def __new__(cls, c):
"""
"""
c = c.view(matlib.matrix).reshape(1, -1)
T = matlib.vstack((
matlib.hstack((matlib.identity(c.size), c)),
matlib.hstack((matlib.zeros(c.size), matlib.ones(1))),
))
return super().__new__(cls, T)
def __new__(cls, c):
"""
"""
c = c.view(matlib.matrix).reshape(1, -1)
T = matlib.vstack((
matlib.hstack((matlib.identity(c.size), c)),
matlib.hstack((matlib.zeros(c.size), matlib.ones(1))),
))
return super().__new__(cls, T)
def increase_input_size(self, nb_added_input = 1):
"""
Increases the number of inputs the network needs
"""
added_input_weights = self.input_scaling * EchoStateNetwork._rand_matrix(self.reservoir_size, nb_added_input)
self.input_weights = npmat.hstack([self.input_weights, added_input_weights])
self.input_size += nb_added_input
if not self.use_raw_input:
return
self.state_size += nb_added_input
self.state = npmat.vstack([self.state, npmat.zeros((nb_added_input, 1))])
self.output_weights = npmat.hstack([self.output_weights, npmat.zeros((self.output_size, nb_added_input))])
def firstInitElem(m33, sortEigs, alpha, A):
l1 = sortEigs[0]
l2 = sortEigs[1]
l3 = sortEigs[2]
m1 = -m33 + l1 + l2 + l3
m2=-((l2-l3)*(l1**2-l1*l2-l1*l3+l2*l3)*(m33**3-2*m33**2*l1+m33*l1**2-2*m33**2*l2+3*m33*l1*l2-l1**2*l2+m33*l2**2- \
l1*l2**2-2*m33**2*l3+3*m33*l1*l3-l1**2*l3+3*m33*l2*l3-2*l1*l2*l3-l2**2*l3+m33*l3**2-l1*l3**2-l2*l3**2))/ \
(2*m33*l1**2*l2+2*m33**2*l1**2*l2-l1**3*l2-3*m33*l1**3*l2+l1**4*l2-2*m33*l1*l2**2-2*m33**2*l1*l2**2+l1**3*l2**2+ \
l1*l2**3+3*m33*l1*l2**3-l1**2*l2**3-l1*l2**4-2*m33*l1**2*l3-2*m33**2*l1**2*l3+l1**3*l3+3*m33*l1**3*l3-l1**4*l3+ \
2*m33*l2**2*l3+2*m33**2*l2**2*l3-l2**3*l3-3*m33*l2**3*l3+l2**4*l3+2*m33*l1*l3**2+2*m33**2*l1*l3**2-l1**3*l3**2- \
2*m33*l2*l3**2-2*m33**2*l2*l3**2+l2**3*l3**2-l1*l3**3-3*m33*l1*l3**3+l1**2*l3**3+l2*l3**3+3*m33*l2*l3**3-l2**2*l3**3+ \
l1*l3**4-l2*l3**4)
m3=((l2-l3)*(l1**2-l1*l2-l1*l3+l2*l3)*(m33**3+m33**4-m33**2*l1-2*m33**3*l1+m33**2*l1**2-m33**2*l2-2*m33**3*l2+ \
m33*l1*l2+3*m33**2*l1*l2-m33*l1**2*l2+m33**2*l2**2-m33*l1*l2**2-m33**2*l3-2*m33**3*l3+m33*l1*l3+3*m33**2*l1*l3- \
m33*l1**2*l3+m33*l2*l3+3*m33**2*l2*l3-l1*l2*l3-4*m33*l1*l2*l3+l1**2*l2*l3-m33*l2**2*l3+l1*l2**2*l3+m33**2*l3**2- \
m33*l1*l3**2-m33*l2*l3**2+l1*l2*l3**2))/(-2*m33*l1**2*l2-2*m33**2*l1**2*l2-m33**3*l1**2*l2+l1**3*l2+3*m33*l1**3*l2+ \
2*m33**2*l1**3*l2-l1**4*l2-m33*l1**4*l2+2*m33*l1*l2**2+2*m33**2*l1*l2**2+m33**3*l1*l2**2-l1**3*l2**2-2*m33*l1**3*l2**2+ \
l1**4*l2**2-l1*l2**3-3*m33*l1*l2**3-2*m33**2*l1*l2**3+l1**2*l2**3+2*m33*l1**2*l2**3+l1*l2**4+m33*l1*l2**4-l1**2*l2**4+ \
2*m33*l1**2*l3+2*m33**2*l1**2*l3+m33**3*l1**2*l3-l1**3*l3-3*m33*l1**3*l3-2*m33**2*l1**3*l3+l1**4*l3+m33*l1**4*l3- \
2*m33*l2**2*l3-2*m33**2*l2**2*l3-m33**3*l2**2*l3+l2**3*l3+3*m33*l2**3*l3+2*m33**2*l2**3*l3-l2**4*l3-m33*l2**4*l3- \
2*m33*l1*l3**2-2*m33**2*l1*l3**2-m33**3*l1*l3**2+l1**3*l3**2+2*m33*l1**3*l3**2-l1**4*l3**2+2*m33*l2*l3**2+2*m33**2*l2*l3**2+ \
m33**3*l2*l3**2-l2**3*l3**2-2*m33*l2**3*l3**2+l2**4*l3**2+l1*l3**3+3*m33*l1*l3**3+2*m33**2*l1*l3**3-l1**2*l3**3- \
2*m33*l1**2*l3**3-l2*l3**3-3*m33*l2*l3**3-2*m33**2*l2*l3**3+l2**2*l3**3+2*m33*l2**2*l3**3-l1*l3**4-m33*l1*l3**4+ \
l1**2*l3**4+l2*l3**4+m33*l2*l3**4-l2**2*l3**4)
b3 = np.sum(ml.eye(3) - A, 1) / (1 - m3 - m33)
b2 = (-m33 * ml.eye(3) + A) * b3 / (1 - m2)
b1 = (-m3 * b3 + A * b2) / (1 - m1)
B = ml.hstack((b1, b2, b3))
a1 = alpha * b1
return (a1, m1, m2, m3, B)
def __init__(self, x1, x2, fclass=None):
super(Graph, self).__init__()
self.xres = x1.shape[0]
self.yres = x1.shape[1]
self.fclass = fclass
n = 2 * x1.size - 2 * self.yres
self.add_nodes_from(range(1, n + 1))
x1 = np.vstack((x1.reshape((-1, 1)), x1[-2:0:-1].reshape((-1, 1))))
x2 = np.vstack((x2.reshape((-1, 1)), x2[-2:0:-1].reshape((-1, 1))))
self.x = np.hstack((x1, x2))
self.pos = dict(
zip(range(1, n + 1), zip(x1.T.tolist()[0],
x2.T.tolist()[0])))
self.full = nx.DiGraph(self)
for i in range(1, n + 1):
for j in self.column(i):
dx1 = np.absolute(self.pos[j][0] - self.pos[i][0])
dx2 = np.absolute(self.pos[j][1] - self.pos[i][1])
if dx2 <= dx1 / _GRAPH_ASPECT_RATIO:
self.full.add_edge(i, j)
tmp = 1
self.basic = set()
step = self.yres / _GRAPH_YRES_BASIC
while tmp < n + 1:
self.basic = self.basic | set(self.column(tmp)[0::step])
tmp = tmp + self.yres
self.update_active()
def __init__(self, x1, x2, fclass=None):
super(Graph, self).__init__()
self.xres = x1.shape[0]
self.yres = x1.shape[1]
self.fclass = fclass
n = 2*x1.size - 2*self.yres
self.add_nodes_from(range(1, n+1))
x1 = np.vstack((x1.reshape((-1, 1)), x1[-2:0:-1].reshape((-1, 1))))
x2 = np.vstack((x2.reshape((-1, 1)), x2[-2:0:-1].reshape((-1, 1))))
self.x = np.hstack((x1, x2))
self.pos = dict(zip(range(1, n+1),
zip(x1.T.tolist()[0], x2.T.tolist()[0])))
self.full = nx.DiGraph(self)
for i in range(1, n+1):
for j in self.column(i):
dx1 = np.absolute(self.pos[j][0] - self.pos[i][0])
dx2 = np.absolute(self.pos[j][1] - self.pos[i][1])
if dx2 <= dx1 / _GRAPH_ASPECT_RATIO:
self.full.add_edge(i, j)
tmp = 1
self.basic = set()
step = self.yres/_GRAPH_YRES_BASIC
while tmp < n+1:
self.basic = self.basic | set(self.column(tmp)[0::step])
tmp = tmp + self.yres
self.update_active()
def firstInitElem(m33,sortEigs,alpha,A):
l1=sortEigs[0]
l2=sortEigs[1]
l3=sortEigs[2]
m1=-m33+l1+l2+l3
m2=-((l2-l3)*(l1**2-l1*l2-l1*l3+l2*l3)*(m33**3-2*m33**2*l1+m33*l1**2-2*m33**2*l2+3*m33*l1*l2-l1**2*l2+m33*l2**2- \
l1*l2**2-2*m33**2*l3+3*m33*l1*l3-l1**2*l3+3*m33*l2*l3-2*l1*l2*l3-l2**2*l3+m33*l3**2-l1*l3**2-l2*l3**2))/ \
(2*m33*l1**2*l2+2*m33**2*l1**2*l2-l1**3*l2-3*m33*l1**3*l2+l1**4*l2-2*m33*l1*l2**2-2*m33**2*l1*l2**2+l1**3*l2**2+ \
l1*l2**3+3*m33*l1*l2**3-l1**2*l2**3-l1*l2**4-2*m33*l1**2*l3-2*m33**2*l1**2*l3+l1**3*l3+3*m33*l1**3*l3-l1**4*l3+ \
2*m33*l2**2*l3+2*m33**2*l2**2*l3-l2**3*l3-3*m33*l2**3*l3+l2**4*l3+2*m33*l1*l3**2+2*m33**2*l1*l3**2-l1**3*l3**2- \
2*m33*l2*l3**2-2*m33**2*l2*l3**2+l2**3*l3**2-l1*l3**3-3*m33*l1*l3**3+l1**2*l3**3+l2*l3**3+3*m33*l2*l3**3-l2**2*l3**3+ \
l1*l3**4-l2*l3**4)
m3=((l2-l3)*(l1**2-l1*l2-l1*l3+l2*l3)*(m33**3+m33**4-m33**2*l1-2*m33**3*l1+m33**2*l1**2-m33**2*l2-2*m33**3*l2+ \
m33*l1*l2+3*m33**2*l1*l2-m33*l1**2*l2+m33**2*l2**2-m33*l1*l2**2-m33**2*l3-2*m33**3*l3+m33*l1*l3+3*m33**2*l1*l3- \
m33*l1**2*l3+m33*l2*l3+3*m33**2*l2*l3-l1*l2*l3-4*m33*l1*l2*l3+l1**2*l2*l3-m33*l2**2*l3+l1*l2**2*l3+m33**2*l3**2- \
m33*l1*l3**2-m33*l2*l3**2+l1*l2*l3**2))/(-2*m33*l1**2*l2-2*m33**2*l1**2*l2-m33**3*l1**2*l2+l1**3*l2+3*m33*l1**3*l2+ \
2*m33**2*l1**3*l2-l1**4*l2-m33*l1**4*l2+2*m33*l1*l2**2+2*m33**2*l1*l2**2+m33**3*l1*l2**2-l1**3*l2**2-2*m33*l1**3*l2**2+ \
l1**4*l2**2-l1*l2**3-3*m33*l1*l2**3-2*m33**2*l1*l2**3+l1**2*l2**3+2*m33*l1**2*l2**3+l1*l2**4+m33*l1*l2**4-l1**2*l2**4+ \
2*m33*l1**2*l3+2*m33**2*l1**2*l3+m33**3*l1**2*l3-l1**3*l3-3*m33*l1**3*l3-2*m33**2*l1**3*l3+l1**4*l3+m33*l1**4*l3- \
2*m33*l2**2*l3-2*m33**2*l2**2*l3-m33**3*l2**2*l3+l2**3*l3+3*m33*l2**3*l3+2*m33**2*l2**3*l3-l2**4*l3-m33*l2**4*l3- \
2*m33*l1*l3**2-2*m33**2*l1*l3**2-m33**3*l1*l3**2+l1**3*l3**2+2*m33*l1**3*l3**2-l1**4*l3**2+2*m33*l2*l3**2+2*m33**2*l2*l3**2+ \
m33**3*l2*l3**2-l2**3*l3**2-2*m33*l2**3*l3**2+l2**4*l3**2+l1*l3**3+3*m33*l1*l3**3+2*m33**2*l1*l3**3-l1**2*l3**3- \
2*m33*l1**2*l3**3-l2*l3**3-3*m33*l2*l3**3-2*m33**2*l2*l3**3+l2**2*l3**3+2*m33*l2**2*l3**3-l1*l3**4-m33*l1*l3**4+ \
l1**2*l3**4+l2*l3**4+m33*l2*l3**4-l2**2*l3**4)
b3=np.sum(ml.eye(3)-A,1)/(1-m3-m33)
b2=(-m33*ml.eye(3)+A)*b3/(1-m2)
b1=(-m3*b3+A*b2)/(1-m1)
B=ml.hstack((b1,b2,b3))
a1=alpha*b1
return (a1,m1,m2,m3,B)
def get_u(self):
points = npmat.hstack([s.center_position for s in self.solids])
center = npmat.asmatrix(npmat.average(points, axis = 1)).T
centered_points = points - center
correlation_matrix = (centered_points * centered_points.T)/self.nb_solids
u = iterate(correlation_matrix, self.NB_ITERATIONS)
return u
def __new__(cls, m, n):
"""
"""
if not all((
isinstance(m, int),
isinstance(n, int),
)):
raise ValueError
data = matlib.hstack([
matlib.vstack((
matlib.hstack((matlib.identity(n, dtype=int), matlib.matrix(p, dtype=int).T)),
matlib.hstack((matlib.matrix(p, dtype=int), matlib.ones(1))),
))
for p in product(range(m), repeat=n)
])
return super().__new__(cls, data, dtype=int).view(cls)
def world_to_view(self, point):
i = self.top.cross(self.normal).normalize()
k = self.normal.normalize()
j = k.cross(i)
s = hstack([i.vec3d(), j.vec3d(), k.vec3d()])
p1 = s.T * point.vec3d()
or1 = s.T * self.origin.vec3d()
return p1 - or1
def world_to_view(self, point):
i = self.top.cross(self.normal).normalize()
k = self.normal.normalize()
j = k.cross(i)
s = hstack([i.vec3d(), j.vec3d(), k.vec3d()])
p1 = s.T * point.vec3d()
or1 = s.T * self.origin.vec3d()
return p1 - or1
def colliding_boxes(self):
nb_solids = self.nb_solids
points = npmat.hstack([s.center_position for s in self.solids])
center = npmat.asmatrix(npmat.average(points, axis = 1)).T
centered_points = points - center
correlation_matrix = (centered_points * centered_points.T) / nb_solids
u = iterate(correlation_matrix, self.NB_ITERATIONS)
if self.projected_bounds is None:
self.projected_bounds = []
for i in range(nb_solids):
self.projected_bounds.append((i, 0, 0.))
self.projected_bounds.append((i, 1, 0.))
min_max_projections = npmat.empty((nb_solids, 2))
for solid_id in range(nb_solids):
solid_AABB_corners = self.solids[solid_id].AABB_corners()
corners_projections = u.T * solid_AABB_corners
min_max_projections[solid_id, 0] = np.min(corners_projections)
min_max_projections[solid_id, 1] = np.max(corners_projections)
for i in range(2 * nb_solids):
solid_id, begin_or_end_id, value = self.projected_bounds[i]
new_value = min_max_projections[solid_id, begin_or_end_id]
self.projected_bounds[i] = (solid_id, begin_or_end_id, new_value)
# TODO: linear sorting
self.projected_bounds.sort(key = lambda x : x[2])
output = []
active_solids = []
for i in range(2 * nb_solids):
solid_id, begin_or_end_id, value = self.projected_bounds[i]
if begin_or_end_id == 0:
for active_solid_id in active_solids:
if self.solids[active_solid_id].AABB_intersect_with(self.solids[solid_id]):
output.append((active_solid_id, solid_id))
active_solids.append(solid_id)
else:
active_solids.remove(solid_id)
return output
def __init__(self, points, masses):
"""
Inputs:
points: list of 3 dimensional points
masses: list of float (same length as points)
"""
self.nb_points = len(points)
if self.nb_points == 0:
raise (Exception, "Provide at least one point")
if len(masses) != self.nb_points:
raise (Exception,
"Same number of points and masses must be provided")
self.masses = masses
self.initial_points = [Vect(p) for p in points]
# Center of mass p and its velocity
self.center_position = npmat.sum(self.initial_points,
axis=0) / self.nb_points
self.center_velocity = npmat.zeros((3, 1))
# Position of the points relative to the center of mass, stacked into a big matrix
self.centered_initial_points_hstack = npmat.hstack(
self.initial_points) - self.center_position
# Total mass and its inverse
self.total_mass = npmat.sum(self.masses)
self.total_mass_inverse = 1. / self.total_mass
# Inertia matrix and its inverse
self.initial_inertia_matrix = Solid._inertia_matrix(
self.masses, self.centered_initial_points_hstack)
self.initial_inertia_matrix_inverse = npmat.linalg.pinv(
self.initial_inertia_matrix)
# Rotation quaternion (used to construct matrix representing orientation) and angular momentum (sigma)
self.rotation_quaternion = Vect([1., 0., 0., 0.])
self.rotation_matrix = rotation_matrix_from_quaternion(
self.rotation_quaternion)
self.angular_momentum = npmat.zeros((3, 1))
self._compute_AABB()
def _compute_AABB(self):
points = self.points_hstack()
self.AABB = npmat.hstack([points.min(axis=1), points.max(axis=1)])
def GeneralFluidSolve (Q, R, Q0=[], prec=1e-14):
"""
Returns the parameters of the matrix-exponentially
distributed stationary distribution of a general
Markovian fluid model, where the fluid rates associated
with the states of the background process can be
arbitrary (zero is allowed as well).
