A hierarchical approach for resource allocation in hybrid cloud environments

Abstract

Cloud computing is a key technology for online service providers. However, current online service systems experience performance degradation due to the heterogeneous and time-variant incoming of user requests. To address this kind of diversity, we propose a hierarchical approach for resource management in hybrid clouds, where local private clouds handle routine requests and a powerful third-party public cloud is responsible for the burst of sudden incoming requests. Our goal is to answer (1) from the online service provider’s perspective, how to decide the local private cloud resource allocation, and how to distribute the incoming requests to private and/or public clouds; and (2) from the public cloud provider’s perspective, how to decide the optimal prices for these public cloud resources so as to maximize its profit. We use a Stackelberg game model to capture the complex interactions between users, online service providers and public cloud providers, based on which we analyze the resource allocation in private clouds and pricing strategy in public cloud. Furthermore, we design efficient online algorithms to determine the public cloud provider’s and the online service provider’s optimal decisions. Simulation results validate the effectiveness and efficiency of our proposed approach.

Appendices

Appendix 1: Derive the infimum of $\mathcal {F}_{j}\left( \mathcal {N}_{j},p_{cj}\right)$

In Sect. 4, we combine Eqs. (16), (17) and (18), and thus we obtain each dimensional $\mathcal {F}_{j}\left( \mathcal {N}_{j},p_{cj}\right)$ in Eq. (35). Note that we divide the $\mathcal {F}_{j}\left( \mathcal {N}_{j},p_{cj}\right)$ into three individual parts, i.e., $\mathbf {A}$, $\mathbf {B}$ and $\mathbf {C}$.

$$\begin{aligned} \mathcal {F}_{j}= & {} \int _{t\in \left[ 0,T\right] }\int _{s_{j}^{t}}\left[ APP_{j}\left( \mathcal {N}_{j},\mathcal {S}_{j}^{t}\right) p_{cj} -\mathcal {N}_{j}p_{lj}\right] \mathcal {G}_j\left( s_{j}^{t}\right) \text {d}s_{j}^{t}\text {d}t\nonumber \\= & {} \int _0^{T}\int _{\sum \limits _{i}k_{imin}}^{\mathcal {N}_{j}}\left( s_{j}^{t}p_{cj}-\mathcal {N}_{j}p_{lj}\right) \mathcal {G}_j\left( s_{j}^{t}\right) \text {d}s_{j}^{t}\text {d}t +\int _0^{T}\int _{\mathcal {N}_{j}}^{\sum \limits _{i}k_{imax}}\left( \mathcal {N}_{j}p_{cj}-\mathcal {N}_{j}p_{lj}\right) \mathcal {G}_j\left( s_{j}^{t}\right) \text {d}s_{j}^{t}\text {d}t\nonumber \\= & {} \int _0^{T}\int _{\sum \limits _{i}k_{imin}}^{\mathcal {N}_{j}}\left[ s_{j}^{t}p_{cj}-\mathcal {N}_{j}p_{lj}\right] \cdot \frac{1}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}} \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\text {d}t\nonumber \\&+\int _0^{T}\int _{\mathcal {N}_{j}}^{\sum \limits _{i}k_{imax}}\left[ \mathcal {N}_{j}p_{cj}-\mathcal {N}_{j}p_{lj}\right] \cdot \frac{1}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}} \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\text {d}t\nonumber \\= & {} \int _0^{T}\frac{p_{cj}}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\sum \limits _{i}k_{imin}}^{\mathcal {N}_{j}}s_{j}^{t} \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\text {d}t\nonumber \\&+\int _0^{T}\frac{\mathcal {N}_{j}p_{cj}}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\mathcal {N}_{j}}^{\sum \limits _{i}k_{imax}} \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\text {d}t\nonumber \\&-\int _0^{T}\frac{\mathcal {N}_{j}p_{lj}}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\sum \limits _{i}k_{imin}}^{\sum \limits _{i}k_{imax}} \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\text {d}t\nonumber \\= & {} \int _0^{T}\mathbf {A}\text {d}t+\int _0^{T}\mathbf {B}\text {d}t-\int _0^{T}\mathbf {C}\text {d}t. \end{aligned}$$

(35)

