def departure(self) -> MarkovArrival:
arrival, service = self._get_casted_arrival_and_service()
# Aliasing matrices from arrival MAP and service PH
d0 = arrival.d0
d1 = arrival.d1
w = arrival.order
iw = np.eye(w)
s = service.s
tau = service.init_probs
v = service.order
iv = np.eye(v)
ev = np.ones((v, 1))
m = self.capacity - 1
b = v * w
ob = np.zeros((b, b))
# Building blocks
d0_iv = np.kron(d0, iv)
d1_iv = np.kron(d1, iv)
d0_s = np.kron(d0, iv) + np.kron(iw, s)
ct = np.kron(-s.dot(ev), tau)
iw_ct = np.kron(iw, ct)
r0 = np.kron(d1, np.kron(tau, ev))
ra = np.kron(d0 + d1, iv) + np.kron(iw, s)
# Building departure D0 and D1
d0_dep = cbdiag(self.capacity, ((0, d0_s), (1, d1_iv)))
d0_dep[m * b:, m * b:] = ra
d0_left_col = np.vstack((d0_iv, ) + (ob, ) * self.capacity)
d0_top_row = np.hstack((r0, ) + (ob, ) * m)
d0_dep = np.hstack((d0_left_col, np.vstack((d0_top_row, d0_dep))))
D1_dep = cbdiag(self.capacity + 1, ((-1, iw_ct), ))
return MarkovArrival(d0_dep, D1_dep)
def departure(self):
# Aliasing matrices from arrival MAP and service PH
D0 = self.arrival.D0
D1 = self.arrival.D1
W = self.arrival.order
Iw = np.eye(W)
S = self.service.subgenerator
tau = self.service.pmf0
V = self.service.order
Iv = np.eye(V)
Ev = np.ones((V, 1))
M = self.capacity - 1
B = V * W
Ob = np.zeros((B, B))
# Building blocks
D0_Iv = np.kron(D0, Iv)
D1_Iv = np.kron(D1, Iv)
D0_S = np.kron(D0, Iv) + np.kron(Iw, S)
Ct = np.kron(-S.dot(Ev), tau)
Iw_Ct = np.kron(Iw, Ct)
R0 = np.kron(D1, np.kron(tau, Ev))
Ra = np.kron(D0 + D1, Iv) + np.kron(Iw, S)
# Building departure D0 and D1
D0_dep = cbdiag(self.capacity, ((0, D0_S), (1, D1_Iv)))
D0_dep[M * B:, M * B:] = Ra
D0_left_col = np.vstack((D0_Iv, ) + (Ob, ) * self.capacity)
D0_top_row = np.hstack((R0, ) + (Ob, ) * M)
D0_dep = np.hstack((D0_left_col, np.vstack((D0_top_row, D0_dep))))
D1_dep = cbdiag(self.capacity + 1, ((-1, Iw_Ct), ))
return MAP(D0_dep, D1_dep)
def test_cbdiag_2x2(self):
b1 = [[1, 1], [1, 1]]
b2 = [[2, 2], [2, 2]]
b3 = [[3, 3], [3, 3]]
m1 = qumo.cbdiag(1, [(0, b1), (1, b2), (-1, b3)])
m2 = qumo.cbdiag(2, [(0, b1), (1, b2), (-1, b3)])
m3 = qumo.cbdiag(3, [(0, b1), (1, b2), (-1, b3)])
m4 = qumo.cbdiag(3, [(0, b2), (1, b3), (2, b1)])
np.testing.assert_allclose(m1, [[1, 1], [1, 1]])
np.testing.assert_allclose(m2, [[1, 1, 2, 2], [1, 1, 2, 2],
[3, 3, 1, 1], [3, 3, 1, 1]])
np.testing.assert_allclose(m3, [[1, 1, 2, 2, 0, 0],
[1, 1, 2, 2, 0, 0],
[3, 3, 1, 1, 2, 2],
[3, 3, 1, 1, 2, 2],
[0, 0, 3, 3, 1, 1],
[0, 0, 3, 3, 1, 1]])
np.testing.assert_allclose(m4, [[2, 2, 3, 3, 1, 1],
[2, 2, 3, 3, 1, 1],
[0, 0, 2, 2, 3, 3],
[0, 0, 2, 2, 3, 3],
[0, 0, 0, 0, 2, 2],
[0, 0, 0, 0, 2, 2]])
def departure(self):
n = self.capacity
a = self.arrival_rate
b = self.service_rate
d0 = cbdiag(n + 1, [(0, [[-(a + b)]]), (1, [[a]])])
d0[0, 0] += b
d0[n, n] += a
d1 = cbdiag(n + 1, [(-1, [[b]])])
return MAP(d0, d1)
def departure(self):
n = self.capacity
a = self.arrival.rate
b = self.service.rate
d0 = cbdiag(n + 1, [(0, np.asarray([[-(a + b)]])),
(1, np.asarray([[a]]))])
d0[0, 0] += b
d0[n, n] += a
d1 = cbdiag(n + 1, [(-1, np.asarray([[b]]))])
return MarkovArrival(d0, d1)
def erlang(shape, rate):
"""MAP representation of Erlang process with the given shape and rate.
