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Intelligent Simulation for the EstimationIntelligent Simulation for the Estimation
of the Uplink Outage Probabilities in CDMA Networksof the Uplink Outage Probabilities in CDMA Networks
Felisa J. Vázquez-Abad and Irina Baltcheva
Département d’informatique et recherche opérationnelleUniversité de Montréal, CANADA
andThe ARC Special Research Centre for Ultra-Broadband Information Networks
(CUBIN)Department of Electrical and Electronic Engineering
The University of Melbourne, AUSTRALIA
email: fvazquez,baltchei [email protected], [email protected]
6th WODES, Zaragoza, October 2-4, 2002.
Felisa J. Vázquez-Abad and Irina Baltcheva 1
CDMA Mobile Networks
reference basestation {0}
mobile
power station
� Users (mobiles): each cell k has a Poisson(�k) number of usersuniformly distributed in the space
Felisa J. Vázquez-Abad and Irina Baltcheva 2
CDMA Mobile Networks
reference basestation {0}
mobile
power station
Gamma(i,k)
� Users (mobiles): each cell k has a Poisson(�k) number of usersuniformly distributed in the space
� Attenuation factor:
�ik(l) =
1dik(l)410Zik(l); fZik(l)g � N (0; �2)
Felisa J. Vázquez-Abad and Irina Baltcheva 3
CDMA Mobile Networks
power station and
reference basestation {0}
neighborhood
mobile
Gamma(i,0)
Gamma(i,k)
� Neighborhood V(k): set of base stations (BS) where search forconnection is performed
Felisa J. Vázquez-Abad and Irina Baltcheva 4
CDMA Mobile Networks
power station and
reference basestation {0}
neighborhood
mobile
Gamma(i,0)
Gamma(i,k)
� Neighborhood V(k): set of base stations (BS) where search forconnection is performed
� Perfect Power Control: choose BS with smallest attenuation andadjust mobile’s transmission power so that it is received at unitpower: Ci;k = maxl2V(k)f�ik(l)g; �(i) = argmaxf�ik(l) : l 2 V(k)g
Felisa J. Vázquez-Abad and Irina Baltcheva 5
CDMA Mobile Networks
power station and
reference basestation {0}
neighborhood
mobile
Gamma(i,0)
Gamma(i,k)
� Neighborhood V(k): set of base stations (BS) where search forconnection is performed
� Perfect Power Control: choose BS with smallest attenuation andadjust mobile’s transmission power so that it is received at unitpower: Ci;k = maxl2V(k)f�ik(l)g; �(i) = argmaxf�ik(l) : l 2 V(k)g
� Interference: caused by mobile i in cell k to the base station 0:
�i;k(0)=Ci;k
Felisa J. Vázquez-Abad and Irina Baltcheva 6
CDMA Mobile Networks
power station and
reference basestation {0}
neighborhood
mobile
Gamma(i,0)
Gamma(i,k)
� S/N: signal to noise ratio is the sum of all particular interferences
� Outage: the event when the signal to noise ratio at the referencebase station 0 is smaller than a threshold �:
B =8<
:KX
k=1
NkXi=1
�i;k(0)
Ci;k
> ��19=
; :
Felisa J. Vázquez-Abad and Irina Baltcheva 7
CDMA Mobile Networks
power station and
reference basestation {0}
neighborhood
mobile
Gamma(i,0)
Gamma(i,k)
Goal: estimate outage probablility p = E(1fBg)
B =8<
:KX
k=1
NkXi=1
�i;k(0)
Ci;k
> ��19=
; ; Nk � Poisson(�k)
�ik(l) =
1dik(l)410Zik(l); Zik(l) iid � N (0; �2)
Felisa J. Vázquez-Abad and Irina Baltcheva 8
Rare Event Estimation
Let A be an event of interest.
P(A) � 10�3 ) 1000 samples/observation
CLT ) � confidence level of the form:P
(p 2 ^YN � z1��=2
rp(1� p)
N
)� 1� �
Felisa J. Vázquez-Abad and Irina Baltcheva 9
Rare Event Estimation
Let A be an event of interest.
