Dynamic bidding in spot market for profit maximization in the public cloud

Abstract

In public cloud domain, some cloud providers sell surplus resources in the form of spot instances to improve their profits. The average spot price is much cheaper than the on-demand options, which drives more and more cloud users to use spot instances to accelerate their services. However, the cloud user faces a challenge in determining the instance rental and management policy due to the unconventional pricing structure in the spot market and fluctuations in workload and spot price. In this work, we propose a dynamic bidding and resource management algorithm for the cloud user to cut down the instance rental bill in the spot market. The proposed algorithm operates in two timescales, i.e., it determines the bidding, instance rental, and job dispatching policy hourly and makes instances allocation decision in a finer granularity. The advantages of our approach are that it has a simple structure and needs no a priori statistical information of spot price and workload. We prove the optimality of the proposed algorithm and use extensive simulations to study its performance. It is shown that D-bid can save up to almost 70% of the rental cost compared to a baseline nonwork-conserving strategy.

Appendices

Appendix 1: Proof of Theorem 1

Squaring the queueing dynamic (1) and using the fact that $(\max \{x,0\})^2\le x^2$, we have

$$\begin{aligned} Q_{ij}^2(t+1)\le&\left( Q_{ij}(t)-\sum _{k}r_{ijk}(t)\right) ^2+2Q_{ij}(t)a_{ij}(t)+a_{ij}^2(t)\\ \le&Q_{ij}^2(t)+2Q_{ij}(t)\left( a_{ij}(t)-\sum _{k}r_{ijk}(t)\right) +r_{\max }^2+a_{\max }^2. \end{aligned}$$

Summing up the above equation over $[t, t+T-1]$ we have

$$\begin{aligned} Q_{ij}^2(t+T)-Q_{ij}^2(t)\le \sum _{\tau =t}^{t+T-1}2Q_{ij}(\tau )\left( a_{ij}(\tau )-\sum _{k}r_{ijk}(\tau )\right) +T\left( r_{\max }^2+a_{\max }^2\right) .\nonumber \\ \end{aligned}$$

(30)

Using a same reasoning we also have

$$\begin{aligned} q_{ij}^2(t+T)-q_{ij}^2(t)\le \sum _{\tau =t}^{t+T-1}2q_{ij}(\tau )\left( \sum _{k}r_{ikj}(\tau )-y_{ij}(\tau )s_i\right) +T\left( y_{\max }^2+n^2r_{\max }^2\right) .\nonumber \\ \end{aligned}$$

(31)

Summing up Eqs. (30) and (31) over i and j, and taking expectation on both sides conditioned on queue state $\theta (t)$, we have

$$\begin{aligned} \varDelta _T(t)\le&\,\,\mathbb {E}\left\{ \sum _{\tau =t}^{t+T-1}\sum _i\sum _j Q_{ij}(\tau )\left( a_{ij}(\tau )-\sum _{k}r_{ijk}(\tau )\right) |\theta (t)\right\} \\&+\mathbb {E}\left\{ \sum _{\tau =t}^{t+T-1}\sum _i\sum _j q_{ij}(\tau )\left( \sum _{k}r_{ikj}(\tau )-y_{ij}(\tau )s_i\right) |\theta (t)\right\} \\&+{Tnk\left( a_{\max }^2+y_{\max }^2+(1+n^2)r_{\max }^2\right) \over 2}. \end{aligned}$$

Now adding $V\mathbb {E}\left\{ \sum _j\left( x_j^\mathrm{o}(t)c_j^\mathrm{o}+x_j^\mathrm{s}(t)c_j^\mathrm{s}(t)1_j^{\mathrm{oob}}(t)\right) |\theta (t)\right\}$ to both sides completes the proof.

