Energy constrained scheduling of stochastic tasks

Abstract

Energy-efficient scheduling of stochastic tasks is considered in this paper. The main characteristic of a stochastic task is that its execution time is a random variable whose actual value is not known in advance, but only its probability distribution. Our performance measures are the probability that the total execution time does not exceed a given bound and the probability that the total energy consumption does not exceed a given bound. Both probabilities need to be maximized. However, maximizations of the two performance measures are conflicting objectives. Our strategy is to fix one and maximize the other. Our investigation includes the following two aspects, with the purpose of maximizing the probability for the total execution time not to exceed a given bound, under the constraint that the probability for the total energy consumption not to exceed a given bound is at least certain value. First, we explore the technique of optimal processor speed setting for a given set of stochastic tasks on a processor with variable speed. It is found that the simple equal speed method (in which all tasks are executed with the same speed) yields high quality solutions. Second, we explore the technique of optimal stochastic task scheduling for a given set of stochastic tasks on a multiprocessor system, assuming that the equal speed method is used. We propose and evaluate the performance of several heuristic stochastic task scheduling algorithms. Our simulation studies identify the best methods among the proposed heuristic methods.

Appendices

Appendix 1: Derivation of $\partial F_T/\partial s_i$ and $\partial F_E/\partial s_i$

Notice that

$$\begin{aligned} {\partial F_T(s_1,s_2,\ldots ,s_n,T^*)\over \partial s_i} =\int _{-\infty }^{T^*}{\partial f_T(x)\over \partial s_i}\hbox {d}x. \end{aligned}$$

Furthermore, we have

$$\begin{aligned}&{\partial f_T(x)\over \partial s_i}\\&\quad ={1\over \sqrt{2\pi }}\left( -{1\over \sigma _T^2}{\partial \sigma _T \over \partial s_i}\hbox {e}^{-(x-\mu _T)^2/2\sigma _T^2}\right. \\&\qquad \left. +{1\over \sigma _T}\hbox {e}^{-(x-\mu _T)^2/2\sigma _T^2}\left( -{1\over 2}\right) 2\left( {x-\mu _T\over \sigma _T}\right) {-\partial \mu _T/\partial s_i\cdot \sigma _T-(x-\mu _T)\partial \sigma _T/\partial s_i\over \sigma _T^2}\right) \\&\quad =-{1\over \sqrt{2\pi }\sigma _T^2}\hbox {e}^{-(x-\mu _T)^2/2\sigma _T^2} \left( {\partial \sigma _T\over \partial s_i} -\left( {x-\mu _T\over \sigma _T^2}\right) \left( \sigma _T{\partial \mu _T\over \partial s_i}+(x-\mu _T){\partial \sigma _T\over \partial s_i}\right) \right) \\&\quad = -{1\over \sqrt{2\pi }\sigma _T^2}\hbox {e}^{-(x-\mu _T)^2/2\sigma _T^2} \left( {\partial \sigma _T\over \partial s_i} -{\partial \mu _T\over \partial s_i}\left( {x-\mu _T\over \sigma _T}\right) -{\partial \sigma _T\over \partial s_i}\left( {x-\mu _T\over \sigma _T}\right) ^2 \right) . \end{aligned}$$

