Erlang Distribution Research Articles

Characterization and Information Measures of Weighted Erlang Distribution

Characterization and Information Measures of Weighted Erlang Distribution

Modelling censored losses using splicing: A global fit strategy with mixed Erlang and extreme value distributions

In risk analysis, a global fit that appropriately captures the body and the tail of the distribution of losses is essential. Modelling the whole range of the losses using a standard distribution is usually very hard and often impossible due to the specific characteristics of the body and the tail of the loss distribution. A possible solution is to combine two distributions in a splicing model: a light-tailed distribution for the body which covers light and moderate losses, and a heavy-tailed distribution for the tail to capture large losses. We propose a splicing model with a mixed Erlang (ME) distribution for the body and a Pareto distribution for the tail. This combines the flexibility of the ME distribution with the ability of the Pareto distribution to model extreme values. We extend our splicing approach for censored and/or truncated data. Relevant examples of such data can be found in financial risk analysis. We illustrate the flexibility of this splicing model using practical examples from risk measurement.

Discrete and Continuous Distributed Delays in Replicator Dynamics

In this paper, we study evolutionary games and we examine the stability of the evolutionarily stable strategy (ESS) in the continuous-time replicator dynamics with distributed time delays. In many examples, the interactions between individuals take place instantaneously, but their impacts are not immediate and take a certain amount of time which is usually random. In this paper, we study the consequences of distributed delays on the stability of the replicator dynamics. The main results are that (i) under the exponential delay distribution, the ESS is asymptotically stable for any value of the rate parameter; (ii) the necessary and sufficient conditions for the asymptotic stability of the ESS under uniform and Erlang distributions; and (iii) for the discrete distributed delays, we derive a necessary and sufficient delay-independent stability conditions. We illustrate our results with numerical examples from the Hawk–Dove game.

Reliability optimization for non-repairable series-parallel systems with a choice of redundancy strategies: Erlang time-to-failure distribution

This article deals with a new redundancy allocation model for non-repairable series-parallel systems with multiple strategy choices. The proposed model simultaneously determines the type of components, number of active and standby components to maximize system reliability subject to design constraints. Traditionally, due to complexity and difficulty in obtaining the closed form version of system reliability, a convenient lower-bound on system reliability has been widely applied to approximate it. Assuming that switching mechanism time-to-failure is exponentially distributed, the closed form version of the reliability of subsystems with cold standby redundancy is derived analytically for the first time. This is successfully performed using Markov process and solving the relevant set of differential-difference equations. With respect to the obtained formulation, a semi-analytical expression for the reliability of subsystems with mixed redundancy strategy is also extracted. Component time-to-failure is assumed to follow an Erlang distribution which is suitable for most engineering design problems. The presented model is linear and in the form of standard zero-one integer programs and thus using integer programming algorithms guarantees optimal solutions. The computational results of solving a well-known example indicate the high performance of the proposed model in improving system reliability.

Erlang-based desynchronized urban traffic simulation for high-performance computing systems

The problem of simulating micro-scale urban traffic in large environments provides a great opportunity for the utilization of HPC systems. Parallel implementation of this kind of computation (where complex, data-intensive processing must be synchronized) is not straightforward. The simulation presented in this paper is based on the concept of controlled desynchronization of the computations, which does not violate the model. The implementation in the Erlang language uses Erlang distribution mechanisms for building and managing the computing cluster. The presented method makes it possible to simulate 11.5M cars traversing a road system composed of 240K crossroads and 144M possible locations. It provides a linear scalability of up to 19,200 computing cores (where simulation speeds reach 160 steps per second).

Risk Measures of the ERNB Distribution Generated by G-NB Family

This paper provides VaR and CVaR risk measures, calculated for the Erlang-Negative Binomial (ERNB) distribution. The Erlang and negative binomial distributions are given and then ERNB distribution is obtained using a family of univariate distributions which is called G-Negative Binomial (G-NB). It is defined as compounding the negative binomial distribution (NB) with any continuous distribution (G). Here, we use the ERNB distribution obtained as taking Erlang distribution instead of G. In this paper, we focus on the estimation of VaR and CVaR risk measures for this distribution in closed form and the explicit expressions are also presented for some parameter values. The results are portrayed in the figures. In additionally, numerical examples are given to illustrate changing of the risk measure according to some parameters on a real data set of automobile insurance policies.

