Motivation and Problem statement

3.5 Motivation and Problem statement

With Thm. 3.7 and Pay Burst Only Once principle, a pipelined system composed of n stages each of which offers service curve βi can be con-sidered as a black-box system which offers an aggregate service curve of β = β1⊗^β2, · · · ^{, β}n. Then, the end-to-end delay bound can be cal-culated sequentially. The Pay-Burst-Only-Once principle points out that the result derived from Thm. 3.7 better bounds the real end-to-end delay than the straightforward method does.

In this chapter, our goal is to calculate the lower service curve with given end-to-end deadline instead of to obtain the end-to-end delay with given service curve. Therefore, Thm. 3.7 and PBOO principle can be used reversely to get a better approximated lower service curve to the actual service curve demanded for the given end-to-end deadline. Since better approximation indicates lower service curve, lower peak temperature for the system is expected, which is the motivation of our work.

In our work, the bounded delay function is widely used to simply the analysis and thus is worth introducing.

Definition 3.9 (Bounded delay function) A Bounded delay function (BDF) specified by constants ρand b is defined as:

bd f(∆) = max[0,ρ(∆−^b)]. (3.17)

3.5.1 Motivation Example

In this chapter, we deploy Periodic Thermal Management to manage the temperature of the chip by periodically switching each stage between two power consumption modes with an individual pair of (t^on,t^{o f f}). Therefore, two vectors,t^off = (t^{o f f}₁ ,t^{o f f}₂ ,· · · ^{,to f f}n )andt^on= (t^on₁ ,t^on₂ ,· · · ^,tonn ), should be determined offline to specify the PTMs deployed on the sys-tem. For the details of PTM, we refer to [1]. Now, we present a moti-vation example to illustrate the advantages of applying Pay-Burst-Only-Once for thermal optimization. For comparison, the PTM schemes are derived from two approaches: the PBOO based approach (PBOO) and the one which partitions the end-to-end deadline into sub-stage dead-lines for each stage, namely SDP (Sub-Deadline Partition).

In this example, an event stream with arrival curve α = 0.15∆+2 and deadline D = 35ms passes through a two-stage pipelined system. For simplicity, the worst-case execution times of the event stream in first and

0

Figure 3.4: An example of calculating PTM schemes by methods SDP and PBOO.

second stage are set asc₁ =c₂ =1ms, respectively. We discuss the case that t^off = (5, 13)ms and compare t^ons generated by the two methods.

Fig. 3.4 graphically illustrates the derivation process corresponding to the two methods.

Let’s first examine the strategy of Sub-Deadline Partition. We divide the deadline D into two sub-deadlines, D1 =10ms and D2 = 25ms, in this case.

To simplify the process of calculating t^on, we adopt the bounded-delay function as an assistant service curve. For instance, for the first stage, with the deadlineD1 =10ms, the service demand should beβ1=α(∆− 10). Denote the service curve provided by first state as β^tdma₁ , which is is a TDMA curve specified by t^{o f f}₁ and t^on₁ . We can guarantee β^tdma₁ ≥ β₁, if β^tdma₁ ≥ ^{bd f}1 ≥ ^β1. Since t^{o f f}₁ = 5ms, the b in bd f₁ is set as b₁ = t^{o f f}₁ = 5ms. Then, bd f₁ ≥ ^β1 yields the minimal slope of bd f₁ as ρ₁ = (2−⁰)/(10−⁵) = 0.4 and finally we have t^on₁ = 3.3ms, which is

3.5. Motivation and Problem statement derived from ^t1on

t^on₁ +t^{o f f}₁ = c1∗^ρ1 = 0.4 to ensure β^tdma₁ ≥ ^{bd f}1. To derive the service demand for the second stage, the output arrive curve ˆαfrom the first stage is needed. From Thm. 3.6,

ˆ

α=α^βtdma1 =0.15∆+2.7 (3.18) In the same way, we have the slope of the BDF in second stage as ρ2 = 0.225 and finally t^on₂ =3.7ms.

Now, the PBOO method is utilized to get t^on. Unlike the procedure in SDP, the total service demand is first obtained as βtot = α(∆−³⁵) and then a BDF bd ftot = max[0,ρ(∆−^b)] should be determined such that the deadline constraint can be met as long as β^srv₁ ⊗^βsrv₂ ≥ ^{bd f}tot ≥ α^u(∆−^D). From the analysis discussed in Section 3.7, we can set btotas its minimum: btot =t^{o f f}₁ +t^{o f f}₂ +c₁+c₂=20ms, then the slope ofbd ftot

can be determined: ρtot = 0.15, which is much smaller than ρ₁ and ρ₂, therefore, resulting in smaller t^on₁ =0.9ms and t^on₂ =2.3ms.

The pessimism in the SDP method comes from paying an additional burst and delay when ˆα is calculated for the second stage, as PBOO principle points out [60]. Moreover, as the stage number increases, this effect is accumulated and then causes more pessimistic results. On the other hand, method PBOO directly calculates the total service demand and then retrieves t^on for every stage, which pays the burst only once and gets better results. Since lower partition of t^on means lower temper-ature of the processor, it is expected that employing PBOO will achieve lower peak temperature than using SDP. Therefore, by reversely using PBOO principle, we can avoid paying the burst repeatedly and therefore better optimize the peak temperature for pipelined systems, especially for which having large number of stages.

3.5.2 Problem Statement

Now our problem is defined as follows:

Given an n-stage pipelined platform specified by the above hardware and ther-mal models, an event stream with end-to-end deadline D, and the WCETs c = (c₁,c₂,· · · ^,cn), our goal is to find the PTM schemes characterized by t^off and t^on such that the peak temperature is minimized while the deadline constraint is satisfied.

To deal with this problem, the first question is how to get the peak temperature with known Periodic Thermal Management, which is dis-cussed in next section.