Using the returned 4 parameters the stationary
solution can be obtained as follows.
The probability that the fluid level is zero while
being in different states of the background process
is given by vector mass0.
The density that the fluid level is x while being in
different states of the background process is
.. math::
\pi(x)=ini\cdot e^{K x}\cdot clo.
Parameters
----------
Q : matrix, shape (N,N)
The generator of the background Markov chain
R : diagonal matrix, shape (N,N)
The diagonal matrix of the fluid rates associated
with the different states of the background process
Q0 : matrix, shape (N,N), optional
The generator of the background Markov chain at
level 0. If not provided, or empty, then Q0=Q is
assumed. The default value is empty.
precision : double, optional
Numerical precision for computing the fundamental
matrix. The default value is 1e-14
Returns
-------
mass0 : matrix, shape (1,Np+Nm)
The stationary probability vector of zero level
ini : matrix, shape (1,Np)
The initial vector of the stationary density
K : matrix, shape (Np,Np)
The matrix parameter of the stationary density
clo : matrix, shape (Np,Np+Nm)
The closing matrix of the stationary density
"""
N = Q.shape[0]
# partition the state space according to zero, positive and negative fluid rates
ix = np.arange(N)
ixz = ix[np.abs(np.diag(R))<=prec]
ixp = ix[np.diag(R)>prec]
ixn = ix[np.diag(R)<-prec]
Nz = len(ixz)
Np = len(ixp)
Nn = len(ixn)
# permutation matrix that converts between the original and the partitioned state ordering
P = ml.zeros((N,N))
for i in range(Nz):
P[i,ixz[i]]=1
for i in range(Np):
P[Nz+i,ixp[i]]=1
for i in range(Nn):
P[Nz+Np+i,ixn[i]]=1
iP = P.I
Qv = P*Q*iP
Rv = P*R*iP
# new fluid process censored to states + and -
iQv00 = la.pinv(-Qv[:Nz,:Nz])
Qbar = Qv[Nz:, Nz:] + Qv[Nz:,:Nz]*iQv00*Qv[:Nz,Nz:]
absRi = Diag(np.abs(1./np.diag(Rv[Nz:,Nz:])))
Qz = absRi * Qbar
Psi, K, U = FluidFundamentalMatrices (Qz[:Np,:Np], Qz[:Np,Np:], Qz[Np:,:Np], Qz[Np:,Np:], "PKU", prec)
# closing matrix
Pm = np.hstack((ml.eye(Np), Psi)) * absRi
iCn = absRi[Np:,Np:]
iCp = absRi[:Np,:Np]
clo = np.hstack(((iCp*Qv[Nz:Nz+Np,:Nz]+Psi*iCn*Qv[Nz+Np:,:Nz])*iQv00, Pm))
if len(Q0)==0: # regular boundary behavior
clo = clo * P # go back the the original state ordering
# calculate boundary vector
Ua = iCn*Qv[Nz+Np:,:Nz]*iQv00*ml.ones((Nz,1)) + iCn*ml.ones((Nn,1)) + Qz[Np:,:Np]*la.inv(-K)*clo*ml.ones((Nz+Np+Nn,1))
pm = Linsolve (ml.hstack((U,Ua)).T, ml.hstack((ml.zeros((1,Nn)),ml.ones((1,1)))).T).T
# create the result
mass0 = ml.hstack((pm*iCn*Qv[Nz+Np:,:Nz]*iQv00, ml.zeros((1,Np)), pm*iCn))*P
ini = pm*Qz[Np:,:Np]
else:
# solve a linear system for ini(+), pm(-) and pm(0)
Q0v = P*Q0*iP
M = ml.vstack((-clo*Rv, Q0v[Nz+Np:,:], Q0v[:Nz,:]))
Ma = ml.vstack((np.sum(la.inv(-K)*clo,1), ml.ones((Nz+Nn,1))))
sol = Linsolve (ml.hstack((M,Ma)).T, ml.hstack((ml.zeros((1,N)),ml.ones((1,1)))).T).T;
ini = sol[:,:Np]
clo = clo * P
mass0 = ml.hstack((sol[:,Np+Nn:], ml.zeros((1,Np)), sol[:,Np:Np+Nn]))*P
return mass0, ini, K, clo
def FluidSolve (Fpp, Fpm, Fmp, Fmm, prec=1e-14):
"""
Returns the parameters of the matrix-exponentially
distributed stationary distribution of a canonical
Markovian fluid model.
The canonical Markov fluid model is defined by the
matrix blocks of the generator of the background Markov
chain partitioned according to the sign of the
associated fluid rates (i.e., there are "+" and "-" states).
Using the returned 4 parameters the stationary
solution can be obtained as follows.
The probability that the fluid level is zero while
being in different states of the background process
is given by vector mass0.
The density that the fluid level is x while being in
different states of the background process is
.. math::
\pi(x)=ini\cdot e^{K x}\cdot clo.
Parameters
----------
Fpp : matrix, shape (Np,Np)
The matrix of transition rates between states
having positive fluid rates
Fpm : matrix, shape (Np,Nm)
The matrix of transition rates where the source
state has a positive, the destination has a
negative fluid rate associated.
Fpm : matrix, shape (Nm,Np)
The matrix of transition rates where the source
state has a negative, the destination has a
positive fluid rate associated.
Fpp : matrix, shape (Nm,Nm)
The matrix of transition rates between states
having negative fluid rates
precision : double, optional
Numerical precision for computing the fundamental
matrix. The default value is 1e-14
Returns
-------
mass0 : matrix, shape (1,Np+Nm)
The stationary probability vector of zero level
ini : matrix, shape (1,Np)
The initial vector of the stationary density
K : matrix, shape (Np,Np)
The matrix parameter of the stationary density
clo : matrix, shape (Np,Np+Nm)
The closing matrix of the stationary density
"""
Psi, K, U = FluidFundamentalMatrices (Fpp, Fpm, Fmp, Fmm, "PKU", prec)
mass0 = CTMCSolve(U)
nr = np.sum(mass0) + 2.0*np.sum(mass0*Fmp*-K.I)
return ml.hstack((ml.zeros((1,Fpp.shape[0])),mass0/nr)), mass0*Fmp/nr, K, ml.hstack((ml.eye(Fpp.shape[0]), Psi))
def MMAPPH1PRPR(D, sigma, S, *argv):
"""
Returns various performane measures of a MMAP[K]/PH[K]/1
preemptive resume priority queue, see [1]_.
Parameters
----------
D : list of matrices of shape (N,N), length (K+1)
The D0...DK matrices of the arrival process.
D1 corresponds to the lowest, DK to the highest priority.
sigma : list of row vectors, length (K)
The list containing the initial probability vectors of the service
time distributions of the various customer types. The length of the
vectors does not have to be the same.
S : list of square matrices, length (K)
The transient generators of the phase type distributions representing
the service time of the jobs belonging to various types.
further parameters :
The rest of the function parameters specify the options
and the performance measures to be computed.
The supported performance measures and options in this
function are:
+----------------+--------------------+----------------------------------------+
| Parameter name | Input parameters | Output |
+================+====================+========================================+
| "ncMoms" | Number of moments | The moments of the number of customers |
+----------------+--------------------+----------------------------------------+
| "ncDistr" | Upper limit K | The distribution of the number of |
| | | customers from level 0 to level K-1 |
+----------------+--------------------+----------------------------------------+
| "stMoms" | Number of moments | The sojourn time moments |
+----------------+--------------------+----------------------------------------+
| "stDistr" | A vector of points | The sojourn time distribution at the |
| | | requested points (cummulative, cdf) |
+----------------+--------------------+----------------------------------------+
| "prec" | The precision | Numerical precision used as a stopping |
| | | condition when solving the Riccati and |
| | | the matrix-quadratic equations |
+----------------+--------------------+----------------------------------------+
| "erlMaxOrder" | Integer number | The maximal Erlang order used in the |
| | | erlangization procedure. The default |
| | | value is 200. |
+----------------+--------------------+----------------------------------------+
| "classes" | Vector of integers | Only the performance measures |
| | | belonging to these classes are |
| | | returned. If not given, all classes |
| | | are analyzed. |
+----------------+--------------------+----------------------------------------+
(The quantities related to the number of customers in
the system include the customer in the server, and the
sojourn time related quantities include the service
times as well)
Returns
-------
Ret : list of the performance measures
Each entry of the list corresponds to a performance
measure requested. Each entry is a matrix, where the
columns belong to the various job types.
If there is just a single item,
then it is not put into a list.
References
----------
.. [1] G. Horvath, "Efficient analysis of the MMAP[K]/PH[K]/1
priority queue", European Journal of Operational
Research, 246(1), 128-139, 2015.
"""
K = len(D) - 1
# parse options
eaten = []
erlMaxOrder = 200
precision = 1e-14
classes = np.arange(0, K)
for i in range(len(argv)):
if argv[i] == "prec":
precision = argv[i + 1]
eaten.append(i)
eaten.append(i + 1)
elif argv[i] == "erlMaxOrder":
erlMaxOrder = argv[i + 1]
eaten.append(i)
eaten.append(i + 1)
elif argv[i] == "classes":
classes = np.array(argv[i + 1]) - 1
eaten.append(i)
eaten.append(i + 1)
if butools.checkInput and not CheckMMAPRepresentation(D):
raise Exception(
'MMAPPH1PRPR: The arrival process is not a valid MMAP representation!'
)
if butools.checkInput:
for k in range(K):
if not CheckPHRepresentation(sigma[k], S[k]):
raise Exception(
'MMAPPH1PRPR: the vector and matrix describing the service times is not a valid PH representation!'
)
# some preparation
D0 = D[0]
N = D0.shape[0]
I = ml.eye(N)
sD = ml.zeros((N, N))
for Di in D:
sD += Di
s = []
M = np.empty(K)
for i in range(K):
s.append(np.sum(-S[i], 1))
M[i] = sigma[i].size
Ret = []
for k in classes:
# step 1. solution of the workload process of the system
# ======================================================
sM = np.sum(M[k:K])
Qwmm = ml.matrix(D0)
for i in range(k):
Qwmm += D[i + 1]
Qwpm = ml.zeros((N * sM, N))
Qwmp = ml.zeros((N, N * sM))
Qwpp = ml.zeros((N * sM, N * sM))
kix = 0
for i in range(k, K):
Qwmp[:, kix:kix + N * M[i]] = np.kron(D[i + 1], sigma[i])
Qwpm[kix:kix + N * M[i], :] = np.kron(I, s[i])
Qwpp[kix:kix + N * M[i], :][:,
kix:kix + N * M[i]] = np.kron(I, S[i])
kix += N * M[i]
# calculate fundamental matrices
Psiw, Kw, Uw = FluidFundamentalMatrices(Qwpp, Qwpm, Qwmp, Qwmm, 'PKU',
precision)
# calculate boundary vector
Ua = ml.ones((N, 1)) + 2 * np.sum(Qwmp * (-Kw).I, 1)
pm = Linsolve(
ml.hstack((Uw, Ua)).T,
ml.hstack((ml.zeros((1, N)), ml.ones((1, 1)))).T).T
Bw = ml.zeros((N * sM, N))
Bw[0:N * M[k], :] = np.kron(I, s[k])
kappa = pm * Qwmp / np.sum(pm * Qwmp * (-Kw).I * Bw)
if k < K - 1:
# step 2. construct fluid model for the remaining sojourn time process
# ====================================================================
# (for each class except the highest priority)
Qsmm = ml.matrix(D0)
for i in range(k + 1):
Qsmm += D[i + 1]
Np = Kw.shape[0]
Qspm = ml.zeros((Np + N * np.sum(M[k + 1:]), N))
Qsmp = ml.zeros((N, Np + N * np.sum(M[k + 1:])))
Qspp = ml.zeros(
(Np + N * np.sum(M[k + 1:]), Np + N * np.sum(M[k + 1:])))
Qspp[:Np, :Np] = Kw
Qspm[:Np, :N] = Bw
kix = Np
for i in range(k + 1, K):
Qsmp[:, kix:kix + N * M[i]] = np.kron(D[i + 1], sigma[i])
Qspm[kix:kix + N * M[i], :] = np.kron(I, s[i])
Qspp[kix:kix + N * M[i], kix:kix + N * M[i]] = np.kron(I, S[i])
kix += N * M[i]
inis = ml.hstack((kappa, ml.zeros((1, N * np.sum(M[k + 1:])))))
Psis = FluidFundamentalMatrices(Qspp, Qspm, Qsmp, Qsmm, 'P',
precision)
# step 3. calculate the performance measures
# ==========================================
argIx = 0
while argIx < len(argv):
if argIx in eaten:
argIx += 1
continue
elif type(argv[argIx]) is str and argv[argIx] == "stMoms":
# MOMENTS OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfSTMoms = argv[argIx + 1]
Pn = [Psis]
rtMoms = []
for n in range(1, numOfSTMoms + 1):
A = Qspp + Psis * Qsmp
B = Qsmm + Qsmp * Psis
C = -2 * n * Pn[n - 1]
bino = 1
for i in range(1, n):
bino = bino * (n - i + 1) / i
C += bino * Pn[i] * Qsmp * Pn[n - i]
P = la.solve_sylvester(A, B, -C)
Pn.append(P)
rtMoms.append(np.sum(inis * P * (-1)**n) / 2**n)
Ret.append(rtMoms)
argIx += 1
elif type(argv[argIx]) is str and argv[argIx] == "stDistr":
# DISTRIBUTION OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
stCdfPoints = argv[argIx + 1]
res = []
for t in stCdfPoints:
L = erlMaxOrder
lambd = L / t / 2
Psie = FluidFundamentalMatrices(
Qspp - lambd * ml.eye(Qspp.shape[0]), Qspm, Qsmp,
Qsmm - lambd * ml.eye(Qsmm.shape[0]), 'P',
precision)
Pn = [Psie]
pr = np.sum(inis * Psie)
for n in range(1, L):
A = Qspp + Psie * Qsmp - lambd * ml.eye(
Qspp.shape[0])
B = Qsmm + Qsmp * Psie - lambd * ml.eye(
Qsmm.shape[0])
C = 2 * lambd * Pn[n - 1]
for i in range(1, n):
C += Pn[i] * Qsmp * Pn[n - i]
P = la.solve_sylvester(A, B, -C)
Pn.append(P)
pr += np.sum(inis * P)
res.append(pr)
Ret.append(np.array(res))
argIx += 1
elif type(argv[argIx]) is str and argv[argIx] == "ncMoms":
# MOMENTS OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLMoms = argv[argIx + 1]
# first calculate it at departure instants
QLDPn = [Psis]
dqlMoms = []
for n in range(1, numOfQLMoms + 1):
A = Qspp + Psis * Qsmp
B = Qsmm + Qsmp * Psis
C = n * QLDPn[n - 1] * D[k + 1]
bino = 1
for i in range(1, n):
bino = bino * (n - i + 1) / i
C = C + bino * QLDPn[i] * Qsmp * QLDPn[n - i]
P = la.solve_sylvester(A, B, -C)
QLDPn.append(P)
dqlMoms.append(np.sum(inis * P))
dqlMoms = MomsFromFactorialMoms(dqlMoms)
# now calculate it at random time instance
pi = CTMCSolve(sD)
lambdak = np.sum(pi * D[k + 1])
QLPn = [pi]
qlMoms = []
iTerm = (ml.ones((N, 1)) * pi - sD).I
for n in range(1, numOfQLMoms + 1):
sumP = np.sum(inis * QLDPn[n]) + n * (
inis * QLDPn[n - 1] - QLPn[n - 1] * D[k + 1] /
lambdak) * iTerm * np.sum(D[k + 1], 1)
P = sumP * pi + n * (QLPn[n - 1] * D[k + 1] - inis *
QLDPn[n - 1] * lambdak) * iTerm
QLPn.append(P)
qlMoms.append(np.sum(P))
qlMoms = MomsFromFactorialMoms(qlMoms)
Ret.append(qlMoms)
argIx += 1
elif type(argv[argIx]) is str and argv[argIx] == "ncDistr":
# DISTRIBUTION OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLProbs = argv[argIx + 1]
sDk = ml.matrix(D0)
for i in range(k):
sDk += D[i + 1]
# first calculate it at departure instants
Psid = FluidFundamentalMatrices(Qspp, Qspm, Qsmp, sDk, 'P',
precision)
Pn = [Psid]
dqlProbs = inis * Psid
for n in range(1, numOfQLProbs):
A = Qspp + Psid * Qsmp
B = sDk + Qsmp * Psid
C = Pn[n - 1] * D[k + 1]
for i in range(1, n):
C += Pn[i] * Qsmp * Pn[n - i]
P = la.solve_sylvester(A, B, -C)
Pn.append(P)
dqlProbs = ml.vstack((dqlProbs, inis * P))
# now calculate it at random time instance
pi = CTMCSolve(sD)
lambdak = np.sum(pi * D[k + 1])
iTerm = -(sD - D[k + 1]).I
qlProbs = lambdak * dqlProbs[0, :] * iTerm
for n in range(1, numOfQLProbs):
P = (qlProbs[n - 1, :] * D[k + 1] + lambdak *
(dqlProbs[n, :] - dqlProbs[n - 1, :])) * iTerm
qlProbs = ml.vstack((qlProbs, P))
qlProbs = np.sum(qlProbs, 1).A.flatten()
Ret.append(qlProbs)
argIx += 1
else:
raise Exception("MMAPPH1PRPR: Unknown parameter " +
str(argv[argIx]))
argIx += 1
elif k == K - 1:
# step 3. calculate the performance measures
# ==========================================
argIx = 0
while argIx < len(argv):
if argIx in eaten:
argIx += 1
continue
elif type(argv[argIx]) is str and argv[argIx] == "stMoms":
# MOMENTS OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfSTMoms = argv[argIx + 1]
rtMoms = []
for i in range(1, numOfSTMoms + 1):
rtMoms.append(
np.sum(
math.factorial(i) * kappa * (-Kw).I**(i + 1) *
Bw))
Ret.append(rtMoms)
argIx += 1
elif type(argv[argIx]) is str and argv[argIx] == "stDistr":
# DISTRIBUTION OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
stCdfPoints = argv[argIx + 1]
rtDistr = []
for t in stCdfPoints:
rtDistr.append(
np.sum(kappa * (-Kw).I *
(ml.eye(Kw.shape[0]) - la.expm(Kw * t)) *
Bw))
Ret.append(np.array(rtDistr))
argIx += 1
elif type(argv[argIx]) is str and (argv[argIx] == "ncMoms" or
argv[argIx] == "ncDistr"):
L = np.kron(sD - D[k + 1], ml.eye(M[k])) + np.kron(
ml.eye(N), S[k])
B = np.kron(ml.eye(N), s[k] * sigma[k])
F = np.kron(D[k + 1], ml.eye(M[k]))
L0 = np.kron(sD - D[k + 1], ml.eye(M[k]))
R = QBDFundamentalMatrices(B, L, F, 'R', precision)
p0 = CTMCSolve(L0 + R * B)
p0 = p0 / np.sum(p0 * (ml.eye(R.shape[0]) - R).I)
if argv[argIx] == "ncMoms":
# MOMENTS OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLMoms = argv[argIx + 1]
qlMoms = []
for i in range(1, numOfQLMoms + 1):
qlMoms.append(
np.sum(
math.factorial(i) * p0 * R**i *
(ml.eye(R.shape[0]) - R).I**(i + 1)))
Ret.append(MomsFromFactorialMoms(qlMoms))
elif argv[argIx] == "ncDistr":
# DISTRIBUTION OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLProbs = argv[argIx + 1]
qlProbs = [np.sum(p0)]
for i in range(1, numOfQLProbs):
qlProbs.append(np.sum(p0 * R**i))
Ret.append(np.array(qlProbs))
argIx += 1
else:
raise Exception("MMAPPH1PRPR: Unknown parameter " +
str(argv[argIx]))
argIx += 1
if len(Ret) == 1:
return Ret[0]
else:
return Ret
def CanonicalFromDPH3(alpha, A, prec=1e-14):
"""
Returns the canonical form of an order-3 discrete phase-type
distribution.
