Closed Form Expressions for the Quantile Function of the Erlang Distribution Used in Engineering Models

Quantile function is heavily utilized in modeling, simulation, reliability analysis and random number generation. The use is often limited if the inversion method fails to estimate it from the cumulative distribution function (CDF). As a result, approximation becomes the other option. The failure of the inversion method is often due to the intractable nature of the CDF of the distribution. Erlang distribution belongs to those classes of distributions. The distribution is a particular case of the gamma distribution. Little is known about the quantile approximation of the Erlang distribution. This is due to the fact that researchers prefer to work with the gamma distribution of which the Erlang is a particular case. This work applied the quantile mechanics approach, power series method and cubic spline interpolation to obtain the approximate of the quantile function of the Erlang distribution for degrees of freedom from one to two. The approximate values compares favorably with the exact ones. Consequently, the result in this paper improved the existing results on the extreme tails of the distribution. The closed form expression for the quantile function obtained here is very useful in modeling physical and engineering systems that are completely described by or fitted with the Erlang distribution.

[1]  Miloš Ivić,et al.  Fuzzy renewal theory about forecasting mistakes done by a locomotive driver: a serbian railway case study , 2012 .

[2]  Asad Munir,et al.  Series representations and approximation of some quantile functions appearing in finance , 2012, 1203.5729.

[3]  Yuyang Zhang,et al.  APPLICATION OF QUEUING THEORY IN HIGHWAY TOLL STATION DESIGN , 2013 .

[4]  Andreas Kleefeld,et al.  A statistical application of the quantile mechanics approach: MTM estimators for the parameters of t and gamma distributions , 2012, European Journal of Applied Mathematics.

[5]  Pedro Jodrá.  Computing the Asymptotic Expansion of the Median of the Erlang Distribution , 2012 .

[6]  Serkan Eryilmaz,et al.  Computing optimal replacement time and mean residual life in reliability shock models , 2017, Comput. Ind. Eng..

[7]  Liuxin Chen,et al.  Dynamic Pricing and Inventory Control in a Make-to-Stock Queue With Information on the Production Status , 2011, IEEE Transactions on Automation Science and Engineering.

[8]  Ali Ghafarian Salehi Nezhad.  An Approach to Consider Uncertain Components' Failure Rates in Series-Parallel Reliability Systems with Redundancy Allocation , 2014 .

[9]  F. Jiménez,et al.  On the computer generation of the Erlang and negative binomial distributions with shape parameter equal to two , 2009, Math. Comput. Simul..

[10]  Feng Zhai,et al.  A model combining discrete event system simulation and genetic algorithm for buffer allocation in unreliable large production lines , 2004 .

[11]  Vladimir Rykov,et al.  On Sensitivity of Reliability Models to the Shape of Life and Repair Time Distributions , 2014, 2014 Ninth International Conference on Availability, Reliability and Security.

[12]  Mohamed Salah El-Sherbeny,et al.  Stochastic Behavior of a Two-Unit Cold Standby Redundant System Under Poisson Shocks , 2017 .

[13]  Gary Ulrich,et al.  A method for computer generation of variates from arbitrary continuous distributions , 1987 .

[14]  Daniël Reijsbergen,et al.  Probabilistic Modelling of the Impact on Bus Punctuality of a Speed Limit Proposal in Edinburgh (Extended Version) , 2015, ArXiv.

[15]  R. Fisher,et al.  148: Moments and Cumulants in the Specification of Distributions. , 1938 .

[16]  A. W. Davis,et al.  Generalized Asymptotic Expansions of Cornish-Fisher Type , 1968 .

[17]  Xu Han-qing.  Research on Traffic Characteristics and Traffic Conflicts of One-way Closed Work Zone on Expressway , 2013 .

[18]  F. Safaei,et al.  The impacts of dynamic failures on the resilience and fragility of P2P networks , 2012, 20th Iranian Conference on Electrical Engineering (ICEE2012).

[19]  E. Cornish,et al.  The Percentile Points of Distributions Having Known Cumulants , 1960 .

[20]  Michael Forde,et al.  THE USE OF A DISCRETE-EVENT SIMULATION MODEL WITH ERLANG PROBABILITY DISTRIBUTIONS IN THE ESTIMATION OF EARTHMOVING PRODUCTION , 1996 .

[21]  Meng Zhang,et al.  Simplified calculation on the time performance of high efficiency frame generation algorithm in Advanced Orbiting Systems , 2013, 2013 5th IEEE International Symposium on Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications.

[22]  Asad Munir,et al.  Dependency without copulas or ellipticity , 2009 .

[23]  Shih-Cheng Horng,et al.  Embedding Advanced Harmony Search in Ordinal Optimization to Maximize Throughput Rate of Flow Line , 2017, Arabian Journal for Science and Engineering.

[24]  Hilary I. Okagbue,et al.  Ordinary differential equations of probability functions of convoluted distributions , 2018, International Journal of ADVANCED AND APPLIED SCIENCES.

[25]  Gaojie Chen,et al.  Outage probability analysis of an amplify-and-forward cooperative communication system with multi-path channels and max??min relay selection , 2013, IET Commun..

[26]  Majid Naderi,et al.  Heavy-tail and voice over internet protocol traffic: queueing analysis for performance evaluation , 2011, IET Commun..

[27]  T. Luu,et al.  Quantile Mechanics II: Changes of Variables in Monte Carlo Methods and GPU-Optimized Normal Quantiles , 2009, 0901.0638.

[28]  Ali S. Al-Ghamdi,et al.  Analysis of Time Headways on Urban Roads: Case Study from Riyadh , 2001 .

[29]  Valentine A. Aalo,et al.  Distribution of random sum cell dwell times in wireless network , 2008 .

[30]  Friedrich Pillichshammer,et al.  A Method for Approximate Inversion of the Hyperbolic CDF , 2002, Computing.

[31]  Mehmet Savsar,et al.  Effects of kanban withdrawal policies and other factors on the performance of JIT systems : a simulation study , 1996 .

[32]  T. Afullo,et al.  Fractal analysis of rainfall event duration for microwave and millimetre networks: rain queueing theory approach , 2015 .

[33]  Reza Tavakkoli-Moghaddam,et al.  Interval programming for the redundancy allocation with choices of redundancy strategy and component type under uncertainty: Erlang time to failure distribution , 2014, Appl. Math. Comput..