We consider the error function in the probability analysis: $erf\left( z\right) =\frac{2}{\sqrt{\pi }}\int _{0}^{z}\exp \left( -x^2\right) \text {d}x$. Then $\mathbf {A}$, $\mathbf {B}$ and $\mathbf {C}$ can be computed as Eq. (36–38), respectively.

$$\begin{aligned} \mathbf {A}= & {} \frac{p_{cj}}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\sum \limits _{i}k_{imin}}^{\mathcal {N}_{j}}s_{j}^{t} \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\nonumber \\= & {} \frac{p_{cj}}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\sum \limits _{i}k_{imin}}^{\mathcal {N}_{j}}\left( s_{j}^{t}-\hat{\mu }_{j}^{t}+\hat{\mu }_{j}^{t}\right) \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\nonumber \\= & {} \frac{p_{cj}}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\sum \limits _{i}k_{imin}}^{\mathcal {N}_{j}}(s_{j}^{t}- \hat{\mu }_{j}^{t})\exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t} +\frac{p_{cj}}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\sum \limits _{i}k_{imin}}^{\mathcal {N}_{j}}\hat{\mu }_{j}^{t}\exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\nonumber \\= & {} \frac{p_{cj}\hat{\sigma }_{j}^{t}}{\sqrt{2\pi }}\exp \left[ {-\left( \mathcal {N}_{j}-\hat{\mu }_{j}^{t}\right) ^2/{2\left( {\hat{\sigma }_{j}^t}\right) ^2}}\right] -\frac{p_{cj}\hat{\sigma }_{j}^{t}}{\sqrt{2\pi }}\exp \left[ {-\left( \sum \limits _{i}k_{imin}-\hat{\mu }_{j}^{t}\right) ^2/{2\left( {\hat{\sigma }_{j}^t}\right) ^2}}\right] \nonumber \\&+\frac{p_{cj}\hat{\mu }_{j}^{t}}{2}erf\left[ \left( \mathcal {N}_{j}-\hat{\mu }_{j}^{t}\right) /{\sqrt{2}\hat{\sigma }_{j}^{t}}\right] -\frac{p_{cj}\hat{\mu }_{j}^{t}}{2}erf\left[ \left( {\sum \limits _{i}k_{imin}-\hat{\mu }_{j}^{t}}\right) /{\sqrt{2}\hat{\sigma }_{j}^{t}}\right] , \end{aligned}$$

(36)

$$\begin{aligned} \mathbf {B}= & {} \frac{\mathcal {N}_{j}p_{cj}}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\mathcal {N}_{j}}^{\sum \limits _{i}k_{imax}} \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\nonumber \\= & {} \frac{\mathcal {N}_{j}p_{cj}}{2}\cdot \frac{2}{\sqrt{\pi }}\int _{0}^{\frac{\sum \limits _{i}k_{imax}-\hat{\mu }_{j}^{t}}{\sqrt{2}\hat{\sigma }_{j}^{t}}}\exp \left( -\tau ^2\right) \text {d}\tau -\frac{\mathcal {N}_{j}p_{cj}}{2}\cdot \frac{2}{\sqrt{\pi }}\int _{0}^{\frac{\mathcal {N}_{j}-\hat{\mu }_{j}^{t}}{\sqrt{2}\hat{\sigma }_{j}^{t}}}\exp (-\tau ^2)\text {d}\tau \nonumber \\= & {} \frac{\mathcal {N}_{j}p_{cj}}{2}erf\left[ \left( {\sum \limits _{i}k_{imax}-\hat{\mu }_{j}^{t}}\right) /{\sqrt{2}\hat{\sigma }_{j}^{t}}\right] -\frac{\mathcal {N}_{j}p_{cj}}{2}erf\left[ \left( {\mathcal {N}_{j}-\hat{\mu }_{j}^{t}}\right) /{\sqrt{2}\hat{\sigma }_{j}^{t}}\right] , \end{aligned}$$

