Thermal analysis of stochastic DVFS-enabled multicore real-time systems

Abstract

This paper considers a multicore system, equipped with dynamic voltage/frequency scaling (DVFS), which runs real-time jobs with probabilistic characteristics. The DVFS policy directly affects the performance of this system as well as its power consumption through impacting both dynamic and leakage powers. While power consumption determines the processor thermal behavior, the temperature in turn affects the processor power consumption by impacting the leakage power. Additionally, temperature variation of a core is coupled with the thermal behaviors of the other cores, namely thermal effects of the surrounding components. These inter-effects make sophisticated relations between the nature of the stochastic real-time system and its performance, power consumption, and temperature behavior. In this paper, we first present an exact thermal analysis approach for the specified system considering these inter-effects. We then propose a scalable approximate method for thermal analysis of the system based on the exact analysis. The proposed analytical method outperforms the traditional simulation-based methods in terms of time complexity, by approximately three orders of magnitude, while introducing a relative error of less than 0.8 % for the mentioned setups. Also, we show the efficacy of the proposed analytical method in temperature-aware power minimization, resulting in significant reductions in the system-wide power consumption.

Appendices

Appendix 1

For the proof of Lemma 1, we use an approach similar to [6, 36], which has been proposed for the analysis of a simpler system, namely a system with a single core and a constant speed. In the following, we assume that the system is in its statistical equilibrium. We use $f_x(.)$ to denote the PDF of a random variable x, and $\tau$ to denote a variable that gains its value from the set of non-negative real numbers. We define two random variables $A_{m,n}(t)$ and $B_{m,n}(t)$ as follows:

$$\begin{aligned} A_{m,n}(t)\equiv & {} \mathrm {\ the\ time\ a\ job\ being\ run\ in\ core\ } m \mathrm {\ at\ time\ } t \nonumber \\&\mathrm {\ has\ spent,\ given\ there\ are}\ n \mathrm {\ jobs\ in\ the\ core\ }\nonumber \\&\mathrm {\ queue,} \end{aligned}$$

(41)

$$\begin{aligned} B_{m,n}(t)\equiv & {} \ \mathrm {the\ time\ a\ job\ waiting\ at\ the\ head\ of\ the\ queue} \nonumber \\&\mathrm {of\ core\ } m \mathrm {\ at\ time\ } t\ \mathrm {has\ spent,\ given\ there\ are\ } \nonumber \\&n \mathrm {\ jobs\ in\ the\ core\ queue.} \end{aligned}$$

(42)

According to these definitions, the random variables $A_{m,n}(t)$ and $B_{m,n}(t)$ are well defined for $n>0$ and $n>1$, respectively. Further, we define:

$$\begin{aligned} D_{m,k}\equiv & {} \ \mathrm {the\ time\ of\ the\ } k\text {th} \mathrm {\ departure\ of\ a\ job,}\ k\ \mathrm {=1,2,\ldots ,} \nonumber \\&\mathrm {from\ core\ } m \mathrm {,\ given\ the\ departure\ finds\ } n\ \mathrm { jobs\ in\ }\nonumber \\&\mathrm {\ the\ core\ queue.} \end{aligned}$$

(43)

$$\begin{aligned} C_{m,k}\equiv & {} \ \mathrm {the\ time\ of\ the\ } k\text {th} \mathrm {\ departure\ of\ a\ job\ which\ has\ } \nonumber \\&\mathrm { successfully\ been\ served,}\ k\ \mathrm {=1,2,\ldots ,\ from\ core\ } m {,\ } \nonumber \\&\mathrm {given\ the\ departure\ finds\ } n\ \mathrm { jobs\ in\ the\ core\ queue.} \end{aligned}$$

(44)

For any time t, let $t+$ and $t-$ show a time immediately after and before t, respectively. Assuming statistical equilibrium, we define the following related random variables:

$$\begin{aligned} \tilde{A}_{m,n}= & {} \lim _{t \rightarrow \infty } A_{m,n}(t), \end{aligned}$$

(45)

$$\begin{aligned} A_{m,n}= & {} \lim _{k \rightarrow \infty } A_{m,n}(D_{m,k}+), \end{aligned}$$

(46)

$$\begin{aligned} \hat{A}_{m,n}= & {} \lim _{k \rightarrow \infty } A_{m,n}(C_{m,k}-). \end{aligned}$$

(47)

Using a method similar to which presented in [36], we can obtain the PDF of $\tilde{A}_{m,n}$ as:

$$\begin{aligned} f_{\tilde{A}_{m,n}}(\tau )= & {} 0,\quad \mathrm {if}\ n=0, \nonumber \\ f_{\tilde{A}_{m,n}}(\tau )= & {} \frac{1-G_m(\tau )}{\mu _{m,n}\varPhi _{m,n}(\mu _{m,n})} \left[ \int _0^\tau (1-G_m(x))\hbox {d}x \right] ^{n-1} e^{-\mu _{m,n}\tau }, \nonumber \\&\mathrm {if}\ n>0, \end{aligned}$$

(48)

where $\varPhi _{m,n}(.)$ is defined in (17).

