def delay_prob_sample_exp_dm1(theta: float, t: int, delay: int, lamb: float,
rate: float, a: float,
sample_size: int) -> float:
if 1 / lamb >= rate:
raise ParameterOutOfBounds(
(f"The arrivals' long term rate={1 / lamb} has to be smaller than "
f"the service's long term rate={rate}"))
if theta <= 0:
raise ParameterOutOfBounds(f"theta={theta} must be > 0")
if a <= 1:
raise ParameterOutOfBounds(f"base a={a} must be >0")
res = 0.0
for j in range(sample_size):
sum_i = 0.0
for i in range(t + 1):
sum_i += a**(np.exp(
theta * f_sample(i=i, s=t + delay, t=t, lamb=lamb, rate=rate)))
res += sum_i
res /= sample_size
return res
def delay_prob_taylor_exp_dm1(theta: float, t: int, delay: int, lamb: float,
rate: float, a: float) -> float:
if 1 / lamb >= rate:
raise ParameterOutOfBounds(
("The arrivals' long term rate {0} has to be smaller than"
"the service's long term rate {1}").format(1 / lamb, rate))
if theta <= 0:
raise ParameterOutOfBounds("theta = {0} must be > 0".format(theta))
if a <= 1:
raise ParameterOutOfBounds("base a={0} must be >0".format(a))
sum_j = 0.0
for _i in range(t + 1):
try:
summand = f_exp(
theta=theta, i=_i, s=t + delay, t=t, lamb=lamb, rate=rate, a=a
) + 0.5 * f_double_prime(
theta=theta, i=_i, s=t + delay, t=t, lamb=lamb, rate=rate,
a=a) * var_dm1(delta_time=t - _i, lamb=lamb)
except (FloatingPointError, OverflowError):
summand = inf
sum_j += summand
return log(sum_j) / log(a)
def delay_prob_lower_exp_dm1(theta: float, t: int, delay: int, lamb: float,
rate: float, a: float) -> float:
if 1 / lamb >= rate:
raise ParameterOutOfBounds(
(f"The arrivals' long term rate {1 / lamb} has to be smaller than"
f"the service's long term rate {rate}"))
if theta <= 0:
raise ParameterOutOfBounds(f"theta = {theta} must be > 0")
if a <= 1:
raise ParameterOutOfBounds(f"base a = {a} must be >1")
sum_j = 0.0
for _i in range(t + 1):
try:
summand = f_exp(theta=theta,
i=_i,
s=t + delay,
t=t,
lamb=lamb,
rate=rate,
a=a)
except (FloatingPointError, OverflowError):
summand = inf
sum_j += summand
return np.log(sum_j) / np.log(a)
def stability_check(arr: Arrival,
ser: Server,
theta: float,
indep=True,
p=1.0,
q=1.0) -> None:
if indep:
# if (arr.rho(theta=theta) < 0 or ser.rho(theta=theta) < 0
# or arr.sigma(theta=theta) < 0 or ser.sigma(theta=theta) < 0):
# raise ParameterOutOfBounds("parameters must be positive")
if arr.rho(theta=theta) >= ser.rho(theta=theta):
raise ParameterOutOfBounds(
f"The arrivals' rho={arr.rho(theta=theta)} has to be "
f"smaller than the service's rho={ser.rho(theta=theta)}")
else:
# if (arr.rho(theta=p * theta) < 0 or ser.rho(theta=q * theta) < 0
# or arr.sigma(theta=p * theta) < 0
# or ser.sigma(theta=q * theta) < 0):
# raise ParameterOutOfBounds("parameters must be positive")
if arr.rho(theta=p * theta) >= ser.rho(theta=q * theta):
raise ParameterOutOfBounds(
f"The arrivals' rho={arr.rho(theta=p * theta)} has to be "
f"smaller than the service's rho={ser.rho(theta=q * theta)}")
def rho(self, theta: float) -> float:
if theta <= 0:
raise ParameterOutOfBounds(f"theta = {theta} must be > 0")
if theta >= self.mu:
raise ParameterOutOfBounds(f"theta = {theta} must"
f"be < mu = {self.mu}")
return self.m * self.lamb / (self.mu - theta)
def sigma(self, theta: float) -> float:
if theta <= 0:
raise ParameterOutOfBounds(f"theta={theta} must be > 0")
if theta >= self.decay:
raise ParameterOutOfBounds(f"theta={theta} must "
f"be < decay={self.decay}")
return (self.m / theta) * log(1 + self.factor_m * theta /
(self.decay - theta))
def rho(self, theta: float) -> float:
# here, theta can simply be replaced by l * theta
l_theta = self.l_power * theta
if self.arr.rho(l_theta) < 0 or self.ser.rho(l_theta) < 0:
raise ParameterOutOfBounds("Check rho's sign")
if self.arr.rho(l_theta) >= self.ser.rho(l_theta):
raise ParameterOutOfBounds(
"The arrivals' rho has to be smaller than the service's rho")
return self.arr.rho(l_theta)
def sigma(self, theta: float) -> float:
if theta <= 0:
raise ParameterOutOfBounds(f"theta = {theta} must be > 0")
if theta >= self.decay:
raise ParameterOutOfBounds("theta {0} must be < decay {1}".format(
theta, self.decay))
theta_over_decay = theta / self.decay
return (self.n / theta) * log(
(self.factor_m**theta_over_decay) / (1 - theta_over_decay))
def rho(self, theta: float) -> float:
"""
rho(theta)
:param theta: mgf parameter
"""
if theta <= 0:
raise ParameterOutOfBounds(f"theta = {theta} must be > 0")
if theta >= self.lamb:
raise ParameterOutOfBounds(
f"theta = {theta} must be < lambda = {self.lamb}")
return (self.n / theta) * log(self.lamb / (self.lamb - theta))
def mgf_fbm(theta: float, delta_time: int, lamb: float, sigma: float,
hurst: float) -> float:
if theta <= 0:
raise ParameterOutOfBounds(f"theta = {theta} must be > 0")
if sigma <= 0:
raise ParameterOutOfBounds(f"sigma = {sigma} must be > 0")
if hurst <= 0 or hurst >= 1:
raise ParameterOutOfBounds(f"Hurst = {hurst} must be in (0, 1)")
return exp(lamb * theta * delta_time +
(0.5 * (sigma * theta)**2) * delta_time**(2 * hurst))
def get_p_n(p_list) -> float:
"""
:param p_list: first p_1, ..., p_n in generalized Hoelder inequality
:return: last p_n
"""
inv_p = [0.0] * len(p_list)
for i in range(len(p_list)):
if p_list[i] <= 1.0:
raise ParameterOutOfBounds(f"p={p_list[i]} must be >1")
inv_p[i] = 1.0 / p_list[i]
if sum(inv_p) >= 1:
raise ParameterOutOfBounds(f"p_i's are too small")
return 1.0 / (1.0 - sum(inv_p))
def rho(self, theta: float) -> float:
"""
rho(theta)
:param theta: mgf parameter
"""
if theta <= 0:
raise ParameterOutOfBounds(f"theta = {theta} must be > 0")
if theta >= self.beta_rate:
raise ParameterOutOfBounds(
f"theta = {theta} must be < beta = {self.beta_rate}")
return (self.m * self.alpha_shape / theta) * log(
self.beta_rate / (self.beta_rate - theta))
def backlog(arr: Arrival,
ser: Service,
theta: float,
prob_b: float,
tau=1.0,
indep=True,
p=1.0) -> float:
"""Implements stationary bound method"""
if indep:
p = 1.0
q = get_q(p=p, indep=indep)
rho_a_p = arr.rho(theta=p * theta)
sigma_a_p = arr.sigma(theta=p * theta)
rho_s_q = ser.rho(theta=q * theta)
sigma_s_q = ser.sigma(theta=q * theta)
if rho_a_p >= rho_s_q:
raise ParameterOutOfBounds(
f"The arrivals' rho {rho_a_p} has to be smaller than"
f"the service's rho {rho_s_q}")
rho_arr_ser = rho_a_p - rho_s_q
sigma_arr_ser = sigma_a_p + sigma_s_q
if arr.is_discrete():
log_part = log(prob_b * (1 - exp(theta * rho_arr_ser)))
return sigma_arr_ser - log_part / theta
else:
log_part = log(prob_b * (1 - exp(theta * tau * rho_arr_ser)))
return tau * rho_a_p + sigma_arr_ser - log_part / theta