:param shape number of phases
:param rate rate at each phase
"""
d0 = cbdiag(shape, [(0, [[-rate]]), (1, [[rate]])])
d1 = np.zeros((shape, shape))
d1[shape-1, 0] = rate
return MAP(d0, d1)
def erlang(shape: int, rate: float) -> 'MarkovArrival':
"""
MAP representation of Erlang process with the given shape and rate.
Parameters
----------
shape : int
Number of phases
:param : float
Rate at each phase
"""
d0 = cbdiag(shape, [(0, np.asarray([[-rate]])),
(1, np.asarray([[rate]]))])
d1 = np.zeros((shape, shape))
d1[shape - 1, 0] = rate
return MarkovArrival(d0, d1)
def fit_map_horvath(ph, lags):
"""Find D1 matrix using a D0 as subgenerator of the given PH and lags.
Args:
ph (`pyqunet.distributions.PH`): a PH distribution approximating D0
lags (array-like): a list of lag-k auto-correlation coefficients
"""
if len(lags) > 1:
return fit_map_horvath(ph, [lags[0]])
N = ph.order
D0 = ph.subgenerator
En = np.ones((N, 1))
pi = ph.init_probs
num_lags = len(lags)
if num_lags == 0:
D1_row_sum = (-D0).dot(En).reshape(N)
D1 = np.asarray([D1_row_sum[i] * pi for i in range(N)])
return MAP(D0, D1)
ph_moments = stats.moment(ph, 2)
ph_moments, mu = stats.normalize_moments(ph_moments)
D0ni = np.linalg.inv(-D0) / mu
D0ni2 = D0ni.dot(D0ni)
rate = ph.param * mu
lag1 = lags[0]
d = (-D0 * mu).dot(En).reshape((N, 1))
gamma = pi.dot(D0ni).reshape((1, N))
block_gamma = cbdiag(N, [(0, gamma)])
block_eye = np.hstack([np.eye(N)] * N)
A = np.vstack([block_eye, block_gamma])
b = np.vstack([d, pi.reshape((N, 1))])
delta = pow(rate, 2) * pi.dot(D0ni2)
f = D0ni.dot(En)
# noinspection PyTypeChecker
v = lag1 * (2 * pow(rate, 2) * pi.dot(D0ni2).dot(En).reshape(1)[0] - 1) + 1
c = np.hstack([f[i] * delta for i in range(N)])
if num_lags == 1:
A = np.vstack([A, c])
b = np.vstack([b, [[v]]]).reshape(2 * N + 1)
ret = scipy.optimize.lsq_linear(A,
b, (0, np.inf),
tol=1e-10,
method='bvls')
# noinspection PyUnresolvedReferences
x = ret.x
assert isinstance(x, np.ndarray)
D1 = x.reshape((N, N)).transpose() / mu
try:
return MAP(D0, D1, rtol=1e-3, atol=1e-4)
except ValueError:
if np.abs(lags[0] < 1e-5):
return fit_map_horvath(ph, [])
else:
return fit_map_horvath(ph, [lags[0] * 0.5])
else:
def residual(xi, n, input_lags):
d1i = xi.reshape(n, n).transpose() / mu
m = MAP(D0, d1i, check=False)
estimated_lags = [m.lag(i + 1) for i in range(len(input_lags))]
system_diff = np.asarray(A.dot(xi) - b).flatten() * 1000
lags_diff = np.asarray(input_lags - estimated_lags)
diff = np.hstack((system_diff, lags_diff))
return diff
lags = np.asarray(lags)
params = {
'fun': residual,
'x0':
np.asarray([d[i] * pi for i in range(N)]).transpose().flatten(),
'bounds': (0, np.inf),
'kwargs': {
'input_lags': lags,
'n': N,
},
}
result = scipy.optimize.least_squares(**params)
# noinspection PyUnresolvedReferences
D1 = result.x.reshape(N, N).transpose() / mu
return MAP(D0, D1, check=False)
def erlang(shape, rate):
s = cbdiag(shape, ((0, [[-rate]]), (1, [[rate]])))
pmf0 = np.asarray((1.0, ) + (0.0, ) * (shape - 1))
return PhaseType(s, pmf0)