Overall algorithm to minimize peak temperature . 74

With formulation (3.41), Algo. 5 provides the pseudo-code of our ap-proach. It’s worth noting thatbshould vary in a feasible region[b_min,bmax], where b_min = _∑iⁿ₌₁(t^{swo f f}_i +t^swon_i +c_i) since t^{o f f}_i is lower bounded by t^{swo f f}_i , andbmax =max

b: max(0,∆−^b) ≥^αu(∆−^D), ∀∆≥^{0 , which} is obtained from [1].

3.8 Solving the sub-problem

This section presents two algorithms to solve the sub-problem (3.41).

Each algorithm can be called individually by Algo. 5 to minimize the peak temperature and find the optimal PTM schemes.

Lemma 3.19 For problem (3.41), R_i can be obtained safely by the following equation:

R_i =c_iρ (3.42)

3.8. Solving the sub-problem Algorithm 5Peak Temperature Optimization

Input: TM, α, D, n, c,t^swon,t^swoff, search step η Output: T_min^? , t^on, t^off

1: calculate[bmin,bmax]

2: T_min^? ←^{∞, R}←^{0, toff}←⁰

3: foreachb =b_min tobmax with step η do

4: get correspondingρ from (3.38)

5: solve sub-problem (3.41) and get

6: T^?, R, t^off

7: ifT^? < T_min^? then

8: T_min^? ←^{T?, R}← ^R,

9: t^off←^toff 10: end if

11: end for

12: calculatet^onwith t_i^on = (R_it^{o f f}_i +t^swon_i )/(1−^Ri)

Proof From the definition ofRi, one can derivet^on_i = (Rit^{o f f}_i +t^swon_i )/(1− R_i). Then, we have ^dt_dR^oni

i >0, which means a larger R_iresults in a higher partition of t^on_i , that is, a higher peak temperature. ThereforeR_i should equal its lower bound ciρ such that the peak temperature won’t be

ele-vated unnecessarily.

From Lem. 3.19, T^?(R,t^off) in (3.41) can be transformed to a function of t^off , T^?(c·^ρ,toff), with the constraint:

∑

n i=1

(t^{o f f}_i +t^swon_i +c_i)≤^{b (3.43)}

Next we present two fast algorithms to solve the sub-problem (3.41) corresponding to the two peak temperature algorithms proposed in Sec-tion 3.6.2.

3.8.1 Algorithm FBGD to solve the FBPT based sub-problem

In this case,T^?(R,t^off)is calculated with the FBPT algorithm. Intuitively, one can brutally search the whole exploring space to find the optimal solution. However, as the stage numbernincreases, the exploring space expends (approximated) exponentially and this approach will finally be

infeasible. For example, ifn =8 and everyt_{o f f} has 50 candidates in the exploring space, there will be 50⁸=3.90625×¹⁰¹³ combinations in total.

The brutally searching algorithm needs more than 120 years to finish if the computer can check 10000 combinations per second. Therefore, a more clever algorithm is needed to solve sub-problem (3.41). We first introduce the following conjecture obtained from a set of systematic experiments.

Conjecture 3.20 (Unimodal Peak Temperature) Given ρ and R = c·^ρ, the peak temperature calculated by FBPTis a unimodal function of t^{o f f}_i in the feasible region of t^{o f f}_i .

An example is shown in Fig. 3.8, the obtained peak temperature first decrease and then increase ast^{o f f}_i varies from its lower bound to upper bound while othert^{o f f}s stay constant.

From Conjecture 3.20, the local minima is also the global minima of the peak temperature. Therefore, inspired by the gradient descent method, we propose an Fast-Bounding-based adaptive-step-Gradient-Descent(FBGD) algorithm to find the optimalt^offin the exploring space.

Algo. 6 outlines the pseudo-code of the algorithm. It takes the thermal model, the stage number, mode-switching overhead vectors, b, ρ, c, a initial step size ξ, and a minimal step size ξ_min as input. The iteration starts at the initial point t^swoff (line 3). In every iteration, it first checks whether feeding one current step size ξ to t^off will violate the limit of b (line 5). If so, the current step size should shrink to a smaller value which satisfies constraint 3.43. In this algorithm, the golden section scale is utilized to obtain the new step size ξ (line 6). Then all the possible directions are checked (line 11) to find the direction which leads to the current steepest descent. If the the minimal gradient results in a lower peak temperature, the corresponding direction will be selected to update t^offby addingξtot^off(line 16). Otherwise, we adapt the step to a smaller size which is also determined by the golden section rule (line 18). the algorithm executes the iteration until the minimal step size is reached (line 8).

