Optimal Downlink Scheduling for Heterogeneous Traffic Types in LTE-A Based on MDP and Chance-Constrained Approaches

Abstract

The current mobile broadband market experiences major growth in data demand and average revenue loss. To remain profitable from the perspective of a service provider (SP), one needs to maximize revenue as much as possible by making subscribers satisfied within the limited budget. On the other hand, traffic demands are moving toward supporting the wide range of heterogeneous applications with different quality of service (QoS) requirements. In this paper, we consider two related packet scheduling problems, i.e., long-term and short-term approaches in the 4th generation partnership project (3GPP) long term evolution-advanced (LTE-A) system. In the long-term approach, the long-term average revenue of SP subject to the long-term QoS constraints for heterogeneous traffic demands is optimized. The problem is first formulated as a constrained Markov decision process (CMDP) problem, of which the optimal control policy is achieved by utilizing the channel and queue information simultaneously. Subsequently, in the short-term approach, we consider the short-term revenue optimization problem which stochastically guarantees the short-term QoS for heterogeneous traffic demands through a set of chance constraints. To make the proposed chance-constrained programming problem computationally tractable, we use the Bernstein approximation technique to analytically approximate the chance constraint as a convex conservative constraint. Finally, the proposed packet scheduling schemes and solution methods are validated via numerical simulations.

Appendix

1.1 Proof of proposition 2

To apply the Bernstein approximation for the constraint in Eq. 18, based on our commonly used assumption of backlogged traffic model we can omit the [⋅]⁺ operator form the constraint. In the backlogged traffic model, as the name suggests, each user has always packet to transmit. Accordingly, the inequality $q_{i}(n)-{\sum }_{j\in \mathcal {R}}{\sum }_{m\in \mathcal {M}} r_{ij}^{m}(n)x_{ij}^{m}+a_{i}(n)>\beta _{i}$ can be equivalently expressed as

$$ F_{i}(\mathbf{x},\mathbf{\zeta_{i}})> 0, $$

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where ζ _i =(r _i ,a _i ) and

$$ F_{i}(\mathbf{x},\mathbf{\zeta_{i}})\triangleq q_{i}(n)-\sum\limits_{j\in\mathcal{R}}\sum\limits_{m\in\mathcal{M}} r_{ij}^{m}(n)x_{ij}^{m}+a_{i}(n)-\beta_{i}. $$

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F _i (x,ζ _i ) is in the form of affine chance constraint which involves linear form of the random variables ζ _i =(r _i ,a _i ). Based on the Bernstein approximation, constraint (18) can be approximated by

$$ \inf_{\varrho_{i}>0}\{{\Psi}_{i}(\mathbf{x},\mathbf{\zeta_{i}})-\varrho_{i}\log \epsilon_{i}\} \leq 0, \forall i $$

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where

$$\begin{array}{@{}rcl@{}} {\Psi}_{i}(\mathbf{x},\mathbf{\zeta_{i}}) &=&\varrho_{i}\log\mathbb{E}[\exp(\varrho_{i}^{-1}(F_{i}(\mathbf{x},\mathbf{\zeta_{i}}))]\\ &=&q_{i}+\varrho_{i}\sum\limits_{j\in\mathcal{R}}\sum\limits_{m\in\mathcal{M}} {\Lambda}_{r_{ij}^{m}}(-\varrho_{i}^{-1}x_{ij}^{m})+\varrho_{i}{\Lambda}_{a_{i}}(\varrho_{i}^{-1})-\beta_{i}, \end{array} $$

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where ${\Lambda }_{r_{ij}^{m}}$ and ${\Lambda }_{a_{i}}$ are the cumulant (log-moment) generating function of the data rate and arrival process, respectively.

In the sequel, we derive the cumulant generating function of the random variable $r_{ij}^{m}$, which is a function of the channel gain random variable. The moment generating function (MGF) of $r(\gamma _{ij}^{m})=W\log (1+\gamma _{ij}^{m})$, where $\gamma _{ij}^{m}=\frac {pg_{ij}}{{\Delta }_{m} \sigma ^{2}}$, is given in [7] as follows

$$ M_{r_{ij}^{m}}(y)=\mathbb{E}[\exp(-yr(\gamma_{ij}^{m}))]=1+{\int}_{0}^{+\infty}Q(\ln(2)y,\xi)M_{\gamma_{ij}^{m}}^{(1)}(\xi)\text{d}\xi, $$

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where Q(a,u)=Γ(a,u)/Γ(a) is the regularized Gamma function¹ and $M_{\gamma _{ij}^{m}}^{(1)}(\cdot )$ is the first derivative of the MGF of $\gamma _{ij}^{m}$. $M_{\gamma _{ij}^{m}}(\cdot )$ is known in closed-form for many fading distributions [15]. Without loss of generality, consider the channel gain as an exponentially distributed random variable with PDF given by $f_{g_{ij}}(\xi )=\frac {1}{\mu _{i}}\exp (-\frac {\xi }{\mu _{i}})$, where μ _i is the mean slow fading channel gain for user i and can be characterized based on path-loss and shadowing effects. In practice, the difference between different resource blocks is indistinguishable for slow fading channel parameter μ _i . The value of μ _i oughts to be updated at the beginning of each time window t. Hence, the cumulant generating function of the data rate can be achieved as

$$\Lambda_{r_{ij}^{m}}(y)=\log\left[\int_{0}^{\infty}(1+\frac{p\xi}{\Delta_{m} \sigma^{2}})^{\frac{Wy}{\ln2}}.\frac{1}{\mu_{i}}\exp(-\frac{\xi}{\mu_{i}})\text{d}\xi\right].$$

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To calculate the cumulant generating function of the arrival process, e.g., ${\Lambda }_{a_{i}}$, without loss of generality consider the arrival process for the traffic of the user i follows Poisson distribution with parameter as $f_{a_{i}}(k)=\frac {\bar {a}_{i}^{k} e^{-\bar {a}_{i}}}{k!}$, where $\bar {a}_{i}$ represents the average rate. ${\Lambda }_{a_{i}}$ can be computed as

$$ {\Lambda}_{a_{i}}(y)=\log\mathbb{E}[e^{ya_{i}}]=\log(e^{\bar{a_{i}}(e^{y}-1)})=\frac{\bar{a_{i}}(e^{y}-1)}{\ln10}. $$

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