Parameters
----------
alpha : matrix, shape (1,3)
Initial vector of the discrete phase-type distribution
A : matrix, shape (3,3)
Transition probability matrix of the discrete phase-type
distribution
prec : double, optional
Numerical precision for checking the input, default value
is 1e-14
Returns
-------
beta : matrix, shape (1,3)
The initial probability vector of the canonical form
B : matrix, shape (3,3)
Transition probability matrix of the canonical form
"""
if butools.checkInput and not CheckMGRepresentation(alpha, A, prec):
raise Exception(
"CanonicalFromDPH3: Input is not a valid DPH representation!")
if butools.checkInput and (A.shape[0] != 3 or A.shape[1] != 3):
raise Exception("CanonicalFromDPH3: Dimension must be 3!")
ev = la.eigvals(A)
ix = np.argsort(-np.abs(np.real(ev)))
lambd = ev[ix]
eye = ml.eye(3)
a0 = -lambd[0] * lambd[1] * lambd[2]
a1 = lambd[0] * lambd[1] + lambd[0] * lambd[2] + lambd[1] * lambd[2]
a2 = -lambd[0] - lambd[1] - lambd[2]
e = ml.matrix([[1, 1, 1]]).T
if np.real(lambd[0]) > 0 and np.real(lambd[1]) >= 0 and np.real(
lambd[2]) >= 0:
#PPP case
alphaout, A2 = CanonicalFromPH3(alpha, A - eye, prec)
Aout = A2 + eye
elif np.real(lambd[0]) > 0 and np.real(lambd[1]) >= 0 and np.real(
lambd[2]) < 0:
#PPN case
x1 = lambd[0]
x2 = lambd[1] + lambd[2]
x3 = lambd[1] * lambd[2] / (lambd[1] + lambd[2] - 1)
Aout = ml.matrix([[x1, 1 - x1, 0], [0, x2, 1 - x2], [0, x3, 0]])
b3 = 1 / (1 - x3) * (e - A * e)
b2 = 1 / (1 - x2) * A * b3
b1 = e - b2 - b3
B = ml.hstack((b1, b2, b3))
alphaout = alpha * B
elif np.real(lambd[0]) > 0 and np.real(lambd[1]) < 0 and np.real(
lambd[2]) >= 0:
#PNP case
x1 = -a2
x2 = (a0 - a1 * a2) / (a2 * (1 + a2))
x3 = a0 * (1 + a2) / (a0 - a2 - a1 * a2 - a2**2)
Aout = ml.matrix([[x1, 1 - x1, 0], [x2, 0, 1 - x2], [0, x3, 0]])
b3 = 1 / (1 - x3) * (e - A * e)
b2 = 1 / (1 - x2) * A * b3
b1 = e - b2 - b3
if alpha * b1 >= 0:
B = ml.hstack((b1, b2, b3))
alphaout = alpha * B
else:
#Set the initial vector first element to 0
x33 = 0
a1 = -1
while x33 <= 1:
[a1, x1, x2, x3, B] = firstInitElem(x33, lambd, alpha, A)
if a1 >= 0 and x1 >= 0 and x2 >= 0 and x3 >= 0 and x3 + x33 < 1:
break
x33 = x33 + 0.01
if a1 >= 0:
Aout = ml.matrix([[x1, 1 - x1, 0], [x2, 0, 1 - x2],
[0, x3, x33]])
alphaout = alpha * B
else:
#PNP+
x1 = lambd[2]
x2 = lambd[0] + lambd[1]
x3 = lambd[0] * lambd[1] / (lambd[0] + lambd[1] - 1)
Aout = ml.matrix([[x1, 0, 0], [0, x2, 1 - x2], [0, x3, 0]])
p1 = alpha * (e - A * e)
p2 = alpha * A * (e - A * e)
d1 = (1 - lambd[0]) * ((1 - lambd[1]) * (1 - lambd[2]) +
(-1 + lambd[1] + lambd[2]) * p1 -
p2) / ((lambd[0] - lambd[1]) *
(lambd[0] - lambd[2]))
d2 = (lambd[1] - 1) * ((1 - lambd[0]) * (1 - lambd[2]) +
(-1 + lambd[0] + lambd[2]) * p1 -
p2) / ((lambd[0] - lambd[1]) *
(lambd[1] - lambd[2]))
d3 = (lambd[2] - 1) * ((1 - lambd[0]) * (1 - lambd[1]) +
(-1 + lambd[0] + lambd[1]) * p1 -
p2) / ((lambd[1] - lambd[2]) *
(lambd[2] - lambd[0]))
alphaout = ml.matrix([[
d3 / (1 - lambd[2]),
(d1 * lambd[0] + d2 * lambd[1]) / ((1 - lambd[0]) *
(1 - lambd[1])),
(d1 + d2) * (1 - lambd[0] - lambd[1]) / ((1 - lambd[0]) *
(1 - lambd[1]))
]])
if np.min(alphaout) < 0 or np.min(np.min(Aout)) < 0:
raise Exception("DPH3Canonical: Unhandled PNP case!")
elif np.real(lambd[0]) > 0 and np.real(lambd[1]) < 0 and np.real(
lambd[2]) < 0:
#PNN case
if np.all(np.isreal(lambd)) or np.abs(
lambd[1])**2 <= 2 * lambd[0] * (-np.real(lambd[1])):
x1 = -a2
x2 = -a1 / (1 + a2)
x3 = -a0 / (1 + a1 + a2)
Aout = ml.matrix([[x1, 1 - x1, 0], [x2, 0, 1 - x2], [x3, 0, 0]])
b3 = 1 / (1 - x3) * (e - A * e)
b2 = 1 / (1 - x2) * A * b3
b1 = e - b2 - b3
B = ml.hstack((b1, b2, b3))
alphaout = alpha * B
else:
[alphaout, A2] = CanonicalFromPH3(alpha, A - eye, prec)
Aout = A2 + eye
return (np.real(alphaout), np.real(Aout))
def SPIRIT(streams, energyThresh, lamb, evalMetrics):
# Make
if type(streams) == np.ndarray:
streams_iter = iter(streams)
# Max No. Streams
if streams.ndim == 1:
streams = np.expand_dims(streams, axis=1)
num_streams = streams.shape[1]
else:
num_streams = streams.shape[1]
count_over = 0
count_under = 0
#===============================================================================
# Initalise k, w and d, lamb
#===============================================================================
k = 1 # Hidden Variables, initialise to one
# Weights
pc_weights = npm.zeros(num_streams)
pc_weights[0, 0] = 1
# initialise outputs
res = {}
all_weights = []
k_hist = []
anomalies = []
x_dash = npm.zeros((1,num_streams))
Eng = mat([0.00000001, 0.00000001])
E_xt = 0 # Energy of X at time t
E_rec_i = mat([0.000000000000001]) # Energy of reconstruction
Y = npm.zeros(num_streams)
timeSteps = streams.shape[0]
#===============================================================================
# Main Loop
#===============================================================================
for t in range(1, timeSteps + 1): # t = 1,...,200
k_hist.append(k)
x_t_plus_1 = mat(streams_iter.next()) # Read in next signals
d_i = E_rec_i * t
# Step 1 - Update Weights
pc_weights, y_t_i, error = track_W(x_t_plus_1,
k, pc_weights, d_i,
num_streams,
lamb)
# Record hidden variables
padding = num_streams - k
y_bar_t = npm.hstack((y_t_i, mat([nan] * padding)))
Y = npm.vstack((Y,y_bar_t))
# Record Weights
all_weights.append(pc_weights)
# Record reconstrunted z and RSRE
x_dash = npm.vstack((x_dash, y_t_i * pc_weights))
# Record RSRE
if t == 1:
top = 0.0
bot = 0.0
top = top + (norm(x_t_plus_1 - x_dash) ** 2 )
bot = bot + (norm(x_t_plus_1) ** 2)
new_RSRE = top / bot
if t == 1:
RSRE = new_RSRE
else:
RSRE = npm.vstack((RSRE, new_RSRE))
### FOR EVALUATION ###
#deviation from truth
if evalMetrics == 'T' :
Qt = pc_weights.T
if t == 1 :
res['subspace_error'] = npm.zeros((timeSteps,1))
res['orthog_error'] = npm.zeros((timeSteps,1))
res['angle_error'] = npm.zeros((timeSteps,1))
Cov_mat = npm.zeros([num_streams,num_streams])
# Calculate Covarentce Matrix of data up to time t
Cov_mat = lamb * Cov_mat + npm.dot(x_t_plus_1, x_t_plus_1.T)
# Get eigenvalues and eigenvectors
W , V = eig(Cov_mat)
# Use this to sort eigenVectors in according to deccending eigenvalue
eig_idx = W.argsort() # Get sort index
eig_idx = eig_idx[::-1] # Reverse order (default is accending)
# v_r = highest r eigen vectors (accoring to thier eigenvalue if sorted).
V_k = V[:, eig_idx[:k]]
# Calculate subspace error
C = npm.dot(V_k , V_k.T) - npm.dot(Qt , Qt.T)
res['subspace_error'][t-1,0] = 10 * np.log10(npm.trace(npm.dot(C.T , C))) #frobenius norm in dB
# Calculate angle between projection matrixes
D = npm.dot(npm.dot(npm.dot(V_k.T, Qt), Qt.T), V_k)
eigVal, eigVec = eig(D)
angle = npm.arccos(np.sqrt(max(eigVal)))
res['angle_error'][t-1,0] = angle
# Calculate deviation from orthonormality
F = npm.dot(Qt.T , Qt) - npm.eye(k)
res['orthog_error'][t-1,0] = 10 * np.log10(npm.trace(npm.dot(F.T , F))) #frobenius norm in dB
# Step 2 - Update Energy estimate
E_xt = ((lamb * (t-1) * E_xt) + norm(x_t_plus_1) ** 2) / t
for i in range(k):
E_rec_i[0, i] = ((lamb * (t-1) * E_rec_i[0, i]) + (y_t_i[0, i] ** 2)) / t
# Step 3 - Estimate the retained energy
E_retained = npm.sum(E_rec_i,1)
# Record Energy
Eng_new = npm.hstack((E_xt, E_retained[0,0]))
Eng = npm.vstack((Eng, Eng_new))
if E_retained < energyThresh[0] * E_xt:
if k != num_streams:
k = k + 1
# Initalise Ek+1 <-- 0
E_rec_i = npm.hstack((E_rec_i, mat([0])))
# Initialise W_i+1
new_weight_vec = npm.zeros(num_streams)
new_weight_vec[0, k-1] = 1
pc_weights = npm.vstack((pc_weights, new_weight_vec))
anomalies.append(t -1)
else:
count_over += 1
elif E_retained > energyThresh[1] * E_xt:
if k > 1 :
k = k - 1
# discard w_k and error
pc_weights = delete(pc_weights, -1, 0)
# Discard E_rec_i[k]
E_rec_i = delete(E_rec_i, -1)
else:
count_under += 1
# Data Stores
res2 = {'hidden' : Y, # Array for hidden Variables
'weights' : all_weights,
'E_t' : Eng[:,0], # total energy of data
'E_dash_t' : Eng[:,1], # hidden var energy
'e_ratio' : np.divide(Eng[:,1], Eng[:,0]), # Energy ratio
'RSRE' : RSRE, # Relative squared Reconstruction error
'recon' : x_dash, # reconstructed data
'r_hist' : k_hist, # history of r values
'anomalies' : anomalies}
res.update(res2)
return res, all_weights
def MMAPPH1FCFS(D, sigma, S, *argv):
"""
Returns various performane measures of a MMAP[K]/PH[K]/1
first-come-first-serve queue, see [1]_.
Parameters
----------
D : list of matrices of shape (N,N), length (K+1)
The D0...DK matrices of the arrival process.
sigma : list of row vectors, length (K)
The list containing the initial probability vectors of the service
time distributions of the various customer types. The length of the
vectors does not have to be the same.
S : list of square matrices, length (K)
The transient generators of the phase type distributions representing
the service time of the jobs belonging to various types.
further parameters :
The rest of the function parameters specify the options
and the performance measures to be computed.
The supported performance measures and options in this
function are:
+----------------+--------------------+----------------------------------------+
| Parameter name | Input parameters | Output |
+================+====================+========================================+
| "ncMoms" | Number of moments | The moments of the number of customers |
+----------------+--------------------+----------------------------------------+
| "ncDistr" | Upper limit K | The distribution of the number of |
| | | customers from level 0 to level K-1 |
+----------------+--------------------+----------------------------------------+
| "stMoms" | Number of moments | The sojourn time moments |
+----------------+--------------------+----------------------------------------+
| "stDistr" | A vector of points | The sojourn time distribution at the |
| | | requested points (cummulative, cdf) |
+----------------+--------------------+----------------------------------------+
| "stDistrME" | None | The vector-matrix parameters of the |
| | | matrix-exponentially distributed |
| | | sojourn time distribution |
+----------------+--------------------+----------------------------------------+
| "stDistrPH" | None | The vector-matrix parameters of the |
| | | matrix-exponentially distributed |
| | | sojourn time distribution, converted |
| | | to a continuous PH representation |
+----------------+--------------------+----------------------------------------+
| "prec" | The precision | Numerical precision used as a stopping |
| | | condition when solving the Riccati |
| | | equation |
+----------------+--------------------+----------------------------------------+
| "classes" | Vector of integers | Only the performance measures |
| | | belonging to these classes are |
| | | returned. If not given, all classes |
| | | are analyzed. |
+----------------+--------------------+----------------------------------------+
(The quantities related to the number of customers in
the system include the customer in the server, and the
sojourn time related quantities include the service
times as well)
Returns
-------
Ret : list of the performance measures
Each entry of the list corresponds to a performance
measure requested. Each entry is a matrix, where the
columns belong to the various job types.
If there is just a single item,
then it is not put into a list.
References
----------
.. [1] Qiming He, "Analysis of a continuous time
SM[K]/PH[K]/1/FCFS queue: Age process, sojourn times,
and queue lengths", Journal of Systems Science and
Complexity, 25(1), pp 133-155, 2012.
"""
K = len(D) - 1
# parse options
eaten = []
precision = 1e-14
classes = np.arange(0, K)
for i in range(len(argv)):
if argv[i] == "prec":
precision = argv[i + 1]
eaten.append(i)
eaten.append(i + 1)
elif argv[i] == "classes":
classes = np.array(argv[i + 1]) - 1
eaten.append(i)
eaten.append(i + 1)
if butools.checkInput and not CheckMMAPRepresentation(D):
raise Exception(
'MMAPPH1FCFS: The arrival process is not a valid MMAP representation!'
)
if butools.checkInput:
for k in range(K):
if not CheckPHRepresentation(sigma[k], S[k]):
raise Exception(
'MMAPPH1FCFS: the vector and matrix describing the service times is not a valid PH representation!'
