derived Bayesian estimators based on a C0-C1 h-C4 h sampling schedule (best LSS) accurately estimated CsA AUC(0,12 h) in the validation group (n = 15; nonmem: bias (mean ± SD)/RMSE 2.05% ± 13.31%/13.02%; ITS: 4.61% ± 10.56%/11.20%; Pmetrics: 0.30% ± 10.12%/10.47%). The dose chosen confronting the three results led to a pertinent dose proposal. The developed Bayesian estimators were all able to predict ciclosporin AUC(0,12 h) in HSCT patients using only three blood with minimal bias and may be combined to increase the reliability of CsA dose adjustment in routine.

Recursive Calculation of Ruin Probabilities at or Before Claim Instants for Non-Identically Distributed Claims

In this paper, we consider a continuous time risk process for which the claim number process extends the classical Poisson process, and the claim sizes are independent, Erlang random variables, but not necessarily identically distributed. For this process, we obtain recursive formulas that allow us to evaluate the ruin probability at or before a certain claim instant. We illustrate the resulting algorithm in a numerical study.

Interval programming for the redundancy allocation with choices of redundancy strategy and component type under uncertainty: Erlang time to failure distribution

Engineering design problems involve meeting system’s performance requirements such as reliability which is in close relation to appropriate selection of components and determination of a system level architecture. In this paper a new nonlinear redundancy allocation model with the choice of redundancy strategy and component types is presented. The redundancy strategies can be in forms of active and cold standby. Also there is a choice for no redundancy strategy. We assume that time to failure follows Erlang distribution which is parameterized in terms of shape and scale parameters which are usually determined deterministically. In this paper we release the deterministic assumption of the scale parameter since in reality the estimation of this parameter accompanies with uncertainty. Also, the cost and weights are not known exactly due to estimation uncertainties and fluctuations. The uncertain proposed model maximizes the system’s reliability, which is converted into a deterministic multi-objective model maximizing left bound and center of intervals. Finally, a numerical example is presented to show the performance of the proposed method.

Finding optimal hyperpaths in large transit networks with realistic headway distributions

This paper implements and tests a label-setting algorithm for finding optimal hyperpaths in large transit networks with realistic headway distributions. It has been commonly assumed in the literature that headway is exponentially distributed. To validate this assumption, the empirical headway data archived by Chicago Transit Agency are fitted into various probabilistic distributions. The results suggest that the headway data fit much better with Loglogistic, Gamma and Erlang distributions than with the exponential distribution. Accordingly, we propose to model headway using the Erlang distribution in the proposed algorithm, because it best balances realism and tractability. When headway is not exponentially distributed, finding optimal hyperpaths may require enumerating all possible line combinations at each transfer stop, which is tractable only for a small number of alternative lines. To overcome this difficulty, a greedy method is implemented as a heuristic and compared to the brute-force enumeration method. The proposed algorithm is tested on a large scale CTA bus network that has over 10,000 stops. The results show that (1) the assumption of exponentially distributed headway may lead to sub-optimal route choices and (2) the heuristic greedy method provides near optimal solutions in all tested cases.

STATISTICAL RESEARCH WITHIN THE PERFORMANCE TESTING IN THE CRITICAL TRANSPORT INFRASTRUCTURE

In the last years crisis management is dealing with many important problems. The current threats of countries, inhabitants, property and natural environment belongs to the most important issues. The extreme weather negative impacts and adverse consequences caused by accidents in technological processes and transport have become more and more frequent and intense events. In addition danger of extremism and terrorism, aimed at important object, is also expanding. In developed countries these objects are parts of critical infrastructure. Critical infrastructure sectors in different countries are not unified. In most of the countries the critical infrastructure includes information technologies, energy and transportation. In Slovakia the transportation sector consists of subsectors – road, water, aviation and railway transport. The railway transport is an important subsector of the critical infrastructure. Its importance is unquestionable especially in the field of mass transportation and transportation of bulk substrata. Traditional methods, which were used in the past, were based on experience. The actual possibilities of computer technique enable to explore the railway transport performance through expert information systems. They are based on detailed statistical research. The paper will present the results of the researchers of the Faculty of Special Engineering, University of Žilina focused on the selection and testing the suitable distribution for stochastic events in railway transport. Based on the extensive statistical research the authors demonstrate that the driving time of trains in line sections is more suitable to be considered as random variable with Erlang distribution. The mentioned theoretical findings were used in development of software tool ASTRA that simulates railway transport in emergency situations. Keywwords: Critical transport infrastructure, statistical research, risk management.