Throughput-Delay Trade-off Scheduling in Multi-channel Downlink Wireless Networks

Abstract

This paper designs a down-link resource scheduling algorithm for wireless cellular networks to achieve throughput-delay trade-off. The available bandwidth of the down-link is divided into some parallel sub-channels, and each sub-channel can be allocated to one potential user in every time-slot. This problem is modeled as a multi-user multi-server discrete-time queuing system with a time-varying connectivity. For this system, it is well-known that the classical MaxWeight algorithm has the optimal throughput, but its delay performance is very poor. To overcome this issue, we use the Lyapunov Optimization technique to design a throughput-utility maximizing algorithm that provides a good trade-off between the throughput and delay performance. Our approach is verified by both theoretical analysis and simulation evaluations.

Appendices

Appendix 1: Proofs of the Lemma 1 and Lemma 2

The proof of Lemma 1: let $\tau \in [0,1, \cdots ,t - 1]$, squaring both sides of the queueing dynamic (1) and using the fact that for any $x \in \mathbb {R}, (max[x,0])^2 \le x^2$, we have:

$$\begin{aligned} {[{Q_i}(\tau + 1)]^2} \le {[{Q_i}(\tau )]^2} + {[{u_i}(\tau )]^2} + {[{A_i}(\tau )]^2} - 2{Q_i}(\tau )[{u_i}(\tau ) - {A_i}(\tau )] \end{aligned}$$

(23)

Summing the above over $i = 1, \cdots ,n$ and using the fact that ${A_i}(t) \le 1,{u_i}(t) = \sum \nolimits _{j = 1}^m {{M_{ij}}(t){Y_{ij}}(t)} = \sum \nolimits _{j = 1}^m {{x_{ij}}(t){c_{ij}}(t)} \le m$, we have:

$$\begin{aligned} \sum \limits _{i = 1}^n {\left( {{{[{Q_i}(\tau + 1)]}^2} - {{[{Q_i}(\tau )]}^2}} \right) } \le n({m^2} + 1) - \sum \limits _{i = 1}^n {2{Q_i}(\tau )[{x_{ij}}(\tau ){c_{ij}}(\tau ) - {A_i}(\tau )]} \end{aligned}$$

(24)

Now multiply the above by $\frac{1}{2}$ and taking expectations conditioning on Q(t), we get:

$$\begin{aligned} \Delta (\tau ) \le B - E\{ \sum \limits _{i = 1}^n {{Q_i}(\tau )[{x_{ij}}(\tau ){c_{ij}}(\tau ) - {A_i}(\tau )]|Q(t)} \} \end{aligned}$$

(25)

Here $B = n(m^2+1)$, summing over $\tau = 0,1, \cdots ,t - 1$ and using the definition of $\Delta (t)$, we have:

$$\begin{aligned} \Delta (t) \le B + \sum \limits _{i = 1}^n {{Q_i}(t)E\{ {A_i}(t)|Q(t)\} } - \sum \limits _{i = 1}^n {{Q_i}(t)} \sum \limits _{j = 1}^m {E\{ {x_{ij}}(t){c_{ij}}(t)|Q(t)\} } \end{aligned}$$

(26)

The Lemma 1 is proven.

The proof of Lemma 2: plug (6), (7) and (8) to (15), we get:

$$\Delta (t) - VE \{ g(y(t)) | Q(t) \} \le B + \sum \limits_{i = 1}^{n} Q_{i} (t) E \{ A_{i}(t) | Q(t) \} - \sum \limits_{i = 1}^{n} Q_{i}(t) \sum \limits_{j = 1}^{m} E\{ x_{ij} (t) c_{ij} (t) | Q(t) \} - VE \left\{ \sum \limits_{i = 1}^{n} \log (1 + \sum \limits_{j = 1}^{m} x_{ij} (t) c_{ij} (t) ) | Q(t) \right\}$$

(27)

The Lemma 2 is proven.

