ESTIMATION OF JOINT PROBABILITY DENSITY FUNCTION OF DELAY AND LEAKAGE POWER WITH VARIABLE SKEWNESS 1
Mohammad Ansari1, Mohsen Imani*1, Hossein Aghababa2, Behjat Forouzandeh1 School of Electrical and Computer engineering, University of Tehran, Tehran, Iran 2
[email protected]
School of Engineering Sciences, University of Tehran, Tehran, Iran
[email protected] method for Statistical estimation of leakage power such as Wilkinson approach [6]. They introduced a statistical method that estimates lognormal distribution moments. Also some methods that model the leakage power with process variation have been introduced in [7, 8]. In 2010 some algorithm based methods are proposed which doesn’t present an analytical function for delay and power[1, 9]. In [10] a linear model to estimate the delay is proposed considering the process variation which is not accurate for some types of variation. The model for transistors delay with process variation is introduced [11, 12]. Determining a closed form expression for Joint Probability Distribution Function of delay and leakage could model the process variation and avoid intensive simulations for exploring the power-delay variability characteristics. Hence, a lot of new researches focus on modeling the JPDF of power and delay. In [13] in this paper, the JPDF of delay and logarithm of power are expressed by bivariate normal distribution. Also has modeled the delay and power distributions using a skew normal bivariate JPDF. Indeed we introduce a new JPDF for estimating delay-power distribution with linear and logarithmic skewness. The rest of the paper is organized as follows. Part 2 brings the related mathematical relationship for finding the JPDF of delay and power. In part 3, our proposed model is introduced and verified with the realistic power-delay distribution exported from SPICE simulations in Nangate 45nm standard cell library. Finally, the last part concludes the paper
ABSTRACT This paper introduces a new joint probability density function (JPDF) for estimating delay-power distribution. Linear and logarithmic skewness factors have been used for estimating the accurate JPDF. Both proposed models are compared to bivariate normal model for NAND2, NAND3, NOR2, NOR3 circuits and ISCAS85-C432We verified the accuracy of our proposed model using Nangate 45nm standard cell library. The results indicate that making use of logarithmic skewness, results in a better modeling compared to linear and bivariate models. Employing linear and logarithmic skewnesses, results in 23.3X and 38.5X improvement in R-Squares in respect with constant and bivariate model. Also, using logarithmic skewness reduces the Root Mean Squares Error (RMSE) and Sum of Squared Errors (SSE) by 14.6% and 26.2% respectively. Keywords— Manufacturing process variation, PowerDelay joint PDF, Nano-CMOS technology. 1. INTRODUCTION Nowadays the variation in nano-scale CMOS technologies has introduced a new critical challenge in circuit design and device fabrication process. A lot of parameters can cause the variation; systematic and non-systematic variations. The sources that cause systematic variation come from interaction between fabrication process and design characteristics which consist the main source of variation[2]. The Second type of variation refers to ability for fabricating the devices with higher accuracy in new technologies due to scaling of technology such as error in mask overlay and acceptable tolerance[3]. The third and most important reason is the random behavior of silicon in new technologies which occurs due to decreasing the number of electrons in device channel[4]. Process variation can severely affect the performance and power of circuits. Leakage power is 18% and 54% of total power in 130nm and 65nm technologies respectively. Therefore, the variation cannot be neglected in new devices[5]. Researcher have been introduced new classic
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2. JOINT PDF MODELING In order to model the delay and power distributions simultaneously, the joint PDF function should be defined. The exponential behavior of the leakage power due to process variation and relation between delay and power makes their distributions skewing from the normal distributions. This paper introduces a new bivariate skewnormal model for estimating the JPDF of delay and power in advantages against the bivariate normal modeling. Having the range of delay and power, the circuit efficiency is defined as:
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ICECCO 2013
P ( D d D0 , P d P0 )
Y
equations directly. The linear relation between skewness and independent variables is defined by: Linear Model 1:
(1) Where, Y shows the probability that the delay and power are below than D0 and P0 respectively. For finding this probability, the JPDF is required. In [10] the delay and power are distributed as normal equation and the joint probability density as: (x Px )
V x2
exp(
2 U ( x P x )( y P y )
V xV y
2
2(1 U )
f xy ( x, y )
γd
(2) Where x, y and ρ are delay, logarithm of the leakage and the correlation coefficient respectively. We assume the JPDF of delay and logarithm of leakage power as Bivariate SkewNormal: 1
I (
* V xV y
x Px
V xl
,
y Py
Vr
[I (
V yl
).I ( f , P x , x ).I ( P y , f, y )
I(
V yr
x Px y P y , ).I ( P x , f, x ).I ( P y , f,, y )]
V xr
V yr
Where exp(
x 2 2 U xy y 2
I ( x, y ) *
S 2
)
1 U2
(J x
I ( a , b, x )
V xl
2(1 U 2 )
1
Jx
)(J y
1 ® ¯0
Vx , V yl Jx
1
Jy
a γd u D b γd
,
γp
a γp u P b γp
is considered as:
γd
a γd u Log P b γ d
γp
a γ p u Log ( D ) b γ p
γd
a γ d u Log D b γ d
γp
a γ p u Log ( P ) b γ p
Note that in above equations, the values of P and D are the normalized power and delay respectively. The comparison of the linear model1 and linear model2 against the simulation results for NAND2 gate is demonstrated in Fig 1. As the top view of JPDF shows, the mode1 more accurately is fitted to the simulation probability distribution. This corresponds to dependency of the delay skewness to the power instead of delay and wise versa. The Fig 2 shows the JPDF of logarithmic model 1 and 2. This figure verifies the previous mentioned reason which cross dependency of delay and power with their skewnesses results in better modeling.
(3)
x Px y P y I ( , ).I ( f , P x , x ).I ( f , P y , y )
V xr
a γp u D b γp
Logarithmic model 2:
x Px y P y , ).I ( f, P x , x ).I ( f, P y , y )
V xl
γp
Also the logarithmic approximation of Logarithmic model 1:
)
2SV xV y 1 U 2
g xy ( x, y )
,
Linear model 2:
(y Py)
V y2
a γd u P b γd
γd
(4) ) (J x
1
Jx
)(J y
1
Jy
1
) tan (
U 1 U2
)
(5)
ad xdb Othewise
Vy Jy
, V xr
(6) V x , V yr
V y .J x
(7) γx and γy are skewness factor and Logarithm of leakage power respectively. Also γ>1 and γ
