{\sqrt{2\epsilon}}\right)\right]^+,
\end{split}$$ where $\kappa$ is a sufficiently small positive constant, known as the step size and $n$ is the iteration index in time slot $t$.

Since $\Lambda_i$ is a linear function of $D_i^d(t)$, the optimum of (\[eq\_24\]) depends on the monotonicity of $\Lambda_i$. Algorithm 2 states how to choose $D_i^d(t)$, where $M$ is the greatest number of iterations and $\xi$ is the convergence criterion. The following result characterizes the effects of constant step size on the convergence property.

By the above iteration (\[eq\_25\]) with constant stepsize, the dual function (22) is guaranteed to converge to a suboptimal value within a finite number of steps, *i.e.,* $\varphi _{t}^{\text{*}}\left( \vec{\lambda}\right)\!-\!\varphi _{n,t}^{\text{best}}\left( \vec{\lambda}\left( m,t\right) \right)\!<\!\kappa
L\max_{l}\left\{ P_{l}^{2}\!+\!\left( D_{l}^{\max }\!+\!\mathbf{E}\left(
N_{l}\right)\!+\!\frac{\sigma _{N_{l}}}{\sqrt{2\epsilon }}\right) ^{2}\right\} $ within at most $\frac{\left\Vert \vec{\lambda}\left( 0,t\right)\!-\!\vec{\lambda}^{\ast }\left( t\right) \right\Vert _{2}^{2}}{\kappa ^{2}L\max_{l}\left\{
P_{l}^{2}\!+\!\left( D_{{}}^{\max }\!+\!\mathbf{E}\left( N_{l}\right)\!+\!\frac{\sigma
_{N_{l}}}{\sqrt{2\epsilon }}\right) ^{2}\right\} }$ steps, where $\varphi
_{t}^{\ast }\left( \vec{\lambda}\right) $ is the optimal value of (22) and $\varphi _{n,t}^{\text{best}}\left( \vec{\lambda}\left( m,t\right)
\right) \hat{=}\max_{m\in \left\{ 1,\cdots ,n\right\} }\varphi _{m,t}\left(
\vec{\lambda}\left( m,t\right) \right) $ is the best value of (22) till now, $\vec{\lambda}\left( 0,t\right) $ is the initial value of Lagrangian multiplier vector and $\vec{\lambda}^{\ast }\left( t\right) $ is the optimal value vector. The above 