def LagkJointMomentsFromMMAP(D, K=0, L=1):
"""
Returns the lag-L joint moments of a marked Markovian
arrival process.
Parameters
----------
D : list/cell of matrices of shape(M,M), length(N)
The D0...DN matrices of the MMAP to check
K : int, optional
The dimension of the matrix of joint moments to
compute. If K=0, the MxM joint moments will be
computed. The default value is 0
L : int, optional
The lag at which the joint moments are computed.
The default value is 1
prec : double, optional
Numerical precision to check if the input is valid.
The default value is 1e-14
Returns
-------
Nm : list/cell of matrices of shape(K+1,K+1), length(L)
The matrices containing the lag-L joint moments,
starting from moment 0.
"""
if butools.checkInput and not CheckMMAPRepresentation(D):
raise Exception(
"LagkJointMomentsFromMMAP: Input is not a valid MMAP representation!"
)
return LagkJointMomentsFromMRAP(D, K, L)
def MarginalMomentsFromMMAP(D, K=0):
"""
Returns the moments of the marginal distribution of a
marked Markovian arrival process.
Parameters
----------
D : list/cell of matrices of shape(M,M), length(N)
The D0...DN matrices of the MMAP
K : int, optional
Number of moments to compute. If K=0, 2*M-1 moments
are computed. The default value is K=0.
precision : double, optional
Numerical precision for checking if the input is valid.
The default value is 1e-14
Returns
-------
moms : row vector of doubles, length K
The vector of moments.
"""
if butools.checkInput and not CheckMMAPRepresentation(D):
raise Exception(
"MarginalMomentsFromMMAP: Input is not a valid MMAP representation!"
)
alpha, A = MarginalDistributionFromMMAP(D)
return MomentsFromPH(alpha, A, K)
def MarginalDistributionFromMMAP(D):
"""
Returns the phase type distributed marginal distribution
of a marked Markovian arrival process.
Parameters
----------
D : list/cell of matrices of shape(M,M), length(N)
The D0...DN matrices of the MMAP
precision : double, optional
Numerical precision for checking if the input is valid.
The default value is 1e-14
Returns
-------
alpha : matrix, shape (1,M)
The initial probability vector of the phase type
distributed marginal distribution
A : matrix, shape (M,M)
The transient generator of the phase type distributed
marginal distribution
"""
if butools.checkInput and not CheckMMAPRepresentation(D):
raise Exception(
"MarginalDistributionFromMMAP: Input is not a valid MMAP representation!"
)
return MarginalDistributionFromMRAP(D)
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
def SamplesFromMMAP(D, k, initial=None, prec=1e-14):
"""
Generates random samples from a marked Markovian
arrival process.
Parameters
----------
D : list of matrices of shape(M,M), length(N)
The D0...DN matrices of the MMAP
K : integer
The number of samples to generate.
prec : double, optional
Numerical precision to check if the input MMAP is
valid. The default value is 1e-14.
Returns
-------
x : matrix, shape(K,2)
The random samples. Each row consists of two
columns: the inter-arrival time and the type of the
arrival.
"""
if butools.checkInput and not CheckMMAPRepresentation(D):
raise Exception(
"SamplesFromMMAP: Input is not a valid MMAP representation!")
N = D[0].shape[0]
if initial == None:
# draw initial state according to the stationary distribution
stst = CTMCSolve(SumMatrixList(D)).A.flatten()
cummInitial = np.cumsum(stst)
r = rand()
state = 0
while cummInitial[state] <= r:
state += 1
else:
state = initial
# auxilary variables
sojourn = -1.0 / np.diag(D[0])
nextpr = ml.matrix(np.diag(sojourn)) * D[0]
nextpr = nextpr - ml.matrix(np.diag(np.diag(nextpr)))
for i in range(1, len(D)):
nextpr = np.hstack((nextpr, np.diag(sojourn) * D[i]))
nextpr = np.cumsum(nextpr, 1)
if len(D) > 2:
x = np.empty((k, 2))
else:
x = np.empty(k)
for n in range(k):
time = 0
# 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
if len(D) > 2:
x[n, 0] = time
x[n, 1] = state // N
else:
x[n] = time
state = state % N
return x
def ImageFromMMAP(D, outFileName="display", prec=1e-13):
"""
Depicts the given marked Markovian arrival process, and
either displays it or saves it to file.
Parameters
----------
D : list of matrices of shape(M,M), length(N)
The D0...DN matrices of the MMAP
outFileName : string, optional
If it is not provided, or equals to 'display', the
image is displayed on the screen, otherwise it is
written to the file. The file format is deduced
from the file name.
prec : double, optional
Transition rates less then prec are considered to
be zero and are left out from the image. The
default value is 1e-13.
Notes
-----
The 'graphviz' software must be installed and available
in the path to use this feature.
"""
if butools.checkInput and not CheckMMAPRepresentation(D):
raise Exception(
"ImageFromMMAP: Input is not a valid MMAP representation!")
if outFileName == "display":
outputFile = ".result.png"
displ = True
else:
outputFile = outFileName
displ = False
inputFile = "_temp.dot"
f = open(inputFile, "w")
f.write("digraph G {\n")
f.write("\trankdir=LR;\n")
f.write('\tnode [shape=circle,width=0.3,height=0.3,label=""];\n')
N = D[0].shape[0]
# transitions without arrivals
Dx = D[0]
for i in range(N):
for j in range(N):
if i != j and abs(Dx[i, j]) > prec:
f.write('\tn{0} -> n{1} [label="{2}"];\n'.format(
i, j, Dx[i, j]))
# transitions with arrivals
for k in range(1, len(D)):
Dx = D[k]
for i in range(N):
for j in range(N):
if abs(Dx[i, j]) > prec:
if len(D) == 2:
f.write(
'\tn{0} -> n{1} [style="dashed",label="{2}"];\n'.
format(i, j, Dx[i, j]))
else:
f.write(
'\tn{0} -> n{1} [style="solid",fontcolor="/dark28/{2}",color="/dark28/{3}",label="{4}"];\n'
.format(i, j, min(k - 1, 8), min(k - 1, 8), Dx[i,
j]))
f.write('}\n')
f.close()
ext = os.path.splitext(outputFile)[1]
call(['dot', '-T' + ext[1:], "_temp.dot", '-o', outputFile])
os.remove(inputFile)
if displ:
from IPython.display import Image
i = Image(filename=outputFile)
os.remove(outputFile)
return i