)
# some preparation
D0 = D[0]
N = D0.shape[0]
Ia = ml.eye(N)
Da = ml.zeros((N, N))
for q in range(K):
Da += D[q + 1]
theta = CTMCSolve(D0 + Da)
beta = [CTMCSolve(S[k] + ml.sum(-S[k], 1) * sigma[k]) for k in range(K)]
lambd = [np.sum(theta * D[k + 1]) for k in range(K)]
mu = [np.sum(beta[k] * (-S[k])) for k in range(K)]
Nsk = [S[k].shape[0] for k in range(K)]
ro = np.sum(np.array(lambd) / np.array(mu))
alpha = theta * Da / sum(lambd)
D0i = (-D0).I
Sa = S[0]
sa = [ml.zeros(sigma[0].shape)] * K
sa[0] = sigma[0]
ba = [ml.zeros(beta[0].shape)] * K
ba[0] = beta[0]
sv = [ml.zeros((Nsk[0], 1))] * K
sv[0] = ml.sum(-S[0], 1)
Pk = [D0i * D[q + 1] for q in range(K)]
for k in range(1, K):
Sa = la.block_diag(Sa, S[k])
for q in range(K):
if q == k:
sa[q] = ml.hstack((sa[q], sigma[k]))
ba[q] = ml.hstack((ba[q], beta[k]))
sv[q] = ml.vstack((sv[q], -np.sum(S[k], 1)))
else:
sa[q] = ml.hstack((sa[q], ml.zeros(sigma[k].shape)))
ba[q] = ml.hstack((ba[q], ml.zeros(beta[k].shape)))
sv[q] = ml.vstack((sv[q], ml.zeros((Nsk[k], 1))))
Sa = ml.matrix(Sa)
P = D0i * Da
iVec = ml.kron(D[1], sa[0])
for k in range(1, K):
iVec += ml.kron(D[k + 1], sa[k])
Ns = Sa.shape[0]
Is = ml.eye(Ns)
# step 1. solve the age process of the queue
# ==========================================
# solve Y0 and calculate T
Y0 = FluidFundamentalMatrices(ml.kron(Ia, Sa), ml.kron(Ia, -ml.sum(Sa, 1)),
iVec, D0, "P", precision)
T = ml.kron(Ia, Sa) + Y0 * iVec
# calculate pi0 and v0
pi0 = ml.zeros((1, T.shape[0]))
for k in range(K):
pi0 += ml.kron(theta * D[k + 1], ba[k] / mu[k])
pi0 = -pi0 * T
iT = (-T).I
oa = ml.ones((N, 1))
# step 2. calculate performance measures
# ======================================
Ret = []
for k in classes:
argIx = 0
clo = iT * ml.kron(oa, sv[k])
while argIx < len(argv):
if argIx in eaten:
argIx += 1
continue
elif type(argv[argIx]) is str and argv[argIx] == "stMoms":
numOfSTMoms = argv[argIx + 1]
rtMoms = []
for m in range(1, numOfSTMoms + 1):
rtMoms.append(
math.factorial(m) * np.sum(pi0 * iT**m * clo /
(pi0 * clo)))
Ret.append(rtMoms)
argIx += 1
elif type(argv[argIx]) is str and argv[argIx] == "stDistr":
stCdfPoints = argv[argIx + 1]
cdf = []
for t in stCdfPoints:
pr = 1 - np.sum(pi0 * la.expm(T * t) * clo / (pi0 * clo))
cdf.append(pr)
Ret.append(np.array(cdf))
argIx += 1
elif type(argv[argIx]) is str and argv[argIx] == "stDistrME":
Bm = SimilarityMatrixForVectors(clo / (pi0 * clo),
ml.ones((N * Ns, 1)))
Bmi = Bm.I
A = Bm * T * Bmi
alpha = pi0 * Bmi
Ret.append(alpha)
Ret.append(A)
elif type(argv[argIx]) is str and argv[argIx] == "stDistrPH":
vv = pi0 * iT
ix = np.arange(N * Ns)
nz = ix[vv.flat > precision]
delta = Diag(vv[:, nz])
cl = -T * clo / (pi0 * clo)
alpha = cl[nz, :].T * delta
A = delta.I * T[nz, :][:, nz].T * delta
Ret.append(alpha)
Ret.append(A)
elif type(argv[argIx]) is str and argv[argIx] == "ncDistr":
numOfQLProbs = argv[argIx + 1]
argIx += 1
values = np.empty(numOfQLProbs)
jm = ml.zeros((Ns, 1))
jm[np.sum(Nsk[0:k]):np.sum(Nsk[0:k + 1]), :] = 1
jmc = ml.ones((Ns, 1))
jmc[np.sum(Nsk[0:k]):np.sum(Nsk[0:k + 1]), :] = 0
LmCurr = la.solve_sylvester(T, ml.kron(D0 + Da - D[k + 1], Is),
-ml.eye(N * Ns))
values[0] = 1 - ro + np.sum(pi0 * LmCurr * ml.kron(oa, jmc))
for i in range(1, numOfQLProbs):
LmPrev = LmCurr
LmCurr = la.solve_sylvester(
T, ml.kron(D0 + Da - D[k + 1], Is),
-LmPrev * ml.kron(D[k + 1], Is))
values[i] = np.sum(pi0 * LmCurr * ml.kron(oa, jmc) +
pi0 * LmPrev * ml.kron(oa, jm))
Ret.append(values)
elif type(argv[argIx]) is str and argv[argIx] == "ncMoms":
numOfQLMoms = argv[argIx + 1]
argIx += 1
jm = ml.zeros((Ns, 1))
jm[np.sum(Nsk[0:k]):np.sum(Nsk[0:k + 1]), :] = 1
ELn = [
la.solve_sylvester(T, ml.kron(D0 + Da, Is),
-ml.eye(N * Ns))
]
qlMoms = []
for n in range(1, numOfQLMoms + 1):
bino = 1
Btag = ml.zeros((N * Ns, N * Ns))
for i in range(n):
Btag += bino * ELn[i]
bino *= (n - i) / (i + 1)
ELn.append(
la.solve_sylvester(T, ml.kron(D0 + Da, Is),
-Btag * ml.kron(D[k + 1], Is)))
qlMoms.append(
np.sum(pi0 * ELn[n]) +
np.sum(pi0 * Btag * ml.kron(oa, jm)))
Ret.append(qlMoms)
else:
raise Exception("MMAPPH1FCFS: Unknown parameter " +
str(argv[argIx]))
argIx += 1
if len(Ret) == 1:
return Ret[0]
else:
return Ret
def MMAPPH1NPPR(D, sigma, S, *argv):
"""
Returns various performane measures of a continuous time
MMAP[K]/PH[K]/1 non-preemptive priority queue, see [1]_.
Parameters
----------
D : list of matrices of shape (N,N), length (K+1)
The D0...DK matrices of the arrival process.
D1 corresponds to the lowest, DK to the highest priority.
sigma : list of row vectors, length (K)
The list containing the initial probability vectors of the service
time distributions of the various customer types. The length of the
vectors does not have to be the same.
S : list of square matrices, length (K)
The transient generators of the phase type distributions representing
the service time of the jobs belonging to various types.
further parameters :
The rest of the function parameters specify the options
and the performance measures to be computed.
The supported performance measures and options in this
function are:
+----------------+--------------------+----------------------------------------+
| Parameter name | Input parameters | Output |
+================+====================+========================================+
| "ncMoms" | Number of moments | The moments of the number of customers |
+----------------+--------------------+----------------------------------------+
| "ncDistr" | Upper limit K | The distribution of the number of |
| | | customers from level 0 to level K-1 |
+----------------+--------------------+----------------------------------------+
| "stMoms" | Number of moments | The sojourn time moments |
+----------------+--------------------+----------------------------------------+
| "stDistr" | A vector of points | The sojourn time distribution at the |
| | | requested points (cummulative, cdf) |
+----------------+--------------------+----------------------------------------+
| "prec" | The precision | Numerical precision used as a stopping |
| | | condition when solving the Riccati and |
| | | the matrix-quadratic equations |
+----------------+--------------------+----------------------------------------+
| "erlMaxOrder" | Integer number | The maximal Erlang order used in the |
| | | erlangization procedure. The default |
| | | value is 200. |
+----------------+--------------------+----------------------------------------+
| "classes" | Vector of integers | Only the performance measures |
| | | belonging to these classes are |
| | | returned. If not given, all classes |
| | | are analyzed. |
+----------------+--------------------+----------------------------------------+
(The quantities related to the number of customers in
the system include the customer in the server, and the
sojourn time related quantities include the service
times as well)
Returns
-------
Ret : list of the performance measures
Each entry of the list corresponds to a performance
measure requested. Each entry is a matrix, where the
columns belong to the various job types.
If there is just a single item,
then it is not put into a list.
References
----------
.. [1] G. Horvath, "Efficient analysis of the MMAP[K]/PH[K]/1
priority queue", European Journal of Operational
Research, 246(1), 128-139, 2015.
"""
K = len(D) - 1
# parse options
eaten = []
erlMaxOrder = 200
precision = 1e-14
classes = np.arange(0, K)
for i in range(len(argv)):
if argv[i] == "prec":
precision = argv[i + 1]
eaten.append(i)
eaten.append(i + 1)
elif argv[i] == "erlMaxOrder":
erlMaxOrder = argv[i + 1]
eaten.append(i)
eaten.append(i + 1)
elif argv[i] == "classes":
classes = np.array(argv[i + 1]) - 1
eaten.append(i)
eaten.append(i + 1)
if butools.checkInput and not CheckMMAPRepresentation(D):
raise Exception(
'MMAPPH1PRPR: The arrival process is not a valid MMAP representation!'
)
if butools.checkInput:
for k in range(K):
if not CheckPHRepresentation(sigma[k], S[k]):
raise Exception(
'MMAPPH1PRPR: the vector and matrix describing the service times is not a valid PH representation!'
)
# some preparation
D0 = D[0]
N = D0.shape[0]
I = ml.eye(N)
sD = ml.zeros((N, N))
for Di in D:
sD += Di
s = []
M = np.empty(K)
for i in range(K):
s.append(np.sum(-S[i], 1))
M[i] = sigma[i].size
# step 1. solution of the workload process of the joint queue
# ===========================================================
sM = np.sum(M)
Qwmm = ml.matrix(D0)
Qwpm = ml.zeros((N * sM, N))
Qwmp = ml.zeros((N, N * sM))
Qwpp = ml.zeros((N * sM, N * sM))
kix = 0
for i in range(K):
Qwmp[:, kix:kix + N * M[i]] = np.kron(D[i + 1], sigma[i])
Qwpm[kix:kix + N * M[i], :] = np.kron(I, s[i])
Qwpp[kix:kix + N * M[i], :][:, kix:kix + N * M[i]] = np.kron(I, S[i])
kix += N * M[i]
# calculate fundamental matrices
Psiw, Kw, Uw = FluidFundamentalMatrices(Qwpp, Qwpm, Qwmp, Qwmm, 'PKU',
precision)
# calculate boundary vector
Ua = ml.ones((N, 1)) + 2 * np.sum(Qwmp * (-Kw).I, 1)
pm = Linsolve(
ml.hstack((Uw, Ua)).T,
ml.hstack((ml.zeros((1, N)), ml.ones((1, 1)))).T).T
ro = ((1.0 - np.sum(pm)) / 2.0) / (
np.sum(pm) + (1.0 - np.sum(pm)) / 2.0
) # calc idle time with weight=1, and the busy time with weight=1/2
kappa = pm / np.sum(pm)
pi = CTMCSolve(sD)
lambd = []
for i in range(K):
lambd.append(np.sum(pi * D[i + 1]))
Psiw = []
Qwmp = []
Qwzp = []
Qwpp = []
Qwmz = []
Qwpz = []
Qwzz = []
Qwmm = []
Qwpm = []
Qwzm = []
for k in range(K):
# step 2. construct a workload process for classes k...K
# ======================================================
Mlo = np.sum(M[:k])
Mhi = np.sum(M[k:])
Qkwpp = ml.zeros((N * Mlo * Mhi + N * Mhi, N * Mlo * Mhi + N * Mhi))
Qkwpz = ml.zeros((N * Mlo * Mhi + N * Mhi, N * Mlo))
Qkwpm = ml.zeros((N * Mlo * Mhi + N * Mhi, N))
Qkwmz = ml.zeros((N, N * Mlo))
Qkwmp = ml.zeros((N, N * Mlo * Mhi + N * Mhi))
Dlo = ml.matrix(D0)
for i in range(k):
Dlo = Dlo + D[i + 1]
Qkwmm = Dlo
Qkwzp = ml.zeros((N * Mlo, N * Mlo * Mhi + N * Mhi))
Qkwzm = ml.zeros((N * Mlo, N))
Qkwzz = ml.zeros((N * Mlo, N * Mlo))
kix = 0
for i in range(k, K):
kix2 = 0
for j in range(k):
bs = N * M[j] * M[i]
bs2 = N * M[j]
Qkwpp[kix:kix + bs,
kix:kix + bs] = np.kron(I, np.kron(ml.eye(M[j]), S[i]))
Qkwpz[kix:kix + bs,
kix2:kix2 + bs2] = np.kron(I,
np.kron(ml.eye(M[j]), s[i]))
Qkwzp[kix2:kix2 + bs2,
kix:kix + bs] = np.kron(D[i + 1],
np.kron(ml.eye(M[j]), sigma[i]))
kix += bs
kix2 += bs2
for i in range(k, K):
bs = N * M[i]
Qkwpp[kix:kix + bs, :][:, kix:kix + bs] = np.kron(I, S[i])
Qkwpm[kix:kix + bs, :] = np.kron(I, s[i])
Qkwmp[:, kix:kix + bs] = np.kron(D[i + 1], sigma[i])
kix += bs
kix = 0
for j in range(k):
bs = N * M[j]
Qkwzz[kix:kix + bs, kix:kix +
bs] = np.kron(Dlo, ml.eye(M[j])) + np.kron(I, S[j])
Qkwzm[kix:kix + bs, :] = np.kron(I, s[j])
kix += bs
if Qkwzz.shape[0] > 0:
Psikw = FluidFundamentalMatrices(
Qkwpp + Qkwpz * (-Qkwzz).I * Qkwzp,
Qkwpm + Qkwpz * (-Qkwzz).I * Qkwzm, Qkwmp, Qkwmm, 'P',
precision)
else:
Psikw = FluidFundamentalMatrices(Qkwpp, Qkwpm, Qkwmp, Qkwmm, 'P',
precision)
Psiw.append(Psikw)
Qwzp.append(Qkwzp)
Qwmp.append(Qkwmp)
Qwpp.append(Qkwpp)
Qwmz.append(Qkwmz)
Qwpz.append(Qkwpz)
Qwzz.append(Qkwzz)
Qwmm.append(Qkwmm)
Qwpm.append(Qkwpm)
Qwzm.append(Qkwzm)
# step 3. calculate Phi vectors
# =============================
lambdaS = sum(lambd)
phi = [(1 - ro) * kappa * (-D0) / lambdaS]
q0 = [[]]
qL = [[]]
for k in range(K - 1):
sDk = ml.matrix(D0)
for j in range(k + 1):
sDk = sDk + D[j + 1]
# pk
pk = sum(lambd[:k + 1]) / lambdaS - (1 - ro) * kappa * np.sum(
sDk, 1) / lambdaS
# A^(k,1)
Qwzpk = Qwzp[k + 1]
vix = 0
Ak = []
for ii in range(k + 1):
bs = N * M[ii]
V1 = Qwzpk[vix:vix + bs, :]
Ak.append(
np.kron(I, sigma[ii]) *
(-np.kron(sDk, ml.eye(M[ii])) - np.kron(I, S[ii])).I *
(np.kron(I, s[ii]) + V1 * Psiw[k + 1]))
vix += bs
# B^k
Qwmpk = Qwmp[k + 1]
Bk = Qwmpk * Psiw[k + 1]
ztag = phi[0] * ((-D0).I * D[k + 1] * Ak[k] - Ak[0] + (-D0).I * Bk)
for i in range(k):
ztag += phi[i + 1] * (Ak[i] - Ak[i + 1]) + phi[0] * (
-D0).I * D[i + 1] * Ak[i]
Mx = ml.eye(Ak[k].shape[0]) - Ak[k]
Mx[:, 0] = ml.ones((N, 1))
phi.append(
ml.hstack((pk, ztag[:, 1:])) *
Mx.I) # phi(k) = Psi^(k)_k * p(k). Psi^(k)_i = phi(i) / p(k)
q0.append(phi[0] * (-D0).I)
qLii = []
for ii in range(k + 1):
qLii.append((phi[ii + 1] - phi[ii] + phi[0] *
(-D0).I * D[ii + 1]) * np.kron(I, sigma[ii]) *
(-np.kron(sDk, ml.eye(M[ii])) - np.kron(I, S[ii])).I)
qL.append(ml.hstack(qLii))
# step 4. calculate performance measures
# ======================================
Ret = []
for k in classes:
sD0k = ml.matrix(D0)
for i in range(k):
sD0k += D[i + 1]
if k < K - 1:
# step 4.1 calculate distribution of the workload process right
# before the arrivals of class k jobs
# ============================================================
if Qwzz[k].shape[0] > 0:
Kw = Qwpp[k] + Qwpz[k] * (
-Qwzz[k]).I * Qwzp[k] + Psiw[k] * Qwmp[k]
else:
Kw = Qwpp[k] + Psiw[k] * Qwmp[k]
BM = ml.zeros((0, 0))
CM = ml.zeros((0, N))
DM = ml.zeros((0, 0))
for i in range(k):
BM = la.block_diag(BM, np.kron(I, S[i]))
CM = ml.vstack((CM, np.kron(I, s[i])))
DM = la.block_diag(DM, np.kron(D[k + 1], ml.eye(M[i])))
if k > 0:
Kwu = ml.vstack((ml.hstack(
(Kw, (Qwpz[k] + Psiw[k] * Qwmz[k]) * (-Qwzz[k]).I * DM)),
ml.hstack((ml.zeros(
(BM.shape[0], Kw.shape[1])), BM))))
Bwu = ml.vstack((Psiw[k] * D[k + 1], CM))
iniw = ml.hstack(
(q0[k] * Qwmp[k] + qL[k] * Qwzp[k], qL[k] * DM))
pwu = q0[k] * D[k + 1]
else:
Kwu = Kw
Bwu = Psiw[k] * D[k + 1]
iniw = pm * Qwmp[k]
pwu = pm * D[k + 1]
norm = np.sum(pwu) + np.sum(iniw * (-Kwu).I * Bwu)
pwu = pwu / norm
iniw = iniw / norm