Appendix 2: Proof of Theorem 2

First we give a bound on future queue length in (15). Based on (1) and (8) we have

$$\begin{aligned} Q_{ij}(t)-Tr_{\max }\le Q_{ij}(t+T)\le Q_{ij}(t)+Ta_{\max }. \end{aligned}$$

Further, from constraint (8) we have $\sum _{k}r_{ikj}\le nr_{\max }$. Combining this with (2) we have

$$\begin{aligned} q_{ij}(t)-Ty_{\max }\le q_{ij}(t+T)\le q_{ij}(t)+Tnr_{\max }. \end{aligned}$$

Hence,

$$\begin{aligned}&\sum _{\tau =t}^{t+T-1} Q_{ij}(\tau )\left( a_{ij}(\tau )-\sum _{k}r_{ijk}(\tau )\right) \\&\quad \le \sum _{\tau =t}^{t+T-1} \left( Q_{ij}(t)+Ta_{\max }\right) a_{ij}(\tau )-\sum _{\tau =t}^{t+T-1} \left( Q_{ij}(t)-Tr_{\max }\right) \sum _{k}r_{ijk}(\tau )\\&\quad \le \sum _{\tau =t}^{t+T-1} Q_{ij}(t)\left( a_{ij}(\tau )-\sum _{k}r_{ijk}(\tau )\right) +T^2\left( a_{\max }^2+r_{\max }^2\right) . \end{aligned}$$

Similarly we have

$$\begin{aligned}&\sum _{\tau =t}^{t+T-1} q_{ij}(\tau )\left( \sum _{k}r_{ikj}(\tau )-y_{ij}(\tau )s_i\right) \\&\quad \le \sum _{\tau =t}^{t+T-1} q_{ij}(t)\left( \sum _{k}r_{ikj}(\tau )-y_{ij}(\tau )s_i\right) +T^2\left( n^2r_{\max }^2+y_{\max }^2\right) . \end{aligned}$$

Taking the above two equations back to (15) we have

$$\begin{aligned}&\varDelta _T(t)+V\mathbb {E}\left\{ \sum _j\left( x_j^\mathrm{o}(t)c_j^\mathrm{o}+x_j^\mathrm{s}(t)c_j^\mathrm{s}(t)1_j^{\mathrm{oob}}(t)\right) |\theta (t)\right\} \nonumber \\&\quad \le \mathbb {E}\left\{ \sum _{\tau =t}^{t+T-1}\sum _i\sum _j Q_{ij}(t)\left( a_{ij}(\tau )-\sum _{k}r_{ijk}(\tau )\right) |\theta (t)\right\} \nonumber \\&\qquad +\mathbb {E}\left\{ \sum _{\tau =t}^{t+T-1}\sum _i\sum _j q_{ij}(t)\left( \sum _{k}r_{ikj}(\tau )-y_{ij}(\tau )s_i\right) |\theta (t)\right\} \nonumber \\&\qquad +V\mathbb {E}\left\{ \sum _j\left( x_j^\mathrm{o}(t)c_j^\mathrm{o}+x_j^\mathrm{s}(t)c_j^\mathrm{s}(t)1_j^{\mathrm{oob}}(t)\right) |\theta (t)\right\} +B_2T. \end{aligned}$$

(32)

Rearranging the R.H.S. of (32) we obtain the result.

