butools.ph.APH3rdMomentLowerBound¶

butools.ph.APH3rdMomentLowerBound()¶

Matlab:	m3 = APH3rdMomentLowerBound(m1, m2, n)
Mathematica:	m3 = APH3rdMomentLowerBound[m1, m2, n]
Python/Numpy:	m3 = APH3rdMomentLowerBound(m1, m2, n)

Returns the lower bound of the third moment of acyclic phase-type (APH) distributions of order n.

Parameters:	m1 : double The first moment m2 : double The second moment n : int Number of states
Returns:	m3 : double The lowest third moment an order-n APH can have with the given first and second moment.

References

[R2]	A. Bobbio, A. Horvath, M. Telek, “Matching three moments with minimal acyclic phase type distributions,” Stochastic models, pp. 303-326, 2005.

Examples

For Matlab:

>>> mean = 1.9;
>>> mom2 = 5;
>>> n = 3;
>>> mom3lower = APH3rdMomentLowerBound(mean, mom2, n);
>>> disp(mom3lower);
       16.577
>>> mom3upper = APH3rdMomentUpperBound(mean, mom2, n);
>>> disp(mom3upper);
       17.081
>>> mean = 1.9;
>>> mom2 = 5;
>>> n = 4;
>>> mom3lower = APH3rdMomentLowerBound(mean, mom2, n);
>>> disp(mom3lower);
       16.079
>>> mom3upper = APH3rdMomentUpperBound(mean, mom2, n);
>>> disp(mom3upper);
   Inf

For Mathematica:

>>> mean = 1.9;
>>> mom2 = 5;
>>> n = 3;
>>> mom3lower = APH3rdMomentLowerBound[mean, mom2, n];
>>> Print[mom3lower];
16.577458090899114
>>> mom3upper = APH3rdMomentUpperBound[mean, mom2, n];
>>> Print[mom3upper];
17.081405365964272
>>> mean = 1.9;
>>> mom2 = 5;
>>> n = 4;
>>> mom3lower = APH3rdMomentLowerBound[mean, mom2, n];
>>> Print[mom3lower];
16.079377140256383
>>> mom3upper = APH3rdMomentUpperBound[mean, mom2, n];
>>> Print[mom3upper];
Infinity

For Python/Numpy:

>>> mean = 1.9
>>> mom2 = 5
>>> n = 3
>>> mom3lower = APH3rdMomentLowerBound(mean, mom2, n)
>>> print(mom3lower)
16.577458090899107
>>> mom3upper = APH3rdMomentUpperBound(mean, mom2, n)
>>> print(mom3upper)
17.081405365964276
>>> mean = 1.9
>>> mom2 = 5
>>> n = 4
>>> mom3lower = APH3rdMomentLowerBound(mean, mom2, n)
>>> print(mom3lower)
16.079377140256387
>>> mom3upper = APH3rdMomentUpperBound(mean, mom2, n)
>>> print(mom3upper)
inf