3.8.2 Algorithm ANSA to solve the ANPT based sub-problem

Obtained from algorithm ANPT, the peak temperature varies irregularly in the space oft^off. As shown in Fig. 3.8, there exist several cliffs as t^{o f f}_i

3.8. Solving the sub-problem Algorithm 6FBGD: solve the FBPT based sub-problem with givenband

ρ

Input: TM, n, b, ρ, c, t^swon, t^swoff, initial step size ξ, minimal step size ξ_min

Output: T^?, R, t^off

1: e₁ ←(1, 0,· · · ^{, 0}), e₂ ←(0, 1,· · · ^{, 0}), e_n ←(0, 0,· · · ^{, 1}),g←⁰1,n 2: R ←^ρ·^c,toff ←^tswoff

3: T_last^? ←^T?(R,t^off) calculated by Algo. 3

4: whiletruedo

5: if∑ⁿ_i=1(t^{o f f}_i +t^swon_i +c_i) +ξ >b then

6: ξ ←^0.618∗(b−∑ⁿi=1(t^{o f f}_i +t^swon_i +ci))

7: end if

8: ifξ ≤^ξmin then

9: break

10: end if

11: foreach 1≤ⁱ ≤^{n do}

12: gi ←(T^?(R, t^off+ξe_i)−^T_last^? )

13: end for

14: if min{^g}<0then

15: findi whereg==min{^g}^,toff ←^toff+ξe_i,

16: T_last^? ← ^T_last^? +min{^g}

17: else

18: ξ ←^0.618∗^ξ

19: end if

20: end while

21: T^? ← ^T_last^{? , R} ← ^R,toff ←^toff

increases, indicating the gradient descent method is not suitable any more. Therefore, we determine to deploy an heuristic algorithm to find the sub-optimal solution of sub-problem (3.41) rather than the global minimum, which can only be safely discovered by the brutally search-ing method. The well-known Simulated Annealsearch-ing (SA) algorithm is adopted to solve (3.41) in our approach and is denoted as ANSA. Note that the typical SA algorithm searches in the space oft^off without linear constraints. Therefore, the section of updating a new candidate in the SA algorithm needs to be revised to guarantee the new candidate meets the linear constraints in (3.41). Algo. 7 presents the pseudo code of the revised section.

Algo. 7 uses the current t^off, b, n, c, t^swon, the current simulated

an-0 50 100 150 200 250 300 377

378 379 380 381 382

t^off_i /ms

PeakTemperature/K FBPT

ANPT

Figure 3.8: Peak temperature obtained by FBPT and ANPT change as t^{o f f}_i increases, while othert^{o f f}s keep constant.

Algorithm 7Update t^offwith linear constraint (3.40) for SA

Input: t^off, b, n, c, t^swon, Current SA Temperature T^SA, Initial SA Tem-perature T_init^SA,

1: lower boundbd^l, upper boundbd^u Output: t^off

2: if ∑ⁿ_i=1(t_i^{o f f} +t^swon_i +c_i) ≥^{b then}

3: t^off ←^bdl

4: return

5: end if

6: k ←an uniformly distributed random integer between 1 andn

7: temp upper boundbd^† ←^min[bd^u_k,b−∑ⁿi=1(t^{o f f}_i +t^swon_i +c_i) +t^{o f f}_k ]

8: step ←an uniformly distributed random number between−^{1 and 1}

9: η ←^T_k^SA/T_init^SA

10: t^{o f f}_k ←^{to f f}_k +η∗(bd^†−^bdl_k)∗^step

11: t^off ←^max[t^off,bd^l], t^off←^min[t^off,bd^u]

3.9. Case Studies nealing temperature T^SA, the initial SA temperature, and the lower and

upper bounds of t^off to update the new candidate. bd^l and bd^u are the strict constraints thatt^offmust not violate, which can be set ast^swoffand (b−∑ⁿi=1(t_i^{swo f f} +t^swon_i +ci))·[1,· · · ^{, 1}]⁰, respectively. The algorithm first check if the current t^off satisfies the constraint (3.43) (line 2). If (3.43) is violated, t^off will be assigned to its lower bound, indicating an error t^off is given (line 3). Otherwise, a t^{o f f}_i in t^off is chosen randomly to be updated by adding a random number between −^η ∗(tbd−^bdl_k) and η∗(tbd−^bdl_k) to it (line 10). η is a control coefficient to ensure the convergence of the algorithm. Finally, the new t^off is checked with bd^l and bd^u and revised to the safe region if necessary (line 11).

3.9 Case Studies

We evaluate the effectiveness and feasibility of our proposed approaches in this section. Four approaches are compared: (1) our FBPT based adaptive-step Gradient-Descent algorithm (FBGD), (2) the ANPT based Simulated Annealing algorithm (ANSA) (3) Brutally Searching based PBOO algorithm (BS), and (4) the brutally searching Sub-Deadline Parti-tions algorithm introduced in Section 3.5 (SDP).

Im Dokument System Level Periodic Thermal Management for Hard Real-Time Systems  (Seite 92-97)