t\right) }\right.\!\right\}\\
&\!-\!\sum\limits_{i=1}^{I}\mathbf{E}\left\{
\sum\limits_{j=1}^{J_{i}}r_{ij}\left( t\right) \left(
\sum\limits_{k=1}^{K}E_{k}\left( t\right) W_{k,ij}\left( t\right)
\!+\!H_{i}\left( t\right) \eta ^{-}\right) \cdot 1_{\left\{ Q_{ij}\left(
t\right) =0 \sum\limits_{j=1}^{J_{i}}r_{ij}\left( t\right) \geq
U_{i}\left( t\right) \right\} }\left\vert \overrightarrow{\theta \left(
t\right) }\right. \right\}\\
&\!+\!\sum\limits_{i=1}^{I}\!\mathbf{E}\!\left\{ H_{i}\left( t\right) R_{i}\left(
t\right) \eta ^{+}\!\cdot\!1_{\left\{ \sum\limits_{j=1}^{J_{i}}r_{ij}\left(
t\right)\!<U_{i}\left( t\right)\!\right\} }\!\left\vert\! \overrightarrow{\theta
\left( t\right) }\right. \right\}\!+\!\sum\limits_{i=1}^{I}\mathbf{E}\!\left\{ H_{i}\left( t\right) \eta ^{-}U_{i}\left( t\right)\!\cdot\!1_{\left\{
\sum\limits_{j=1}^{J_{i}}r_{ij}\left( t\right) \geq U_{i}\left( t\right)
\right\} }\left\vert \overrightarrow{\theta \left( t\right) }\right.
\right\}\\
&+\sum\limits_{i=1}^{I}\mathbf{E}\left\{
D_{i}^{d}\left( t\right) \left( H_{i}\left( t\right) \eta ^{-}\!+\!Vc\left(
t\right) \right) \cdot 1_{\left\{ \sum\limits_{j=1}^{J_{i}}r_{ij}\left(
t\right) \geq U_{i}\left( t\right) \right\} }\left\vert \overrightarrow{\theta \left( t\right) }\right. \right\},
\end{split}$$where $\delta_{max} \!\!= \!\!\sum_{i=1}^{I}J_i\alpha_{max}\!\! +\!\! I\beta_{max}\!\! <\!\! \infty$, and $V$ is an important parameter in regulating the trade-off between system cost and storage capacity. By adopting the framework of Lyapunov optimization, two sub-problems only relying on current information are derived by minimizing the right hand side of (\[eq\_15\]) approximately subject to the constraints in $\widetilde\textbf{P1}$. Later it is proven that the proposed algorithm can satisfy the constraints of (\[eq\_5\]). Notice that (\[eq\_1\]) has been replaced by (\[eq\_9\]). $$\begin{aligned}
\label{eq_16}
&\textbf{P2:}~~\text{If}~~Q_{ij}(t)\!>\!0 \notag\\
&\sum\limits_{i=1}^{I}\!\left\{\!\sum\limits_{j=1}^{J_{i}}r_{ij}\left(
t\right) \left( Q_{ij}\left( t\right)\!+\!H_{i}\left( t\right) \eta ^{-}\right)
\!\cdot\!1_{\left\{ Q_{ij}\left( t\right)>0\right\}}\!+\!\sum\limits_{j=1}^{J_{i}}r_{ij}\left( t\right)\!\left(
\!\sum\limits_{k=1}^{K}\!E_{k}\left( t\right) W_{k,ij}\left( t\right)
\!+\!H_{i}\left( t\right) \eta ^{-}\right)\!\cdot\!1_{\left\{ Q_{ij}\left(
t\right) =0\right\}}\right.\notag\\
&\left.\!-\!D_{i}^{d}\left( t\right) \left( H_{i}\left( t\right)
\eta ^{-}\!+\!Vc\left( t\right) 