Energy-efficient adaptive networked datacenters for the QoS support of real-time applications

Abstract

In this paper, we develop the optimal minimum-energy scheduler for the adaptive joint allocation of the task sizes, computing rates, communication rates and communication powers in virtualized networked data centers (VNetDCs) that operate under hard per-job delay-constraints. The considered VNetDC platform works at the Middleware layer of the underlying protocol stack. It aims at supporting real-time stream service (such as, for example, the emerging big data stream computing (BDSC) services) by adopting the software-as-a-service (SaaS) computing model. Our objective is the minimization of the overall computing-plus-communication energy consumption. The main new contributions of the paper are the following ones: (i) the computing-plus-communication resources are jointly allotted in an adaptive fashion by accounting in real-time for both the (possibly, unpredictable) time fluctuations of the offered workload and the reconfiguration costs of the considered VNetDC platform; (ii) hard per-job delay-constraints on the overall allowed computing-plus-communication latencies are enforced; and, (iii) to deal with the inherently nonconvex nature of the resulting resource optimization problem, a novel solving approach is developed, that leads to the lossless decomposition of the afforded problem into the cascade of two simpler sub-problems. The sensitivity of the energy consumption of the proposed scheduler on the allowed processing latency, as well as the peak-to-mean ratio (PMR) and the correlation coefficient (i.e., the smoothness) of the offered workload is numerically tested under both synthetically generated and real-world workload traces. Finally, as an index of the attained energy efficiency, we compare the energy consumption of the proposed scheduler with the corresponding ones of some benchmark static, hybrid and sequential schedulers and numerically evaluate the resulting percent energy gaps.

Appendices

Appendix A: Derivations of Eqs. (21a)–(23)

Since the constraint in (10g) is already accounted for by the feasibility condition (18b), without loss of optimality, we may directly focus on the resolution of optimization problem in (16) under the constraints in (10b)–(10e). Since this problem is strictly convex and all its constraints are linear, the Slater’s qualification conditions hold [52, chap.5], so that the Karush–Khun–Tucker (KKT) conditions [52, chap.4] are both necessary and sufficient for analytically characterizing the corresponding unique optimal global solution. Before applying these conditions, we observe that each power-rate function in (17) is increasing for $L_i\ge 0$, so that, without loss of optimality, we may replace the equality constraint in (10c) by the following equivalent one: $\sum _{i=1}^M L_i\ge L_\mathrm{tot}$. In doing so, the Lagrangian function of the afforded problem reads as in

$$\begin{aligned}&\mathcal {L}\left( \left\{ L_i,f_i,\nu _i,\mu \right\} \right) \equiv \mathcal {Z}\left( \left\{ L_i,f_i\right\} \right) \sum _{i=1}^M \nu _i\left( L_i-f_i\Delta +L_b(i)\right) +\mu \left( L_t-\sum _{i=1}^ML_i\right) , \end{aligned}$$

(A.1)

where $\mathcal {Z}\left( \left\{ L_i,f_i\right\} \right)$ indicates the objective function in (16), $\nu _i$’s and $\mu$ are nonnegative Lagrange multipliers and the box constraints in (10d), (10e) are managed as implicit ones. The partial derivatives of $\mathcal {L}(.;.)$ with respect to $f_i$, $L_i$ are given by

$$\begin{aligned} \frac{\partial \mathcal {L}(.)}{\partial f_i}=\frac{\mathcal {E}_i^\mathrm{max}}{f_i^\mathrm{max}} \frac{\partial \Phi _i(f_i/f_i^\mathrm{max})}{\partial \eta _i}+2k_e\left( f_i-f_i^0\right) -\nu _i\Delta , \end{aligned}$$

(A.2)

$$\begin{aligned} \frac{\partial \mathcal {L}(.)}{\partial L_i}= 2 \frac{\partial }{\partial R_i}P_i^\mathrm{net} \left( \frac{2L_i}{T_t-\Delta }\right) +\nu _i-\mu , \end{aligned}$$

(A.3)

