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

Morteza Mohaqeqi; Mehdi Kargahi

doi:10.1007/s11227-015-1562-1

The Journal of Supercomputing

December 2015, Volume 71, Issue 12, pp 4594–4622 | Cite as

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

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Authors and affiliations

Morteza Mohaqeqi
Mehdi KargahiEmail author

Article

First Online: 11 November 2015

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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.

Keywords

Dynamic voltage and frequency scaling (DVFS) Markovian model Multicore processor Performance analysis Power management Real-time systems Thermal analysis

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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

Symbol	Definition
M	Number of processing cores
$\lambda _m$	Arrival rate of jobs entering to core m
$K_m$	Capacity of the job queue of core m
$v_{m,n}$	The voltage of core m when n jobs reside in the core job queue
$f_{m,n}$	The frequency of core m when n jobs reside in the core job queue
$P^\mathrm{dyn}_m(v,f)$	The dynamic power of core m when it works with voltage v and frequency f
$P^\mathrm{leak}_m(v,T)$	The leakage power of core m when it works with voltage v and temperature T
$P_m(v,f,T)$	Total power consumption of core m working with voltage v, frequency f, and temperature T
$T_m(t)$	The temperature of core m at time t
$P_m(t)$	The power consumption of core m at time t
$\mathbf T (t)$	$[T_1(t),T_2(t),\ldots ,T_M(t),\ldots ,T_{N}(t)]^T$, instantaneous temperature of the thermal nodes
$\mathbf P (t)$	$[P_1(t),P_2(t),\ldots ,P_M(t),0,\ldots ,0]^T$, power consumption of the thermal node
c	The job execution demand
$\bar{c}_m$	Mean execution demand of jobs belonging to the stream of core m
$\theta $	Job’s relative deadline
$\bar{\theta }_m$	Mean of $\theta $
$G_m(x)$	Cumulative distribution function (CDF) of relative deadline of the jobs
${\varvec{s}}[m]$	The number of jobs of class m in the job queue of core m
${\varvec{s}}$	$({\varvec{s}}[1],{\varvec{s}}[2],\ldots , {\varvec{s}}[M])$, state of the system
${\varvec{s}}_{[m^+]}$	$({\varvec{s}}[1],\ldots ,{\varvec{s}}[m]+1,\ldots , {\varvec{s}}[M])$, adjacent states of ${\varvec{s}}$ with respect to core m
${\varvec{s}}_{[m^-]}$	$({\varvec{s}}[1],\ldots ,{\varvec{s}}[m]-1,\ldots ,{\varvec{s}}[M])$, adjacent states of ${\varvec{s}}$ with respect to core m
S	The set of all system states
$\pi _{\varvec{s}}$	The probability that the system is in state ${\varvec{s}}\in S$ in steady-state
$\alpha _{b,m}$	The probability that an arriving job to core m experiences full job queue
$\alpha _{d,m}$	The probability of a job deadline miss when it is residing in the job queue of core m
$\alpha _m $	The total loss probability for jobs associated with core m
$\overline{{P}}$	The long-run average power consumption of the system
$\mu _{m,n}$	The service rate when n jobs are present in the job queue of core m
$\gamma _{m,n}$	The deadline miss rate when n jobs are present in the job queue of core m
$\omega _{m,n}$	$\mu _{m,n} + \gamma _{m,n}$, death rates of the Birth–Death Process (BDP)
$\pi _{m,n}$	The probability that there are n jobs in the job queue of core m in long term
$\sigma _{{\varvec{s}} \rightarrow {\varvec{s}}'}$	The rate of transition from state ${\varvec{s}}$ to ${\varvec{s}}'$
$W_{\varvec{s}}$	The sojourn time for state ${\varvec{s}}$
$f_{W_{\varvec{s}}}(t)$	The probability density function (PDF) of $W_{\varvec{s}}$
${\varvec{T}}_{{\varvec{s}}}^{en}$	The value of T(t) at the instant of entering state ${\varvec{s}}$
${\varvec{T}}_{\varvec{s}}^{ex}$	The value of $\mathbf T (\textit{t})$ at the instant of exiting state ${\varvec{s}}$

