Erlang Distribution Research Articles

MODELING OF PASSENGER FLOW AT CITY RAILWAY TRANSPORT STATIONS

In this article, model of passenger flow distribution in regard to their waiting time at city railway transport stations is constructed. For modelling, developed method is used, in which is fill that the distribution of passengers in regard to waiting time is consistent with function of Erlang distribution of k order. Taken as a basis the results of experiment that was carried out on the existing network of city railway transport in Kiev. The matching of the empirical distribution with the theoretical is carried out, to obtain validity check of practical results and appropriateness of using the considered method. We determined the value of waiting time for most users of this type of public transport with the help of accumulative graphic the passenger flow distribution.

Stochastic Model Based Opportunistic Channel Access in Dynamic Spectrum Access Networks

We present a stochastic model-based opportunistic channel access and transmission scheme for cognitive radio enabled secondary users for single data channel. We refer to this scheme as Residual Idle Time Distribution based Scheme (RIBS). In this scheme, the SU randomly senses the channel and if the channel is sensed idle, then it uses residual idle time distribution to estimate its transmission duration such that the probability of its interference with PU is below a predefined threshold. We derive analytical formulae for various performance metrics of SU and validate them through simulations. Simulation experiments are conducted for two different scenarios. In the first scenario, we use synthetically generated channel occupancy data due to PU transmissions using two standard distributions (2-phase Erlang and Uniform distribution). In the second scenario, we use data collected by simulating a TDMA-based PU network that runs realistic applications (VoIP and Web browsing). A pair of sender and receiver SU uses RIBS to opportunistically transmit on the channel. Our simulation experiments show that SU can use RIBS to opportunistically transmit on the channel without violating the interference probability constraint. We also list some of the challenges in using RIBS in realistic scenarios, and provide a comprehensive methodology to use RIBS in such scenarios.

Failure Time of Non Homogeneous Gamma Process

We consider the non homogeneous gamma process as a degradation model. The hitting time of a deterministic or random level is studied here. We provide its distribution (both cdf and pdf) explicitly in the first case and in the second case when the threshold is exponentially or gamma distributed. The general case for a random threshold can be approximated by considering mixtures of Erlang distributions. Aging properties are also discussed in this article.

COORDINATED PRICING AND INVENTORY CONTROL WITH BATCH PRODUCTION AND ERLANG LEADTIMES

This paper addresses a joint pricing and inventory control problem for a batch production system with random leadtimes. Assume that demand arrives according to a Poisson process with a price-dependent arrival rate. Each replenishment order contains a single batch of a fixed lot size. The replenishment leadtime follows an Erlang distribution, with the number of completed phases recording the delivery state of outstanding orders. The objective is to determine an optimal inventory-pricing policy that maximizes total expected discounted profit or long-run average profit. We first show that when there is at most one order outstanding at any point in time and that excess demand is lost, the optimal reorder policy can be characterized by a critical stock level and the optimal pricing decision is decreasing in the inventory level and delivery state. We then extend the analysis to mixed-Erlang leadtime distribution which can be used to approximate any random leadtime to any degree of accuracy. We further extend the analysis to allowing three outstanding orders where the optimal reorder point becomes state-dependent: the closer an outstanding order is to its arrival or the more orders are outstanding, the lower selling price is charged and the lower reorder point is chosen. Finally, we address the backlog case and show that the monotone pricing structure may not be true when the optimal reorder point is negative.

Bayesian and Bühlmann credibility for phase-type distributions with a univariate risk parameter

Credibility theory is a statistical tool to calculate the premium for the next period based on past claims experience and the manual rate. Each contract is characterized by a risk parameter. A phase-type (or PH) random variable, which is defined as the time until absorption in a continuous-time Markov chain, is fully characterized by two sets of parameters from that Markov chain: the initial probability vector and transition intensity matrix. In this article, we identify an interpretable univariate risk parameter from amongst the many candidate parameters, by means of uniformization. The resulting density form is then expressed as an infinite mixture of Erlang distributions. These results are used to obtain a tractable likelihood function by a recursive formula. Then the best estimator for the next premium, i.e. the Bayesian premium, as well as its approximation by the Bühlmann credibility premium are calculated. Finally, actuarial calculations for the Bühlmann and Bayesian premiums are investigated in the context of a gamma prior, and illustrated by simulated data in a series of examples.

