def PdfFromME(alpha, A, x):
"""
Returns the probability density function of a matrix-
exponential distribution.
Parameters
----------
alpha : vector, shape (1,M)
The initial vector of the matrix-exponential
distribution.
A : matrix, shape (M,M)
The matrix parameter of the matrix-exponential
distribution.
x : vector of doubles
The density function will be computed at these points
prec : double, optional
Numerical precision to check if the input ME
distribution is valid. The default value is 1e-14.
Returns
-------
pdf : column vector of doubles
The values of the density function at the
corresponding "x" values
"""
if butools.checkInput and not CheckMERepresentation(alpha, A):
raise Exception("PdfFromME: Input is not a valid ME representation!")
y = [np.sum(alpha * expm(A * xv) * (-A)) for xv in x]
return np.array(y)
def CdfFromME(alpha, A, x):
"""
Returns the cummulative distribution function of a
matrix-exponential distribution.
Parameters
----------
alpha : matrix, shape (1,M)
The initial vector of the matrix-exponential
distribution.
A : matrix, shape (M,M)
The matrix parameter of the matrix-exponential
distribution.
x : vector of doubles
The cdf will be computed at these points
Returns
-------
cdf : column vector of doubles
The values of the cdf at the corresponding "x" values
"""
if butools.checkInput and not CheckMERepresentation(alpha, A):
raise Exception("CdfFromME: Input is not a valid ME representation!")
return np.array([1.0 - np.sum(alpha * expm(A.A * xv)) for xv in x])
def CanonicalFromPH3(alpha, A, prec=1e-14):
"""
Returns the canonical form of an order-3 phase-type
distribution.
Parameters
----------
alpha : matrix, shape (1,3)
Initial vector of the phase-type distribution
A : matrix, shape (3,3)
Transient generator of the phase-type distribution
prec : double, optional
Numerical precision, default value is 1e-14
Returns
-------
beta : matrix, shape (1,3)
The initial probability vector of the canonical form
B : matrix, shape (3,3)
Transient generator matrix of the canonical form
Notes
-----
This procedure calculates 5 moments of the input and
calls 'PH3From5Moments'.
"""
if butools.checkInput and not CheckMERepresentation(alpha, A):
raise Exception(
"CanonicalFromPH3: Input isn''t a valid ME distribution!")
if A.shape[0] != 3:
raise Exception("CanonicalFromPH3: Dimension is not 3!")
return PH3From5Moments(MomentsFromPH(alpha, A, 5), prec)
def PHFromME (alpha, A, precision=1e-14):
"""
Obtains a Markovian representation of a matrix
exponential distribution of the same size, if possible.
Parameters
----------
alpha : vector, shape (1,M)
The initial vector of the matrix-exponential
distribution.
A : matrix, shape (M,M)
The matrix parameter of the matrix-exponential
distribution.
precision : double, optional
A representation is considered to be a Markovian one
if it is closer than the precision. The default value
is 1e-14.
Returns
-------
beta : vector, shape (1,M)
The initial probability vector of the Markovian
monocyclic representation
B : matrix, shape (M,M)
Transient generator matrix of the Markovian
monocyclic representation
References
----------
.. [1] G Horváth, M Telek, "A minimal representation of
Markov arrival processes and a moments matching
method," Performance Evaluation 64:(9-12) pp.
1153-1168. (2007)
"""
def transfun (orep, B):
ao, Ao = orep
return (ao*B, la.inv(B)*Ao*B)
def evalfun (orep, k=0):
ao, Ao = orep
av= np.sum(-Ao,1)
Ad = Ao-np.diag(np.diag(Ao))
if k%2 == 0:
return -min(np.min(ao), np.min(av), np.min(Ad))
else:
return -np.sum(ao[ao<0]) - np.sum(av[av<0]) - np.sum(Ad[Ad<0])
if butools.checkInput and not CheckMERepresentation (alpha, A):
raise Exception("PHFromME: Input is not a valid ME representation!")
return FindMarkovianRepresentation ((alpha,A), transfun, evalfun, precision)
def CheckMEPositiveDensity (alpha, A, maxSize=100, prec=None):
"""
Checks if the given matrix-exponential distribution has
positive density.
Parameters
----------
alpha : matrix, shape (1,M)
Initial vector of the matrix-exponential distribution
to check
A : matrix, shape (M,M)
Matrix parameter of the matrix-exponential distribution
to check
maxSize : int, optional
The procedure tries to transform the ME distribution
to phase-type up to order maxSize. The default value
is 100.
prec : double, optional
Numerical precision. The default value is 1e-14.
Returns
-------
r : bool
True, if the given matrix-exponential distribution has
a positive density
Notes
-----
This procedure calls MonocyclicPHFromME, and can be time
consuming.
"""
if prec==None:
prec=butools.checkPrecision
try:
beta, B = MonocyclicPHFromME (alpha, A, maxSize, prec)
r = CheckMERepresentation (beta, B, prec)
except Exception:
r = False
return r
def MomentsFromME(alpha, A, K=0):
"""
Returns the first K moments of a matrix-exponential
distribution.