Modeling Censored Losses Using Splicing: A Global Fit Strategy with Mixed Erlang and Extreme Value Distributions

In risk analysis, a global fit that appropriately captures the body and the tail of the distribution of losses is essential. Modeling the whole range of the losses using a standard distribution is usually very hard and often impossible due to the specific characteristics of the body and the tail of the loss distribution. A possible solution is to combine two distributions in a splicing model: a light-tailed distribution for the body which covers light and moderate losses, and a heavy-tailed distribution for the tail to capture large losses. We propose a splicing model with a mixed Erlang (ME) distribution for the body and a Pareto distribution for the tail. This combines the flexibility of the ME distribution with the ability of the Pareto distribution to model extreme values. We extend our splicing approach for censored and/or truncated data.Relevant examples of such data can be found in financial risk analysis. We illustrate the flexibility of this splicing model using practical examples from risk measurement.

Stochastic Behavior of a Two-Unit Cold Standby Redundant System Under Poisson Shocks

This article investigates the influence of some system parameters on a system consisting of a 2-unit cold standby system with a single repair person. The system failure is due to an external factor like Poisson shocks that occur in different times. The shocks arrive according to a Poisson process, in addition to the fact that its magnitudes are independent random variable that follows a known distribution. The measures of reliability are determined by using a supplementary variable technique and Laplace transform. The sensitivity and relative sensitivity analysis of the reliability characteristics in terms of the system parameters are calculated. There is also a comparison between two-stage Erlang and exponential distribution as a special case.

Preferred time headway of drivers on two-lane highways with heterogeneous traffic

This paper presents an investigation on the adequacy of one of the major performance indicators for two-lane highways, the percent-time-spent-following (PTSF) under changing driver’s behavior in choosing headways. Field study was conducted on a two-lane highway in India that exhibits heterogeneity in its traffic composition and a wide range of flow levels was covered while collecting data. The appropriate headway distributions were selected based on goodness-of-fit test. Exponential or Erlang distributions exhibit their aptness in describing headways up to a volume to capacity ratio of 0.3 and gamma and exponential distributions were compatible for headway data corresponding to a volume to capacity ratio of 0.4 and above. Around 40% drivers were observed to adopt headways 2 s or less at heavy flow when a car-following situation prevails; these headways are considered unsafe in several instances. Further, speed–headway relationships were established to assess a headway threshold beyond which vehicles could be considered free. Empirical investigation indicates a value of about 6 s under such traffic unlike 3 s as considered in PTSF estimation. Another major implication on PTSF is a few drivers move in a platoon by choice even if they have a gap of more than 6 s and passing opportunities.

Stable scheduling of single machine with probabilistic parameters

Abstract We consider a stochastic variant of the single machine total weighted tardiness problem jobs parameters are independent random variables with normal or Erlang distributions. Since even deterministic problem is NP-hard, it is difficult to find global optimum for large instances in the reasonable run time. Therefore, we propose tabu search metaheuristics in this work. Computational experiments show that solutions obtained by the stochastic version of metaheuristics are more stable (i.e. resistant to data disturbance) than solutions generated by classic, deterministic version of the algorithm.

Analytical modelling of multiservice switching networks with multiservice sources and resource management mechanisms

This article proposes a new analytical method of determining occupancy distribution and blocking probability in multiservice switching networks with admission control mechanisms and resource management mechanisms. The proposed method is a universal one, which allows for determining blocking probability using the same model, regardless of the type of admission control mechanisms and resource management mechanisms. The versatility of the method is demonstrated in the article for two selected mechanisms, i.e. a threshold mechanism and a resource reservation mechanisms. The switching networks under consideration are offered multiservice traffic streams, generated by the so-called multiservice sources, according to Erlang, Engset and Pascal distributions. The proposed method is based on a generalised way of determining traffic characteristics for inter-stage links of switching networks with resource and call management mechanisms. Particular attention is given to determining the influence of the introduced mechanisms in outgoing links of the switching network on the traffic properties of inter-stage links of the network. The results of analytical calculations are then compared with the results obtained in digital simulation for a number of selected switching networks.