Therefore, we get

$$\begin{aligned}&{\partial F_T(s_1,s_2,\ldots ,s_n,T^*)\over \partial s_i}\\&\quad =\int _{-\infty }^{T^*} -{1\over \sqrt{2\pi }\sigma _T^2}\hbox {e}^{-(x-\mu _T)^2/2\sigma _T^2} \left( {\partial \sigma _T\over \partial s_i} -{\partial \mu _T\over \partial s_i}\left( {x-\mu _T\over \sigma _T}\right) -{\partial \sigma _T\over \partial s_i}\left( {x-\mu _T\over \sigma _T}\right) ^2 \right) \hbox {d}x\\&\quad = -{1\over \sqrt{2\pi }\sigma _T^2} \left( {\partial \sigma _T\over \partial s_i}\int _{-\infty }^{T^*}\hbox {e}^{-(x-\mu _T)^2/2\sigma _T^2}\hbox {d}x -{\partial \mu _T\over \partial s_i}\int _{-\infty }^{T^*}\left( {x-\mu _T\over \sigma _T}\right) \hbox {e}^{-(x-\mu _T)^2/2\sigma _T^2}\hbox {d}x \right. \\&\qquad \left. -{\partial \sigma _T\over \partial s_i}\int _{-\infty }^{T^*} \left( {x-\mu _T\over \sigma _T}\right) ^2\hbox {e}^{-(x-\mu _T)^2/2\sigma _T^2}\hbox {d}x \right) . \end{aligned}$$

By letting

$$\begin{aligned} y={x-\mu _T\over \sigma _T}, \end{aligned}$$

we have

$$\begin{aligned}&{\partial F_T(s_1,s_2,\ldots ,s_n,T^*)\over \partial s_i}\\&\quad = -{1\over \sqrt{2\pi }\sigma _T} \left( {\partial \sigma _T\over \partial s_i}\int _{-\infty }^{(T^*-\mu _T)/\sigma _T}\hbox {e}^{-y^2/2}\hbox {d}y -{\partial \mu _T\over \partial s_i}\int _{-\infty }^{(T^*-\mu _T)/\sigma _T}y\hbox {e}^{-y^2/2}\hbox {d}y \right. \\&\qquad \left. -{\partial \sigma _T\over \partial s_i}\int _{-\infty }^{(T^*-\mu _T)/\sigma _T}y^2\hbox {e}^{-y^2/2}\hbox {d}y \right) . \end{aligned}$$

Since

$$\begin{aligned} \int y\hbox {e}^{-y^2/2}\hbox {d}y=-\hbox {e}^{-y^2/2}, \end{aligned}$$

and

$$\begin{aligned} \int y^2\hbox {e}^{-y^2/2}\hbox {d}y=-y\hbox {e}^{-y^2/2}+\int \hbox {e}^{-y^2/2}\hbox {d}y, \end{aligned}$$

we obtain

$$\begin{aligned}&{\partial F_T(s_1,s_2,\ldots ,s_n,T^*)\over \partial s_i}\\&\quad = -{1\over \sqrt{2\pi }\sigma _T} \left( {\partial \sigma _T\over \partial s_i}\sqrt{2\pi }F_Y\left( {T^*-\mu _T\over \sigma _T}\right) -{\partial \mu _T\over \partial s_i}\left( -\hbox {e}^{-y^2/2}\right) \bigg |_{-\infty }^{(T^*-\mu _T)/\sigma _T} \right. \\&\qquad \left. -{\partial \sigma _T\over \partial s_i} \left( -y\hbox {e}^{-y^2/2}\bigg |_{-\infty }^{(T^*-\mu _T)/\sigma _T} +\sqrt{2\pi }F_y\left( {T^*-\mu _T\over \sigma _T}\right) \right) \right) \\&\quad = -{1\over \sqrt{2\pi }\sigma _T} \left( {\partial \sigma _T\over \partial s_i}\sqrt{2\pi }F_Y\left( {T^*-\mu _T\over \sigma _T}\right) +{\partial \mu _T\over \partial s_i}\hbox {e}^{-(T^*-\mu _T)^2/2\sigma _T^2} \right. \\&\qquad \left. +{\partial \sigma _T\over \partial s_i} \left( \left( {T^*-\mu _T\over \sigma _T}\right) \hbox {e}^{-(T^*-\mu _T)^2/2\sigma _T^2} -\sqrt{2\pi }F_Y\left( {T^*-\mu _T\over \sigma _T}\right) \right) \right) \\&\quad = -{1\over \sqrt{2\pi }\sigma _T} \left( {\partial \mu _T\over \partial s_i}\hbox {e}^{-(T^*-\mu _T)^2/2\sigma _T^2} +{\partial \sigma _T\over \partial s_i}\left( {T^*-\mu _T\over \sigma _T}\right) \hbox {e}^{-(T^*-\mu _T)^2/2\sigma _T^2} \right) \\&\quad = -{1\over \sqrt{2\pi }\sigma _T}\hbox {e}^{-(T^*-\mu _T)^2/2\sigma _T^2} \left( {\partial \mu _T\over \partial s_i} +\left( {T^*-\mu _T\over \sigma _T}\right) {\partial \sigma _T\over \partial s_i} \right) \\&\quad = -f_T(T^*) \left( {\partial \mu _T\over \partial s_i} +\left( {T^*-\mu _T\over \sigma _T}\right) {\partial \sigma _T\over \partial s_i} \right) . \end{aligned}$$