P(A) � 10�3 ) 1000 samples/observation
CLT ) � confidence level of the form:P
(p 2 ^YN � z1��=2
rp(1� p)
N
)� 1� �
If required precision is �p) � �q
1N
(1�p)p
) N �(1� p)
p
! +1 when p! 0
How to estimate p using simulation, without needing a sample sizedepending on p?
Felisa J. Vázquez-Abad and Irina Baltcheva 10
Rare Event Estimation
Let A be an event of interest.
P(A) � 10�3 ) 1000 samples/observation
CLT ) � confidence level of the form:P
(p 2 ^YN � z1��=2
rp(1� p)
N
)� 1� �
If required precision is �p) � �q
1N
(1�p)p
) N �(1� p)
p
! +1 when p! 0
How to estimate p using simulation, without needing a sample sizedepending on p?
� Problem: complex models, no analytical solutions
Felisa J. Vázquez-Abad and Irina Baltcheva 11
Rare Event Estimation
Let A be an event of interest.
P(A) � 10�3 ) 1000 samples/observation
CLT ) � confidence level of the form:P
(p 2 ^YN � z1��=2
rp(1� p)
N
)� 1� �
If required precision is �p) � �q
1N
(1�p)p
) N �(1� p)
p
! +1 when p! 0
How to estimate p using simulation, without needing a sample sizedepending on p?
� Problem: complex models, no analytical solutions
� Approximations: limits, continuous approximation, no indication oferror
Felisa J. Vázquez-Abad and Irina Baltcheva 12
Rare Event Estimation
Let A be an event of interest.
P(A) � 10�3 ) 1000 samples/observation
CLT ) � confidence level of the form:
P(
p 2 ^YN � z1��=2r
p(1� p)
N
)� 1� �
If required precision is �p) � �q
1N
(1�p)p
) N �(1� p)
p
! +1 when p! 0
How to estimate p using simulation, without needing a sample sizedepending on p?
� Problem: complex models, no analytical solutions
� Approximations: limits, continuous approximation, no indication oferror
� Solution: Monte Carlo simulation
Felisa J. Vázquez-Abad and Irina Baltcheva 13
Importance sampling
S
A
A
S
change of measure
� Idea: sample mostly from the important event, by “twisting” theoriginal measure, then compensate by appropriate weightZ
�(x)f(x)dx =Z
�(x)f(x)
f�(x)f�(x)dx
E[�(X)] = E�[�(X)L�(X)]
(S;B(S);P) �! (S;B(S);P�); P
Felisa J. Vázquez-Abad and Irina Baltcheva 14
Change of measure
In our model, we have:
1. Attenuation factor: �ik(l) = 1dik(l)410Zik(l); fZik(l)g � N (0; �2)
(lognormal)
2. Poisson number of callers: Nk � Poisson(�k).
� Problem: direct application of theory requires tilting a heavy tailedrandom variable, for which the moment generating function doesnot exist.
Felisa J. Vázquez-Abad and Irina Baltcheva 15
Change of measure
In our model, we have:
1. Attenuation factor: �ik(l) = 1dik(l)410Zik(l); fZik(l)g � N (0; �2)
(lognormal)
2. Poisson number of callers: Nk � Poisson(�k).
� Problem: direct application of theory requires tilting a heavy tailedrandom variable, for which the moment generating function doesnot exist.
� Solution: different mean for the Poisson number Nk of callers percell and an exponential tilt of the shadowing factors Zi;k(0):
– �k ! �k, �k > �k, 8k;
– Zi;k(0)! Z�i;k(0), Z�
i;k(0) iid N (�k; �2).
Felisa J. Vázquez-Abad and Irina Baltcheva 16
Change of measure
Lemma Let N = (N1; : : : ; NK), where fNkg are independent Poissonrandom variables with means �k, and let Zi;k(0); : : : ZNk;k(0) be iid
N (0; �2), conditionally independent of N . For any value of theparameter (�k; �k; k = 1; : : : ; K) 2 R 2K and for any real valued function
�(N ;Z):E[�(N ;Z)] = E[L(N�;Z�)�(N�;Z�)];
where N �k �Poisson(�k), Z�
i;k � N (�k; �2) are independent normal
random variables, and:
L(N ;Z) = exp(
KXk=1
�(�k � �k) +Nk�2
k
2�2��k
�2kBNk;k�) KY
k=1�
�k�k
�Nk
with BNk;k =PNk
i=1Zi;k.