Headway Distributions Based on Empirical Erlang and Pearson Type III Time Methods Compared

The study is aimed at exploring empirical headway derived from Erlang and Pearson type III distribution and comparing the resultant distributions. The Erlang distribution is a simplified form of the Pearson type III. In circumstance where vehicle overtaking is very limited and the ensuing headway values are small, Pearson type IIIor the Erlang distribution may be used to represent the headway frequency distribution. Based on the hypothesis that headways derived from Erlang and Pearson type III distribution will not differ significantly, headway study was carried out in Kulai, Malaysia under dry and daylight conditions. Twenty four hour volume, speed and headways and vehicle type data were collected continuously for eight weeks. The survey data were supplemented with highway design information culled from the Malaysian Design Manual. Empirical headway data were computed using Pearson type III and Erlang distribution methods. Results show that the difference between estimated headways using both techniques is very marginal. The study concluded that in cases where overtaking induces minimal headways, the Pearson type III or the Erlang distribution may be used to represent the headway frequency distribution. However, Pearson type III distribution has to be modified in circumstances where the probability of observing headway increases.

Unified description of transverse momentum spectrums contributed by soft and hard processes in high-energy nuclear collisions

The transverse momentum spectrums of π −, π +, K −, K +, $$\bar p$$ , and p produced in p-Pb collisions at √s NN = 502 TeV measured by the CMS Collaboration and in Pb-Pb collisions at √s NN = 2.76 TeV measured by the ALICE Collaboration are described by a two-component Erlang distribution. The first component corresponds to “soft” excitation process and contributes in the low transverse momentum region, which is contributed by 2–5 partons (sea quarks and gluons) with strong interactions. The second component corresponds to “hard” scattering process and contributes in the high transverse momentum region, which is contributed by 2 partons (valent quarks) with violent head-on collision. Each parton source contributes an exponential transverse momentum spectrum. Both the soft and hard processes result in an Erlang distribution. The transverse momentum spectrums of final-state charged particles are then described by the two-component Erlang distribution. The contribution ratio (30–40%) of the hard process extracted from nuclear collisions at the Large Hadron Collider is consistent with that (17–46%) obtained from other methods.

THE NEGATIVE BINOMIAL-ERLANG DISTRIBUTION WITH APPLICATIONS

This paper introduces a new three-parameter of the mixed negative binomial distribution which is called the negative binomial-Erlang distribution. This distribution obtained by mixing the negative binomial distribution with the Erlang distribution. The negative binomial-Erlang distribution can be used to describe count data with a large number of zeros. The negative binomial- exponential is presented as special cases of the negative binomial-Erlang distri- bution. In addition, we present some properties of the negative binomial-Erlang distribution including factorial moments, mean, variance, skewness and kurto- sis. The parameter estimation for negative binomial-Erlang distribution by the maximum likelihood estimation are provided. Applications of the the nega- tive binomial-Erlang distribution are carried out two real count data sets. The result shown that the negative binomial-Erlang distribution is better than fit when compared the Poisson and negative binomial distributions.

