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

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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.

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References

  1. 1.

    Feng, M., Mao, S., & Jiang, T. (2015). Joint duplex mode selection, channel allocation, and power control for full-duplex cognitive femtocell networks. Digital Communications and Networks, 1(1), 30–44.

  2. 2.

    Wang, S., & Wang, C. (2015). Joint optimization of spectrum and energy efficiency in cognitive radio networks. Digital Communications and Networks, 1(3), 161–170.

  3. 3.

    3GPP TR 25.913. (2006) Requirements for Evolved UTRA (E-UTRA) and Evolved UTRAN (E-UTRAN).

  4. 4.

    WiMax Forum, Mobile WiMAX Part I. (2006) “A technical overview and performance evaluation” White Paper.

  5. 5.

    Tassiulas, L., & Ephremides, A. (1993). Dynamic server allocation to parallel queues with randomly varying connectivity. IEEE Transactions on Information Theory, 39(2), 466–478.

  6. 6.

    Ying, L., Srikant, R., Eryilmaz, A., & Dullerud, G. (2006). A large deviations analysis of scheduling in wireless networks. IEEE Transactions on Information Theory, 52(11), 5088–5098.

  7. 7.

    Stolyar, A. (2008). Large deviations of queues sharing a randomly time-varying server. Queueing Systems, 59, 1–35.

  8. 8.

    Venkataramanan, V. J., & Lin, X. (2007) “Structural properties of LDP for queue-length based wireless scheduling algorithms. In ” Proceedings on Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL.

  9. 9.

    Bui, L., Srikant, R., & Stolyar, A. (2009) “Novel architectures and algorithms for delay reduction in back-pressure scheduling and routing”. In Proceedings on IEEE Infocom.

  10. 10.

    Gupta, P. & Javidi, T. (2007) “Towards throughput and delay-optimal routing for wireless ad-hoc networks”. In Asilomar Conference on Signals,Systems and Computers.

  11. 11.

    Ying, L., Shakkottai, S., & Reddy, A. (2009). “On combining shortestpath and back-pressure routing over multihop wireless networks”. In Proceedings on IEEE Infocom.

  12. 12.

    Georgiadis, L., Neely, M. J., & Tassiulas, L. (2006). Resource allocation and cross-layer control in wireless networks. Foundations and Trends in Networking, 1(1), 1–144.

  13. 13.

    Neely, M. J., Modiano, E., & Li, C. (2005). Fairness and optimal stochastic control for heterogeneous networks. Proceedings on IEEE Infocom, 3, 1723–1734.

  14. 14.

    Neely, M. J. (2003) “Dynamic power allocation and routing for satellite and wireless networks with time varying channels” Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, MA, LIDS.

  15. 15.

    Bertsekas, D. P., Nedic, A., & Ozdaglar, A. E. (2003). Convex analysis and optimization. Convex Analysis and Optimization, 12, 67.

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Acknowledgments

This work is supported by the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (No. 2015D07) and the Special Fund of Chongqing Key Laboratory (CSTC).

Author information

Correspondence to Xiaoling Xiong.

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

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Li, Y., Xia, S., Xiong, X. et al. Throughput-Delay Trade-off Scheduling in Multi-channel Downlink Wireless Networks. Wireless Pers Commun 92, 681–694 (2017). https://doi.org/10.1007/s11277-016-3571-7

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Keywords

  • Scheduling
  • Lyapunov optimization
  • Utility maximization