# step 4.2 create the fluid model whose first passage time equals the
# WAITING time of the low prioroity customers
# ==================================================================
KN = Kwu.shape[0]
Qspp = ml.zeros(
(KN + N * np.sum(M[k + 1:]), KN + N * np.sum(M[k + 1:])))
Qspm = ml.zeros((KN + N * np.sum(M[k + 1:]), N))
Qsmp = ml.zeros((N, KN + N * np.sum(M[k + 1:])))
Qsmm = sD0k + D[k + 1]
kix = 0
for i in range(k + 1, K):
bs = N * M[i]
Qspp[KN + kix:KN + kix + bs, :][:, KN + kix:KN + kix +
bs] = np.kron(I, S[i])
Qspm[KN + kix:KN + kix + bs, :] = np.kron(I, s[i])
Qsmp[:, KN + kix:KN + kix + bs] = np.kron(D[i + 1], sigma[i])
kix += bs
Qspp[:KN, :][:, :KN] = Kwu
Qspm[:KN, :] = Bwu
inis = ml.hstack((iniw, ml.zeros((1, N * np.sum(M[k + 1:])))))
# calculate fundamental matrix
Psis = FluidFundamentalMatrices(Qspp, Qspm, Qsmp, Qsmm, 'P',
precision)
# step 4.3. calculate the performance measures
# ==========================================
argIx = 0
while argIx < len(argv):
if argIx in eaten:
argIx += 1
continue
elif type(argv[argIx]) is str and argv[argIx] == "stMoms":
# MOMENTS OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfSTMoms = argv[argIx + 1]
# calculate waiting time moments
Pn = [Psis]
wtMoms = []
for n in range(1, numOfSTMoms + 1):
A = Qspp + Psis * Qsmp
B = Qsmm + Qsmp * Psis
C = -2 * n * Pn[n - 1]
bino = 1
for i in range(1, n):
bino = bino * (n - i + 1) / i
C += bino * Pn[i] * Qsmp * Pn[n - i]
P = la.solve_sylvester(A, B, -C)
Pn.append(P)
wtMoms.append(np.sum(inis * P * (-1)**n) / 2**n)
# calculate RESPONSE time moments
Pnr = [np.sum(inis * Pn[0]) * sigma[k]]
rtMoms = []
for n in range(1, numOfSTMoms + 1):
P = n * Pnr[n - 1] * (-S[k]).I + (-1)**n * np.sum(
inis * Pn[n]) * sigma[k] / 2**n
Pnr.append(P)
rtMoms.append(
np.sum(P) + np.sum(pwu) * math.factorial(n) *
np.sum(sigma[k] * (-S[k]).I**n))
Ret.append(rtMoms)
argIx += 1
elif type(argv[argIx]) is str and argv[argIx] == "stDistr":
# DISTRIBUTION OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
stCdfPoints = argv[argIx + 1]
res = []
for t in stCdfPoints:
L = erlMaxOrder
lambdae = L / t / 2
Psie = FluidFundamentalMatrices(
Qspp - lambdae * ml.eye(Qspp.shape[0]), Qspm, Qsmp,
Qsmm - lambdae * ml.eye(Qsmm.shape[0]), 'P',
precision)
Pn = [Psie]
pr = (np.sum(pwu) + np.sum(inis * Psie)) * (1 - np.sum(
sigma[k] *
(ml.eye(S[k].shape[0]) - S[k] / 2 / lambdae).I**L))
for n in range(1, L):
A = Qspp + Psie * Qsmp - lambdae * ml.eye(
Qspp.shape[0])
B = Qsmm + Qsmp * Psie - lambdae * ml.eye(
Qsmm.shape[0])
C = 2 * lambdae * Pn[n - 1]
for i in range(1, n):
C += Pn[i] * Qsmp * Pn[n - i]
P = la.solve_sylvester(A, B, -C)
Pn.append(P)
pr += np.sum(inis * P) * (
1 - np.sum(sigma[k] *
(np.eye(S[k].shape[0]) -
S[k] / 2 / lambdae).I**(L - n)))
res.append(pr)
Ret.append(np.array(res))
argIx += 1
elif type(argv[argIx]) is str and (argv[argIx] == "ncMoms" or
argv[argIx] == "ncDistr"):
W = (-np.kron(sD - D[k + 1], ml.eye(M[k])) -
np.kron(I, S[k])).I * np.kron(D[k + 1], ml.eye(M[k]))
iW = (ml.eye(W.shape[0]) - W).I
w = np.kron(ml.eye(N), sigma[k])
omega = (-np.kron(sD - D[k + 1], ml.eye(M[k])) -
np.kron(I, S[k])).I * np.kron(I, s[k])
if argv[argIx] == "ncMoms":
# MOMENTS OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLMoms = argv[argIx + 1]
# first calculate it at departure instants
Psii = [Psis]
QLDPn = [inis * Psii[0] * w * iW]
for n in range(1, numOfQLMoms + 1):
A = Qspp + Psis * Qsmp
B = Qsmm + Qsmp * Psis
C = n * Psii[n - 1] * D[k + 1]
bino = 1
for i in range(1, n):
bino = bino * (n - i + 1) / i
C = C + bino * Psii[i] * Qsmp * Psii[n - i]
P = la.solve_sylvester(A, B, -C)
Psii.append(P)
QLDPn.append(n * QLDPn[n - 1] * iW * W +
inis * P * w * iW)
for n in range(numOfQLMoms + 1):
QLDPn[n] = (QLDPn[n] +
pwu * w * iW**(n + 1) * W**n) * omega
# now calculate it at random time instance
QLPn = [pi]
qlMoms = []
iTerm = (ml.ones((N, 1)) * pi - sD).I
for n in range(1, numOfQLMoms + 1):
sumP = np.sum(QLDPn[n]) + n * np.sum(
(QLDPn[n - 1] - QLPn[n - 1] * D[k + 1] /
lambd[k]) * iTerm * D[k + 1])
P = sumP * pi + n * (
QLPn[n - 1] * D[k + 1] -
QLDPn[n - 1] * lambd[k]) * iTerm
QLPn.append(P)
qlMoms.append(np.sum(P))
qlMoms = MomsFromFactorialMoms(qlMoms)
Ret.append(qlMoms)
argIx += 1
elif argv[argIx] == "ncDistr":
# DISTRIBUTION OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLProbs = argv[argIx + 1]
Psid = FluidFundamentalMatrices(
Qspp, Qspm, Qsmp, sD0k, 'P', precision)
Pn = [Psid]
XDn = inis * Psid * w
dqlProbs = (XDn + pwu * w) * omega
for n in range(1, numOfQLProbs):
A = Qspp + Psid * Qsmp
B = sD0k + Qsmp * Psid
C = Pn[n - 1] * D[k + 1]
for i in range(1, n):
C += Pn[i] * Qsmp * Pn[n - i]
P = la.solve_sylvester(A, B, -C)
Pn.append(P)
XDn = XDn * W + inis * P * w
dqlProbs = ml.vstack(
(dqlProbs, (XDn + pwu * w * W**n) * omega))
# now calculate it at random time instance
iTerm = -(sD - D[k + 1]).I
qlProbs = lambd[k] * dqlProbs[0, :] * iTerm
for n in range(1, numOfQLProbs):
P = (qlProbs[n - 1, :] * D[k + 1] + lambd[k] *
(dqlProbs[n, :] - dqlProbs[n - 1, :])) * iTerm
qlProbs = ml.vstack((qlProbs, P))
qlProbs = np.sum(qlProbs, 1).A.flatten()
Ret.append(qlProbs)
argIx += 1
else:
raise Exception("MMAPPH1NPPR: Unknown parameter " +
str(argv[argIx]))
argIx += 1
elif k == K - 1:
# step 3. calculate the performance measures
# ==========================================
argIx = 0
while argIx < len(argv):
if argIx in eaten:
argIx += 1
continue
elif type(argv[argIx]) is str and (argv[argIx] == "stMoms" or
argv[argIx] == "stDistr"):
Kw = Qwpp[k] + Qwpz[k] * (
-Qwzz[k]).I * Qwzp[k] + Psiw[k] * Qwmp[k]
AM = ml.zeros((0, 0))
BM = ml.zeros((0, 0))
CM = ml.zeros((0, 1))
DM = ml.zeros((0, 0))
for i in range(k):
AM = la.block_diag(
AM,
np.kron(ml.ones((N, 1)),
np.kron(ml.eye(M[i]), s[k])))
BM = la.block_diag(BM, S[i])
CM = ml.vstack((CM, s[i]))
DM = la.block_diag(DM, np.kron(D[k + 1], ml.eye(M[i])))
Z = ml.vstack((ml.hstack(
(Kw, ml.vstack((AM, ml.zeros(
(N * M[k], AM.shape[1])))))),
ml.hstack((ml.zeros(
(BM.shape[0], Kw.shape[1])), BM))))
z = ml.vstack((ml.zeros(
(AM.shape[0], 1)), np.kron(ml.ones((N, 1)), s[k]), CM))
iniw = ml.hstack((q0[k] * Qwmp[k] + qL[k] * Qwzp[k],
ml.zeros((1, BM.shape[0]))))
zeta = iniw / np.sum(iniw * (-Z).I * z)
if argv[argIx] == "stMoms":
# MOMENTS OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfSTMoms = argv[argIx + 1]
rtMoms = []
for i in range(1, numOfSTMoms + 1):
rtMoms.append(
np.sum(
math.factorial(i) * zeta *
(-Z).I**(i + 1) * z))
Ret.append(rtMoms)
argIx += 1
if argv[argIx] == "stDistr":
# DISTRIBUTION OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
stCdfPoints = argv[argIx + 1]
rtDistr = []
for t in stCdfPoints:
rtDistr.append(
np.sum(zeta * (-Z).I *
(ml.eye(Z.shape[0]) - la.expm(Z * t)) *
z))
Ret.append(np.array(rtDistr))
argIx += 1
elif type(argv[argIx]) is str and (argv[argIx] == "ncMoms" or
argv[argIx] == "ncDistr"):
L = ml.zeros((N * np.sum(M), N * np.sum(M)))
B = ml.zeros((N * np.sum(M), N * np.sum(M)))
F = ml.zeros((N * np.sum(M), N * np.sum(M)))
kix = 0
for i in range(K):
bs = N * M[i]
F[kix:kix + bs, :][:, kix:kix + bs] = np.kron(
D[k + 1], ml.eye(M[i]))
L[kix:kix + bs, :][:, kix:kix + bs] = np.kron(
sD0k, ml.eye(M[i])) + np.kron(I, S[i])
if i < K - 1:
L[kix:kix + bs, :][:,
N * np.sum(M[:k]):] = np.kron(
I, s[i] * sigma[k])
else:
B[kix:kix + bs, :][:,
N * np.sum(M[:k]):] = np.kron(
I, s[i] * sigma[k])
kix += bs
R = QBDFundamentalMatrices(B, L, F, 'R', precision)
p0 = ml.hstack((qL[k], q0[k] * np.kron(I, sigma[k])))
p0 = p0 / np.sum(p0 * (ml.eye(R.shape[0]) - R).I)
if argv[argIx] == "ncMoms":
# MOMENTS OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLMoms = argv[argIx + 1]
qlMoms = []
for i in range(1, numOfQLMoms + 1):
qlMoms.append(
np.sum(
math.factorial(i) * p0 * R**i *
(ml.eye(R.shape[0]) - R).I**(i + 1)))
Ret.append(MomsFromFactorialMoms(qlMoms))
elif argv[argIx] == "ncDistr":
# DISTRIBUTION OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLProbs = argv[argIx + 1]
qlProbs = [np.sum(p0)]
for i in range(1, numOfQLProbs):
qlProbs.append(np.sum(p0 * R**i))
Ret.append(np.array(qlProbs))
argIx += 1
else:
raise Exception("MMAPPH1NPPR: Unknown parameter " +
str(argv[argIx]))
argIx += 1
if len(Ret) == 1:
return Ret[0]
else:
return Ret
def test9():
#####################################
## STANDARD COOLEY HANSE CIA MODEL ##
#####################################
# Load and solve the manually linearized model
rbc1 = pm.newMOD(models.abcs_rbcs.cooley_hansen_cia_linear,mesg=True)
rbc1.modsolvers.pyuhlig.solve()
rbc1.modsolvers.forklein.solve()
# Load and solve the automatically linearized model
rbc2 = pm.newMOD(models.abcs_rbcs.cooley_hansen_cia_cf,mesg=True)
rbc2.modsolvers.forkleind.solve()
# Check equivalence of steady states
for keyo in rbc1.sstate.keys():
if keyo in rbc2.sstate.keys():
assert round(rbc1.sstate[keyo],5) == round(rbc2.sstate[keyo],5)
# Check equivalence of results
nexo = len(rbc2.vardic['exo']['var'])
nendo = len(rbc2.vardic['endo']['var'])
modlin1 = MAT.hstack((rbc1.modsolvers.pyuhlig.Q,rbc1.modsolvers.pyuhlig.P))
modlin1 = [round(modlin1[0,i1],5) for i1 in range(modlin1.shape[1])]
modnlin1 = rbc2.modsolvers.forkleind.P[-nendo:,:]
modnlin1 = [round(modnlin1[0,i1],5) for i1 in range(modnlin1.shape[1])]
print 'Comparison: Standard CIA model'
print "Linear is: ",modlin1
print '----------------------'
print "Nonlinear is: ",modnlin1
assert modlin1 == modnlin1
modlin2 = MAT.hstack((rbc1.modsolvers.pyuhlig.S,rbc1.modsolvers.pyuhlig.R))
modlin2 = [[round(modlin2[i2,i1],5) for i1 in range(modlin2.shape[1])] for i2 in range(modlin2.shape[0])]
modnlin2 = rbc2.modsolvers.forkleind.F
modnlin2 = [[round(modnlin2[i2,i1],5) for i1 in range(modnlin2.shape[1])] for i2 in range(modnlin2.shape[0])]
print modlin2
print '----------------------'
print modnlin2
assert modlin2 == modnlin2
#####################################################
## STANDARD COOLEY HANSE CIA MODEL WITH SEIGNORAGE ##
#####################################################
# Load and solve the manually linearized model
rbc1 = pm.newMOD(models.abcs_rbcs.cooley_hansen_cia_seignorage_linear,mesg=True)
rbc1.modsolvers.pyuhlig.solve()
rbc1.modsolvers.forklein.solve()
# Load and solve the automatically linearized model
rbc2 = pm.newMOD(models.abcs_rbcs.cooley_hansen_cia_seignorage_cf,mesg=True)
rbc2.modsolvers.forkleind.solve()
# Check equivalence of steady states
for keyo in rbc1.sstate.keys():
if keyo in rbc2.sstate.keys():
assert round(rbc1.sstate[keyo],5) == round(rbc2.sstate[keyo],5)
# Check equivalence of results
nexo = len(rbc2.vardic['exo']['var'])
nendo = len(rbc2.vardic['endo']['var'])
modlin1 = MAT.hstack((rbc1.modsolvers.pyuhlig.Q,rbc1.modsolvers.pyuhlig.P))
modlin1 = [round(modlin1[0,i1],5) for i1 in range(modlin1.shape[1])]
modnlin1 = rbc2.modsolvers.forkleind.P[-nendo:,:]
modnlin1 = [round(modnlin1[0,i1],5) for i1 in range(modnlin1.shape[1])]
print 'Comparison: Standard CIA model with seignorage'
print "Linear is: ",modlin1
print '----------------------'
print "Nonlinear is: ",modnlin1
assert modlin1 == modnlin1
modlin2 = MAT.hstack((rbc1.modsolvers.pyuhlig.S,rbc1.modsolvers.pyuhlig.R))
modlin2 = [[round(modlin2[i2,i1],5) for i1 in range(modlin2.shape[1])] for i2 in range(modlin2.shape[0])]
modnlin2 = rbc2.modsolvers.forkleind.F
modnlin2 = [[round(modnlin2[i2,i1],5) for i1 in range(modnlin2.shape[1])] for i2 in range(modnlin2.shape[0])]
print modlin2
print '----------------------'
print modnlin2
assert modlin2 == modnlin2
#####################################################################
## STANDARD COOLEY HANSE CIA MODEL WITH SEIGNORAGE AND CES UTILITY ##
#####################################################################
# Be careful, here in the log-linearized version we have two endogenous states, k and mg
# But in the automatically linearized version using the Jacobian we only have one, k
# Load and solve the manually linearized model
rbc1 = pm.newMOD(models.abcs_rbcs.cooley_hansen_cia_seignorage_ces_linear,mesg=True)
rbc1.modsolvers.pyuhlig.solve()
rbc1.modsolvers.forklein.solve()
# Load and solve the automatically linearized model
rbc2 = pm.newMOD(models.abcs_rbcs.cooley_hansen_cia_seignorage_ces_cf,mesg=True)
rbc2.modsolvers.forkleind.solve()
# Check equivalence of steady states
for keyo in rbc1.sstate.keys():
if keyo in rbc2.sstate.keys():
assert round(rbc1.sstate[keyo],5) == round(rbc2.sstate[keyo],5)
# Check equivalence of results
nexo = len(rbc2.vardic['exo']['var'])
nendo = len(rbc2.vardic['endo']['var'])
modlin1 = MAT.hstack((rbc1.modsolvers.pyuhlig.Q[:nendo,:],rbc1.modsolvers.pyuhlig.P[:nendo,:nendo]))
modlin1 = [[round(modlin1[i2,i1],5) for i1 in range(modlin1.shape[1])] for i2 in range(modlin1.shape[0])]
modnlin1 = rbc2.modsolvers.forkleind.P[-nendo:,:]
modnlin1 = [[round(modnlin1[i2,i1],5) for i1 in range(modnlin1.shape[1])] for i2 in range(modnlin1.shape[0])]
print 'Comparison: Standard CIA model with seignorage and CES utility'
print "Linear is: ",modlin1
print '----------------------'
print "Nonlinear is: ",modnlin1
assert modlin1 == modnlin1
modlin2 = MAT.hstack((rbc1.modsolvers.pyuhlig.S[:,:],rbc1.modsolvers.pyuhlig.R[:,:nendo]))
modlin2 = [[round(modlin2[i2,i1],5) for i1 in range(modlin2.shape[1])] for i2 in range(modlin2.shape[0])]
modnlin2 = rbc2.modsolvers.forkleind.F[:-1,:]
modnlin2 = [[round(modnlin2[i2,i1],5) for i1 in range(modnlin2.shape[1])] for i2 in range(modnlin2.shape[0])]
print modlin2
print '----------------------'
print modnlin2
assert modlin2 == modnlin2
def CanonicalFromDPH3 (alpha,A,prec=1e-14):
"""
Returns the canonical form of an order-3 discrete phase-type
distribution.