형상모수를 고려한 소프트웨어 신뢰성 비용 모형에 관한 비교 연구

본 연구에서는 소프트웨어 제품 테스팅 과정에서 고장 수명분포의 형상모수를 고려한 소프트웨어 신뢰성 비용 모형에 대하여 연구 하였다. 신뢰성 분야에서 많이 사용되는 어랑 분포와 로그-로지스틱 모형을 이용한 형상모수를 반영한 문제를 제시하였다. 소프트웨어 고장모형은 유한고장 비동질적인 포아송과정을 이용하고 모수추정법은 최우추정법을 이용 하였다. 따라서 본 논문에서는 형상모수를 고려한 소프트웨어 비용모형 분석을 위하여 소프트웨어 고장 시간 자료를 적용하여 비교 분석하였다. 본 연구에서 사용된 어랑 분포와 로그-로지스틱분포에 근거한 소프트웨어 비용 모델을 비교한 결과 어랑 모형은 최적의 소프트웨어 방출 시간을 예측 할 수 있지만 로그-로지스틱 모형은 방출시간을 예측 할 수 없기 때문에 로그-로지스틱 보다 어랑 모형이 보다 효율적으로 나타나고 있다. 이 연구를 통하여 소프트웨어 개발자들은 소프트웨어 개발 비용을 파악 하는데 어느 정도 도움을 줄 수 있을 것으로 사료된다. In this study, reliability software cost model considering shape parameter based on life distribution from the process of software product testing was studied. The shape parameter using the Erlang and Log-logistic model that is widely used in the field of reliability problems presented. The software failure model was used finite failure non-homogeneous Poisson process model, the parameters estimation using maximum likelihood estimation was conducted. In comparison result of software cost model based on the Erlang distribution and the log-logistic distribution software cost model, because Erlang model is to predict the optimal release time can be software, but the log-logistic model to predict to optimal release time can not be, Erlang distribution than the log-logistic distribution appears to be effective. In this research, software developers to identify software development cost some extent be able to help is considered.

Fitting Mixtures of Erlangs to Uncensored and Untruncated Data Using the EM Algorithm - Addendum

In this addendum, we present the EM algorithm of Lee and Lin (2010) customized for fitting mixtures of Erlang distributions with a common scale parameter to uncensored and untruncated data. We work out the details with zero-one component indicators inspired by McLachlan and Peel (2001) and Lee and Scott (2012).

The Gerber-Shiu Discounted Penalty Function of Sparre Andersen Risk Model with a Constant Dividend Barrier

This paper constructs a Sparre Andersen risk model with a constant dividend barrier in which the claim interarrival distribution is a mixture of an exponential distribution and an Erlang(n) distribution. We derive the integro-differential equation satisfied by the Gerber-Shiu discounted penalty function of this risk model. Finally, we provide a numerical example.

Modeling trailer-traffic flow on a Kenyan highway

Traffic flow modelling is studied in order to ease congestion on roads. However, congestion is also caused by irregular occurrences, such as traffic accidents, poor roads, vehicle disablement, spilled loads and hazardous materials. This study explored the area of poor roads which was considered to be as a result poor planning for road network repairs. Traffic flow was categorized to be either in low, intermediate, or high volume. Modelling for every flow required a different distribution namely the Exponential, Pearson type III and Normal distributions for low, intermediate and high traffic flow volume respectively. However, we modelled the trailers traffic flow using a model that covered all the 3 states, in this case,the Pearson type III distribution. Further,we investigated the shape of the probability distribution function assumed by the trailers. After data collection,extraction and analysis, Pearson type III distribution model was calibrated to fit. Lastly, Kolmogorov-Smirnov and Chi-square tests of goodness of fit was run on the observed data. Although, the data was also fitted into Normal, Erlang, Exponential, Beta and Gamma Distributions, Pearson type III distribution provided the best fit. Mathematics Subject Classification: 49K05 Keywords: Traffic flow, Modelling, Congestion, Road network, Trailers, Probability distribution Language: en

Double-Differential Production Cross Sections of Charged Pions in Charged Pion Induced Nuclear Reactions at High Momentums

The double-differentialπ±production cross sections in interactions of charged pions on targets at high momentums are analyzed by using a multicomponent Erlang distribution which is obtained in the framework of a multisource thermal model. The calculated results are compared and found to be in agreement with the experimental data at the incident momentums of 3, 5, 8, and 12 GeV/c measured by the HARP Collaboration. It is found that the source contributions to the mean momentum of charged particles and to the distribution width of particle momentums decrease with increase of the emission angle, and the source number and temperature do not show an obvious dependence on the emission angle of the considered particle.