It is clear that

$$\begin{aligned} {\partial \mu _T\over \partial s_i}=-{\mu _i\over s_i^2}. \end{aligned}$$

Since

$$\begin{aligned} \sigma _T=\left( {\sigma _1^2\over s_1^2}+{\sigma _2^2\over s_2^2}+\cdots +{\sigma _n^2\over s_n^2}\right) ^{1/2}, \end{aligned}$$

we get

$$\begin{aligned} {\partial \sigma _T\over \partial s_i} ={1\over 2}\left( {\sigma _1^2\over s_1^2}+{\sigma _2^2\over s_2^2}+\cdots +{\sigma _n^2\over s_n^2}\right) ^{-1/2} \sigma _i^2\left( -{2\over s_i^3}\right) =-{1\over \sigma _T}\cdot {\sigma _i^2\over s_i^3}. \end{aligned}$$

Consequently, we get

$$\begin{aligned} {\partial F_T(s_1,s_2,\ldots ,s_n,T^*)\over \partial s_i} =f_T(T^*) \left( {\mu _i\over s_i^2}+{1\over \sigma _T}\left( {T^*-\mu _T\over \sigma _T}\right) {\sigma _i^2\over s_i^3}\right) . \end{aligned}$$

In a similar way, we can also derive

$$\begin{aligned} {\partial F_E(s_1,s_2,\ldots ,s_n,E^*)\over \partial s_i} =-f_E(E^*) \left( {\partial \mu _E\over \partial s_i} +\left( {E^*-\mu _E\over \sigma _E}\right) {\partial \sigma _E\over \partial s_i} \right) . \end{aligned}$$

It is clear that

$$\begin{aligned} {\partial \mu _E\over \partial s_i}=(\alpha -1)\mu _is_i^{\alpha -2}, \end{aligned}$$

and

$$\begin{aligned} {\partial \sigma _E\over \partial s_i}=2(\alpha -1)\sigma _i^2s_i^{2\alpha -3}. \end{aligned}$$

Consequently, we get

$$\begin{aligned} {\partial F_E(s_1,s_2,\ldots ,s_n,E^*)\over \partial s_i} =-(\alpha -1)f_E(E^*) \left( \mu _is_i^{\alpha -2} +2\left( {E^*-\mu _E\over \sigma _E}\right) \sigma _i^2s_i^{2\alpha -3} \right) . \end{aligned}$$

Appendix 2: Calculation of $\partial F_i(\mathbf{y})/\partial y_j$

First, we have

$$\begin{aligned} {\partial F_0(\mathbf{y})\over \partial y_0}={\partial F_0(\mathbf{y})\over \partial \phi }=0, \end{aligned}$$

and

$$\begin{aligned}&{\partial F_0(\mathbf{y})\over \partial y_j} ={\partial F_0(\mathbf{y})\over \partial s_j} ={\partial F_E(s_1,s_2,\ldots ,s_n,E^*)\over \partial s_j}\\&\quad =-(\alpha -1)f_E(E^*) \left( \mu _js_j^{\alpha -2} +2\left( {E^*-\mu _E\over \sigma _E}\right) \sigma _j^2s_j^{2\alpha -3} \right) , \end{aligned}$$