def output_power(arr: Arrival,
ser: Service,
theta: float,
delta_time: int,
l_power=1.0) -> float:
"""Implements stationary bound method"""
if l_power < 1.0:
l_power = 1.0
# raise ParameterOutOfBounds("l must be >= 1")
l_theta = l_power * theta
if arr.rho(theta=l_theta) >= ser.rho(theta=l_theta):
raise ParameterOutOfBounds(
f"The arrivals' rho {arr.rho(theta)} has to be smaller than"
f"the service's rho {ser.rho(theta)}")
sigma_l_arr_ser = arr.sigma(theta=l_theta) + ser.sigma(theta=l_theta)
rho_l_arr_ser = arr.rho(theta=l_theta) - ser.rho(theta=l_theta)
if arr.is_discrete():
numerator = exp(theta * arr.rho(theta=l_theta) * delta_time) * exp(
theta * sigma_l_arr_ser)
denominator = (1 - exp(l_theta * rho_l_arr_ser))**(1 / l_power)
else:
numerator = exp(theta * arr.rho(theta=l_theta) *
(delta_time + 1)) * exp(theta * sigma_l_arr_ser)
denominator = (1 - exp(l_theta * rho_l_arr_ser))**(1 / l_power)
try:
return numerator / denominator
except ZeroDivisionError:
return inf
def rho(self, theta: float) -> float:
if theta <= 0:
raise ParameterOutOfBounds(f"theta = {theta} must be > 0")
bb = theta * self.burst - self.mu - self.lamb
return 0.5 * self.n * (bb + sqrt(
(bb**2) + 4 * self.mu * theta * self.burst)) / theta
def rho(self, theta: float) -> float:
"""
:param theta: mgf parameter
:return: rho(theta)
"""
p_theta = self.p * theta
q_theta = self.q * theta
if self.arr.rho(p_theta) < 0 or self.ser.rho(q_theta) < 0:
raise ParameterOutOfBounds("The rhos must be >= 0")
if self.arr.rho(p_theta) >= self.ser.rho(q_theta):
raise ParameterOutOfBounds(
"The arrivals' rho has to be smaller than the service's rho")
return self.arr.rho(p_theta)
def rho(self, theta):
p_theta = self.p * theta
q_theta = self.q * theta
if self.ser.rho(q_theta) < 0 or self.arr.rho(p_theta) < 0:
raise ParameterOutOfBounds("The rhos must be > 0")
return self.ser.rho(q_theta) - self.arr.rho(p_theta)
def rho(self, theta: float) -> float:
arr_1_rho_p_theta = self.arr1.rho(self.p * theta)
arr_2_rho_q_theta = self.arr2.rho(self.q * theta)
if arr_1_rho_p_theta < 0 or arr_2_rho_q_theta < 0:
raise ParameterOutOfBounds("The rhos must be >= 0")
return arr_1_rho_p_theta + arr_2_rho_q_theta
def rho(self, theta: float) -> float:
"""
rho(theta)
:param theta: mgf parameter
"""
if theta <= 0:
raise ParameterOutOfBounds(f"theta = {theta} must be > 0")
return (self.m / theta) * self.lamb * (exp(theta) - 1)
def rho(self, theta: float) -> float:
if theta <= 0:
raise ParameterOutOfBounds(f"theta = {theta} must be > 0")
off_on = self.stay_off + self.stay_on * exp(theta * self.burst)
sqrt_part = sqrt(off_on**2 - 4 * (self.stay_off + self.stay_on - 1) *
exp(theta * self.burst))
return log(0.5 * (off_on + sqrt_part)) / theta
def rho(self, theta: float) -> float:
"""
rho(theta)
:param theta: mgf parameter
"""
if theta <= 0:
raise ParameterOutOfBounds(f"theta = {theta} must be > 0")
return self.m * self.rho_single
def rho(self, theta: float) -> float:
for i in range(len(self.arr_list)):
if self.arr_list[i].rho(self.p_list[i] * theta) < 0:
raise ParameterOutOfBounds("The rhos must be > 0")
res = 0.0
for i in range(len(self.arr_list)):
res += self.arr_list[i].rho(self.p_list[i] * theta)
return res
def rho(self, theta):
if (isinstance(self.ser, RateLatencyServer)
and isinstance(self.cross_arr, DetermTokenBucket)):