Parameters
----------
alpha : matrix, shape (1,3)
Initial vector of the discrete phase-type distribution
A : matrix, shape (3,3)
Transition probability matrix of the discrete phase-type
distribution
prec : double, optional
Numerical precision for checking the input, default value
is 1e-14
Returns
-------
beta : matrix, shape (1,3)
The initial probability vector of the canonical form
B : matrix, shape (3,3)
Transition probability matrix of the canonical form
"""
if butools.checkInput and not CheckMGRepresentation (alpha, A, prec):
raise Exception("CanonicalFromDPH3: Input is not a valid DPH representation!")
if butools.checkInput and (A.shape[0]!=3 or A.shape[1]!=3):
raise Exception("CanonicalFromDPH3: Dimension must be 3!")
ev = la.eigvals(A)
ix = np.argsort(-np.abs(np.real(ev)))
lambd = ev[ix]
eye = ml.eye(3);
a0 = -lambd[0]*lambd[1]*lambd[2]
a1 = lambd[0]*lambd[1]+lambd[0]*lambd[2]+lambd[1]*lambd[2]
a2 = -lambd[0]-lambd[1]-lambd[2]
e = ml.matrix([[1,1,1]]).T
if np.real(lambd[0])>0 and np.real(lambd[1])>=0 and np.real(lambd[2])>=0:
#PPP case
alphaout,A2 = CanonicalFromPH3(alpha,A-eye,prec)
Aout = A2+eye
elif np.real(lambd[0])>0 and np.real(lambd[1])>=0 and np.real(lambd[2])<0:
#PPN case
x1 = lambd[0]
x2 = lambd[1]+lambd[2]
x3=lambd[1]*lambd[2]/(lambd[1]+lambd[2]-1)
Aout=ml.matrix([[x1,1-x1,0],[0,x2,1-x2],[0,x3,0]])
b3=1/(1-x3)*(e-A*e)
b2=1/(1-x2)*A*b3
b1=e-b2-b3
B=ml.hstack((b1,b2,b3))
alphaout=alpha*B
elif np.real(lambd[0])>0 and np.real(lambd[1])<0 and np.real(lambd[2])>=0:
#PNP case
x1=-a2
x2=(a0-a1*a2)/(a2*(1+a2))
x3=a0*(1+a2)/(a0-a2-a1*a2-a2**2)
Aout=ml.matrix([[x1,1-x1,0],[x2,0,1-x2],[0,x3,0]])
b3=1/(1-x3)*(e-A*e)
b2=1/(1-x2)*A*b3
b1=e-b2-b3
if alpha*b1>=0:
B=ml.hstack((b1,b2,b3))
alphaout = alpha*B
else:
#Set the initial vector first element to 0
x33=0
a1=-1
while x33 <= 1:
[a1,x1,x2,x3,B]=firstInitElem(x33,lambd,alpha,A)
if a1 >= 0 and x1 >= 0 and x2 >= 0 and x3 >= 0 and x3+x33 < 1:
break
x33=x33+0.01
if a1 >= 0:
Aout=ml.matrix([[x1,1-x1,0],[x2,0,1-x2],[0,x3,x33]])
alphaout=alpha*B
else:
#PNP+
x1=lambd[2]
x2=lambd[0]+lambd[1]
x3=lambd[0]*lambd[1]/(lambd[0]+lambd[1]-1)
Aout=ml.matrix([[x1,0,0],[0,x2,1-x2],[0,x3,0]])
p1=alpha*(e-A*e)
p2=alpha*A*(e-A*e)
d1=(1-lambd[0])*((1-lambd[1])*(1-lambd[2])+(-1+lambd[1]+lambd[2])*p1-p2)/((lambd[0]-lambd[1])*(lambd[0]-lambd[2]))
d2=(lambd[1]-1)*((1-lambd[0])*(1-lambd[2])+(-1+lambd[0]+lambd[2])*p1-p2)/((lambd[0]-lambd[1])*(lambd[1]-lambd[2]))
d3=(lambd[2]-1)*((1-lambd[0])*(1-lambd[1])+(-1+lambd[0]+lambd[1])*p1-p2)/((lambd[1]-lambd[2])*(lambd[2]-lambd[0]))
alphaout=ml.matrix([[d3/(1-lambd[2]),(d1*lambd[0]+d2*lambd[1])/((1-lambd[0])*(1-lambd[1])),(d1+d2)*(1-lambd[0]-lambd[1])/((1-lambd[0])*(1-lambd[1]))]])
if np.min(alphaout) < 0 or np.min(np.min(Aout)) < 0:
raise Exception("DPH3Canonical: Unhandled PNP case!")
elif np.real(lambd[0])>0 and np.real(lambd[1])<0 and np.real(lambd[2])<0:
#PNN case
if np.all(np.isreal(lambd)) or np.abs(lambd[1])**2 <= 2*lambd[0]*(-np.real(lambd[1])):
x1=-a2
x2=-a1/(1+a2)
x3=-a0/(1+a1+a2)
Aout=ml.matrix([[x1,1-x1,0],[x2,0,1-x2],[x3,0,0]])
b3=1/(1-x3)*(e-A*e)
b2=1/(1-x2)*A*b3
b1=e-b2-b3
B=ml.hstack((b1,b2,b3))
alphaout=alpha*B
else:
[alphaout,A2]=CanonicalFromPH3(alpha,A-eye,prec)
Aout=A2+eye
return (np.real(alphaout),np.real(Aout))
def MMAPPH1NPPR(D, sigma, S, *argv):
"""
Returns various performane measures of a continuous time
MMAP[K]/PH[K]/1 non-preemptive priority queue, see [1]_.
Parameters
----------
D : list of matrices of shape (N,N), length (K+1)
The D0...DK matrices of the arrival process.
D1 corresponds to the lowest, DK to the highest priority.
sigma : list of row vectors, length (K)
The list containing the initial probability vectors of the service
time distributions of the various customer types. The length of the
vectors does not have to be the same.
S : list of square matrices, length (K)
The transient generators of the phase type distributions representing
the service time of the jobs belonging to various types.
further parameters :
The rest of the function parameters specify the options
and the performance measures to be computed.
The supported performance measures and options in this
function are:
+----------------+--------------------+----------------------------------------+
| Parameter name | Input parameters | Output |
+================+====================+========================================+
| "ncMoms" | Number of moments | The moments of the number of customers |
+----------------+--------------------+----------------------------------------+
| "ncDistr" | Upper limit K | The distribution of the number of |
| | | customers from level 0 to level K-1 |
+----------------+--------------------+----------------------------------------+
| "stMoms" | Number of moments | The sojourn time moments |
+----------------+--------------------+----------------------------------------+
| "stDistr" | A vector of points | The sojourn time distribution at the |
| | | requested points (cummulative, cdf) |
+----------------+--------------------+----------------------------------------+
| "prec" | The precision | Numerical precision used as a stopping |
| | | condition when solving the Riccati and |
| | | the matrix-quadratic equations |
+----------------+--------------------+----------------------------------------+
| "erlMaxOrder" | Integer number | The maximal Erlang order used in the |
| | | erlangization procedure. The default |
| | | value is 200. |
+----------------+--------------------+----------------------------------------+
| "classes" | Vector of integers | Only the performance measures |
| | | belonging to these classes are |
| | | returned. If not given, all classes |
| | | are analyzed. |
+----------------+--------------------+----------------------------------------+
(The quantities related to the number of customers in
the system include the customer in the server, and the
sojourn time related quantities include the service
times as well)
Returns
-------
Ret : list of the performance measures
Each entry of the list corresponds to a performance
measure requested. Each entry is a matrix, where the
columns belong to the various job types.
If there is just a single item,
then it is not put into a list.
References
----------
.. [1] G. Horvath, "Efficient analysis of the MMAP[K]/PH[K]/1
priority queue", European Journal of Operational
Research, 246(1), 128-139, 2015.
"""
K = len(D)-1
# parse options
eaten = []
erlMaxOrder = 200;
precision = 1e-14;
classes = np.arange(0,K)
for i in range(len(argv)):
if argv[i]=="prec":
precision = argv[i+1]
eaten.append(i)
eaten.append(i+1)
elif argv[i]=="erlMaxOrder":
erlMaxOrder = argv[i+1]
eaten.append(i)
eaten.append(i+1)
elif argv[i]=="classes":
classes = np.array(argv[i+1])-1
eaten.append(i)
eaten.append(i+1)
if butools.checkInput and not CheckMMAPRepresentation(D):
raise Exception('MMAPPH1PRPR: The arrival process is not a valid MMAP representation!')
if butools.checkInput:
for k in range(K):
if not CheckPHRepresentation(sigma[k],S[k]):
raise Exception('MMAPPH1PRPR: the vector and matrix describing the service times is not a valid PH representation!')
# some preparation
D0 = D[0]
N = D0.shape[0]
I = ml.eye(N)
sD = ml.zeros((N,N))
for Di in D:
sD += Di
s = []
M = np.empty(K)
for i in range(K):
s.append(np.sum(-S[i],1))
M[i] = sigma[i].size
# step 1. solution of the workload process of the joint queue
# ===========================================================
sM = np.sum(M)
Qwmm = ml.matrix(D0)
Qwpm = ml.zeros((N*sM, N))
Qwmp = ml.zeros((N, N*sM))
Qwpp = ml.zeros((N*sM, N*sM))
kix = 0
for i in range(K):
Qwmp[:,kix:kix+N*M[i]] = np.kron(D[i+1], sigma[i])
Qwpm[kix:kix+N*M[i],:] = np.kron(I,s[i])
Qwpp[kix:kix+N*M[i],:][:,kix:kix+N*M[i]] = np.kron(I,S[i])
kix += N*M[i]
# calculate fundamental matrices
Psiw, Kw, Uw = FluidFundamentalMatrices (Qwpp, Qwpm, Qwmp, Qwmm, 'PKU', precision)
# calculate boundary vector
Ua = ml.ones((N,1)) + 2*np.sum(Qwmp*(-Kw).I,1)
pm = Linsolve (ml.hstack((Uw,Ua)).T, ml.hstack((ml.zeros((1,N)),ml.ones((1,1)))).T).T
ro = ((1.0-np.sum(pm))/2.0)/(np.sum(pm)+(1.0-np.sum(pm))/2.0) # calc idle time with weight=1, and the busy time with weight=1/2
kappa = pm/np.sum(pm)
pi = CTMCSolve (sD)
lambd = []
for i in range(K):
lambd.append(np.sum(pi*D[i+1]))
Psiw = []
Qwmp = []
Qwzp = []
Qwpp = []
Qwmz = []
Qwpz = []
Qwzz = []
Qwmm = []
Qwpm = []
Qwzm = []
for k in range(K):
# step 2. construct a workload process for classes k...K
# ======================================================
Mlo = np.sum(M[:k])
Mhi = np.sum(M[k:])
Qkwpp = ml.zeros((N*Mlo*Mhi+N*Mhi, N*Mlo*Mhi+N*Mhi))
Qkwpz = ml.zeros((N*Mlo*Mhi+N*Mhi, N*Mlo))
Qkwpm = ml.zeros((N*Mlo*Mhi+N*Mhi, N))
Qkwmz = ml.zeros((N, N*Mlo))
Qkwmp = ml.zeros((N, N*Mlo*Mhi+N*Mhi))
Dlo = ml.matrix(D0)
for i in range(k):
Dlo = Dlo + D[i+1]
Qkwmm = Dlo
Qkwzp = ml.zeros((N*Mlo, N*Mlo*Mhi+N*Mhi))
Qkwzm = ml.zeros((N*Mlo, N))
Qkwzz = ml.zeros((N*Mlo, N*Mlo))
kix = 0
for i in range(k,K):
kix2 = 0
for j in range(k):
bs = N*M[j]*M[i]
bs2 = N*M[j]
Qkwpp[kix:kix+bs,kix:kix+bs] = np.kron(I,np.kron(ml.eye(M[j]),S[i]))
Qkwpz[kix:kix+bs,kix2:kix2+bs2] = np.kron(I,np.kron(ml.eye(M[j]),s[i]))
Qkwzp[kix2:kix2+bs2,kix:kix+bs] = np.kron(D[i+1],np.kron(ml.eye(M[j]), sigma[i]))
kix += bs
kix2 += bs2
for i in range(k,K):
bs = N*M[i]
Qkwpp[kix:kix+bs,:][:,kix:kix+bs] = np.kron(I,S[i])
Qkwpm[kix:kix+bs,:] = np.kron(I,s[i])
Qkwmp[:,kix:kix+bs] = np.kron(D[i+1],sigma[i])
kix += bs
kix = 0
for j in range(k):
bs = N*M[j]
Qkwzz[kix:kix+bs,kix:kix+bs] = np.kron(Dlo, ml.eye(M[j])) + np.kron(I, S[j])
Qkwzm[kix:kix+bs,:] = np.kron(I, s[j])
kix += bs
if Qkwzz.shape[0]>0:
Psikw = FluidFundamentalMatrices (Qkwpp+Qkwpz*(-Qkwzz).I*Qkwzp, Qkwpm+Qkwpz*(-Qkwzz).I*Qkwzm, Qkwmp, Qkwmm, 'P', precision)
else:
Psikw = FluidFundamentalMatrices (Qkwpp, Qkwpm, Qkwmp, Qkwmm, 'P', precision)
Psiw.append(Psikw)
Qwzp.append(Qkwzp)
Qwmp.append(Qkwmp)
Qwpp.append(Qkwpp)
Qwmz.append(Qkwmz)
Qwpz.append(Qkwpz)
Qwzz.append(Qkwzz)
Qwmm.append(Qkwmm)
Qwpm.append(Qkwpm)
Qwzm.append(Qkwzm)
# step 3. calculate Phi vectors
# =============================
lambdaS = sum(lambd)
phi = [(1-ro)*kappa*(-D0) / lambdaS]
q0 = [[]]
qL = [[]]
for k in range(K-1):
sDk = ml.matrix(D0)
for j in range(k+1):
sDk = sDk + D[j+1]
# pk
pk = sum(lambd[:k+1])/lambdaS - (1-ro)*kappa*np.sum(sDk,1)/lambdaS
# A^(k,1)
Qwzpk = Qwzp[k+1]
vix = 0
Ak = []
for ii in range(k+1):
bs = N*M[ii]
V1 = Qwzpk[vix:vix+bs,:]
Ak.append (np.kron(I,sigma[ii]) * (-np.kron(sDk,ml.eye(M[ii]))-np.kron(I,S[ii])).I * (np.kron(I,s[ii]) + V1*Psiw[k+1]))
vix += bs
# B^k
Qwmpk = Qwmp[k+1]
Bk = Qwmpk * Psiw[k+1]
ztag = phi[0]*((-D0).I*D[k+1]*Ak[k] - Ak[0] + (-D0).I*Bk)
for i in range(k):
ztag += phi[i+1]*(Ak[i]-Ak[i+1]) + phi[0]*(-D0).I*D[i+1]*Ak[i]
Mx = ml.eye(Ak[k].shape[0])-Ak[k]
Mx[:,0] = ml.ones((N,1))
phi.append(ml.hstack((pk, ztag[:,1:]))*Mx.I) # phi(k) = Psi^(k)_k * p(k). Psi^(k)_i = phi(i) / p(k)
q0.append(phi[0]*(-D0).I)
qLii = []
for ii in range(k+1):
qLii.append((phi[ii+1] - phi[ii] + phi[0]*(-D0).I*D[ii+1]) * np.kron(I,sigma[ii]) * (-np.kron(sDk,ml.eye(M[ii]))-np.kron(I,S[ii])).I)
qL.append(ml.hstack(qLii))
# step 4. calculate performance measures
# ======================================
Ret = []
for k in classes:
sD0k = ml.matrix(D0)
for i in range(k):
sD0k += D[i+1]
if k<K-1:
# step 4.1 calculate distribution of the workload process right
# before the arrivals of class k jobs
# ============================================================
if Qwzz[k].shape[0]>0:
Kw = Qwpp[k]+Qwpz[k]*(-Qwzz[k]).I*Qwzp[k] + Psiw[k]*Qwmp[k]
else:
Kw = Qwpp[k] + Psiw[k]*Qwmp[k]
BM = ml.zeros((0,0))
CM = ml.zeros((0,N))
DM = ml.zeros((0,0))
for i in range(k):
BM = la.block_diag(BM,np.kron(I,S[i]))
CM = ml.vstack((CM, np.kron(I,s[i])))
DM = la.block_diag(DM,np.kron(D[k+1],ml.eye(M[i])))
if k>0:
Kwu = ml.vstack((ml.hstack((Kw, (Qwpz[k]+Psiw[k]*Qwmz[k])*(-Qwzz[k]).I*DM)), ml.hstack((ml.zeros((BM.shape[0],Kw.shape[1])), BM))))
Bwu = ml.vstack((Psiw[k]*D[k+1], CM))
iniw = ml.hstack((q0[k]*Qwmp[k]+qL[k]*Qwzp[k], qL[k]*DM))
pwu = q0[k]*D[k+1]
else:
Kwu = Kw
Bwu = Psiw[k]*D[k+1]
iniw = pm*Qwmp[k]
pwu = pm*D[k+1]
norm = np.sum(pwu) + np.sum(iniw*(-Kwu).I*Bwu)
pwu = pwu / norm
iniw = iniw / norm
# step 4.2 create the fluid model whose first passage time equals the
# WAITING time of the low prioroity customers
# ==================================================================
KN = Kwu.shape[0]
Qspp = ml.zeros((KN+N*np.sum(M[k+1:]), KN+N*np.sum(M[k+1:])))
Qspm = ml.zeros((KN+N*np.sum(M[k+1:]), N))
Qsmp = ml.zeros((N, KN+N*np.sum(M[k+1:])))
Qsmm = sD0k + D[k+1]
kix = 0
for i in range(k+1,K):
bs = N*M[i]