Comparing Multicomponent Erlang Distribution and Lévy Distribution of Particle Transverse Momentums

The transverse momentum spectrums of final-state products produced in nucleus-nucleus and proton-proton collisions at different center-of-mass energies are analyzed by using a multicomponent Erlang distribution and the Lévy distribution. The results calculated by the two models are found in most cases to be in agreement with experimental data from the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC). The multicomponent Erlang distribution that resulted from a multisource thermal model seems to give a better description as compared with the Lévy distribution. The temperature parameters of interacting system corresponding to different types of final-state products are obtained. Light particles correspond to a low temperature emission, and heavy particles correspond to a high temperature emission. Extracted temperature from central collisions is higher than that from peripheral collisions.

Probability Law and Flow Function of Brownian Motion Driven by a Generalized Telegraph Process

We consider a standard Brownian motion whose drift alternates randomly between a positive and a negative value, according to a generalized telegraph process. We first investigate the distribution of the occupation time, i.e. the fraction of time when the motion moves with positive drift. This allows to obtain explicitly the probability law and the flow function of the random motion. We discuss three special cases when the times separating consecutive drift changes have (i) exponential distribution with constant rates, (ii) Erlang distribution, and (iii) exponential distribution with linear rates. In conclusion, in view of an application in environmental sciences we evaluate the density of a Wiener process with infinitesimal moments alternating at inverse Gaussian distributed random times.

Fractional Moments of Solutions to Stochastic Recurrence Equations

In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equationXt=AtXt−1+Bt,t∈Z, where ((At,Bt))t∈Zis an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X0|p,p∈R. Special attention is given to the case whenBthas an Erlang distribution. We provide various approximations to the moments E|X0|pand show their performance in a small numerical study.

Fractional Moments of Solutions to Stochastic Recurrence Equations

In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equationXt=AtXt−1+Bt,t∈Z, where ((At,Bt))t∈Zis an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X0|p,p∈R. Special attention is given to the case whenBthas an Erlang distribution. We provide various approximations to the moments E|X0|pand show their performance in a small numerical study.

Research on the Operational Characteristics of Diverging Area on Expressway Off-ramp based on Different Restricted Strategy of Lane

In order to find the operational characteristics of diverging area on expressway off-ramp with Vehicle Type Restricted Strategy(VTRS) and Vehicle Type Unrestricted Strategy(VTUS) in different lane, this paper analyzed the distribution of vehicle type in the each lane, in use of the data which is investigated from diverging area on the two expressway off-ramp with different operational strategy of lane. Furthermore, the χ 2 test technology was used to do the hypothesis test on the headway distribution. The result shows that the shift-negative- exponential distribution can well reflect the headway distribution of deceleration lane, and the first-order or thesecond-order Erlang distribution can well reflect the headway distribution of mainline. Finally, the conclusion can provide an important theoretical basis for the freeway off-ramp planning, designing, and management.

Fitting the Pht/Mt/s/c Time-Dependent Departure Process for Use in Tandem Queueing Networks

This paper considers time-dependent Pht/Mt/s/c queueing nodes and small tandem networks of such nodes. We examine characteristics of the departure processes from a multiserver queueing node; in particular, we focus on solving for the first two time-dependent moments of the departure-count process. A finite set of partial moment differential equations is developed to numerically solve for the departure-count moments over specified intervals of time [ti, ti + τi). We also present a distribution fitting algorithm to match these key characteristics with a [Formula: see text] process serving as the approximate departure process. A distribution fitting algorithm is presented for time-dependent point processes where a two-level balanced mixture of Erlang distribution is used to serve as the approximating process. We then use the [Formula: see text] approximating departure process as the approximate composite arrival process to downstream node(s) in a network of tandem queues.

Sampling of piecewise constant stochastic processes with smooth distributions of stay time in states

In the solution of sampling problem, a requirement is that the probability of skipping a state must not exceed a prescribed value. A general solution is obtained that makes it possible to construct approximate procedures of various accuracy. Formulas for the Erlang distribution of the stay times and formulas based on the recalculation of the first two moments are obtained. Computed characteristics are presented and an illustrative example is discussed.