def CdfFromPH(alpha, A, x):
"""
Returns the cummulative distribution function of a
continuous phase-type distribution.
Parameters
----------
alpha : matrix, shape (1,M)
The initial probability vector of the phase-type
distribution.
A : matrix, shape (M,M)
The transient generator matrix of the phase-type
distribution.
x : vector of doubles
The cdf will be computed at these points
Returns
-------
cdf : column vector of doubles
The values of the cdf at the corresponding "x" values
"""
if butools.checkInput and not CheckPHRepresentation(alpha, A):
raise Exception("CdfFromPH: Input is not a valid PH representation!")
return CdfFromME(alpha, A, x)
def MomentsFromPH(alpha, A, K=0):
"""
Returns the first K moments of a continuous phase-type
distribution.
Parameters
----------
alpha : vector, shape (1,M)
The initial probability vector of the phase-type
distribution.
A : matrix, shape (M,M)
The transient generator matrix of the phase-type
distribution.
K : int, optional
Number of moments to compute. If K=0, 2*M-1 moments
are computed. The default value is K=0.
prec : double, optional
Numerical precision for checking the input.
The default value is 1e-14.
Returns
-------
moms : row vector of doubles
The vector of moments.
"""
if butools.checkInput and not CheckPHRepresentation(alpha, A):
raise Exception(
"MomentsFromPH: Input is not a valid PH representation!")
return MomentsFromME(alpha, A, K)
def PdfFromPH(alpha, A, x):
"""
Returns the probability density function of a continuous
phase-type distribution.
Parameters
----------
alpha : vector, shape (1,M)
The initial probability vector of the phase-type
distribution.
A : matrix, shape (M,M)
The transient generator matrix of the phase-type
distribution.
x : vector of doubles
The density function will be computed at these points
prec : double, optional
Numerical precision to check if the input ME
distribution is valid. The default value is 1e-14.
Returns
-------
pdf : column vector of doubles
The values of the density function at the
corresponding "x" values
"""
if butools.checkInput and not CheckPHRepresentation(alpha, A):
raise Exception("PdfFromPH: Input is not a valid PH representation!")
return PdfFromME(alpha, A, x)
def SamplesFromPH(a, A, k):
"""
Generates random samples from a phase-type distribution.
Parameters
----------
alpha : matrix, shape (1,M)
The initial probability vector of the phase-type
distribution.
A : matrix, shape (M,M)
The transient generator matrix of the phase-type
distribution.
K : integer
The number of samples to generate.
prec : double, optional
Numerical precision to check if the input phase-type
distribution is valid. The default value is 1e-14.
Returns
-------
x : vector, length(K)
The vector of random samples
"""
if butools.checkInput and not CheckPHRepresentation(a, A):
raise Exception(
"SamplesFromPH: input is not a valid PH representation!")
# auxilary variables
a = a.A.flatten()
N = len(a)
cummInitial = np.cumsum(a)
sojourn = -1.0 / np.diag(A)
nextpr = ml.matrix(np.diag(sojourn)) * A
nextpr = nextpr - ml.matrix(np.diag(np.diag(nextpr)))
nextpr = np.hstack((nextpr, 1.0 - np.sum(nextpr, 1)))
nextpr = np.cumsum(nextpr, 1)
x = np.zeros(k)
for n in range(k):
time = 0
# draw initial distribution
r = rand()
state = 0
while cummInitial[state] <= r:
state += 1
# play state transitions
while state < N:
time += -np.log(rand()) * sojourn[state]
r = rand()
nstate = 0
while nextpr[state, nstate] <= r:
nstate += 1
state = nstate
x[n] = time
return x
def AcyclicPHFromME (alpha, A, maxSize=100, precision=1e-14):
"""
Transforms an arbitrary matrix-exponential representation
to an acyclic phase-type representation. (see [1]_).
Parameters
----------
alpha : matrix, shape (1,N)
Initial vector of the distribution
A : matrix, shape (N,N)
Matrix parameter of the distribution
maxSize : int, optional
The maximum number of phases for the result.
The default value is 100.
precision : double, optional
Vector and matrix entries smaller than the precision
are considered to be zeros. The default value is 1e-14.
Returns
-------
beta : matrix, shape (1,M)
The initial probability vector of the Markovian
acyclic representation
B : matrix, shape (M,M)
Transient generator matrix of the Markovian
acyclic representation
Notes
-----
Raises an error if no Markovian acyclic representation
has been found.
References
----------
.. [1] Mocanu, S., Commault, C.: "Sparse representations of
phase-type distributions," Stoch. Models 15, 759-778
(1999)
"""
G = TransformToAcyclic (A, maxSize, precision)
# find transformation matrix
T = SimilarityMatrix (A, G)
gamma = np.real(alpha*T)
if np.min(gamma) <= -precision:
gamma, G = ExtendToMarkovian (gamma, G, maxSize, precision)
if not CheckPHRepresentation (gamma, G, precision):
raise Exception("AcyclicPHFromME: No acyclic representation found up to the given size and precision!")
else:
return (gamma, G)
def IntervalPdfFromPH(alpha, A, intBounds):
"""
Returns the approximate probability density function of a
continuous phase-type distribution, based on the
probability of falling into intervals.
Parameters
----------
alpha : vector, shape (1,M)
The initial probability vector of the phase-type
distribution.
A : matrix, shape (M,M)
The transient generator matrix of the phase-type
distribution.
intBounds : vector, shape (K)
The array of interval boundaries. The pdf is the
probability of falling into an interval divided by
the interval length.
If the size of intBounds is K, the size of the result is K-1.
prec : double, optional
Numerical precision to check if the input is a valid
phase-type distribution. The default value is 1e-14
Returns
-------
x : matrix of doubles, shape(K-1,1)
The points at which the pdf is computed. It holds the center of the
intervals defined by intBounds.
y : matrix of doubles, shape(K-1,1)
The values of the density function at the corresponding "x" values
Notes
-----
This method is more suitable for comparisons with empirical
density functions than the exact one (given by PdfFromPH).
"""
if butools.checkInput and not CheckPHRepresentation(alpha, A):
raise Exception(
"IntervalPdfFromPH: Input is not a valid PH representation!")
steps = len(intBounds)
x = [(intBounds[i + 1] + intBounds[i]) / 2.0 for i in range(steps - 1)]
y = [(np.sum(alpha * expm2((A * intBounds[i]).A)) - np.sum(alpha * expm2(
(A * intBounds[i + 1]).A))) / (intBounds[i + 1] - intBounds[i])
for i in range(steps - 1)]
return (np.array(x), np.array(y))
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 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 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