(37)

and with the last part

$$\begin{aligned} \mathbf {C}= & {} \frac{\mathcal {N}_{j}p_{lj}}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\sum \limits _{i}k_{imin}}^{\sum \limits _{i}k_{imax}} \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\nonumber \\= & {} \frac{\mathcal {N}_{j}p_{lj}}{2}\cdot \frac{2}{\sqrt{\pi }}\int _{0}^{\frac{\sum \limits _{i}k_{imax}-\hat{\mu }_{j}^{t}}{\sqrt{2}\hat{\sigma }_{j}^{t}}}\exp \left( -\tau ^2\right) \text {d}\tau -\frac{\mathcal {N}_{j}p_{lj}}{2}\cdot \frac{2}{\sqrt{\pi }}\int _{0}^{\frac{\sum \limits _{i}k_{imin}-\hat{\mu }_{j}^{t}}{\sqrt{2}\hat{\sigma }_{j}^{t}}}\exp \left( -\tau ^2\right) \text {d}\tau \nonumber \\= & {} \frac{\mathcal {N}_{j}p_{lj}}{2}erf\left[ \left( {\sum \limits _{i}k_{imax}-\hat{\mu }_{j}^{t}}\right) /{\sqrt{2}\hat{\sigma }_{j}^{t}}\right] -\frac{\mathcal {N}_{j}p_{lj}}{2}erf\left[ \left( {\sum \limits _{i}k_{imin}-\hat{\mu }_{j}^{t}}\right) /{\sqrt{2}\hat{\sigma }_{j}^{t}}\right] . \end{aligned}$$

(38)

Conclude Eq. (36) to (38), we can derive Eq. (35) as

$$\begin{aligned} \mathcal {F}_{j}\left( \mathcal {N}_{j},p_{cj}\right) = \int _{0}^{T}\mathbf {A}+\mathbf {B}-\mathbf {C}\text {d}t \end{aligned}$$

(39)

Due to the generality of problem formulation, it is hard to obtain an accurate formulas to Eq. (35). Instead, combined with Eq. (35) and properties of $erf\left( z\right)$, we can obtain the lower bound of $\mathcal {F}_{j}\left( \mathcal {N}_{j},p_{cj}\right)$ as

$$\begin{aligned} \inf \mathcal {F}_{j}\left( \mathcal {N}_{j},p_{cj}\right)= & {} \frac{p_{cj}\hat{\sigma }_{j}^{t}}{\sqrt{2\pi }}\int _{0}^{T}\rho \left( \mathcal {N}_{j},t\right) \text {d}t-\frac{p_{cj}}{2}\int _{0}^{T}\hat{\mu }_{j}^{t}\text {d}t -\frac{p_{cj}+2p_{lj}}{2}\mathcal {N}_{j}T\nonumber \\\leqslant & {} \mathcal {F}_{j}\left( \mathcal {N}_{j},p_{cj}\right) , \end{aligned}$$

(40)

where

$$\begin{aligned} \rho \left( \mathcal {N}_{j},t\right) =\exp \left[ {-\left( \mathcal {N}_{j}-\hat{\mu }_{j}^{t}\right) ^2/{2\left( {\hat{\sigma }_{j}^t}\right) ^2}}\right] -\exp \left[ {-\left( \sum \limits _{i}k_{imin}-\hat{\mu }_{j}^{t}\right) ^2/{2\left( {\hat{\sigma }_{j}^t}\right) ^2}}\right] . \end{aligned}$$

(41)

Note that a natural assumption is $\sum \limits _{i}k_{imin}\leqslant \hat{\mu }_{j}^{t}\leqslant \sum \limits _{i}k_{imax}$. Up till now, we obtain the infimum of $\mathcal {F}_{j}\left( \mathcal {N}_{j},p_{cj}\right)$ in Sect. 4.

Appendix 2: Derive the infimum of $\mathcal {H}_{j}\left( p_{cj}\right)$

In Sect. 5, we formulate each dimensional $\mathcal {H}_{j}\left( p_{cj}\right)$ as Eq. (24). Note that the first term equals 0, because the case $APP_{j}\left( \mathcal {N}_{j},\mathcal {S}_{j}^{t}\right) =\mathcal {S}_{j}^{t}$ implies the OSP does not need to purchase public resources. Thus, we can rewrite Eq. (24) as follows:

$$\begin{aligned} \mathcal {H}_{j}\left( p_{cj}\right)= & {} \int _{0}^{T}\int _{\sum \limits _{i}k_{imin}}^{\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) } \left( s_{j}^{t}-s_{j}^{t}\right) {\mathcal {G}}_j\left( s_{j}^{t}\right) p_{cj}\text {d}s_{j}^{t}\text {d}t\nonumber \\&+\int _{0}^{T}\int _{\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) }^{\sum \limits _{i}k_{imax}}\left[ s_{j}^{t}-\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) \right] {\mathcal {G}}_j\left( s_{j}^{t}\right) p_{cj}\text {d}s_{j}^{t}\text {d}t\nonumber \\= & {}\,0+\int _{0}^{T}\mathbf {D}\text {d}t. \end{aligned}$$

