felisa j. vvazquez/slides/s_wodes02.pdf · felisa j. v´ azquez-abad and irina baltcheva 4 cdma...

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




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)



power station and


neighborhood

mobile

Gamma(i,0)

Gamma(i,k)

� Neighborhood V(k): set of base stations (BS) where search forconnection is performed



power station and


neighborhood

mobile

Gamma(i,0)

Gamma(i,k)


� 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



power station and


neighborhood

mobile

Gamma(i,0)

Gamma(i,k)


� 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



power station and


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=

; :



power station and


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)


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� �






(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?






(p 2 ^YN � z1��=2

rp(1� p)

N

)� 1� �


1N

(1�p)p

) N �(1� p)

p

! +1 when p! 0


� Problem: complex models, no analytical solutions






(p 2 ^YN � z1��=2

rp(1� p)

N

)� 1� �


1N

(1�p)p

) N �(1� p)

p

! +1 when p! 0



� Approximations: limits, continuous approximation, no indication oferror





CLT ) � confidence level of the form:

P(

p 2 ^YN � z1��=2r

p(1� p)

N

)� 1� �


1N

(1�p)p

) N �(1� p)

p

! +1 when p! 0



� Approximations: limits, continuous approximation, no indication oferror

� Solution: Monte Carlo simulation


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


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.


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).


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�

:


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]


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]



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 (�; �)



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 (�).




constant, then

E�

@@�X(�; �)�

=

@@�E [X(�; �)]

E�

@@�X(�; �)�

=

@@�E [X(�; �)]


� Problem: in our case, discontinuities arise because both thenumber of terms M � and the outage event may jump as the valuesof � and � change.




constant, then

E�

@@�X(�; �)�

=

@@�E [X(�; �)]

E�

@@�X(�; �)�

=

@@�E [X(�; �)]


� 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.


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

:




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 �; �.




L(N ;Z) = exp(

K(� � �) +N�2

2�2��

�2NX

i=1Zi(0)

) ��

��N

:


� Thus, for each value of ! 2 in this representation, 1fBg is constant




L(N ;Z) = exp(

K(� � �) +N�2

2�2��

�2NX

i=1Zi(0)

) ��

��N

:


� 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 �; �).



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.




^G� =�

K �M�

��

L2(M�;Z�)1fB(M�;Z�)g (3)

^G� =�

M�� B�M�

�2

�L2(M�;Z�)1fB(M�;Z�)g (4)


Proof :

� Use original measure for unbiasedness,

r�F (�; �) = E[L0

�(N ;Z)1fB(N;Z)g]




^G� =�

K �M�

��

L2(M�;Z�)1fB(M�;Z�)g (5)

^G� =�

M�� B�M�

�2

�L2(M�;Z�)1fB(M�;Z�)g (6)


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�]


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





� 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


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).


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)


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.


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.

felisa j. vvazquez/slides/s_wodes02.pdf · felisa j. v´ azquez-abad and irina baltcheva 4 cdma...

Documents