New a priori and a posteriori probabilistic bounds for robust counterpart optimization: III. Exact and near-exact a posteriori expressions for known probability distributions

The performance of robust optimization is closely connected with probabilistic bounds that determine the probability of constraint violation due to uncertain parameter realizations. In Part I of this work, new a priori and a posteriori probabilistic bounds were developed for cases when robust optimization is applied to uncertain optimization problems with parameters whose probability distributions were unknown. In Part II, the focus shifted to known probability distributions and a priori bounds. In this paper, new, tight a posteriori expressions are developed for constraints containing parameters with specific known distributions, that is, those attributed normal, uniform, discrete, gamma, chi-squared, Erlang, or exponential distributions. The nature of some of the expressions requires efficient implementations, and new algorithmic methods are discussed which greatly improve applicability. These new expressions are much tighter than existing bounds and greatly reduce the conservatism of robust solutions. The theoretical and algorithmic results of Parts I, II, and III allow for wider usage of robust optimization in process synthesis and operations research applications.

Event patterns (particle scatter plots) extracted from charged particle spectra in pp and Pb–Pb collisions at 2.76 TeV

The transverse momentum (pT) and pseudorapidity (η) spectra of charged particles produced in proton–proton (pp) and lead–lead (Pb-Pb) collisions at the Large Hadron Collider (LHC) are described by a hybrid model. In the model, the pT spectrum is described by a two-component distribution which contains an inverse power-law suggested by quantum chromodynamic calculus and an Erlang distribution resulted from a multisource thermal model. The η spectrum is described by a Gaussian rapidity (y) distribution resulted from the Landau hydrodynamic model and the two-component pT distribution, where the conversion between y and η is accurately considered. The modelling results are in agreement with the experimental data measured by the ATLAS Collaboration in pp collisions at center-of-mass energy TeV and in Pb–Pb collisions at center-of-mass energy per nucleon pair TeV. Based on the parameter values extracted from pT and η or y spectra, the event patterns or particle scatter plots in three-dimensional velocity and momentum spaces are obtained.

The Length-Biased Weighted Erlang Distribution

The Length-Biased Weighted Erlang Distribution

The Conditional Poisson Process and the Erlang and Negative Binomial Distributions

It is a well known fact that for the hierarchical model of a Poisson random variable Y whose mean has an Erlang distribution, the unconditional distribution of Y is negative binomial. However, the proofs in the literature [1] [2] provide no intuitive understanding as to why this result should be true. It is the purpose of this manuscript to give a new proof of this result which provides such an understanding. The memoryless property of the exponential distribution allows one to conclude that the events in two independent Poisson processes may be regarded as Bernoulli trials, and this fact is used to achieve the research purpose. Another goal of this manuscript is to give another proof of this last fact which does not rely on the memoryless property.

A Study on Vehicle Connectivity for VANETs Adopting Beamforming Antenna Array

An analytical mobility model based on Erlang distribution is introduced to portray sparse and dense traffic scenarios in the distributions of time headway. Upon the mobility model, the connectivity probability is then derived in two-way highway when adopting store-forward strategy. Moreover, path losses with and without beamforming antenna array are compared to further study the connectivity under different traffic density scenarios. Meanwhile, the effectiveness of beamforming technology in Vehicular Ad hoc Networks (VANETs) is investigated and concluded through analytical study and simulations.

Interval optimality analysis in Erlang bulk arrival model in fuzzy environment

In this paper, we propose a methodology to analyse the best arrival of K-phase Erlang distribution in fuzzy environment. We consider the queue parameters as triangular fuzzy numbers. At first, we construct the inverse membership of the steady state performance measures. Second, we construct the pairs of MINLP to calculate the lower and upper bound of the performance measure at the different possibility level of α via Zadeh extension principle. We analyse the interval of the best arrival optimality using testing of hypothesis. A numerical example is illustrated.

