Analysis of performance trade-offs for an adaptive channel-aware wireless scheduler

Abstract

In this paper, we consider the scheduling problem where data packets from K input-flows need to be delivered to K corresponding wireless receivers over a heterogeneous wireless channel. Our objective is to design a wireless scheduler that achieves good throughput and fairness performance while minimizing the buffer requirement at each wireless receiver. This is a challenging problem due to the unique characteristics of the wireless channel. We propose a novel idea of exploiting both the long-term and short-term error behavior of the wireless channel in the scheduler design. In addition to typical first-order Quality of Service (QoS) metrics such as throughput and average delay, our performance analysis of the scheduler permits the evaluation of higher-order metrics, which are needed to evaluate the buffer requirement. We show that variants of the proposed scheduler can achieve high overall throughput or fairness as well as low buffer requirement when compared to other wireless schedulers that either make use only of the instantaneous channel state or are channel-state independent in a heterogenous channel.

Appendix: proof of Theorem 1

We begin with the derivation of the expressions for per-flow throughput and fairness in terms of $(p_{{\mathcal{D}}}, p_{{\mathcal{S}}^{1}|0} \hbox{and} p_{{\mathcal{S}}^{\eta^+}|0}).$ Let $T^{j|x_{i}}$ denote the throughput of flow j in slot i given x _i . From [7], the per-flow throughput of a K-flow CSD scheduler with uniform arbitration is given as follows:

$$ T^{j|a_{i}}=\left\{ \begin{array}{ll} p_{{\mathcal{S}}|0}, & a_{i}=j;\\ \frac{p_{{\mathcal{S}}|0}\cdot p_{{\mathcal{D}}}(1-p_{{\mathcal{D}}}^{K-1})}{(K-1)(1-p_{{\mathcal{D}}})}, & \hbox{otherwise}, \end{array}\right. $$

where $p_{{\mathcal{S}}|m}$ = Prob(a flow transmits successfully given that m other eligible flows exist). Applying the above expression for $T^{j|a_{i}}$ in our (η+1)-flow CSD scheduling scenario, we obtain the following:

$$ T^{j \in {\bf C}^{1}|a_{i}}=\left\{ \begin{array}{ll} p_{{\mathcal{S}}^{1}|0}, & a_{i}=j;\\ \frac{p_{{\mathcal{S}}^{1}|0}\cdot p_{{\mathcal{D}}}(1-p_{{\mathcal{D}}}^{\eta})}{\eta(1-p_{{\mathcal{D}}})}, & \hbox{otherwise}, \end{array}\right. $$

$$ T^{\eta^+|a_{i}}=\left\{ \begin{array}{ll} p_{{\mathcal{S}}^{\eta^+}|0}, & a_{i}=\eta^+;\\ \frac{p_{{\mathcal{S}}^{\eta^+}|0}\cdot p_{{\mathcal{D}}}(1-p_{{\mathcal{D}}}^{\eta})}{\eta(1-p_{{\mathcal{D}}})}, & \hbox{otherwise}. \end{array}\right. $$

According to Eq. 6, for any i, we have the following:

$$ Prob(a_{i}=j)=\left\{ \begin{array}{ll} \frac{1}{K}, & j \in {\bf C}^{1};\\ \frac{K-\eta}{K}, & j = \eta^+. \end{array}\right. $$

Hence, unconditioning the expressions for $T^{j|a_{i}}$ on a _i , we obtain the following expressions:

$$ \begin{aligned} T^{j \in {\mathbf{C}}^{1}}&=\frac{1}{K}p_{{\mathcal{S}}^{1}|0} + \frac{K-1}{K}\frac{p_{{\mathcal{S}}^{1}|0}\cdot p_{{\mathcal{D}}}(1-p_{{\mathcal{D}}}^{\eta})}{(\eta)(1-p_{{\mathcal{D}}})},\\ T^{\eta^+}&=\frac{K-\eta}{K}p_{{\mathcal{S}}^{\eta^+}|0} + \frac{\eta}{K}\frac{p_{{\mathcal{S}}^{\eta^+}|0}\cdot p_{{\mathcal{D}}}(1-p_{{\mathcal{D}}}^{\eta})}{(\eta)(1-p_{{\mathcal{D}}})}. \end{aligned} $$

Let us consider the aggregate flow, η⁺, which comprises packets of flows η + 1, η + 2,...,K. Since the weight of each flow is identical, the probability that a flow j packet is HOL at any instant is identical and given by $\frac{1}{K-\eta}$ for $\eta+1 \leq j \leq K.$

Hence, for j ∈ C ², we obtain the following:

$$ \begin{aligned} T^{j}&=\frac{1}{K-\eta}T^{\eta^+}\\ &=\frac{p_{{\mathcal{S}}^{\eta^+}|0}}{K} + \frac{p_{{\mathcal{S}}^{\eta^+}|0}\cdot p_{{\mathcal{D}}}(1-p_{{\mathcal{D}}}^{\eta})}{K(K-\eta)(1-p_{{\mathcal{D}}})}. \end{aligned} $$

Substituting the expressions for T ^j into Eq. 3, we obtain the expression for FM as given in Theorem 1.

Next, we derive the expressions for $p_{{\mathcal{D}}},$ $p_{{\mathcal{S}}^{1}|0}$ and $p_{{\mathcal{S}}^{\eta^+}|0}$ in terms of (p _c (0),ε).

According to our transmission algorithm, a flow j will defer its transmission in slot i only if it is not eligible for transmission, i.e., when $\hat{c}^{j}_{i}=1.$ The corresponding probability is given as follows:

$$ \begin{aligned} p_{{\mathcal{D}}}&=\sum_{x=0}^{1}\hbox{Prob}(\hat{c}^{j}_{i}=1|c^{j}_{i-1}=x)\cdot \hbox{Prob}(c^{j}_{i-1}=x)\\&=1-p_{c}(0). \end{aligned} $$

We note that $p_{{\mathcal{D}}}$ is independent of the channel agility, g, and hence, it is the same for all flows.

In the absence of other eligible flows, a flow j will transmit successfully in slot i as long as $\hat{c}^{j}_{i}=c^{j}_{i}=0.$ Therefore, we can evaluate $p_{{\mathcal{S}}^{1}|0}$ as shown below, where $p_{x,y}(X,Y)\equiv\hbox{Prob}(x=X,y=Y)$ and $p_{x|y}(X|Y)\,\equiv\hbox{Prob}(x=X|y=Y):$

$$ \begin{aligned} p_{{\mathcal{S}}^{1}|0}&=\sum_{x=0}^{1}p_{c^{j}_{i},\hat{c}^{j}_{i}|c^{j}_{i-1}}(0,0|x)\cdot p_{c^{j}_{i-1}}(x)\\&=\sum_{x=0}^{1}p_{c^{j}_{i}|c^{j}_{i-1}}(0|x)\cdot p_{\hat{c}^{j}_{i}|c^{j}_{i-1}}(0|x)\\&=p_{c}(0)p_{0|0}. \end{aligned} $$

Substituting for p _0|0 in terms of (p _c (0),ε), we obtain the expressions as given in Theorem 1. The corresponding expression for flow η⁺ is obtained by replacing ε with 1.0.