$i=1,\ldots ,M$, while the complementary conditions [52, chap.4] associated to the constraints present in (A.1) read as in

$$\begin{aligned} \nu _i\left[ L_i-f_i\Delta +L_b(i)\right] =0,\quad i = 1,\ldots ,M;\;\; \mu \left( L_\mathrm{tot}-\sum _{i=1}^M L_i\right) =0. \end{aligned}$$

(A.4)

Hence, by equating (A.2) to zero we directly arrive at (21a), that also accounts for the box constraint: $f_i^{\min }\le f_i\le f_i^{\max }$ through the corresponding projector operator. Moreover, a direct exploitation of the last complementary condition in (A.4) allows us to compute the optimal $\mu ^*$ by solving the algebraic equation in (23). To obtain the analytical expressions for $L_i^*$ and $\nu _i^*$, we proceed to consider the two cases of $\nu _i^*>0$ and $\nu _i^*=0$. Specifically, when $\nu _i^*$ is positive, the $i$th constraint in (10b) is bound [see (A.4)], so that we have

$$\begin{aligned} L_i^*=\Delta f_i^*-L_b(i),\;\; \text{ at } \nu _i^*>0. \end{aligned}$$

(A.5)

Hence, after equating (A.3) to zero, we obtain the following expression for the corresponding optimal $\nu _i^*$:

$$\begin{aligned} \nu _i^*=\mu ^*-2\left[ \frac{\partial P_i^\mathrm{net}}{\partial R_i}\left( \frac{2L_i^*}{T_t-\Delta }\right) \right] , \quad \text{ at } \nu _i^*>0. \end{aligned}$$

(A.6)

Since $L_i^*$ must fall into the closed interval $[0,\Delta f_i^*-L_b(i)]$ for feasible CPOPs [see (10b), (10e)], at $\nu _i^*=0$, we must have: $L_i^*=0$ or $0<L_i^*<\Delta f_i^*-L_b(i)$. Specifically, we observe that, by definition, vanishing $L_i^*$ is optimal when $\left[ \partial \mathcal {L}/\partial L_i\right] _{L_i=0}\ge 0$. Therefore, by imposing that the derivative in (A.3) is nonnegative at $L_i^*=\nu _i^*=0$, we obtain the following condition for the resulting optimal $\mu ^*$:

$$\begin{aligned} \mu ^*\le 2 \left[ \partial P_i^{net}(R_i)/\partial R_i\right] _{R_i=0}\triangleq TH(i),\;\; \text{ at } \nu _i^*=L_i^*=0. \end{aligned}$$

(A.7)

Passing to consider the case of $\nu _i^*=0$ and $L_i^*\in ]0,\Delta f_i^*-L_b(i)[$, we observe that the corresponding KKT condition is unique, it is necessary and sufficient for the optimality and requires that (A.3) vanishes [52, chap.4]. Hence, the application of this condition leads to the following expression for the optimal $L_i^*$ [see (A.3)]:

$$\begin{aligned} L_i^*\!=\!\frac{(T_t-\Delta )}{2}\!\left( \partial P_i^{net}(R_i)/\partial R_i\right) ^{-1}(\mu ^*/2),\quad \!\! \text{ at } \! \nu _i^*\!=\!0 \text{ and } 0<L_i^*<\left( \Delta f_i^*-L_b(i)\right) \!. \end{aligned}$$

(A.8)

Equation (A.8) vanishes at $\mu ^*=TH(i)$ [see (A.7)] and this proves that the function: $L_i^*(\mu ^*)$ vanishes at $\mu ^*=$TH($i$). Therefore, since (A.7) already assures that vanishing $L_i$ is optimal at $\nu _i^*=0$ and $\mu ^*\le$ TH($i$), we conclude that the expression in (A.8) for the optimal $L_i^*$ must hold when $\nu _i^*=0$ and $\mu ^*\ge$ TH($i$). This structural property of the optimal scheduler allows us to merge (A.7), (A.8) into the following equivalent expression:

$$\begin{aligned} L_i^*=\frac{(T_t-\Delta )}{2}\left[ \left( \partial P_i^\mathrm{net}(R_i)/\partial R_i\right) ^{-1}(\mu ^*/2)\right] _+, \quad \text{ for } \nu _i^*=0, \end{aligned}$$

(A.9)

so that Equation (21b) directly arises from (A.5), (A.9). Finally, after observing that $\nu _i^*$ cannot be negative by definition, from (A.6) we obtain (22). This completes the proof of Proposition 4.