Parameters
----------
alpha : vector, shape (1,M)
The initial vector of the matrix-exponential
distribution.
A : matrix, shape (M,M)
The matrix parameter of the matrix-exponential
distribution.
K : int, optional
Number of moments to compute. If K=0, 2*M-1 moments
are computed. The default value is K=0.
prec : double, optional
Numerical precision for checking the input.
The default value is 1e-14.
Returns
-------
moms : row vector of doubles
The vector of moments.
"""
if butools.checkInput and not CheckMERepresentation(alpha, A):
raise Exception(
"MomentsFromME: Input is not a valid ME representation!")
if K == 0:
K = 2 * np.size(alpha, 1) - 1
return [
math.factorial(i) * np.sum(alpha * la.inv(-A)**i)
for i in range(1, K + 1)
]
def MEOrder(alpha, A, kind="moment", prec=1e-10):
"""
Returns the order of the ME distribution (which is not
necessarily equal to the size of the representation).
Parameters
----------
alpha : vector, shape (1,M)
The initial vector of the matrix-exponential
distribution.
A : matrix, shape (M,M)
The matrix parameter of the matrix-exponential
distribution.
kind : {'obs', 'cont', 'obscont', 'moment'}, optional
Determines which order is computed. Possibilities:
'obs': observability,
'cont': controllability,
'obscont': the minimum of observability and
controllability order,
'moment': moment order (which is the default).
prec : double, optional
Precision used to detect if the determinant of the
Hankel matrix is zero (in case of kind="moment" only),
or the tolerance for the rank calculation. The
default value is 1e-10.
Returns
-------
order : int
The order of ME distribution
References
----------
.. [1] P. Buchholz, M. Telek, "On minimal representation
of rational arrival processes." Madrid Conference on
Qeueuing theory (MCQT), June 2010.
"""
if butools.checkInput and not CheckMERepresentation(alpha, A):
raise Exception("MEOrder: Input is not a valid ME representation!")
N = alpha.shape[1]
if kind == "cont":
re = np.zeros((N, N))
for n in range(N):
re[n, :] = np.sum(A.T**n, 0)
return nla.matrix_rank(re, prec)
elif kind == "obs":
re = np.zeros((N, N))
for n in range(N):
re[n, :] = alpha * A**n
return nla.matrix_rank(re, prec)
elif kind == "obscont":
re = np.zeros((N, N))
for n in range(N):
re[n, :] = alpha * A**n
obsOrder = nla.matrix_rank(re, prec)
re = np.zeros((N, N))
for n in range(N):
re[n, :] = np.sum(A.T**n, 0)
contOrder = nla.matrix_rank(re, prec)
return min(obsOrder, contOrder)
elif kind == "moment":
return MEOrderFromMoments(MomentsFromME(alpha, A), prec)
else:
raise Exception("Invalid 'kind' parameter!")
def MinimalRepFromME(alpha, A, how="moment", precision=1e-12):
"""
Returns the minimal representation of the given ME
distribution.
Parameters
----------
alpha : vector, shape (1,M)
The initial vector of the matrix-exponential
distribution.
A : matrix, shape (M,M)
The matrix parameter of the matrix-exponential
distribution.
how : {"obs", "cont", "obscont", "moment"}, optional
Determines how the representation is minimized.
Possibilities:
'obs': observability,
'cont': controllability,
'obscont': the minimum of observability and
controllability order,
'moment': moment order (which is the default).
precision : double, optional
Precision used by the Staircase algorithm. The default
value is 1e-12.
Returns
-------
beta : vector, shape (1,N)
The initial vector of the minimal representation
B : matrix, shape (N,N)
The matrix parameter of the minimal representation
References
----------
.. [1] P. Buchholz, M. Telek, "On minimal representation
of rational arrival processes." Madrid Conference on
Qeueuing theory (MCQT), June 2010.
"""
if butools.checkInput and not CheckMERepresentation(alpha, A):
raise Exception(
"MinimalRepFromME: Input is not a valid ME representation!")
if how == "cont":
H0 = A
H1 = np.sum(-A, 1) * alpha
B, n = MStaircase([H0, H1], ml.ones((A.shape[0], 1)), precision)
return ((alpha * B)[0, 0:n], (la.inv(B) * A * B)[0:n, 0:n])
elif how == "obs":
H0 = A
H1 = np.sum(-A, 1) * alpha
G = [H0.T, H1.T]
B, n = MStaircase(G, alpha.T, precision)
return ((alpha * B)[:, 0:n], (la.inv(B) * A * B)[0:n, 0:n])
elif how == "obscont":
alphav, Av = MinimalRepFromME(alpha, A, "cont", precision)
return MinimalRepFromME(alphav, Av, "obs", precision)
elif how == "moment":
N = MEOrder(alpha, A, "moment", precision)
moms = MomentsFromME(alpha, A, 2 * N - 1)
return MEFromMoments(moms)