Quantile Approximation of the Erlang Distribution using Differential Evolution Algorithm
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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.
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QA Mathematics