Appendix 3: Proof of Theorem 4

D-bid seeks to minimize the R.H.S. of (32), therefore

$$\begin{aligned}&\varDelta _T(t)+V\mathbb {E}\left\{ \sum _j\left( x_j^\mathrm{o}(t)c_j^\mathrm{o}+x_j^\mathrm{s}(t)c_j^\mathrm{s}(t)1_j^{\mathrm{oob}}(t)\right) |\theta (t)\right\} \nonumber \\&\quad \le \mathbb {E}\left\{ \sum _{\tau =t}^{t+T-1}\sum _i\sum _j Q_{ij}(t)\left( a_{ij}(\tau )-\sum _{k}(r_{ijk})^{\pi }(\tau )\right) |\theta (t)\right\} \nonumber \\&\qquad +\mathbb {E}\left\{ \sum _{\tau =t}^{t+T-1}\sum _i\sum _j q_{ij}(t)\left( \sum _{k}(r_{ikj})^{\pi }(\tau )-(y_{ij})^{\pi }(\tau )s_i\right) |\theta (t)\right\} \nonumber \\&\qquad +V\mathbb {E}\left\{ \sum _j\left( (x_j^\mathrm{o})^{\pi }(t)c_j^\mathrm{o}+(x_j^\mathrm{s})^{\pi }(t)c_j^\mathrm{s}(t)1_j^{\mathrm{oob}}(t)\right) |\theta (t)\right\} +B_2T, \end{aligned}$$

(33)

where $\pi$ is any alternative policy other than D-bid which depends only on $a_{ij}(t)$ and $c_j^\mathrm{s}(t)$. Now letting $\pi =\pi ^*$ and taking expectation over $\theta (t)$ we have

$$\begin{aligned}&L(\theta (t+T))-L(\theta (t))+V\mathbb {E}\left\{ \sum _j\left( x_j^\mathrm{o}(t)c_j^\mathrm{o}+x_j^\mathrm{s}(t)c_j^\mathrm{s}(t)1_j^{\mathrm{oob}}(t)\right) \right\} \\&\quad \le B_2T+\delta \sum _i\sum _j\left( Q_{ij}(t)+q_{ij}(t)\right) +V(o^*+T\delta ). \end{aligned}$$

Summing up the above equation over $t=0, T,\ldots , (M-1)T$ and letting $\delta \rightarrow 0$ we have

$$\begin{aligned}&\sum _{m=0}^{M-1}V\mathbb {E}\left\{ \sum _j\left( x_j^\mathrm{o}(mt)c_j^\mathrm{o}+x_j^\mathrm{s}(mt)c_j^\mathrm{s}(mt)1_j^{\mathrm{oob}}(mt)\right) \right\} \\&\quad \le MB_2T+MVo^* +L(\theta (0))+L(\theta (MT)). \end{aligned}$$

Dividing both sides by MV and letting $M\rightarrow \infty$ proves part 1).

To prove part 2), observe $\lambda +\epsilon {\mathbf {1}}\in \varLambda$ indicates the following Slater’s condition:

$$\begin{aligned} \mathbb {E}\left\{ o^{\pi '}(mT)\right\} =&\,\, o^{\pi '}_{\lambda +\epsilon {\mathbf {1}}}, \\ \sum _{\tau =mT}^{(m+1)T-1}\mathbb {E}\left\{ a_{ij}(\tau )\right\} \le&\sum _{\tau =mT}^{(m+1)T-1}\mathbb {E}\left\{ \sum _{k} (r_{ijk})^{\pi '}(\tau )\right\} -\epsilon ,\\ \sum _{\tau =mT}^{(m+1)T-1}\mathbb {E}\left\{ \sum _{k} (r_{ikj})^{\pi '}(\tau )\right\} =&\sum _{\tau =mT}^{(m+1)T-1}\mathbb {E}\left\{ (y_{ij})^{\pi '}(\tau )s_i\right\} -\epsilon , \end{aligned}$$

where $o^{\pi '}_{\lambda +\epsilon {\mathbf {1}}}$ denotes the value of the object function under arrival matrix $\lambda +\epsilon {\mathbf {1}}$ and control policy $\pi '$. Plugging the above equations into the R.H.S. of (33) by letting $\pi =\pi '$ we have

$$\begin{aligned}&\varDelta _T(t)+V\mathbb {E}\left\{ \sum _j\left( x_j^\mathrm{o}(t)c_j^\mathrm{o}+x_j^\mathrm{s}(t)c_j^\mathrm{s}(t)1_j^{\mathrm{oob}}(t)\right) |\theta (t)\right\} \\&\quad \le B_2T-T\epsilon \sum _i\sum _j \left( Q_{ij}(t)+q_{ij}(t)\right) +Vo^{\pi '}_{\lambda +\epsilon {\mathbf {1}}}. \end{aligned}$$