(42)

Now we compute the term $\mathbf {D}$ as Eq. (43).

$$\begin{aligned} \mathbf {D}= & {} \int _{\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) }^{\sum \limits _{i}k_{imax}}\left[ s_{j}^{t}-\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) \right] {\mathcal {G}}_j\left( s_{j}^{t}\right) p_{cj}\text {d}s_{j}^{t}\nonumber \\= & {} \frac{p_{cj}}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) }^{\sum \limits _{i}k_{imax}}\left[ s_{j}^{t}-\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) \right] \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\nonumber \\= & {} \frac{p_{cj}}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) }^{\sum \limits _{i}k_{imax}}\left[ s_{j}^{t}-\hat{\mu }_{j}^{t}\right] \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\nonumber \\&+\frac{p_{cj}}{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) }^{\sum \limits _{i}k_{imax}}\left[ \hat{\mu }_{j}^{t}-\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) \right] \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\nonumber \\= & {} \frac{p_{cj}\hat{\sigma }_{j}^{t}}{\sqrt{2\pi }}\int _{-\frac{\left( \hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) -\hat{\mu }_{j}^{t}\right) ^2}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}} ^{-\frac{\left( \sum \limits _{i}k_{imax}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}}\exp \left( \tau \right) \text {d}\tau +\frac{p_{cj}\left[ \hat{\mu }_{j}^{t}-\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) \right] }{\sqrt{2\pi }\hat{\sigma }_{j}^{t}}\int _{\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) }^{\sum \limits _{i}k_{imax}} \exp \left[ -\frac{\left( s_{j}^{t}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \text {d}s_{j}^{t}\nonumber \\= & {} \frac{p_{cj}\hat{\sigma }_{j}^{t}}{\sqrt{2\pi }}\exp \left[ -\frac{\left( \sum \limits _{i}k_{imax}-\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] -\frac{p_{cj}\hat{\sigma }_{j}^{t}}{\sqrt{2\pi }}\exp \left[ -\frac{\left( \hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) -\hat{\mu }_{j}^{t}\right) ^{2}}{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] \nonumber \\&+\frac{p_{cj}\left[ \hat{\mu }_{j}^{t}-\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) \right] }{2} erf\left( \frac{\sum \limits _{i}k_{imax}-\hat{\mu }_{j}^{t}}{\sqrt{2}\hat{\sigma }_{j}^{t}}\right) -\frac{p_{cj}\left[ \hat{\mu }_{j}^{t}-\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) \right] }{2}erf\left( \frac{\hat{\mathcal {N}}_{j}^{*}-\hat{\mu }_{j}^{t}}{\sqrt{2}\hat{\sigma }_{j}^{t}}\right) . \end{aligned}$$

(43)

Similar to the previous derivations in Appendix 1, we can have

$$\begin{aligned} \inf \mathcal {H}_{j}\left( p_{cj}\right)= & {} \frac{p_{cj}}{\sqrt{2\pi }}\int _{0}^{T}{\hat{\sigma }_{j}^t}\omega \left( p_{cj},t\right) \text {d}t +\frac{p_{cj}}{2}\int _{0}^{T}\hat{\mu }_{j}^t\text {d}t-\frac{1}{2}\hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) p_{cj}T\nonumber \\\leqslant & {} \mathcal {H}_{j}\left( p_{cj}\right) , \end{aligned}$$

(44)

where

$$\begin{aligned} \omega \left( p_{cj},t\right) =\exp \left[ -\left( \sum \limits _{i}k_{imax}-\hat{\mu }_{j}^{t}\right) ^{2}/{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] - \exp \left[ -\left( \hat{\mathcal {N}}_{j}^{*}\left( p_{cj}\right) -\hat{\mu }_{j}^{t}\right) ^{2}/{2\left( {\hat{\sigma }_{j}^t}\right) ^2}\right] . \end{aligned}$$

(45)

Up till now, we obtain the infimum of $\mathcal {H}_{j}\left( p_{cj}\right)$ in Sect. 5.