Appendix 2: Proof of the Theorem 1

From (15), we have

$$\begin{aligned} \Delta (\tau ) \le B + Vg^* - VE\{ g(y(\tau ))|{{\varvec{Q}}}(\tau )\} - \varepsilon \sum \limits _{i = 1}^n {|Q_i(\tau )|} \end{aligned}$$

(28)

We first prove part (b) of Theorem 1. Taking expectations of (28) yields:

$$\begin{aligned} \begin{array}{lll} &{}&{}E\{ L(\tau + 1)\} - E\{ L(\tau )\} \le B + Vg^* - VE\{ g(y(\tau ))\} \\ &{}&{}\quad - \varepsilon \sum \limits _{i = 1}^n {E\{ |Q_i(\tau )|\} } \\ \end{array} \end{aligned}$$

(29)

Summing it over $\tau \in \{ 0,1,...,t - 1\}$ for all time-slot $t>0$ yields:

$$\begin{aligned} \begin{array}{l} E\{ L(t)\} - E\{ L(0)\} \le (B + Vg^*)t - V\sum \limits _{\tau = 0}^{t - 1} {E\{ g(y(\tau ))\} } \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - \varepsilon \sum \limits _{\tau = 0}^{t - 1} {\sum \limits _{i = 1}^n {E\{ |Q_i(\tau )|\} } } \\ \end{array} \end{aligned}$$

(30)

Now assume that $\varepsilon > 0$. Dividing the above inequation by $t\varepsilon$, we have

$$\frac{E \{ L(t) \} - E \{ L(0)\}} {t\varepsilon} \le \frac{B + Vg^{*}} {\varepsilon } - \frac{V} {t\varepsilon} \sum \limits_{\tau = 0}^{t - 1} E\{ g(y(\tau ))\} - \frac{1}{t}\sum \limits_{\tau = 0}^{t - 1} \sum \limits_{i = 1}^{n} E\{ |{Q_i}(\tau )|\}$$

(31)

Rearrange terms of above inequation yields:

$$\begin{array}{l} \frac{1}{t}\sum \limits _{\tau = 0} ^{t - 1} {\sum \limits _{i = 1}^n {E\{ |{Q_i}(\tau = 0)|\} } } \le \frac{{B + V{g^*}}}{\varepsilon } - \frac{V}{{t\varepsilon }}\sum \limits _{\tau = 0}^{t - 1} {E\{ g(y(\tau ))\} } \\ \qquad - \frac{{E\{ L(t)\} }}{{t\varepsilon }} + \frac{{E\{ L(0)\} }}{{t\varepsilon }} \end{array}$$

(32)

Using the fact that $E\{ L(t)\} \ge 0$ and $E\{ g(y(t))\} \ge 0$ we have:

$$\begin{aligned} \begin{array}{l} \frac{1}{t}\sum \limits _{\tau = 0}^{t - 1} {\sum \limits _{i = 1}^n {E\{ |{Q_i}(\tau = 0)|\} } } \le \frac{{B + V{g^*}}}{\varepsilon } + \frac{{E\{ L(0)\} }}{{\varepsilon (t)}} \end{array} \end{aligned}$$

(33)

The above inequation holds for all time-slot $t > 0$. Taking a limit for above inequation as $t \rightarrow \infty$, then part (b) of Theorem 1 is proved.

To prove part (a) of Theorem 1, from (30) we have

$$\begin{aligned} E\{ L(t)\} - E\{ L(0)\} \le (B + Vg^*)t, &&\text {for each time-slot}~ t > 0 \end{aligned}$$

(34)

Then plugging (12) into (34) yields:

$$\begin{aligned} \frac{1}{2}\sum \limits _{i = 1}^n {E\{ Q_i(t)^2 \} } \le (B + Vg^*)t + E\{ L(0)\} \end{aligned}$$

(35)

Therefore, for all $i \in \{ 1,...,n\}$, we have:

$$\begin{aligned} E\{ Q_i(t)^2 \} \le 2(B + Vg^*)t + 2E\{ L(0)\} \end{aligned}$$

(36)

Because $|Q_i(t)|$ cannot be negative, we have $E\{ Q_i(t)^2 \} \ge E\{ |Q_i(t)|\} ^2$. Thus for each time-slot $t > 0$, we have:

$$\begin{aligned} E\{ |Q_i(t)|\} \le \sqrt{2(B + Vg^*)t + 2E\{ L(0)\} } \end{aligned}$$

(37)

Dividing (37) by t and taking a limit as $t \rightarrow \infty$ as follows:

$$\begin{aligned} \mathop {\lim }\limits _{t \rightarrow \infty } \frac{{E\{ |Q_i(t)|\} }}{t} \le \mathop {\lim }\limits _{t \rightarrow \infty } \sqrt{\frac{{2(B + Vg^*)t + 2E\{ L(0)\} }}{{t^2 }}} = 0 \end{aligned}$$

(38)

(38) shows that all queues $Q_i(t)$ are mean rate stable, thus part (a) is proved.