Define the random variable $S_{m,n}(t)$ as:

$$\begin{aligned} S_{m,n}(t)\equiv & {} \ \mathrm {the\ offered\ sojourn\ time\ (previously)\ seen\ } \nonumber \\&\mathrm {(upon\ arrival)\ by\ a\ job\ departing\ from\ core\ } m \nonumber \\&\mathrm {in\ the\ long\ run\ which\ finds\ } n \ \mathrm {\ jobs\ in\ the\ } \nonumber \\&\mathrm {respective\ queue.} \end{aligned}$$

(49)

We are interested in finding the PDF of $S_{m,n}$.

Consider a job in the queue of core m completed its service successfully, while n ($n>0$) jobs exist in the queue. Then, $\hat{A}_{m,n}$, defined in (47), shows the steady-state value of $A_{m,n}(t)$ for the queue right before this service completion. Also, $\hat{A}_{m,n}$ can be viewed as $S_{m,n-1}$, conditioned by the event $\{S_{m,n-1} \le \theta \}$. Hence, for $n>0$, we have

$$\begin{aligned} P(\hat{A}_{m,n} \le \tau ) = P(S_{m,n-1} \le \tau \mid S_{m,n-1} \le \theta ). \end{aligned}$$

(50)

Define $SD_{m,n}(t,t+\epsilon )$ as the event that a job in the queue of core m completes its service successfully during $(t,t+\epsilon ]$, given there are n jobs in the queue at time t. Then, we have

$$\begin{aligned} P(\hat{A}_{m,n} \le \tau ) = \lim _{t \rightarrow \infty } \lim _{\epsilon \rightarrow 0} P(A_{m,n}(t) \le \tau \mid SD_{m,n}(t,t+\epsilon )). \end{aligned}$$

(51)

As the execution demand of the jobs is exponentially distributed, the two events $\{B_{m,n}(t) \le \tau \}$ and $SD_{m,n}(t,t+\epsilon )$ are independent. Therefore, we have

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} P(A_{m,n}(t) \le \tau \mid SD_{m,n}(t,t+\epsilon ))=P(A_{m,n}(t) \le \tau ), \end{aligned}$$

(52)

or

$$\begin{aligned} P(\hat{A}_{m,n} \le \tau ) = \lim _{t \rightarrow \infty } P(A_{m,n}(t) \le \tau )=P(\tilde{A}_{m,n} \le \tau ). \end{aligned}$$

(53)

Thus, from (50), we have

$$\begin{aligned} P(\tilde{A}_{m,n} \le \tau ) = P(S_{m,n-1} \le \tau \mid S_{m,n-1} \le \theta ). \end{aligned}$$

(54)

Then, using $f_{\tilde{A}_{m,n}}(.)$ in (48) above, the PDF of $f_{S_{m,n}}(\tau )$ can be obtained as

$$\begin{aligned} f_{S_{m,n}}(\tau ) =&\frac{1}{\varPhi _{m,n}(\mu _{m,n+1})} \left[ \int _0^\tau (1-G_m(x))\hbox {d}x \right] ^{n} e^{-\mu _{m,n+1}\tau }, \end{aligned}$$

(55)

where $\varPhi _{m,n}(\mu )$ is defined in (17).

Consider a job in the queue of core m which is going to depart immediately. The probability that this job meets its deadline can be expressed as $P(S_{m,n-1} \le \theta )$. Using the formulation of $f_{S_{m,n}}(.)$ provided in (55), we immediately get

$$\begin{aligned} P(S_{m,n-1} \le \theta ) = \frac{\mu _{m,n} \varPhi _{m,n}(\mu _{m,n})}{n \varPhi _{m,n-1}(\mu _{m,n})} , \quad \mathrm {if} \ n>0. \end{aligned}$$

(56)

Finally, for calculating $\gamma _{m,n}$, consider a job that immediately departs from the queue of core m, while it contains n jobs, $n>0$. According to the definition of $\gamma _{m,n}$, the probability that this job meets its deadline is equal to

$$\begin{aligned} \frac{\mu _{m,n}}{\mu _{m,n}+\gamma _{m,n}}. \end{aligned}$$

(57)

Further, this probability can also be obtained as $P(S_{m,n-1} \le \theta )$, where $\theta$ is the relative deadline of the departing job. As a result, we have

$$\begin{aligned} P(S_{m,n-1} \le \theta ) = \frac{\mu _{m,n}}{\mu _{m,n}+\gamma _{m,n}}. \end{aligned}$$

(58)

Using (56) and (58), $\gamma _{m,n}$ is obtained as provided in (18), which completes the proof.

Appendix 2

See Table 5.

Table 5 List of the main notations used throughout the paper