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A novel contention-on-demand design for WiFi hotspots

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Abstract

In widely deployed wireless “hot-spot” networks, nodes frequently join or leave, and inelastic/elastic and saturated/nonsaturated flows coexist. In such dynamic and diverse environments, it is challenging to maximize the channel utilization while providing satisfactory user experiences. In this paper, we propose a novel contention-on-demand (CoD) MAC scheme to address this problem. The CoD scheme consists of a fixed-CW algorithm, a dynamic-CW algorithm, and an admission control rule. The fixed-CW algorithm allows elastic flows to access limited system bandwidth; the dynamic-CW algorithm enables inelastic flows to contend for channel on demand and quickly adapt to network change; and the admission control rule rejects overloaded traffic for providing good user experiences. We then perform an asymptotic analysis to develop a simple and practical admission control rule for homogeneous and heterogeneous traffic; our rule can not only adapt to the change in offered loads and node number, but also maximize the system utilization . Finally, extensive simulations verify that our scheme is very effective and our theoretical result is very accurate.

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Acknowledgments

This work is partially supported by Macao Science and Technology Development Fund under Grant (013/2014/A1, 005/2016/A1 and 049/2016/A2), the NSF of China under Grant 61373027, and NSF of Shandong Province under Grant ZR2012FM023.

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Correspondence to Jiguo Yu.

Appendix

Appendix

In this appendix, we finish the proof of Theorem 2.

Proof of Theorem 2

We first define four types of virtual slots: (1) an idle MAC slot with probability \(1-P_{\mathrm{{b}}}\), (2) one successful transmission time of an LP node with probability \(P_{\mathrm{{s}}_{0}}\), one collision time with probability \(P_{\mathrm{{c}}}\), and one successful transmission time of an HP node with probability \(P_{\mathrm{{s}}_{h}}\,\triangleq\, \sum \nolimits _{i=1}^{I}P_{\mathrm{{s}}_{i}}\), where \(1-P_{\mathrm{{b}}}+P_{\mathrm{{s}}_{0}}+P_{\mathrm{{c}}}+P_{\mathrm{{s}}_{h}}=1 \).

Let \(T_{\mathrm{{o}}}\) denote the interval between when one successful transmission from HP nodes   ends and when the next successful transmission from HP nodes begins, as shown in Fig. 9. Clearly, \(T_{\mathrm{{o}}}\) contains the first three types of virtual slots.

Let \(\overline{T}_{\mathrm{{o}}}\) denote the mean of \(T_{\mathrm{{o}}}\). To maximize the effective bandwidth occupied by HP nodes, we should minimize \(\overline{T}_{\mathrm{{o}}}\).

Below, we first express \(\overline{T}_{\mathrm{{o}}}\) and then find the optimal \(\beta ^{+}\) that minimizes \(\overline{T}_{\mathrm{{o}}}\).

Let \(X^{\circ }\) denote the number of the virtual slots during \(T_{\mathrm{{o}}}\). Since a virtual slot during \(T_{\mathrm{{o}}}\) appears with probability \(1-P_{\mathrm{{s}}_{h}}\), \(X^{\circ }\) follows a geometric distribution with parameter \(P_{\mathrm{{s}}_{h}}\) and therefore its mean \(\overline{X}^{\circ }=1/P_{\mathrm{{s}}_{h}}-1\).

Let \(\varOmega ^{\circ }\) be a random variable representing the length of a virtual slot during \(T_{\mathrm{{o}}}\). \(\varOmega ^{\circ }\) takes three types of values depending on the thee types of virtual slots during \(T_{\mathrm{{o}}}\). In terms of \(P_{\mathrm{{b}}},\) \(P_{\mathrm{{s}}_{0}},\) and \(P_{\mathrm{{c}}}\), we define \(\varOmega ^{\circ }\) as follows:

$$\begin{aligned} \varOmega ^{\circ }=\left\{ \begin{array}{lll} {\upsigma } & \quad \hbox {w.p.} & \frac{1-P_{\mathrm{{b}}}}{1-P_{\mathrm{{b}}}+P_{\mathrm{{s}}_{0}}+P_{\mathrm{{c}}}}, \\ T_{\mathrm{{s}}_{0}} & \quad \hbox {w.p.} & \frac{P_{\mathrm{{s}}_{0}}}{1-P_{\mathrm{{b}}}+P_{\mathrm{{s}}_{0}}+P_{\mathrm{{c}}}}, \\ T_{\mathrm{{c}}} & \quad \hbox {w.p.} & \frac{P_{\mathrm{{c}}}}{1-P_{\mathrm{{b}}}+P_{\mathrm{{s}}_{0}}+P_{\mathrm{{c}}}},\end{array} \ \ \ \ \ \right. \end{aligned}$$
(17)

where \(T_{\mathrm{{s}}_{0}}\), \(T_{\mathrm{{c}}}\), \(P_{\mathrm{{b}}}\), \(P_{\mathrm{{s}}_{0}}\), \(P_{\mathrm{{c}}}\) are defined in (3). Then, the mean \(\overline{\varOmega }^{\circ }\) can be easily calculated by (17).