Homogeneous case (�k = �; �k = �; M =PK
k=1Nk):
L(M �;Z�) = exp8<
:K(� � �) +M� �2
2�2��
�2M�X
i=1Z�i (0)
9=;
��
��M�
:
Felisa J. Vázquez-Abad and Irina Baltcheva 17
Change of measure
Let ^p(�; �) = L(N;Z)1fBg be an unbiased estimator of outage at basestation f0g.
� Efficiency is measured in terms of computational effort required toachieve a certain relative precision
� Improving efficiency � find the values of �k; �k that minimise thevariance Var[^p(�; �)].
Var[^p(�; �)] = E(^p2(�; �))� p2
p is independent of (�; �) )
minVar[^p(�; �)] � min�;�
E[^p(�; �)2]
Felisa J. Vázquez-Abad and Irina Baltcheva 18
Self-Optimised Importance Sampling
Let F (�; �) � E[^p2(�; �)] = E[L21fBg)].� Optimization problem: min
�;�
F (�; �).
� Steepest descent: convexity ) functional estimation
8 8.028.04 8.06
8.08 8.1
0
0.005
0.01
0.039
0.0395
0.04
0.0405
theta
mu
Var
[Xi]
Felisa J. Vázquez-Abad and Irina Baltcheva 19
Self-Optimised Importance Sampling
Let G(�; �) = r�;�E[L2(M�;Z�)1fB(M�;Z�g]. Suppose that ( ^G�(n); ^G�(n)) isan unbiased estimator of the gradient. Consider the recursion:
�(n+ 1) = �(n)� �n ^G�(n)
�(n + 1) = �(n)� �n ^G�(n)
Then �(n)! �� and �(n)! �� a.s.
� Objective: build an estimator ( ^G�(n); ^G�(n)) that satisfiesE�( ^G�(n)) = r�F (�; �)
E�( ^G�(n)) = r�F (�; �)
Felisa J. Vázquez-Abad and Irina Baltcheva 20
Self-Optimised Importance Sampling
Theorem: X 2 (S;B(S);P�). Let X(�; �; !) be a random variable under(;F ;P). If X(�; !) is a.s. Lipschitz continuous with integrable Lipschitz
constant, then
E�
@@�X(�; �)�
=
@@�E [X(�; �)]
E�
@@�X(�; �)�
=
@@�E [X(�; �)]
and the stochastic gradient is unbiased for rF (�).
Felisa J. Vázquez-Abad and Irina Baltcheva 21
Self-Optimised Importance Sampling
Theorem: X 2 (S;B(S);P�). Let X(�; �; !) be a random variable under(;F ;P). If X(�; !) is a.s. Lipschitz continuous with integrable Lipschitz
constant, then
E�
@@�X(�; �)�
=
@@�E [X(�; �)]
E�
@@�X(�; �)�
=
@@�E [X(�; �)]
and the stochastic gradient is unbiased for rF (�).
� Problem: in our case, discontinuities arise because both thenumber of terms M � and the outage event may jump as the valuesof � and � change.
Felisa J. Vázquez-Abad and Irina Baltcheva 22
Self-Optimised Importance Sampling
Theorem: X 2 (S;B(S);P�). Let X(�; �; !) be a random variable under(;F ;P). If X(�; !) is a.s. Lipschitz continuous with integrable Lipschitz
constant, then
E�
@@�X(�; �)�
=
@@�E [X(�; �)]
E�
@@�X(�; �)�
=
@@�E [X(�; �)]
and the stochastic gradient is unbiased for rF (�).
� Problem: in our case, discontinuities arise because both thenumber of terms M � and the outage event may jump as the valuesof � and � change.
