200 research outputs found
Quantile Approximation of the Erlang Distribution using Differential Evolution Algorithm
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
Approximations for the inverse cumulative distribution function of the gamma distribution used in wireless communication
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
Performance Modelling and Optimisation of Multi-hop Networks
A major challenge in the design of large-scale networks is to predict and optimise the
total time and energy consumption required to deliver a packet from a source node to a
destination node. Examples of such complex networks include wireless ad hoc and sensor
networks which need to deal with the effects of node mobility, routing inaccuracies, higher
packet loss rates, limited or time-varying effective bandwidth, energy constraints, and the
computational limitations of the nodes. They also include more reliable communication
environments, such as wired networks, that are susceptible to random failures, security
threats and malicious behaviours which compromise their quality of service (QoS) guarantees.
In such networks, packets traverse a number of hops that cannot be determined
in advance and encounter non-homogeneous network conditions that have been largely
ignored in the literature. This thesis examines analytical properties of packet travel in
large networks and investigates the implications of some packet coding techniques on both
QoS and resource utilisation.
Specifically, we use a mixed jump and diffusion model to represent packet traversal
through large networks. The model accounts for network non-homogeneity regarding
routing and the loss rate that a packet experiences as it passes successive segments of a
source to destination route. A mixed analytical-numerical method is developed to compute
the average packet travel time and the energy it consumes. The model is able to capture
the effects of increased loss rate in areas remote from the source and destination, variable
rate of advancement towards destination over the route, as well as of defending against
malicious packets within a certain distance from the destination. We then consider sending
multiple coded packets that follow independent paths to the destination node so as to
mitigate the effects of losses and routing inaccuracies. We study a homogeneous medium
and obtain the time-dependent properties of the packet’s travel process, allowing us to
compare the merits and limitations of coding, both in terms of delivery times and energy
efficiency. Finally, we propose models that can assist in the analysis and optimisation
of the performance of inter-flow network coding (NC). We analyse two queueing models
for a router that carries out NC, in addition to its standard packet routing function. The
approach is extended to the study of multiple hops, which leads to an optimisation problem
that characterises the optimal time that packets should be held back in a router, waiting
for coding opportunities to arise, so that the total packet end-to-end delay is minimised
NEAR EXACT QUANTILE ESTIMATES OF THE BETA DISTRIBUTION
We propose a near exact equation that links the quantile function and
its quartiles of beta distribution (BD). This is done with the use of
quantile mechanics approach whereby the probability density function
of beta distribution is transformed into second order nonlinear
ordinary differential equations whose solutions give the required
inverse cumulative distribution function of the distribution. The
median can easily be obtained from the proposed equations. Further
efforts are required to refine and simplify the results. The results
from this paper will greatly affect the way, beta distribution is used
in engineering, in general, and wireless communications, in particular.
Furthermore, the results obtained are very close the machine values
Aggregation of Dependent Risks in Mixtures of Exponential Distributions and Extensions
The distribution of the sum of dependent risks is a crucial aspect in actuarial sciences, risk management and in many branches of applied probability. In this paper, we obtain analytic expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of aggregated risks, modeled according to a mixture of exponential distributions. We first review the properties of the multivariate mixture of exponential distributions, to then obtain the analytical formulation for the pdf and the cdf for the aggregated distribution. We study in detail some specific families with Pareto (Sarabia et al, 2016), Gamma, Weibull and inverse Gaussian mixture of exponentials (Whitmore and Lee, 1991) claims. We also discuss briefly the computation of risk measures, formulas for the ruin probability (Albrecher et al., 2011) and the collective risk model. An extension of the basic model based on mixtures of gamma distributions is proposed, which is one of the suggested directions for future research.The authors thanks to the Ministerio de Econom´ıa y Competitividad (projects
ECO2016-76203-C2-1-P, JMS, FP and VJ ECO2013-47092 EGD) for partial
support of this work. In addition, this work is part of the Research Project
APIE 1/2015-17 (JMS, FP, VJ): “New methods for the empirical analysis of
financial markets” of the Santander Financial Institute (SANFI) of UCEIF
Foundation resolved by the University of Cantabria and funded with sponsorship from Banco Santander
Closed-Form Expressions for the Quantile Function of the Chi Square Distribution Using the Hybrid of Quantile Mechanics and Spline Interpolation
Chi square distribution is a continuous probability
distribution primarily used in hypothesis testing,
contingency analysis, and construction of
confidence limits in inferential statistics but not
necessarily in the modeling of real-life phenomena.
The closed-form expression for the quantile
function (QF) of Chi square is not available because
the cumulative distribution function cannot be
transformed to yield the QF and consequently
places limitations on the use of the QF. Researchers
have over the years proposed approximations that
improve over time in terms of speed, computational
efficiency, and precision, and so on. However, most
of the available closed-form expressions (quantile approximation) fail at the extreme tails of the
distribution. This paper used the Quantile
mechanics approach to obtain second-order
nonlinear ordinary differential equations whose
solutions using the power series method yielded
initial approximates in form of series for different
values of the degrees of freedom. The initial
approximate varies with the exact (R software)
values which serve as the reference and the error
between them was minimized by the natural cubic
spline interpolation. The final approximates are
correct up to an average of 8 decimal places, have
small error, and is closer to the exact when
compared with some other results from other
researchers. The upper tail of the distribution was
considered and excellent results were obtained
which is a major improvement over the existing
results in the literature. The approach used in this
work is a hybrid of series expansions and numerical
algorithms. Computer codes can be written for the
application
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