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Energy constrained scheduling of stochastic tasks

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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.

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References

  1. 1.

    Ahmadizar F, Ghazanfari M, Ghomi SMTF (2010) Group shops scheduling with makespan criterion subject to random release dates and processing times. Comput Oper Res 37(1):152–162

  2. 2.

    Ando E, Nakata T, Yamashita M (2009) Approximating the longest path length of a stochastic DAG by a normal distribution in linear time. J Discrete Algorithms 7(4):420–438

  3. 3.

    Burden RL, Faires JD, Reynolds AC (1981) Numerical analysis, 2nd edn. Prindle, Weber & Schmidt, Boston

  4. 4.

    Canon L-C, Jeannot E (2009) Precise evaluation of the efficiency and the robustness of stochastic DAG schedules. Research report RR-6895 INRIA

  5. 5.

    Chandrakasan AP, Sheng S, Brodersen RW (1992) Low-power CMOS digital design. IEEE J Solid State Circuits 27(4):473–484

  6. 6.

    Chen L, Megow N, Rischke R, Stougie L (2015) Stochastic and robust scheduling in the cloud. In: 18th Int’l Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX’15) and 19th Int’l Workshop on Randomization and Computation (RANDOM’15), pp 175–186

  7. 7.

    Chrétienne P, Coffman EG, Lenstra JK, Liu Z (eds) (1995) Scheduling theory and its applications. Wiley, Chichester

  8. 8.

    Dong F, Luo J, Song A, Jin J (2010) Resource load based stochastic DAGs scheduling mechanism for grid environment. In: 12th IEEE International Conference on High Performance Computing and Communications, pp 197–204

  9. 9.

    Furht B, Escalante A (eds) (2010) Handbook of cloud computing. Springer, New York

  10. 10.

    Gu J, Gu X, Gu M (2009) A novel parallel quantum genetic algorithm for stochastic job shop scheduling. J Math Anal Appl 355(1):63–81

  11. 11.

    Jeannot E, Namyst R, Roman J (eds) (2011) Euro-Par 2011 parallel processing. LNCS 6852. Springer, Berlin

  12. 12.

    Li K, Tang X, Li K (2014) Energy-efficient stochastic task scheduling on heterogeneous computing systems. IEEE Trans Parallel Distrib Syst 25(11):2867–2876

  13. 13.

    Li K, Tang X, Veeravalli B, Li K (2015) Scheduling precedence constrained stochastic tasks on heterogeneous cluster systems. IEEE Trans Comput 64(1):191–204

  14. 14.

    Megow N, Uetz M, Vredeveld T (2006) Models and algorithms for stochastic online scheduling. Math Oper Res 31(3):513–525

  15. 15.

    Möhring RH, Schulz AS, Uetz M (1999) Approximation in stochastic scheduling: the power of LP-based priority policies. J ACM 46(6):924–942

  16. 16.

    Peng Z, Cui D, Zuo J, Li Q, Xu B, Lin W (2015) Random task scheduling scheme based on reinforcement learning in cloud computing. Cluster Comput 18(4):1595–1607

  17. 17.

    Rothkopf MH (1966) Scheduling with random service times. Manage Sci 12(9):703–713

  18. 18.

    Sarin SC, Nagarajan B, Liao L (2010) Stochastic scheduling: expectation-variance analysis of a schedule. Cambridge University Press, Cambridge

  19. 19.

    Scharbrodt M, Schickinger T, Steger A (2006) A new average case analysis for completion time scheduling. J ACM 53(1):121–146

  20. 20.

    Shin SY, Gantenbein R, Kuo T-W, Hong J (2011) Reliable and autonomous computational science. Birkhäuser, Basel

  21. 21.

    Skutella M, Uetz M (2005) Stochastic machine scheduling with precedence constraints. SIAM J Comput 34(4):788–802

  22. 22.

    Tang X, Li K, Liao G, Fang K, Wu F (2011) A stochastic scheduling algorithm for precedence constrained tasks on grid. Future Gener Comput Syst 27(8):1083–1091

  23. 23.

    Tongsima S, Sha EHM, Chantrapornchai C, Surma DR, Passos NL (2000) Probabilistic loop scheduling for applications with uncertain execution time. IEEE Trans Comput 49(1):65–80

  24. 24.

    Wang L, Ranjan R, Chen J, Benatallah B (eds) (2012) Cloud computing: methodology, systems, and applications. CRC Press, Boca Raton

  25. 25.

    Weiss G (1992) Turnpike optimality of Smith’s rule in parallel machines stochastic scheduling. Math Oper Res 17(2):255–270

  26. 26.

    Weiss G, Pinedo M (1980) Scheduling tasks with exponential service times on non-identical processors to minimize various cost functions. J Appl Probab 17(1):187–202

  27. 27.

    Xhafa F, Abraham A (eds) (2008) Metaheuristics for scheduling in distributed computing environments. Springer, Berlin

  28. 28.

    Zhai B, Blaauw D, Sylvester D, Flautner K (2004) Theoretical and practical limits of dynamic voltage scaling. In: Proceedings of the 41st Design Automation Conference, pp 868–873

  29. 29.

    Zhang W, Bai E, He H, Cheng AMK (2015) Solving energy-aware real-time tasks scheduling problem with shuffled frog leaping algorithm on heterogeneous platforms. Sensors 15(6):13778–13804

  30. 30.

    Zhang W, Xie H, Cao B, Cheng AMK (2014) Energy-aware real-time task scheduling for heterogeneous multiprocessors with particle swarm optimization algorithm. Math Prob Eng, Article ID 287475

  31. 31.

    Zhang Z, Su S, Zhang J, Shuang K, Xu P (2015) Energy aware virtual network embedding with dynamic demands: online and offline, part 3. Comput Netw 93:448–459

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Acknowledgements

The author deeply appreciates eighteen (18) anonymous reviewers for their corrections, criticism, and comments on the original manuscript.

Author information

Correspondence to Keqin Li.

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 \(1\le j\not =i\le n\).

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Li, K. Energy constrained scheduling of stochastic tasks. J Supercomput 74, 485–508 (2018). https://doi.org/10.1007/s11227-017-2137-0

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Keywords

  • Energy consumption
  • Energy-efficient scheduling
  • Execution time
  • Heuristic algorithm
  • Optimization problem
  • Processor speed setting
  • Stochastic tasks