for all $1\le j\le n$. Next, we have

$$\begin{aligned} {\partial F_i(\mathbf{y})\over \partial y_0} ={\partial F_i(\mathbf{y})\over \partial \phi } =(\alpha -1)f_E(E^*) \left( \mu _is_i^{\alpha -2} +2\left( {E^*-\mu _E\over \sigma _E}\right) \sigma _i^2s_i^{2\alpha -3} \right) , \end{aligned}$$

for all $1\le i\le n$. Recall that

$$\begin{aligned}&{\partial f_T(x)\over \partial s_i}\\&\quad = -{1\over \sqrt{2\pi }\sigma _T^2}\hbox {e}^{-(x-\mu _T)^2/2\sigma _T^2} \left( {\partial \sigma _T\over \partial s_i} -{\partial \mu _T\over \partial s_i}\left( {x-\mu _T\over \sigma _T}\right) -{\partial \sigma _T\over \partial s_i}\left( {x-\mu _T\over \sigma _T}\right) ^2 \right) \\&\quad ={f_T(x)\over \sigma _T} \left( -{\partial \mu _T\over \partial s_i}\left( {x-\mu _T\over \sigma _T}\right) +{\partial \sigma _T\over \partial s_i}\left( 1-\left( {x-\mu _T\over \sigma _T}\right) ^2\right) \right) \\&\quad ={f_T(x)\over \sigma _T} \left( {\mu _i\over s_i^2}\left( {x-\mu _T\over \sigma _T}\right) -{1\over \sigma _T}\cdot {\sigma _i^2\over s_i^3}\left( 1-\left( {x-\mu _T\over \sigma _T}\right) ^2\right) \right) , \end{aligned}$$

which implies that

$$\begin{aligned} {\partial f_T(T^*)\over \partial s_i} ={f_T(T^*)\over \sigma _T} \left( {\mu _i\over s_i^2}\left( {T^*-\mu _T\over \sigma _T}\right) -{1\over \sigma _T}\cdot {\sigma _i^2\over s_i^3}\left( 1-\left( {T^*-\mu _T\over \sigma _T}\right) ^2\right) \right) . \end{aligned}$$

Similarly, we can also get

$$\begin{aligned}&{\partial f_E(x)\over \partial s_i}\\&\quad = -{1\over \sqrt{2\pi }\sigma _E^2}\hbox {e}^{-(x-\mu _E)^2/2\sigma _E^2} \left( {\partial \sigma _E\over \partial s_i} -{\partial \mu _E\over \partial s_i}\left( {x-\mu _E\over \sigma _E}\right) -{\partial \sigma _E\over \partial s_i}\left( {x-\mu _E\over \sigma _E}\right) ^2 \right) \\&\quad ={f_E(x)\over \sigma _E} \left( -{\partial \mu _E\over \partial s_i}\left( {x-\mu _E\over \sigma _E}\right) +{\partial \sigma _E\over \partial s_i}\left( 1-\left( {x-\mu _E\over \sigma _E}\right) ^2\right) \right) \\&\quad ={f_E(x)\over \sigma _E} \left( -(\alpha -1)\mu _is_i^{\alpha -2}\left( {x-\mu _E\over \sigma _E}\right) +2(\alpha -1)\sigma _i^2s_i^{2\alpha -3}\left( 1-\left( {x-\mu _E \over \sigma _E}\right) ^2\right) \right) , \end{aligned}$$

which implies that

$$\begin{aligned} {\partial f_E(E^*)\over \partial s_i}= & {} (\alpha -1){f_E(E^*)\over \sigma _E} \left( -\mu _is_i^{\alpha -2}\left( {E^*-\mu _E\over \sigma _E}\right) \right. \\&\left. +2\sigma _i^2s_i^{2\alpha -3}\left( 1-\left( {E^*-\mu _E\over \sigma _E}\right) ^2\right) \right) . \end{aligned}$$