Then, \(\overline{T}_{\mathrm{{o}}}\) is equal to the mean time of a virtual slot, \(\overline{\varOmega }^{\circ }\), times the mean number of virtual slots \(\overline{X}^{\circ }\). Noting that \(1-P_{\mathrm{{b}}}+P_{\mathrm{{s}}_{0}}+P_{\mathrm{{c}}}+P_{\mathrm{{s}}_{h}}=1\), we have

$$\begin{aligned} \overline{T}_{\mathrm{{o}}}=\overline{\varOmega }^{\circ }\cdot \overline{X}^{\circ }=\frac{\sigma (1-P_{\mathrm{{b}}})+T_{\mathrm{{s}}_{0}}P_{\mathrm{{s}}_{0}}+T_{\mathrm{{c}}}P_{\mathrm{{c}}}}{P_{\mathrm{{s}}_{h}}} \end{aligned}$$
(18)

Define \(\overline{T}_{\mathrm{{o}}}^{+}(\beta ^{+})\,\triangleq\, \lim \limits _{n\rightarrow \infty }\overline{T}_{\mathrm{{o}}}\). Let \(C_{2}\,\triangleq\, \sigma -T_{\mathrm{{c}}}+(T_{\mathrm{{s}}_{0}}-T_{\mathrm{{c}}})C_{1}/C_{0}\). Taking \(n\rightarrow \infty \) for both sides in (18) and applying (6), we have

$$\begin{aligned} \overline{T}_{\mathrm{{o}}}^{+}(\beta ^{+}) & = \frac{\begin{array}{c} \sigma [e^{-\beta ^{+}}C_{0}]+T_{\mathrm{{s}}_{0}}[e^{-\beta ^{+}}C_{1}]+ \\ T_{\mathrm{{c}}}[1-e^{-\beta ^{+}}(C_{0}+C_{1}+\beta ^{+}C_{0})]\end{array}}{\beta ^{+}e^{-\beta ^{+}}C_{0}} \\ & = \frac{C_{2}+T_{\mathrm{{c}}}[\frac{e^{\beta ^{+}}}{C_{0}}-\beta ^{+}]}{\beta ^{+}} \end{aligned}$$

To minimize \(\overline{T}_{\mathrm{{o}}}^{+}(\beta ^{+})\), we set the first derivative of \(\overline{T}_{\mathrm{{o}}}^{+}(\beta ^{+})\) with respect to \(\beta ^{+}\) to zero. This leads to

$$\begin{aligned} T_{\mathrm{{c}}}\left( \frac{e^{\beta ^{+}}}{C_{0}}-1\right) \beta ^{+} & = C_{2}+T_{\mathrm{{c}}}\left( \frac{e^{\beta ^{+}}}{C_{0}}-\beta ^{+}\right) \\ (\beta ^{+}-1)e^{(\beta ^{+}-1)} & = C_{0}C_{2}e^{-1}/T_{\mathrm{{c}}}. \end{aligned}$$

Then \(\beta ^{+}-1=\mathscr {W}_{0}(C_{0}C_{2}e^{-1}/T_{\mathrm{{c}}})\) or \(\mathscr {W}_{-1}(C_{0}C_{2}e^{-1}/T_{\mathrm{{c}}})\). We have \(\beta _{\mathrm{{opt}}}^{+}=\mathscr {W}_{0}(C_{0}C_{2}e^{-1}/T_{\mathrm{{c}}})+1\ge 0\), since only \(\mathscr {W}_{0}(C_{0}C_{2}e^{-1}/T_{\mathrm{{c}}})>-1\) for \(C_{0}C_{2}e^{-1}/T_{\mathrm{{c}}}\in (-1/e,0).\)

As a result, we obtain (12) since \(\frac{C_{0}C_{2}}{eT_{\mathrm{{c}}}}=\frac{C_{0}(\sigma -T_{\mathrm{{c}}})+C_{1}(T_{\mathrm{{s}}_{0}}-T_{\mathrm{{c}}})}{eT_{\mathrm{{c}}}}\).

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Feng, L., Yu, J., Cheng, X. et al. A novel contention-on-demand design for WiFi hotspots. Pers Ubiquit Comput 20, 705–716 (2016). https://doi.org/10.1007/s00779-016-0942-8

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