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[3], a function conserving a Gaussian-like shape
is considered. In essence, the algorithm can
be resumed as indicated below,
> ! start loop
> do q=1,1000 ! distance
> call random x1 ! first aleatory number
> call random x2 ! secondary aleatory number
> f(q) = N exp(-(q-h(x2)/g(x2))**2) ! define
Gaussian parameters
> if(f(q).lt.x1) then ! Monte-Carlo-like
step
> accept q, h(x2), g(x2) endif ! acceptance
> enddo ! end loop
The algorithm collects the ”true” parameters to
be used for the reconstruction of the ”true” PDF
as well. The h(x2) and g(x2) functions would
give information about the shape of the PDF
mainly in their behavior. Physically, one PDF
should fall down as fast as the distance given in
kilometers grows. In left side Fig. 2 is shown up
to four PDF curves after of using the algorithm
above. Mainly all of them come down in a range
between 4 and 6 kilometers. It is actually congruent
with the fact that the handover contains
a radio coverage of 5 Km being assumed in this
report.
B. Transfer Function Selection and Second Order
Integration
A transfer function containing a step-like behavior
is adapted to the I/O scheme and it can be
written as follows,
CIENCIA, CULTURA Y TECNOLOGÍA - UNIVERSIDAD TECNOLÓGICA DEL PERÚ
(7)
where A and B are parameters to be adjusted in
according to integration of Eq. (1). To complete
the stochastic picture of the problem, a random
function have been multiplied to the proposed
transfer function,
(8)
where S is a random function thereby providing
to some extent the stochastic character to the
transfer functions. The parameter is tunned
with the shape of one Gaussian when the calculation
of Eq. (1) is performed. Therefore one
can express the full nonlinear I/O operation as
(9)
IV. RESULTS
In right side Fig. 2 different curves obtained
with the second order generalized I/O integral
(9) are presented. Integrations were performed
numerically. Concretely, an optimized trapezoidal
rule with 1/100 partitions is implemented
in the computation of PSCC. We underlined the
fact that one of them (in black dots) has turned
out to be approximated (to certain extent)
to the one from the Meo-Ajmone model (Fig.
5 Ref. [1]). In our model, for Q( =6.5)=0.9,
(0.98 in Meo- Ajmone Model) yielding a discrepancy
of order of 10%. The blue boxes share
the same morphology up to =7 to the black
dots, but showing a rapid fall by demonstrating
the weakness of the Q( ) curve obtained by
the I/O scheme. Green and red crosses show
a poor manifestation of the I/O model proposal.
Nevertheless we address the question
whether incorrect numbers of the parameters
space were taken into account. The upper limit
in (9) denoted by R plays no relevant role for
the computation of Q( ) as shown during the
computation of the integrals. Values of 15, 25,
30 and 50 Km. were considered. It should be
noted in the Q( ) curve obtained with (9) is
its approximate flat shape for the first 6 kilometers
as observed at the red and blue curves.
A simple fit gives a value of 0.95 over 5 degree
of freedom, value which is consistent with
[1]. Although the model fails for a certain set
of parameter space, the central concept of a
nonlinear I/O correspondence between PDF and
PSCC might be conceivable under certain circumstances
without to appeal to instances inside
pure probabilistic and statistics conceptions.
V. CONCLUSIONS
In this short report we have employed the typical
I/O procedure to obtain a PSCC through
a convolution operation together to an input
function denoted by a PDF obtained from a simulation
procedure. The modeling of that is carried
out by assuming the operativeness of up
to two handovers in a distance of 10 Km. We
have obtained a curve of PSCC against the call
arrival rate which presents a discrepancy of the
order of 10% with respect to the Meo-Ajmone
Model in one of the curves obtained with the
I/O criterion. In the future, would be possible
Fig. 2. (Left) Probability distribution functions
for various random scenarios with their transfer
functions multiplied by random functions. (Right)
The variable Q or PSCC as obtained through Eq.
(9) against the parameter call arrival rate . In all
cases the tails are shorter than the Meo-Ajmone
scenario except the cases of black dots which
cover a portion of the Meo-Ajmone region within a
discrepancy of the order of 10%.
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