� Solution: change back the measure to calculate derivatives.
Felisa J. Vázquez-Abad and Irina Baltcheva 23
Self-Optimised Importance Sampling� Changing back to original measure:
F (�; �) = E[L2(M�;Z�)1fB(M�;Z�g] = E[L(N ;Z)1fB(N;Z)g];
L(N ;Z) = exp(
K(� � �) +N�2
2�2��
�2NX
i=1Zi(0)
) ��
��N
:
Felisa J. Vázquez-Abad and Irina Baltcheva 24
Self-Optimised Importance Sampling� Changing back to original measure:
F (�; �) = E[L2(M�;Z�)1fB(M�;Z�g] = E[L(N ;Z)1fB(N;Z)g];
L(N ;Z) = exp(
K(� � �) +N�2
2�2��
�2NX
i=1Zi(0)
) ��
��N
:
� Key observation: under P, the distribution of (N ;Z1; : : : ; ZN) isindependent of the parameter �; �.
Felisa J. Vázquez-Abad and Irina Baltcheva 25
Self-Optimised Importance Sampling� Changing back to original measure:
F (�; �) = E[L2(M�;Z�)1fB(M�;Z�g] = E[L(N ;Z)1fB(N;Z)g];
L(N ;Z) = exp(
K(� � �) +N�2
2�2��
�2NX
i=1Zi(0)
) ��
��N
:
� Key observation: under P, the distribution of (N ;Z1; : : : ; ZN) isindependent of the parameter �; �.
� Thus, for each value of ! 2 in this representation, 1fBg is constant
Felisa J. Vázquez-Abad and Irina Baltcheva 26
Self-Optimised Importance Sampling� Changing back to original measure:
F (�; �) = E[L2(M�;Z�)1fB(M�;Z�g] = E[L(N ;Z)1fB(N;Z)g];
L(N ;Z) = exp(
K(� � �) +N�2
2�2��
�2NX
i=1Zi(0)
) ��
��N
:
� Key observation: under P, the distribution of (N ;Z1; : : : ; ZN) isindependent of the parameter �; �.
� Thus, for each value of ! 2 in this representation, 1fBg is constant
� For every N ;Z the function L(N ;Z) is differentiable in � and �, andthis derivative has uniformly bounded expectation over compactsets (in �; �).
Felisa J. Vázquez-Abad and Irina Baltcheva 27
Self-Optimised Importance Sampling
Lemma Under the change of measure, the estimators:
^G� =�
K �M�
��
L2(M�;Z�)1fB(M�;Z�)g (1)
^G� =�
M��� B�M�
�2
�L2(M�;Z�)1fB(M�;Z�)g (2)
are unbiased for the derivatives of F (�; �) w.r.t. � and �, respectively.
Felisa J. Vázquez-Abad and Irina Baltcheva 28
Self-Optimised Importance Sampling
Lemma Under the change of measure, the estimators:
^G� =�
K �M�
��
L2(M�;Z�)1fB(M�;Z�)g (3)
^G� =�
M��� B�M�
�2
�L2(M�;Z�)1fB(M�;Z�)g (4)
are unbiased for the derivatives of F (�; �) w.r.t. � and �, respectively.
Proof :
� Use original measure for unbiasedness,
r�F (�; �) = E[L0
�(N ;Z)1fB(N;Z)g]
Felisa J. Vázquez-Abad and Irina Baltcheva 29
Self-Optimised Importance Sampling
Lemma Under the change of measure, the estimators:
^G� =�
K �M�
��
L2(M�;Z�)1fB(M�;Z�)g (5)
^G� =�
M��� B�M�
�2
�L2(M�;Z�)1fB(M�;Z�)g (6)
are unbiased for the derivatives of F (�; �) w.r.t. � and �, respectively.