Hence, we have

$$\begin{aligned}&{\partial F_i(\mathbf{y})\over \partial y_i} ={\partial F_i(\mathbf{y})\over \partial s_i}\\&\quad = {\partial f_T(T^*)\over \partial s_i} \left( {\mu _i\over s_i^2}+\sigma _i^2\left( {T^*-\mu _T\over \sigma _T^2s_i^3}\right) \right) + f_T(T^*) {\partial \over \partial s_i}\left( {\mu _i\over s_i^2} +\sigma _i^2\left( {T^*-\mu _T\over \sigma _T^2s_i^3}\right) \right) \\&\qquad +\phi (\alpha -1) \left( {\partial f_E(E^*)\over \partial s_i} \left( \mu _is_i^{\alpha -2} +2\sigma _i^2s_i^{2\alpha -3}\left( {E^*-\mu _E\over \sigma _E}\right) \right) \right. \\&\qquad \left. +f_E(E^*) {\partial \over \partial s_i} \left( \mu _is_i^{\alpha -2} +2\sigma _i^2s_i^{2\alpha -3}\left( {E^*-\mu _E\over \sigma _E}\right) \right) \right) \\&\quad = {\partial f_T(T^*)\over \partial s_i} \left( {\mu _i\over s_i^2}+\sigma _i^2\left( {T^*-\mu _T\over \sigma _T^2s_i^3} \right) \right) \\&\qquad + f_T(T^*) \left( -{2\mu _i\over s_i^3}+\sigma _i^2\left( {-\partial \mu _T/\partial s_i\cdot \sigma _T^2s_i^3-(T^*-\mu _T)(2\sigma _T\partial \sigma _T/\partial s_i\cdot s_i^3+\sigma _T^23s_i^2)\over \sigma _T^4s_i^6}\right) \right) \\&\qquad +\phi (\alpha -1) \left( {\partial f_E(E^*)\over \partial s_i} \left( \mu _is_i^{\alpha -2} +2\sigma _i^2s_i^{2\alpha -3}\left( {E^*-\mu _E\over \sigma _E}\right) \right) \right. \\&\qquad \left. +f_E(E^*) \left( \mu _i(\alpha -2)s_i^{\alpha -3} +2\sigma _i^2 \left( (2\alpha -3)s_i^{2\alpha -4}\left( {E^*-\mu _E\over \sigma _E}\right) \right. \right. \right. \\&\qquad \left. \left. \left. + s_i^{2\alpha -3}\left( {-\partial \mu _E/\partial s_i\cdot \sigma _E -(E^*-\mu _E)\partial \sigma _E/\partial s_i\over \sigma _E^2}\right) \right) \right) \right) \\&\quad = {\partial f_T(T^*)\over \partial s_i} \left( {\mu _i\over s_i^2}+\sigma _i^2\left( {T^*-\mu _T\over \sigma _T^2s_i^3}\right) \right) \\&\qquad + f_T(T^*) \left( -{2\mu _i\over s_i^3}+\sigma _i^2\left( {\sigma _T^2 \mu _is_i-(T^*-\mu _T)(-2\sigma _i^2+3\sigma _T^2s_i^2)\over \sigma _T^4s_i^6} \right) \right) \\&\qquad +\phi (\alpha -1) \left( {\partial f_E(E^*)\over \partial s_i} \left( \mu _is_i^{\alpha -2} +2\sigma _i^2s_i^{2\alpha -3}\left( {E^*-\mu _E\over \sigma _E}\right) \right) \right. \\&\qquad \left. +f_E(E^*) \left( (\alpha -2)\mu _is_i^{\alpha -3} +2\sigma _i^2 \left( (2\alpha -3)s_i^{2\alpha -4}\left( {E^*-\mu _E\over \sigma _E}\right) \right. \right. \right. \\&\qquad \left. \left. \left. -(\alpha -1)s_i^{2\alpha -3}\left( {\sigma _E\mu _is_i^{\alpha -2} +2(E^*-\mu _E)\sigma _i^2s_i^{2\alpha -3}\over \sigma _E^2}\right) \right) \right) \right) , \end{aligned}$$