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Authors and Affiliations

Morteza Mohaqeqi
- 1
Mehdi Kargahi
- 1
- 2
Email author

1.DRTS Research Lab, School of Electrical and Computer EngineeringUniversity of TehranTehranIran
2.School of Computer ScienceInstitute for Research in Fundamental Sciences (IPM)TehranIran

Symbol	Definition
M	Number of processing cores
\(\lambda _m\)	Arrival rate of jobs entering to core m
\(K_m\)	Capacity of the job queue of core m
\(v_{m,n}\)	The voltage of core m when n jobs reside in the core job queue
\(f_{m,n}\)	The frequency of core m when n jobs reside in the core job queue
\(P^\mathrm{dyn}_m(v,f)\)	The dynamic power of core m when it works with voltage v and frequency f
\(P^\mathrm{leak}_m(v,T)\)	The leakage power of core m when it works with voltage v and temperature T
\(P_m(v,f,T)\)	Total power consumption of core m working with voltage v, frequency f, and temperature T
\(T_m(t)\)	The temperature of core m at time t
\(P_m(t)\)	The power consumption of core m at time t
\(\mathbf T (t)\)	\([T_1(t),T_2(t),\ldots ,T_M(t),\ldots ,T_{N}(t)]^T\), instantaneous temperature of the thermal nodes
\(\mathbf P (t)\)	\([P_1(t),P_2(t),\ldots ,P_M(t),0,\ldots ,0]^T\), power consumption of the thermal node
c	The job execution demand
\(\bar{c}_m\)	Mean execution demand of jobs belonging to the stream of core m
\(\theta \)	Job’s relative deadline
\(\bar{\theta }_m\)	Mean of \(\theta \)
\(G_m(x)\)	Cumulative distribution function (CDF) of relative deadline of the jobs
\({\varvec{s}}[m]\)	The number of jobs of class m in the job queue of core m
\({\varvec{s}}\)	\(({\varvec{s}}[1],{\varvec{s}}[2],\ldots , {\varvec{s}}[M])\), state of the system
\({\varvec{s}}_{[m^+]}\)	\(({\varvec{s}}[1],\ldots ,{\varvec{s}}[m]+1,\ldots , {\varvec{s}}[M])\), adjacent states of \({\varvec{s}}\) with respect to core m
\({\varvec{s}}_{[m^-]}\)	\(({\varvec{s}}[1],\ldots ,{\varvec{s}}[m]-1,\ldots ,{\varvec{s}}[M])\), adjacent states of \({\varvec{s}}\) with respect to core m
S	The set of all system states
\(\pi _{\varvec{s}}\)	The probability that the system is in state \({\varvec{s}}\in S\) in steady-state
\(\alpha _{b,m}\)	The probability that an arriving job to core m experiences full job queue
\(\alpha _{d,m}\)	The probability of a job deadline miss when it is residing in the job queue of core m
\(\alpha _m \)	The total loss probability for jobs associated with core m
\(\overline{{P}}\)	The long-run average power consumption of the system
\(\mu _{m,n}\)	The service rate when n jobs are present in the job queue of core m
\(\gamma _{m,n}\)	The deadline miss rate when n jobs are present in the job queue of core m
\(\omega _{m,n}\)	\(\mu _{m,n} + \gamma _{m,n}\), death rates of the Birth–Death Process (BDP)
\(\pi _{m,n}\)	The probability that there are n jobs in the job queue of core m in long term
\(\sigma _{{\varvec{s}} \rightarrow {\varvec{s}}'}\)	The rate of transition from state \({\varvec{s}}\) to \({\varvec{s}}'\)
\(W_{\varvec{s}}\)	The sojourn time for state \({\varvec{s}}\)
\(f_{W_{\varvec{s}}}(t)\)	The probability density function (PDF) of \(W_{\varvec{s}}\)
\({\varvec{T}}_{{\varvec{s}}}^{en}\)	The value of T(t) at the instant of entering state \({\varvec{s}}\)
\({\varvec{T}}_{\varvec{s}}^{ex}\)	The value of \(\mathbf T (\textit{t})\) at the instant of exiting state \({\varvec{s}}\)

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

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