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Throughput and delay analysis for hybrid radio-frequency and free-space-optical (RF/FSO) networks

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Abstract

In this paper the per-node throughput and end-to-end delay of randomly deployed (i.e. ad-hoc) hybrid radio frequency - free space optics (RF/FSO) networks are studied. The hybrid RF/FSO network consists of an RF ad hoc network of n nodes, f(n) of them, termed ‘super nodes’, are equipped with an additional FSO transceiver with transmission range s(n). Every RF and FSO transceiver is able to transmit at a maximum data rate of W 1 and W 2 bits/sec, respectively. An upper bound on the per node throughput capacity is derived. In order to prove that this upper bound is achievable, a hybrid routing scheme is designed whereby the data traffic is divided into two classes and assigned different forwarding strategies. The capacity improvement with the support of FSO nodes is evaluated and compared against the corresponding results for pure RF wireless networks. Under optimal throughput scaling, the scaling of average end-to-end delay is derived. A significant gain in throughput capacity and a notable reduction in delay will be achieved if \(f(n) = \Upomega\left(\frac{1}{s(n)}\sqrt{\frac{n}{\log n}}\cdot \frac{W_1}{W_2} \right)\). Furthermore, it is found that for fixed W 1, f(n) and n where f(n) < n, there is no capacity incentive to increase the FSO data rate beyond a critical value. In addition, both throughput and delay can achieve linear scaling by properly adjusting the FSO transmission range and the number of FSO nodes.

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Fig. 1

Notes

  1. 1.

    For the sake of simplicity, in this paper we sometimes just write f instead of f(n). The same treatment is also applied to other network parameters e.g. r(n), s(n), λ(n) and so on.

  2. 2.

    The proof technique used here is similar to Lemma 4.9 in [2].

References

  1. 1.

    Wang, D., & Abouzeid, A. A. (2007). Throughput capacity of hybrid radio-frequency and free-space-optical (RF/FSO) multi-hop networks. In 2007 information theory and applications workshop (pp. 1–8). CA, USA: La Jolla.

  2. 2.

    Gupta, P., & Kumar, P. R. (2000). The capacity of wireless networks. IEEE Transactions on Information Theory, 46(2), 388–404.

  3. 3.

    Milner, S. D., & Davis, C. C. (2004). Hybrid free space optical/RF networks for tactical operations. In Proceedings of Military Communications Conference (MILCOM), 1, 409–415, Oct 31–Nov 3.

  4. 4.

    Bloom, S., & Hartley, W. S. (2002). The last mile solution: Hybrid FSO Radio. Airfiber, May 2002.

  5. 5.

    Akella, J. Liu, Chang, Partyka, D., Yuksel, M., Kalyanaraman, S., & Dutta, P. (2005). Building blocks for mobile free-space-optical networks. In Proceedings of Second IFIP International Conference on Wireless and Optical Communications Networks (WOCN), pp. 164–168, 6–8 March 2005.

  6. 6.

    Derenick, J., Thorne, C., & Spletzer, J. (2005). On the deployment of a hybrid free-space optic/radio frequency (FSO/RF) mobile ad-hoc network. In Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3990–3996, Aug 2005.

  7. 7.

    Kashyap, A., & Shayman, M. (2005). Routing and traffic engineering in hybrid RF/FSO networks. In Proceedings of IEEE International Conference on Communications (ICC), 5, 3427–3433, May 2005.

  8. 8.

    Yi, S., Pei, Y., & Kalyanaraman, S. (2003). On the capacity improvement of ad hoc wireless networks using directional antennas. In Proceedings of the 4th ACM International Symposium on Mobile Ad Hoc Networking And Computing (MobiHoc), pp. 108–116.

  9. 9.

    Liu, B., Liu, Z., & Towsley, D. (2003). On the capacity of hybrid wireless networks. In Proceedings of the 22nd Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), 2, 1543–1552.

  10. 10.

    Kozat, U. C., & Tassiulas, L. (2003). Throughput capacity of random Ad Hoc networks with infrastructure support. In Proceedings of the 9th Annual International Conference on Mobile Computing and Networking (MobiCom), September.

  11. 11.

    Agarwal, A., & Kumar, P. R. (2004). Capacity bounds for ad hoc and hybrid wireless networks. ACM SIGCOMM Computer Communication Review, 34(3), July.

  12. 12.

    Zhou, P., Wang, X., & Rao, R. (2008). Asymptotic capacity of infrastructure wireless mesh networks. IEEE Transactions on Mobile Computing, 7(8), August.

  13. 13.

    Li, P., Zhang, C., & Fang, Y. (2009). Capacity and delay of hybrid wireless broadband access networks. IEEE Journal on Selected Areas in Communications, 27(2).

  14. 14.

    Ozgur, A., Leveque, O., & Tse, D. (2007). Hierarchical cooperation achieves optimal capacity scaling in ad hoc networks. IEEE Transactions on Information Theory, 3549–3572.

  15. 15.

    Niesen, U., Gupta, P., & Shah, D. (2007). On capacity scaling in arbitrary wireless networks, In Proceedings of the Information Theory and Applications workshop (ITA), San Diego.

  16. 16.

    Khan, S. A., & Riza, N. A. (2004). Demonstration of 3-dimensional wide-angle no-moving-parts laser beam steering. In Proceedings of SPIE, 5550, pp. 47–59.