Proof :
� Use original measure for unbiasedness,
r�F (�; �) = E[L0
�(N ;Z)1fB(N;Z)g]
� and then change the measure again:
r�F (�; �) = E[L0
�(N ;Z)1fB(N;Z)g]
= E[L(M�;Z�)L0�(M�;Z�)1fB(M�;Z�)g]
= E[ ^G�]
Felisa J. Vázquez-Abad and Irina Baltcheva 30
Stratified Importance Sampling� Stratifying: using different control variables
– (�1; �1): cells inside the neighborhood V(0)
– (�2; �2): cells outside the neighborhood V(0)
Felisa J. Vázquez-Abad and Irina Baltcheva 31
Stratified Importance Sampling� Stratifying: using different control variables
– (�1; �1): cells inside the neighborhood V(0)
– (�2; �2): cells outside the neighborhood V(0)
� Motivations:
– better performance with higher dimension
– computational overhead independent of dimension
Felisa J. Vázquez-Abad and Irina Baltcheva 32
Stratified Importance Sampling� Stratifying: using different control variables
– (�1; �1): cells inside the neighborhood V(0)
– (�2; �2): cells outside the neighborhood V(0)
� Motivations:
– better performance with higher dimension
– computational overhead independent of dimension
Likelihood ratio in terms of the neighborhood:
L(M�; Z�) = L1(M�
1 ; Z�
1)� L2(M�
2 ; Z�
2)
L1(M�
1 ; Z�
1) = exp8<
:7(�1 � �) + M�
1�2
1
2�2��1
�2M�1X
i=1Zi(0)
9=;
��
�1�M�
1
L2(M�
2 ; Z�
2) = exp8<
:(K � 7)(�2 � �) + M�
2�2
2
2�2��2
�2M�2X
i=1Zi(0)
9=;
��
�2�M�
2
Felisa J. Vázquez-Abad and Irina Baltcheva 33
Stratified Importance Sampling
Derivate each component with respect to each variable passingthrough the original measure :
G�1 = E��
7�M�1
�1�
L21(M�;Z�)L22(M�;Z�) 1fB(M�;Z�)g�
G�2 = E��
(K � 7)�M�2
�2�
L21(M�;Z�)L22(M�;Z�) 1fB(M�;Z�)g�
G�1 = E"
M�1�1 �B�
M�1
�2
!L21(M�;Z�)L22(M�;Z�) 1fB(M�;Z�)g#
G�2 = E"
M�2�2 �B�
M�2
�2
!L21(M�;Z�)L22(M�;Z�) 1fB(M�;Z�)g#
with B�M�1
=PM�
1
i=1Z�
i (0) and B�
M�2
=PM�
2
i=1Z�
i (0).
Felisa J. Vázquez-Abad and Irina Baltcheva 34
Results
0 10 20 30 40 50 60 70 80 90 1007.9
8
8.1
8.2
8.3
8.4
time
thet
a
0 10 20 30 40 50 60 70 80 90 100−0.01
0
0.01
0.02
0.03
time
mu
neighborhood
non − stratified
outside neighborhood
neighborhood
non − stratified
outside neighborhood
(Dotted line: optimal value as estimated with functional estimation)
Felisa J. Vázquez-Abad and Irina Baltcheva 35
Remarks� Parameters: �0 = 8:0, �0 = 0:0, � = 0:8, � = 20:0, �n =a�0
n , a constant� Optimal variance: 15% better than without stratification
� CPU time (same parameters):
– functional estimation: 25.75 minutes– intelligent simulation: 5.10 minutes
� Stratifying yields virtially no change of measure for callers outsidethe neighborhood of the base station, while those inside have ahigher intensity and tilted shadowing.
Felisa J. Vázquez-Abad and Irina Baltcheva 36
Remarks� Parameters: �0 = 8:0, �0 = 0:0, � = 0:8, � = 20:0, �n =a�0
n , a constant� Optimal variance: 15% better than without stratification
� CPU time (same parameters):
– functional estimation: 25.75 minutes– intelligent simulation: 5.10 minutes
� Stratifying yields virtially no change of measure for callers outsidethe neighborhood of the base station, while those inside have ahigher intensity and tilted shadowing.
Discussion
� Problem: choice of initial values of parameters and step sizes
� Idea: simulation tree with adaptive grid to start the method
) seek a reasonable step size
) determine an initial region for the parameters.