for all $1\le i\le n$, and

$$\begin{aligned}&{\partial F_i(\mathbf{y})\over \partial y_j} ={\partial F_i(\mathbf{y})\over \partial s_j}\\&\quad = {\partial f_T(T^*)\over \partial s_j} \left( {\mu _i\over s_i^2}+\sigma _i^2\left( {T^*-\mu _T\over \sigma _T^2s_i^3}\right) \right) + f_T(T^*) {\partial \over \partial s_j}\left( {\mu _i\over s_i^2}+ \sigma _i^2\left( {T^*-\mu _T\over \sigma _T^2s_i^3}\right) \right) \\&\qquad +\phi (\alpha -1) \left( {\partial f_E(E^*)\over \partial s_j} \left( \mu _is_i^{\alpha -2} +2\sigma _i^2s_i^{2\alpha -3}\left( {E^*-\mu _E\over \sigma _E}\right) \right) \right. \\&\qquad \left. +f_E(E^*) {\partial \over \partial s_j} \left( \mu _is_i^{\alpha -2} +2\sigma _i^2s_i^{2\alpha -3}\left( {E^*-\mu _E\over \sigma _E}\right) \right) \right) \\&\quad = {\partial f_T(T^*)\over \partial s_j} \left( {\mu _i\over s_i^2}+\sigma _i^2\left( {T^*-\mu _T\over \sigma _T^2s_i^3}\right) \right) \\&\qquad +f_T(T^*) {\sigma _i^2\over s_i^3}\left( {-\partial \mu _T/\partial s_j \cdot \sigma _T^2-(T^*-\mu _T)2\sigma _T\partial \sigma _T/\partial s_j\over \sigma _T^4}\right) \\&\qquad +\phi (\alpha -1) \left( {\partial f_E(E^*)\over \partial s_j} \left( \mu _is_i^{\alpha -2} +2\sigma _i^2s_i^{2\alpha -3}\left( {E^*-\mu _E\over \sigma _E}\right) \right) \right. \\&\qquad \left. +2f_E(E^*)\sigma _i^2s_i^{2\alpha -3}\left( {-\partial \mu _E/ \partial s_j\cdot \sigma _E-(E^*-\mu _E)\partial \sigma _E/\partial s_j \over \sigma _E^2}\right) \right) \\&\quad = {\partial f_T(T^*)\over \partial s_j} \left( {\mu _i\over s_i^2}+\sigma _i^2\left( {T^*-\mu _T\over \sigma _T^2s_i^3} \right) \right) \\&\qquad +f_T(T^*) {\sigma _i^2\over s_i^3}\left( {\sigma _T^2\mu _j/s_j^2+2(T^*-\mu _T) \sigma _j^2/s_j^3\over \sigma _T^4}\right) \\&\qquad +\phi (\alpha -1) \left( {\partial f_E(E^*)\over \partial s_j} \left( \mu _is_i^{\alpha -2} +2\sigma _i^2s_i^{2\alpha -3}\left( {E^*-\mu _E\over \sigma _E}\right) \right) \right. \\&\qquad \left. -2(\alpha -1)f_E(E^*)\sigma _i^2s_i^{2\alpha -3}\left( {\sigma _E \mu _js_j^{\alpha -2}+2(E^*-\mu _E)\sigma _j^2s_j^{2\alpha -3}\over \sigma _E^2}\right) \right) , \end{aligned}$$

for all $1\le i\le n$ and all $1\le j\not =i\le n$.