Appendix 3: Proof of the Theorem 2

From (16), we have:

$$\begin{aligned} \Delta (t) &- VE\{g(y(t))|{{\varvec{Q}}}(t)\} \le B - VE\{ g(y(t))|{{\varvec{Q}}}(t)\} \\ & + \sum \limits _{i = 1}^n {Q_i(t)} E\{ (A_i(t) - \mu _i(t))|{{\varvec{Q}}}(t)\}\end{aligned}$$

(39)

Using $g({{\varvec{y}}}^*) = g^*$ and $E\{ \mu _i^*(t)\} = E\{ A_i(t)\} = y^*$, inequation (39) is rewritten as:

$$\begin{aligned} \Delta (t) &- VE\{ g(y(t))|{{\varvec{Q}}}(t)\} \le B - VE\{ g(y^*(t))|{{\varvec{Q}}}(t)\} \\ & + \sum \limits _{i = 1}^n {Q_i(t)} E\{ (A_i(t) - \mu _i^*(t))|{{\varvec{Q}}}(t)\}\end{aligned}$$

(40)

Taking expectations of (40), we can obtain:

$$\begin{aligned} E\{ L(t + 1)\} - E\{ L(t)\} - VE\{ g(y(t))\} \le B \\ & -VE\{ g(y^*(t))\} + \sum \limits _{i = 1}^n {E\{ Q_i(t)\} E\{ A_i(t) - \mu _i^*(t)\}} \\ \le B-Vg^* \end{aligned}$$

(41)

Summing the above inequation over $\tau \in \{ 0,1,...,t - 1\}$, and dividing it by t, we have the following inequation:

$$\begin{aligned} \frac{1}{t}\{ E[L(t)] - E[L(0)]\} - \frac{V}{t}\sum \limits _{\tau = 0}^{t - 1} {E\{ g(y(\tau ))\} } \le B - Vg^* \end{aligned}$$

(42)

As $L( \bullet ) \ge 0$, realigning terms in (42), we have:

$$\begin{aligned} \frac{1}{t}\sum \limits _{\tau = 0}^{t - 1} {E\{ g(y(\tau ))\} } \ge g^* - \frac{B}{V} - \frac{1}{{Vt}}E\{ L(0)\} \end{aligned}$$

(43)

Using the Jensens inequality [15] for $g( \bullet )$, we have:

$$\begin{aligned} g({\bar{y}}(t)) \ge g^* - \frac{B}{V} - \frac{1}{{Vt}}E\{ L(0)\} \end{aligned}$$

(44)

Taking limits of the above inequation as $t\rightarrow \infty$ yields Eq. (22).

Appendix 4: A List of Parameters and Their Descriptions

Parameters	Description
$Q_i$	The queue(at the BS) of $i{\text{th}}$ user
Q(t)	The packet backlogs of each queue in $t{\text{th}}$ time-slot
$S_j$	The $j{\text{th}}$ server
A(t)	The arrival vector
$c_{ij}(t)$	The connection status between ith queue and jth server in tth time-slot
$x_j(t)$	Transmission vector of server j
$\varPsi _{ij}(x_{ij}(t),c_{ij}(t))$	The probability that the channel at time-slot t is adequate for supporting the transmission of queue i
$M_{ij}(x_{ij}(t),c_{ij}(t))$	The probability that channel at time-slot t is able to transmit $i{\text{th}}$ queue
$M_{ij}(t)$	The amount of packets in queue $Q_i$ that could possibly be served by server $S_j$ in time-slot t
$Y_{ij}(t)$	The server j allocated to queue i
$\mu _i(t)$	Utility variable
$y_i(t)$	The number of user is packets served in time-slot t
$\varvec{\varLambda }$	The network capacity in wireless downlink
$g( \bullet )$	Generic utility function
V	Nonnegative control parameter