Qspp[KN+kix:KN+kix+bs,:][:,KN+kix:KN+kix+bs] = np.kron(I,S[i])
Qspm[KN+kix:KN+kix+bs,:] = np.kron(I,s[i])
Qsmp[:,KN+kix:KN+kix+bs] = np.kron(D[i+1],sigma[i])
kix += bs
Qspp[:KN,:][:,:KN] = Kwu
Qspm[:KN,:] = Bwu
inis = ml.hstack((iniw, ml.zeros((1,N*np.sum(M[k+1:])))))
# calculate fundamental matrix
Psis = FluidFundamentalMatrices (Qspp, Qspm, Qsmp, Qsmm, 'P', precision)
# step 4.3. calculate the performance measures
# ==========================================
argIx = 0
while argIx<len(argv):
if argIx in eaten:
argIx += 1
continue
elif type(argv[argIx]) is str and argv[argIx]=="stMoms":
# MOMENTS OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfSTMoms = argv[argIx+1]
# calculate waiting time moments
Pn = [Psis]
wtMoms = []
for n in range(1,numOfSTMoms+1):
A = Qspp + Psis*Qsmp
B = Qsmm + Qsmp*Psis
C = -2*n*Pn[n-1]
bino = 1
for i in range(1,n):
bino = bino * (n-i+1) / i
C += bino * Pn[i]*Qsmp*Pn[n-i]
P = la.solve_sylvester(A, B, -C)
Pn.append(P)
wtMoms.append(np.sum(inis*P*(-1)**n) / 2**n)
# calculate RESPONSE time moments
Pnr = [np.sum(inis*Pn[0])*sigma[k]]
rtMoms = []
for n in range(1,numOfSTMoms+1):
P = n*Pnr[n-1]*(-S[k]).I + (-1)**n*np.sum(inis*Pn[n])*sigma[k] / 2**n
Pnr.append(P)
rtMoms.append(np.sum(P)+np.sum(pwu)*math.factorial(n)*np.sum(sigma[k]*(-S[k]).I**n))
Ret.append(rtMoms)
argIx += 1
elif type(argv[argIx]) is str and argv[argIx]=="stDistr":
# DISTRIBUTION OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
stCdfPoints = argv[argIx+1]
res = []
for t in stCdfPoints:
L = erlMaxOrder
lambdae = L/t/2
Psie = FluidFundamentalMatrices (Qspp-lambdae*ml.eye(Qspp.shape[0]), Qspm, Qsmp, Qsmm-lambdae*ml.eye(Qsmm.shape[0]), 'P', precision)
Pn = [Psie]
pr = (np.sum(pwu) + np.sum(inis*Psie)) * (1-np.sum(sigma[k]*(ml.eye(S[k].shape[0])-S[k]/2/lambdae).I**L))
for n in range(1,L):
A = Qspp + Psie*Qsmp - lambdae*ml.eye(Qspp.shape[0])
B = Qsmm + Qsmp*Psie - lambdae*ml.eye(Qsmm.shape[0])
C = 2*lambdae*Pn[n-1]
for i in range(1,n):
C += Pn[i]*Qsmp*Pn[n-i]
P = la.solve_sylvester(A, B, -C)
Pn.append(P)
pr += np.sum(inis*P) * (1-np.sum(sigma[k]*(np.eye(S[k].shape[0])-S[k]/2/lambdae).I**(L-n)))
res.append(pr)
Ret.append(np.array(res))
argIx += 1
elif type(argv[argIx]) is str and (argv[argIx]=="ncMoms" or argv[argIx]=="ncDistr"):
W = (-np.kron(sD-D[k+1],ml.eye(M[k]))-np.kron(I,S[k])).I*np.kron(D[k+1],ml.eye(M[k]))
iW = (ml.eye(W.shape[0])-W).I
w = np.kron(ml.eye(N),sigma[k])
omega = (-np.kron(sD-D[k+1],ml.eye(M[k]))-np.kron(I,S[k])).I*np.kron(I,s[k])
if argv[argIx]=="ncMoms":
# MOMENTS OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLMoms = argv[argIx+1]
# first calculate it at departure instants
Psii = [Psis]
QLDPn = [inis*Psii[0]*w*iW]
for n in range(1,numOfQLMoms+1):
A = Qspp + Psis*Qsmp
B = Qsmm + Qsmp*Psis
C = n*Psii[n-1]*D[k+1]
bino = 1
for i in range(1,n):
bino = bino * (n-i+1) / i
C = C + bino * Psii[i]*Qsmp*Psii[n-i]
P = la.solve_sylvester(A, B, -C)
Psii.append(P)
QLDPn.append(n*QLDPn[n-1]*iW*W + inis*P*w*iW)
for n in range(numOfQLMoms+1):
QLDPn[n] = (QLDPn[n] + pwu*w*iW**(n+1)*W**n)*omega
# now calculate it at random time instance
QLPn = [pi]
qlMoms = []
iTerm = (ml.ones((N,1))*pi - sD).I
for n in range(1,numOfQLMoms+1):
sumP = np.sum(QLDPn[n]) + n*np.sum((QLDPn[n-1] - QLPn[n-1]*D[k+1]/lambd[k])*iTerm*D[k+1])
P = sumP*pi + n*(QLPn[n-1]*D[k+1] - QLDPn[n-1]*lambd[k])*iTerm
QLPn.append(P)
qlMoms.append(np.sum(P))
qlMoms = MomsFromFactorialMoms(qlMoms)
Ret.append(qlMoms)
argIx += 1
elif argv[argIx]=="ncDistr":
# DISTRIBUTION OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLProbs = argv[argIx+1]
Psid = FluidFundamentalMatrices (Qspp, Qspm, Qsmp, sD0k, 'P', precision)
Pn = [Psid]
XDn = inis*Psid*w
dqlProbs = (XDn+pwu*w)*omega
for n in range(1,numOfQLProbs):
A = Qspp + Psid*Qsmp
B = sD0k + Qsmp*Psid
C = Pn[n-1]*D[k+1]
for i in range(1,n):
C += Pn[i]*Qsmp*Pn[n-i]
P = la.solve_sylvester(A, B, -C)
Pn.append(P)
XDn = XDn*W + inis*P*w
dqlProbs = ml.vstack((dqlProbs, (XDn+pwu*w*W**n)*omega))
# now calculate it at random time instance
iTerm = -(sD-D[k+1]).I
qlProbs = lambd[k]*dqlProbs[0,:]*iTerm
for n in range(1,numOfQLProbs):
P = (qlProbs[n-1,:]*D[k+1]+lambd[k]*(dqlProbs[n,:]-dqlProbs[n-1,:]))*iTerm
qlProbs = ml.vstack((qlProbs, P))
qlProbs = np.sum(qlProbs,1).A.flatten()
Ret.append(qlProbs)
argIx += 1
else:
raise Exception("MMAPPH1NPPR: Unknown parameter "+str(argv[argIx]))
argIx += 1
elif k==K-1:
# step 3. calculate the performance measures
# ==========================================
argIx = 0
while argIx<len(argv):
if argIx in eaten:
argIx += 1
continue
elif type(argv[argIx]) is str and (argv[argIx]=="stMoms" or argv[argIx]=="stDistr"):
Kw = Qwpp[k]+Qwpz[k]*(-Qwzz[k]).I*Qwzp[k] + Psiw[k]*Qwmp[k]
AM = ml.zeros((0,0))
BM = ml.zeros((0,0))
CM = ml.zeros((0,1))
DM = ml.zeros((0,0))
for i in range(k):
AM = la.block_diag(AM,np.kron(ml.ones((N,1)),np.kron(ml.eye(M[i]),s[k])))
BM = la.block_diag(BM,S[i])
CM = ml.vstack((CM, s[i]))
DM = la.block_diag(DM,np.kron(D[k+1],ml.eye(M[i])))
Z = ml.vstack((ml.hstack((Kw, ml.vstack((AM,ml.zeros((N*M[k],AM.shape[1])))))), ml.hstack((ml.zeros((BM.shape[0],Kw.shape[1])), BM))))
z = ml.vstack((ml.zeros((AM.shape[0],1)), np.kron(ml.ones((N,1)),s[k]), CM))
iniw = ml.hstack((q0[k]*Qwmp[k]+qL[k]*Qwzp[k], ml.zeros((1,BM.shape[0]))))
zeta = iniw/np.sum(iniw*(-Z).I*z)
if argv[argIx]=="stMoms":
# MOMENTS OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfSTMoms = argv[argIx+1]
rtMoms = []
for i in range(1,numOfSTMoms+1):
rtMoms.append(np.sum(math.factorial(i)*zeta*(-Z).I**(i+1)*z))
Ret.append(rtMoms)
argIx += 1
if argv[argIx]=="stDistr":
# DISTRIBUTION OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
stCdfPoints = argv[argIx+1]
rtDistr = []
for t in stCdfPoints:
rtDistr.append (np.sum(zeta*(-Z).I*(ml.eye(Z.shape[0])-la.expm(Z*t))*z))
Ret.append(np.array(rtDistr))
argIx += 1
elif type(argv[argIx]) is str and (argv[argIx]=="ncMoms" or argv[argIx]=="ncDistr"):
L = ml.zeros((N*np.sum(M),N*np.sum(M)))
B = ml.zeros((N*np.sum(M),N*np.sum(M)))
F = ml.zeros((N*np.sum(M),N*np.sum(M)))
kix = 0
for i in range(K):
bs = N*M[i]
F[kix:kix+bs,:][:,kix:kix+bs] = np.kron(D[k+1],ml.eye(M[i]))
L[kix:kix+bs,:][:,kix:kix+bs] = np.kron(sD0k,ml.eye(M[i])) + np.kron(I,S[i])
if i<K-1:
L[kix:kix+bs,:][:,N*np.sum(M[:k]):] = np.kron(I,s[i]*sigma[k])
else:
B[kix:kix+bs,:][:,N*np.sum(M[:k]):] = np.kron(I,s[i]*sigma[k])
kix += bs
R = QBDFundamentalMatrices (B, L, F, 'R', precision)
p0 = ml.hstack((qL[k], q0[k]*np.kron(I,sigma[k])))
p0 = p0/np.sum(p0*(ml.eye(R.shape[0])-R).I)
if argv[argIx]=="ncMoms":
# MOMENTS OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLMoms = argv[argIx+1]
qlMoms = []
for i in range(1,numOfQLMoms+1):
qlMoms.append(np.sum(math.factorial(i)*p0*R**i*(ml.eye(R.shape[0])-R).I**(i+1)))
Ret.append(MomsFromFactorialMoms(qlMoms))
elif argv[argIx]=="ncDistr":
# DISTRIBUTION OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLProbs = argv[argIx+1]
qlProbs = [np.sum(p0)]
for i in range(1,numOfQLProbs):
qlProbs.append(np.sum(p0*R**i))
Ret.append(np.array(qlProbs))
argIx += 1
else:
raise Exception("MMAPPH1NPPR: Unknown parameter "+str(argv[argIx]))
argIx += 1
if len(Ret)==1:
return Ret[0]
else:
return Ret
def test9():
#####################################
## STANDARD COOLEY HANSE CIA MODEL ##
#####################################
# Load and solve the manually linearized model
rbc1 = pm.newMOD(models.abcs_rbcs.cooley_hansen_cia_linear, mesg=True)
rbc1.modsolvers.pyuhlig.solve()
rbc1.modsolvers.forklein.solve()
# Load and solve the automatically linearized model
rbc2 = pm.newMOD(models.abcs_rbcs.cooley_hansen_cia_cf, mesg=True)
rbc2.modsolvers.forkleind.solve()
# Check equivalence of steady states
for keyo in rbc1.sstate.keys():
if keyo in rbc2.sstate.keys():
assert round(rbc1.sstate[keyo], 5) == round(rbc2.sstate[keyo], 5)
# Check equivalence of results
nexo = len(rbc2.vardic['exo']['var'])
nendo = len(rbc2.vardic['endo']['var'])
modlin1 = MAT.hstack(
(rbc1.modsolvers.pyuhlig.Q, rbc1.modsolvers.pyuhlig.P))
modlin1 = [round(modlin1[0, i1], 5) for i1 in range(modlin1.shape[1])]
modnlin1 = rbc2.modsolvers.forkleind.P[-nendo:, :]
modnlin1 = [round(modnlin1[0, i1], 5) for i1 in range(modnlin1.shape[1])]
print 'Comparison: Standard CIA model'
print "Linear is: ", modlin1
print '----------------------'
print "Nonlinear is: ", modnlin1
assert modlin1 == modnlin1
modlin2 = MAT.hstack(
(rbc1.modsolvers.pyuhlig.S, rbc1.modsolvers.pyuhlig.R))
modlin2 = [[round(modlin2[i2, i1], 5) for i1 in range(modlin2.shape[1])]
for i2 in range(modlin2.shape[0])]
modnlin2 = rbc2.modsolvers.forkleind.F
modnlin2 = [[
round(modnlin2[i2, i1], 5) for i1 in range(modnlin2.shape[1])
] for i2 in range(modnlin2.shape[0])]
print modlin2
print '----------------------'
print modnlin2
assert modlin2 == modnlin2
#####################################################
## STANDARD COOLEY HANSE CIA MODEL WITH SEIGNORAGE ##
#####################################################
# Load and solve the manually linearized model
rbc1 = pm.newMOD(models.abcs_rbcs.cooley_hansen_cia_seignorage_linear,
mesg=True)
rbc1.modsolvers.pyuhlig.solve()
rbc1.modsolvers.forklein.solve()
# Load and solve the automatically linearized model
rbc2 = pm.newMOD(models.abcs_rbcs.cooley_hansen_cia_seignorage_cf,
mesg=True)
rbc2.modsolvers.forkleind.solve()
# Check equivalence of steady states
for keyo in rbc1.sstate.keys():
if keyo in rbc2.sstate.keys():
assert round(rbc1.sstate[keyo], 5) == round(rbc2.sstate[keyo], 5)
# Check equivalence of results
nexo = len(rbc2.vardic['exo']['var'])
nendo = len(rbc2.vardic['endo']['var'])
modlin1 = MAT.hstack(
(rbc1.modsolvers.pyuhlig.Q, rbc1.modsolvers.pyuhlig.P))
modlin1 = [round(modlin1[0, i1], 5) for i1 in range(modlin1.shape[1])]
modnlin1 = rbc2.modsolvers.forkleind.P[-nendo:, :]
modnlin1 = [round(modnlin1[0, i1], 5) for i1 in range(modnlin1.shape[1])]
print 'Comparison: Standard CIA model with seignorage'
print "Linear is: ", modlin1
print '----------------------'
print "Nonlinear is: ", modnlin1
assert modlin1 == modnlin1
modlin2 = MAT.hstack(
(rbc1.modsolvers.pyuhlig.S, rbc1.modsolvers.pyuhlig.R))
modlin2 = [[round(modlin2[i2, i1], 5) for i1 in range(modlin2.shape[1])]
for i2 in range(modlin2.shape[0])]
modnlin2 = rbc2.modsolvers.forkleind.F
modnlin2 = [[
round(modnlin2[i2, i1], 5) for i1 in range(modnlin2.shape[1])
] for i2 in range(modnlin2.shape[0])]
print modlin2
print '----------------------'
print modnlin2
assert modlin2 == modnlin2
#####################################################################
## STANDARD COOLEY HANSE CIA MODEL WITH SEIGNORAGE AND CES UTILITY ##
#####################################################################
# Be careful, here in the log-linearized version we have two endogenous states, k and mg
# But in the automatically linearized version using the Jacobian we only have one, k
# Load and solve the manually linearized model
rbc1 = pm.newMOD(models.abcs_rbcs.cooley_hansen_cia_seignorage_ces_linear,
mesg=True)
rbc1.modsolvers.pyuhlig.solve()
rbc1.modsolvers.forklein.solve()
# Load and solve the automatically linearized model
rbc2 = pm.newMOD(models.abcs_rbcs.cooley_hansen_cia_seignorage_ces_cf,
mesg=True)
rbc2.modsolvers.forkleind.solve()
# Check equivalence of steady states
for keyo in rbc1.sstate.keys():
if keyo in rbc2.sstate.keys():
assert round(rbc1.sstate[keyo], 5) == round(rbc2.sstate[keyo], 5)
# Check equivalence of results
nexo = len(rbc2.vardic['exo']['var'])
nendo = len(rbc2.vardic['endo']['var'])
modlin1 = MAT.hstack((rbc1.modsolvers.pyuhlig.Q[:nendo, :],
rbc1.modsolvers.pyuhlig.P[:nendo, :nendo]))
modlin1 = [[round(modlin1[i2, i1], 5) for i1 in range(modlin1.shape[1])]
for i2 in range(modlin1.shape[0])]
modnlin1 = rbc2.modsolvers.forkleind.P[-nendo:, :]
modnlin1 = [[
round(modnlin1[i2, i1], 5) for i1 in range(modnlin1.shape[1])
] for i2 in range(modnlin1.shape[0])]
print 'Comparison: Standard CIA model with seignorage and CES utility'
print "Linear is: ", modlin1
print '----------------------'
print "Nonlinear is: ", modnlin1
assert modlin1 == modnlin1
modlin2 = MAT.hstack((rbc1.modsolvers.pyuhlig.S[:, :],
rbc1.modsolvers.pyuhlig.R[:, :nendo]))
modlin2 = [[round(modlin2[i2, i1], 5) for i1 in range(modlin2.shape[1])]
for i2 in range(modlin2.shape[0])]
modnlin2 = rbc2.modsolvers.forkleind.F[:-1, :]
modnlin2 = [[
round(modnlin2[i2, i1], 5) for i1 in range(modnlin2.shape[1])
] for i2 in range(modnlin2.shape[0])]
print modlin2
print '----------------------'
print modnlin2
assert modlin2 == modnlin2
def MMAPPH1FCFS(D, sigma, S, *argv):
"""
Returns various performane measures of a MMAP[K]/PH[K]/1
first-come-first-serve queue, see [1]_.