Interval optimality analysis in Erlang bulk arrival model in fuzzy environment

In this paper, we propose a methodology to analyse the best arrival of K-phase Erlang distribution in fuzzy environment. We consider the queue parameters as triangular fuzzy numbers. At first, we construct the inverse membership of the steady state performance measures. Second, we construct the pairs of MINLP to calculate the lower and upper bound of the performance measure at the different possibility level of α via Zadeh extension principle. We analyse the interval of the best arrival optimality using testing of hypothesis. A numerical example is illustrated.

어랑분포의 형상모수 변화에 따른 소프트웨어 개발 비용모형에 관한 비교 연구

소프트웨어 개발과정에서 소프트웨어 신뢰성은 매우 중요한 문제 중에 하나이다. 소프트웨어 고장현상을 분석하기 위하여 비동질적인 포아송과정에서 고장 발생 추이를 의미하는 위험함수가 고장시간에 독립적으로 일정하거나, 종속적인 경우, 즉 비-증가 또는, 비-감소하는 속성을 가질 수 있다. 본 연구에서는 소프트웨어 제품 테스팅 과정에서 고장 수명분포로서 어랑분포의 다양한 형상모수를 고려한 소프트웨어 개발 비용 분석에 대하여 연구되었다. 소프트웨어 고장현상을 분석하기 위하여 모수추정은 최우추정법이 사용되었다. 따라서 본 논문에서는 어랑분포의 형상모수를 고려한 소프트웨어 개발비용모형 분석을 위하여 소프트웨어 고장간격 시간자료를 이용하여 비교 및 평가하였다. 그 결과 형상모수에 따른 비용곡선을 비교 하였을 때 형상모형이 작을수록 비용이 많고 소프트웨어 최적 방출시간이 지연 됨을 알 수 있었다. 이 연구를 통하여 소프트웨어 개발자들에게 소프트웨어 형상모수에 따른 개발 비용을 탐색하는데, 기본적으로 도움을 줄 수 있는 사전정보의 역할을 할 수 있을 것으로 판단된다. Software Reliability implemented in software development is one of the most important issues. In finite failure NHPP software reliability models for software failure analysis, the hazard function that means a failure rate may have constant independently for failure time, non-increasing or non-decreasing pattern. In this study, software development cost analysis considering the variable shape parameter of Erlang distribution as the failure life distribution in the software product testing process was studied. The software failure model was applied finite failure Non-Homogeneous Poisson Procedure and the parameters approximation using maximum likelihood estimation was accompanied. Thus, this paper was presented comparative analysis by applying a software failure time data to the software, considering the shape parameter of Erlang distribution for development cost model analysis. When compared to the cost curve in accordance with the shape parameter, the model of smaller shape can be seen that the optimal software release time delay and more cost. Through this study, it is thought that it can serve as a preliminary information which can basically help the software developers to search for development cost according to software shape parameters.

Pulse Jitter, Delay Spread, and Doppler Shift in Mode-Stirred Reverberation

A time-domain characterization and stochastic model are developed for analyzing the evolution of the early randomization of fields inside mode-stirred reverberation chambers. The chamber is excited by a pulse amplitude modulated RF signal and stirred mechanically by a rotational stirrer. Phase noise (jitter) in the arrival times of a received pulse train is found to contain a systematic contribution that depends on stir speed and stir sense, as well as a random contribution governed by the rate of field depolarization. For the systematic component, a model of the phase delay for a uniformly orbiting scatterer inside a cavity is derived. For the random component, a nonhomogeneous Poisson model for arrivals of wave fronts in a nonstationary stir process exhibits good agreement with the empirical distribution of the measured jitter. At increasing stir velocities, the rate of field mixing increases less rapidly. The excess delay approaches asymptotically an Erlang distribution, in the limit of a homogeneous Poisson process for uniformly distributed late arrival times.