Appendix B: Proof of Proposition 5

The reported proof exploits arguments based on the Lyapunov Theory which are similar, indeed, to those already used, for example, in [56, Sects. 3.4, 8.2], [13, Appendix II]. Specifically, after noting that the feasibility and strict convexity of the CPOP in (16) guarantees the existence and uniqueness of the equilibrium point of the iterates in (25) and (26), we note that Proposition 4 assures that, for any assigned $\mu ^{(n)}$ and $\left\{ L_i^{(n-1)}\right\}$, Eqs. (26a)–(26c) give the corresponding optimal values of the primal and dual variables $\left\{ f_i^{(n)},L_i^{(n)},\nu _i^{(n)}\right\}$. Hence, it suffices to prove the global asymptotic convergence of the iteration in (25). To this end, after posing

$$\begin{aligned} U^{(n-1)}\left( \left\{ L_i^{(n-1)}\right\} \right) \equiv U^{(n-1)}\triangleq \left[ \sum _{i=1}^ML_i^{(n-1)}-L_\mathrm{tot}\right] ^2, \end{aligned}$$

(B.1)

we observe that $U^{(n-1)}>0$ for $\left\{ L_i^{(n-1)}\right\} \ne \left\{ L_i^{*}\right\}$ and $U^{(n-1)}=0$ at the optimum, i.e., for $\left\{ L_i^{(n-1)}\right\} = \left\{ L_i^{*}\right\}$ ⁴. Hence, since $U^{(n-1)}(.)$ in (B.1) is also radially unbounded (that is, $U^{(n-1)}(.)\rightarrow \infty$ as $||\sum _{i=1}^M L_i^{(n-1)}-L_\mathrm{tot}||\rightarrow \infty$), we conclude that (B.1) is an admissible Lyapunov’s function for the iterations in (25). Hence, after posing $U^{(n)}\left( \left\{ L_i^{(n)}\right\} \right) =U^{(n)}\triangleq \left[ \sum _{i=1}^ML_i^{(n)}-L_\mathrm{tot}\right] ^2$, according to the Lyapunov’s Theorem [56, Sect. 3.10], we must prove that the following (sufficient) condition for the asymptotic global stability of (25) is met:

$$\begin{aligned} U^{(n)}<U^{(n-1)}, \quad \text{ for } n\rightarrow \infty . \end{aligned}$$

(B.2)

To this end, after assuming $U^{(n-1)}>0$, let us consider, at first, the case of

$$\begin{aligned} \left( \sum _{i=1}^ML_i^{(n-1)}-L_\mathrm{tot}\right) >0. \end{aligned}$$

(B.3)

Hence, since $\alpha ^{(n-1)}$ is positive, we have [see (25)]: $\mu ^{(n)}<\mu ^{(n-1)}$, that, in turn, leads to [see (26c)]: $L_i^{(n)}<L_i^{(n-1)}$, for any $i=1,\ldots ,M$ ⁵. Therefore, to prove (B.2), it suffices to prove that the following inequality holds for large $n$:

$$\begin{aligned} \left( \sum _{i=1}^M L_i^{(n)}-L_\mathrm{tot}\right) \ge 0. \end{aligned}$$

(B.4)

To this end, we observe that: (i) $\left\{ \alpha {(n-1)}\right\}$ in (25) vanishes for $n\rightarrow \infty$; and, (ii) $L_i^{(n)}$ is limited up to $\Delta f_i^{\max }$, for any $i=1,\ldots ,M$ [see the constraints in (10b), (10d)]. As a consequence, the difference: $\left( \mu ^{(n)}-\mu ^{(n-1)}\right)$ may be done vanishing as $n\rightarrow \infty$. Hence, after noting that the functions in (25), (26) are continue by assumption, a direct application of the Sign Permanence Theorem guarantees that (B.4) holds when the difference in (B.3) is positive.

By duality, it is direct to prove that (B.2) is also met when the difference in (B.3) is negative. This completes the proof of Proposition 5.