residual_rate = self.ser.rate - self.cross_arr.arr_rate
if residual_rate <= 0:
raise ParameterOutOfBounds("The residual rate must be > 0")
return residual_rate
arr_rho_p_theta = self.cross_arr.rho(theta=self.p * theta)
ser_rho_q_theta = self.ser.rho(theta=self.q * theta)
residual_rate = ser_rho_q_theta - arr_rho_p_theta
if residual_rate <= 0:
raise ParameterOutOfBounds("The residual rate must be > 0")
return residual_rate
def rho(self, theta: float) -> float:
if isinstance(self.ser1, RateLatencyServer) and isinstance(
self.ser2, RateLatencyServer):
return min(self.ser1.rate, self.ser2.rate)
ser_1_rho_p = self.ser1.rho(self.p * theta)
ser_2_rho_q = self.ser2.rho(self.q * theta)
if ser_1_rho_p < 0 or ser_2_rho_q < 0:
raise ParameterOutOfBounds("The rhos must be > 0")
if not is_equal(ser_1_rho_p, ser_2_rho_q):
return min(ser_1_rho_p, ser_2_rho_q)
else:
if self.delta < 0 or ser_1_rho_p - self.delta <= 0:
raise ParameterOutOfBounds("Residual rate must be > 0")
return ser_1_rho_p - self.delta
def fifo_delay(token_bucket_constant: TokenBucketConstant,
constant_rate: ConstantRate) -> int:
"""DNC FIFO Delay Bound"""
if token_bucket_constant.rho(1.0) >= constant_rate.rate:
raise ParameterOutOfBounds(
"The arrivals' rho {0} has to be smaller than"
" the service's rho {1}".format(token_bucket_constant.rho(1.0),
constant_rate.rate))
return int(ceil(token_bucket_constant.sigma() / constant_rate.rate))
def sigma(self, theta: float) -> float:
if theta <= 0:
raise ParameterOutOfBounds("theta = {0} must be > 0".format(theta))
try:
return log(1.0 + sqrt(2 * pi * self.n * (self.sigma_single**2)) *
theta * exp(0.5 * self.n * (self.sigma_single**2) *
(theta**2))) / theta
except OverflowError:
return inf
def rho(self, theta: float) -> float:
res = 0.0
if self.indep:
for arrival in self.arr_list:
rho_i = arrival.rho(theta)
if rho_i < 0:
raise ParameterOutOfBounds("The rhos must be >= 0")
res += rho_i
else:
for i, arr in enumerate(self.arr_list):
rho_i = arr.rho(self.p_list[i] * theta)
if rho_i < 0:
raise ParameterOutOfBounds("The rhos must be >= 0")
res += rho_i
return res
def rho(self, theta: float) -> float:
if theta <= 0:
raise ParameterOutOfBounds(f"theta = {theta} must be > 0")
if self.stay_on <= 0.0 or self.stay_on >= 1.0:
raise ValueError(f"p_stay_on = {self.stay_on} must be in (0,1)")
if self.stay_off <= 0.0 or self.stay_off >= 1.0:
raise ValueError(f"p_stay_off = {self.stay_off} must be in (0,1)")
off_on = self.stay_off + self.stay_on * exp(theta * self.peak_rate)
sqrt_part = sqrt(off_on**2 - 4 * (self.stay_off + self.stay_on - 1) *
exp(theta * self.peak_rate))
rho_mmoo_disc = log(0.5 * (off_on + sqrt_part))
if rho_mmoo_disc < 0:
raise ParameterOutOfBounds("rho must be >= 0")
return rho_mmoo_disc / theta
def mgf_regulated_arrive(theta: float,
delta_time: int,
sigma_single: float,
rho_single: float,
n=1) -> float:
if theta <= 0:
raise ParameterOutOfBounds(f"theta = {theta} must be > 0")
rho_delta = rho_single * delta_time
return 1 + rho_delta / (sigma_single + rho_delta) * (
exp(theta * n * (sigma_single + rho_delta)) - 1)
def get_q(p: float) -> float:
"""
:param p: Hoelder p
:return: q
"""
if p <= 1.0:
raise ParameterOutOfBounds(f"p={p} must be >1")
# 1/p + 1/q = 1
# 1/q = (p - 1) / p
return p / (p - 1.0)