Taking expectation over $\theta (t)$ and summing over $t=0,T,\ldots ,(M-1)T$, and using the fact that $\mathbb {E}\left\{ \sum _j\left( x_j^\mathrm{o}(mT)c_j^\mathrm{o}+x_j^\mathrm{s}(t)c_j^\mathrm{s}(mT)1_j^{\mathrm{oob}}(mT)\right) \right\} \ge o^*$ yields

$$\begin{aligned}&L(\theta (MT))-L(\theta (0))+MVo^*\\&\quad \le MB_2T-T\epsilon \sum _{m=0}^{M-1}\sum _i\sum _j \left( Q_{ij}(mT)+q_{ij}(mT)\right) +MVo^{\pi '}_{\lambda +\epsilon {\mathbf {1}}}. \end{aligned}$$

Rearraying the terms, dividing both sides by $MT\epsilon$, and letting $M\rightarrow \infty$, we obtain

$$\begin{aligned} {1\over M}\sum _{m=0}^{M-1}\sum _i\sum _j \left( Q_{ij}(mT)+q_{ij}(mT)\right) \le {B_2\over \epsilon }+{V\left( o^{\pi '}_{\lambda +\epsilon {\mathbf {1}}}-o^*\right) \over T\epsilon }. \end{aligned}$$

Since $o^*\ge 0$ and $o^{\pi '}_{\lambda +\epsilon {\mathbf {1}}}\le \sum _j \left( x_{\max }^\mathrm{o} c_j^\mathrm{o}+x_{\max }^\mathrm{s}c_j^{\max }\right)$, part 2) is proved.

Appendix 4: Proof of Theorem 5

Part 1. For a data center j, suppose that $\sum _i q_{ij}(t)\le {\texttt {buffer}}_j$, we show that $\sum _i q_{ij}(t+T)\le {\texttt {buffer}}_j$.

• If $\sum _i q_{ij}(t)\le \sum _i {Vc_j^\mathrm{o}\over Ts_i}$, since there are at most $nkr_{\max }$ jobs can be routed to a data center in this frame, we have $\sum _i q_{ij}(t+T)\le \sum _i {Vc_j^\mathrm{o}\over Ts_i}+nkr_{\max }$

• If $\sum _i q_{ij}(t)> \sum _i {Vc_j^\mathrm{o}\over Ts_i}$, then at least one queue in data center j satisfies $Vc_j^\mathrm{o}-Tq_{ij}(t)s_i<0$. According to the on-demand instance provisioning policy (24), broker j will rent $x_{\max }^\mathrm{o}$ on-demand instances. The assumption $\sum _i r_{\max }s_i\le Tx_{\max }^\mathrm{o} s_{\min }$ indicates that the number of served jobs is no less than the number of arrived jobs given $x_{\max }^\mathrm{o}$ amount of on-demand instances in this frame. Combined with queue dynamic (2), we can conclude that the number of buffered jobs cannot increase in the next frame, which proves (26).

Part 2. The proof is similar to part 1. For broker j, we can see that

• If $\sum _i Q_{ij}(t)\le {\texttt {buffer}}_j$, then $\sum _i Q_{ij}(t+T)\le {\texttt {buffer}}_j+Tka_{\max }$ since there can be at most $a_{\max }$ type-i jobs arrived at broker j in each slot.

• If $\sum _i Q_{ij}(t)>{\texttt {buffer}}_j$, then at least one broker queue satisfies $Q_{ij}(t)-q_{ik}(t)>0$. Using the solution framework in Sect. 5.1, and combining with the assumption $Ta_{\max }\le r_{\max }$ and queue dynamic (1), we have (27).