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

Okagbue, Hilary I.; Adamu, Muminu O.; Anake, T. A.

doi:10.1007/s11277-018-6090-x

Cited by 5 publications

(3 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A comparison with the machine values was done to evaluate the efficiency and the performance of the quantile models obtained. Recently, authors [ 54 ] applied the same methodology proposed in this paper to obtained the near exact quantile values for the Erlang distribution. Unfortunately, the results cannot be applied to the gamma distribution since Erlang is a subset of the gamma distributions.…”

Section: Gamma Distributionmentioning

confidence: 99%

See 1 more Smart Citation

Approximations for the inverse cumulative distribution function of the gamma distribution used in wireless communication

Okagbue

Adamu

Anake

2020

Heliyon

Self Cite

View full text Add to dashboard Cite

The use of quantile functions of probability distributions whose cumulative distribution is intractable is often limited in Monte Carlo simulation, modeling, and random number generation. Gamma distribution is one of such distributions, and that has placed limitations on the use of gamma distribution in modeling fading channels and systems described by the gamma distribution. This is due to the inability to find a suitable closed-form expression for the inverse cumulative distribution function, commonly known as the quantile function (QF). This paper adopted the Quantile mechanics approach to transform the probability density function of the gamma distribution to second-order nonlinear ordinary differential equations (ODEs) whose solution leads to quantile approximation. Closed-form expressions, although complex of the QF, were obtained from the solution of the ODEs for degrees of freedom from one to five. The cases where the degree of freedom is not an integer were obtained, which yielded values closed to the R software values via Monte Carlo simulation. This paper provides an alternative for simulating gamma random variables when the degree of freedom is not an integer. The results obtained are fast, computationally efficient and compare favorably with the machine (R software) values using absolute error and Kullback-Leibler divergence as performance metrics.

show abstract

Section: Gamma Distributionmentioning

confidence: 99%

“…Erlang distribution considers only the cases of gamma, where the shape parameter is a positive integer. The implication is that the result of [ 54 ] cannot be applied when the shape parameter is not a positive integer.…”

Section: Gamma Distributionmentioning

confidence: 99%

Approximations for the inverse cumulative distribution function of the gamma distribution used in wireless communication

Okagbue

Adamu

Anake

2020

Heliyon

Self Cite

View full text Add to dashboard Cite

show abstract

“…Approximation remains the only viable option to use in obtaining a function that closely resembles the closed-form expression for the QF. Approximation in this context is the use of numerical optimization, although other methods such as functional approximation, the use of series expansions are available [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Quantile Approximation of the Erlang Distribution using Differential Evolution Algorithm

Okagbue¹

2020

IJATCSE

View full text Add to dashboard Cite

Erlang distribution is a particular case of the gamma distribution and is often used in modeling queues, traffic congestion in wireless sensor networks, cell residence duration and finding the optimal queueing model to reduce the probability of blocking. The application is limited because of the unavailability of closed-form expression for the quantile (inverse cumulative distribution) function of the distribution. The problem is primarily tackled using approximation since the inversion method cannot be applied. This paper extended a six parameter quantile model earlier proposed to the Nakagami distribution to the Erlang distributions. Consequently, the established relationship between the two distributions is now extended to their quantile functions. The quantile model was used to fit the machine (R software) values with their corresponding quartiles in two ways. Firstly, artificial neural network (ANN) was used to establish that a curve fitting can be achieved. Lastly, differential evolution (DE) algorithm was used to minimize the errors obtained from the curve fitting and hence estimate the values of the six parameters of the quantile model that will ensure the best possible fit, for different values of the parameters that characterize Erlang distribution. Hence, the problem is constrained optimization in nature and the DE algorithm was able to find the different values of the parameters of the quantile model. The simulation result corroborates theoretical findings. The work is a welcome result for the quest for a universal quantile model that can be applied to different distributions.

show abstract

Nature inspired quantile estimates of the Nakagami distribution

et al. 2019

View full text Add to dashboard Cite

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

Cited by 5 publications

References 31 publications

Approximations for the inverse cumulative distribution function of the gamma distribution used in wireless communication

Approximations for the inverse cumulative distribution function of the gamma distribution used in wireless communication

Quantile Approximation of the Erlang Distribution using Differential Evolution Algorithm

Nature inspired quantile estimates of the Nakagami distribution

Contact Info

Product

Resources

About