Parameters
----------
D : list of matrices of shape (N,N), length (K+1)
The D0...DK matrices of the arrival process.
sigma : list of row vectors, length (K)
The list containing the initial probability vectors of the service
time distributions of the various customer types. The length of the
vectors does not have to be the same.
S : list of square matrices, length (K)
The transient generators of the phase type distributions representing
the service time of the jobs belonging to various types.
further parameters :
The rest of the function parameters specify the options
and the performance measures to be computed.
The supported performance measures and options in this
function are:
+----------------+--------------------+----------------------------------------+
| Parameter name | Input parameters | Output |
+================+====================+========================================+
| "ncMoms" | Number of moments | The moments of the number of customers |
+----------------+--------------------+----------------------------------------+
| "ncDistr" | Upper limit K | The distribution of the number of |
| | | customers from level 0 to level K-1 |
+----------------+--------------------+----------------------------------------+
| "stMoms" | Number of moments | The sojourn time moments |
+----------------+--------------------+----------------------------------------+
| "stDistr" | A vector of points | The sojourn time distribution at the |
| | | requested points (cummulative, cdf) |
+----------------+--------------------+----------------------------------------+
| "stDistrME" | None | The vector-matrix parameters of the |
| | | matrix-exponentially distributed |
| | | sojourn time distribution |
+----------------+--------------------+----------------------------------------+
| "stDistrPH" | None | The vector-matrix parameters of the |
| | | matrix-exponentially distributed |
| | | sojourn time distribution, converted |
| | | to a continuous PH representation |
+----------------+--------------------+----------------------------------------+
| "prec" | The precision | Numerical precision used as a stopping |
| | | condition when solving the Riccati |
| | | equation |
+----------------+--------------------+----------------------------------------+
| "classes" | Vector of integers | Only the performance measures |
| | | belonging to these classes are |
| | | returned. If not given, all classes |
| | | are analyzed. |
+----------------+--------------------+----------------------------------------+
(The quantities related to the number of customers in
the system include the customer in the server, and the
sojourn time related quantities include the service
times as well)
Returns
-------
Ret : list of the performance measures
Each entry of the list corresponds to a performance
measure requested. Each entry is a matrix, where the
columns belong to the various job types.
If there is just a single item,
then it is not put into a list.
References
----------
.. [1] Qiming He, "Analysis of a continuous time
SM[K]/PH[K]/1/FCFS queue: Age process, sojourn times,
and queue lengths", Journal of Systems Science and
Complexity, 25(1), pp 133-155, 2012.
"""
K = len(D)-1
# parse options
eaten = []
precision = 1e-14;
classes = np.arange(0,K)
for i in range(len(argv)):
if argv[i]=="prec":
precision = argv[i+1]
eaten.append(i)
eaten.append(i+1)
elif argv[i]=="classes":
classes = np.array(argv[i+1])-1
eaten.append(i)
eaten.append(i+1)
if butools.checkInput and not CheckMMAPRepresentation(D):
raise Exception('MMAPPH1FCFS: The arrival process is not a valid MMAP representation!')
if butools.checkInput:
for k in range(K):
if not CheckPHRepresentation(sigma[k],S[k]):
raise Exception('MMAPPH1FCFS: the vector and matrix describing the service times is not a valid PH representation!')
# some preparation
D0 = D[0]
N = D0.shape[0]
Ia = ml.eye(N);
Da = ml.zeros((N,N))
for q in range(K):
Da += D[q+1]
theta = CTMCSolve(D0+Da)
beta = [CTMCSolve(S[k]+ml.sum(-S[k],1)*sigma[k]) for k in range(K)]
lambd = [np.sum(theta*D[k+1]) for k in range(K)]
mu = [np.sum(beta[k]*(-S[k])) for k in range(K)]
Nsk = [S[k].shape[0] for k in range(K)]
ro = np.sum(np.array(lambd)/np.array(mu))
alpha = theta*Da/sum(lambd)
D0i = (-D0).I
Sa = S[0];
sa = [ml.zeros(sigma[0].shape)]*K
sa[0] = sigma[0]
ba = [ml.zeros(beta[0].shape)]*K
ba[0] = beta[0]
sv = [ml.zeros((Nsk[0],1))]*K
sv[0] = ml.sum(-S[0],1)
Pk = [D0i*D[q+1] for q in range(K)]
for k in range(1,K):
Sa = la.block_diag(Sa, S[k])
for q in range(K):
if q==k:
sa[q] = ml.hstack((sa[q], sigma[k]))
ba[q] = ml.hstack((ba[q], beta[k]))
sv[q] = ml.vstack((sv[q], -np.sum(S[k],1)))
else:
sa[q] = ml.hstack((sa[q], ml.zeros(sigma[k].shape)))
ba[q] = ml.hstack((ba[q], ml.zeros(beta[k].shape)))
sv[q] = ml.vstack((sv[q], ml.zeros((Nsk[k],1))))
Sa = ml.matrix(Sa)
P = D0i*Da
iVec = ml.kron(D[1],sa[0])
for k in range(1,K):
iVec += ml.kron(D[k+1],sa[k])
Ns = Sa.shape[0]
Is = ml.eye(Ns)
# step 1. solve the age process of the queue
# ==========================================
# solve Y0 and calculate T
Y0 = FluidFundamentalMatrices (ml.kron(Ia,Sa), ml.kron(Ia,-ml.sum(Sa,1)), iVec, D0, "P", precision)
T = ml.kron(Ia,Sa) + Y0 * iVec
# calculate pi0 and v0
pi0 = ml.zeros((1,T.shape[0]))
for k in range(K):
pi0 += ml.kron(theta*D[k+1],ba[k]/mu[k])
pi0 = - pi0 * T
iT = (-T).I
oa = ml.ones((N,1))
# step 2. calculate performance measures
# ======================================
Ret = []
for k in classes:
argIx = 0
clo = iT*ml.kron(oa,sv[k])
while argIx<len(argv):
if argIx in eaten:
argIx += 1
continue
elif type(argv[argIx]) is str and argv[argIx]=="stMoms":
numOfSTMoms = argv[argIx+1]
rtMoms = []
for m in range(1,numOfSTMoms+1):
rtMoms.append(math.factorial(m) * np.sum(pi0 * iT**m * clo / (pi0*clo)))
Ret.append(rtMoms)
argIx += 1
elif type(argv[argIx]) is str and argv[argIx]=="stDistr":
stCdfPoints = argv[argIx+1]
cdf = [];
for t in stCdfPoints:
pr = 1 - np.sum(pi0 * la.expm(T*t) * clo / (pi0*clo))
cdf.append(pr)
Ret.append(np.array(cdf))
argIx += 1
elif type(argv[argIx]) is str and argv[argIx]=="stDistrME":
Bm = SimilarityMatrixForVectors(clo/(pi0*clo),ml.ones((N*Ns,1)))
Bmi = Bm.I
A = Bm * T * Bmi
alpha = pi0 * Bmi
Ret.append(alpha)
Ret.append(A)
elif type(argv[argIx]) is str and argv[argIx]=="stDistrPH":
vv = pi0*iT
ix = np.arange(N*Ns)
nz = ix[vv.flat>precision]
delta = Diag(vv[:,nz])
cl = -T*clo/(pi0*clo)
alpha = cl[nz,:].T*delta
A = delta.I*T[nz,:][:,nz].T*delta
Ret.append(alpha)
Ret.append(A)
elif type(argv[argIx]) is str and argv[argIx]=="ncDistr":
numOfQLProbs = argv[argIx+1]
argIx += 1
values = np.empty(numOfQLProbs)
jm = ml.zeros((Ns,1))
jm[np.sum(Nsk[0:k]):np.sum(Nsk[0:k+1]),:] = 1
jmc = ml.ones((Ns,1))
jmc[np.sum(Nsk[0:k]):np.sum(Nsk[0:k+1]),:] = 0
LmCurr = la.solve_sylvester(T, ml.kron(D0+Da-D[k+1],Is), -ml.eye(N*Ns))
values[0] = 1-ro+np.sum(pi0*LmCurr*ml.kron(oa,jmc))
for i in range(1,numOfQLProbs):
LmPrev = LmCurr
LmCurr = la.solve_sylvester(T, ml.kron(D0+Da-D[k+1],Is), -LmPrev*ml.kron(D[k+1],Is))
values[i] = np.sum(pi0*LmCurr*ml.kron(oa,jmc) + pi0*LmPrev*ml.kron(oa,jm));
Ret.append(values)
elif type(argv[argIx]) is str and argv[argIx]=="ncMoms":
numOfQLMoms = argv[argIx+1]
argIx += 1
jm = ml.zeros((Ns,1))
jm[np.sum(Nsk[0:k]):np.sum(Nsk[0:k+1]),:] = 1
ELn = [la.solve_sylvester(T, ml.kron(D0+Da,Is), -ml.eye(N*Ns))]
qlMoms = []
for n in range(1,numOfQLMoms+1):
bino = 1
Btag = ml.zeros((N*Ns,N*Ns))
for i in range(n):
Btag += bino * ELn[i]
bino *= (n-i) / (i+1)
ELn.append(la.solve_sylvester(T, ml.kron(D0+Da,Is), -Btag*ml.kron(D[k+1],Is)))
qlMoms.append(np.sum(pi0*ELn[n]) + np.sum(pi0*Btag*ml.kron(oa,jm)))
Ret.append(qlMoms)
else:
raise Exception("MMAPPH1FCFS: Unknown parameter "+str(argv[argIx]))
argIx += 1
if len(Ret)==1:
return Ret[0]
else:
return Ret
def SPIRIT(streams, energyThresh, lamb, evalMetrics):
# Make
if type(streams) == np.ndarray:
streams_iter = iter(streams)
# Max No. Streams
if streams.ndim == 1:
streams = np.expand_dims(streams, axis=1)
num_streams = streams.shape[1]
else:
num_streams = streams.shape[1]
count_over = 0
count_under = 0
#===============================================================================
# Initalise k, w and d, lamb
#===============================================================================
k = 1 # Hidden Variables, initialise to one
# Weights
pc_weights = npm.zeros(num_streams)
pc_weights[0, 0] = 1
# initialise outputs
res = {}
all_weights = []
k_hist = []
anomalies = []
x_dash = npm.zeros((1, num_streams))
Eng = mat([0.00000001, 0.00000001])
E_xt = 0 # Energy of X at time t
E_rec_i = mat([0.000000000000001]) # Energy of reconstruction
Y = npm.zeros(num_streams)
timeSteps = streams.shape[0]
#===============================================================================
# Main Loop
#===============================================================================
for t in range(1, timeSteps + 1): # t = 1,...,200
k_hist.append(k)
x_t_plus_1 = mat(streams_iter.next()) # Read in next signals
d_i = E_rec_i * t
# Step 1 - Update Weights
pc_weights, y_t_i, error = track_W(x_t_plus_1, k, pc_weights, d_i,
num_streams, lamb)
# Record hidden variables
padding = num_streams - k
y_bar_t = npm.hstack((y_t_i, mat([nan] * padding)))
Y = npm.vstack((Y, y_bar_t))
# Record Weights
all_weights.append(pc_weights)
# Record reconstrunted z and RSRE
x_dash = npm.vstack((x_dash, y_t_i * pc_weights))
# Record RSRE
if t == 1:
top = 0.0
bot = 0.0
top = top + (norm(x_t_plus_1 - x_dash)**2)
bot = bot + (norm(x_t_plus_1)**2)
new_RSRE = top / bot
if t == 1:
RSRE = new_RSRE
else:
RSRE = npm.vstack((RSRE, new_RSRE))
### FOR EVALUATION ###
#deviation from truth
if evalMetrics == 'T':
Qt = pc_weights.T
if t == 1:
res['subspace_error'] = npm.zeros((timeSteps, 1))
res['orthog_error'] = npm.zeros((timeSteps, 1))
res['angle_error'] = npm.zeros((timeSteps, 1))
Cov_mat = npm.zeros([num_streams, num_streams])
# Calculate Covarentce Matrix of data up to time t
Cov_mat = lamb * Cov_mat + npm.dot(x_t_plus_1, x_t_plus_1.T)
# Get eigenvalues and eigenvectors
W, V = eig(Cov_mat)
# Use this to sort eigenVectors in according to deccending eigenvalue
eig_idx = W.argsort() # Get sort index
eig_idx = eig_idx[::-1] # Reverse order (default is accending)
# v_r = highest r eigen vectors (accoring to thier eigenvalue if sorted).
V_k = V[:, eig_idx[:k]]
# Calculate subspace error
C = npm.dot(V_k, V_k.T) - npm.dot(Qt, Qt.T)
res['subspace_error'][t - 1, 0] = 10 * np.log10(
npm.trace(npm.dot(C.T, C))) #frobenius norm in dB
# Calculate angle between projection matrixes
D = npm.dot(npm.dot(npm.dot(V_k.T, Qt), Qt.T), V_k)
eigVal, eigVec = eig(D)
angle = npm.arccos(np.sqrt(max(eigVal)))
res['angle_error'][t - 1, 0] = angle
# Calculate deviation from orthonormality
F = npm.dot(Qt.T, Qt) - npm.eye(k)
res['orthog_error'][t - 1, 0] = 10 * np.log10(
npm.trace(npm.dot(F.T, F))) #frobenius norm in dB
# Step 2 - Update Energy estimate
E_xt = ((lamb * (t - 1) * E_xt) + norm(x_t_plus_1)**2) / t
for i in range(k):
E_rec_i[0, i] = ((lamb * (t - 1) * E_rec_i[0, i]) +
(y_t_i[0, i]**2)) / t
# Step 3 - Estimate the retained energy
E_retained = npm.sum(E_rec_i, 1)
# Record Energy
Eng_new = npm.hstack((E_xt, E_retained[0, 0]))
Eng = npm.vstack((Eng, Eng_new))
if E_retained < energyThresh[0] * E_xt:
if k != num_streams:
k = k + 1
# Initalise Ek+1 <-- 0
E_rec_i = npm.hstack((E_rec_i, mat([0])))
# Initialise W_i+1
new_weight_vec = npm.zeros(num_streams)
new_weight_vec[0, k - 1] = 1
pc_weights = npm.vstack((pc_weights, new_weight_vec))
anomalies.append(t - 1)
else:
count_over += 1
elif E_retained > energyThresh[1] * E_xt:
if k > 1:
k = k - 1
# discard w_k and error
pc_weights = delete(pc_weights, -1, 0)
# Discard E_rec_i[k]
E_rec_i = delete(E_rec_i, -1)
else:
count_under += 1
# Data Stores
res2 = {
'hidden': Y, # Array for hidden Variables
'weights': all_weights,
'E_t': Eng[:, 0], # total energy of data
'E_dash_t': Eng[:, 1], # hidden var energy
'e_ratio': np.divide(Eng[:, 1], Eng[:, 0]), # Energy ratio
'RSRE': RSRE, # Relative squared Reconstruction error
'recon': x_dash, # reconstructed data
'r_hist': k_hist, # history of r values
'anomalies': anomalies
}
res.update(res2)
return res, all_weights