  17. 17.

    Matkin, B. W. (2002). Steered agile beams program support for Army requirements. In Proceedings of SPIE, 4489, 1–12.

  18. 18.

    Song, Q. W., Wang, X. M., Bussjager, R., & Osman, J. (1996). Electro-optic beam-steering device based on a lanthanum-modified lead zirconate titanate ceramic wafer. Applied Optics, 35(17), 3155–3162.

  19. 19.

    McRuer, R., McAdams, L. R., & Goodman, J. W. (1990). Ferroelectric liquid-crystal digital scanner. Optics Letters, 15(23), 1415–1417.

  20. 20.

    Titus, C. M., Bos, P. J., & Lavrentovich, O. D. (1999). Efficient, accurate liquid crystal digital light deflector. In Proceedings of SPIE, 3633, 244.

  21. 21.

    Xue, F., & Kumar, P. R. (2004). The number of neighbors needed for connectivity of wireless networks. Wireless Networks, 10(2), 169–181.

  22. 22.

    El Gamal, A., Mammen, J., Prabhakar, B., & Shah, D. (2006). Optimal throughput-delay scaling in wireless networks—part I: The fluid model. IEEE Transactions on Information Theory, 2568–2592, June.

  23. 23.

    El Gamal, A., Mammen, J., Prabhakar, B., & Shah, D. (2006). Optimal throughput-delay scaling in wireless networks—part II: Constant-size packets, IEEE Transactions on Information Theory, 5111–5116, Nov 2006.

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Acknowledgments

This work was supported by National Science Foundation grants No. 546402. The preliminary version was presented at the 2007 Information Theory and Applications Workshop (ITA’07).

Author information

Correspondence to Di Wang.

Appendices

Appendix 1

Proof of Lemma 1

To prove this lemma we need to show there exists constants c 4 and c 5 such that λ′ is upper bounded by

$$ \lambda' \leq \frac{c_{16}W_2f(n)s(n)}{n} $$
(78)

and there exists a spatial and temporal scheduling scheme such that a per session throughput

$$ \lambda' = \frac{c_{17}W_2f(n)s(n)}{n} $$
(79)

is achievable.

To derive the upper bound for λ′, we notice that the number of simultaneous FSO transmissions is no more than m, thus the total data rate served by the entire FSO network is no more than W 2 f(n). Now let \(\overline{L}\) denote the mean length of a line connecting two independently and uniformly distributed points on S 2. Then the mean length of the path of packets is at least \(\overline{L}-o(1)\). Thus the mean number of hops taken by a packet is at least \(\frac{\overline{L}-o(1)}{s(n)}\). Then the total number of bits per second served by the entire network needs to be at least \(\frac{(\overline{L}-o(1))n\lambda'}{s(n)}\). Then we have the following inequality:

$$ \frac{(\overline{L}-o(1))n \lambda'}{s} \leq W_2f(n) $$
(80)

Therefore we have the following upper bound on λ′:

$$ \lambda' \leq \frac{c_{16}W_2f(n)s(n)}{n} $$
(81)

To show that this upper bound is order tight, we design a routing scheme using similar techniques as in [2]. Let

$$ \rho'(n) := \hbox{radius of a disk of area } \frac{\pi s^2(n)}{64} $$
(82)

Then it is readily known that

$$ s(n)\geq 8\rho'(n) $$
(83)

According to Lemma 4.1 in [2], we can construct a Voronoi tessellation \(\mathcal{V}'_n\) such that

  1. (A1)

    Every Voronoi cell contains a disk of radius ρ′(n);

  2. (A2)

    Every Voronoi cell is contained in a disk of radius 2 ρ′(n);

This setting allows direct communication within a cell and between adjacent cells, i.e., every node in a cell is within this distance s(n) from every other node in its own cell or adjacent cell. The routing strategy forwards the packets along the sequence of hops that approximate the straight line that connects the source and destination, i.e. packets are passing the cells that the straight-line intersects.

Now we show that the number of nodes in each cell is bounded from below with high probability. Let \(\mathcal{F}\) denote the class of disks of area \(\frac{\pi s^2(n)}{64}\). Then it is implied by Vapnik-Chervonenkis Theorem (see [2] and references therein) that

$$ \begin{aligned} &Pr\left\{ \sup_{D \in{\mathcal{F}}}\left|\frac{\hbox{number\,of\,nodes\,in\,D}} {f(n)}-\frac{\pi s^2(n)}{64}\right|\leq\epsilon(n)\right\}\\ &\quad >1-\delta(n) \\ \end{aligned} $$
(84)

whenever

$$ n>\max\left\{\frac{24}{\epsilon(n)}\log\frac{16e} {\epsilon(n)},\frac{4}{\epsilon(n)}\log\frac{2} {\delta(n)}\right\} $$
(85)

Note that s(n) is chosen such that the FSO network is asymptotically connected, and s(n) must be less than the maximum distance between any two points in S 2 which is \(\sqrt\pi/2\), then

$$ c_5\sqrt{\frac{\log f(n)}{f(n)}}\leq s(n)\leq\frac{\sqrt\pi} {2} $$
(86)

then Eq. 85 is satisfied by choosing \(\epsilon(n)={s^2(n)}/{64}\) and δ(n) = 1/n when n is large enough.

Since each cell V in \(\mathcal{V}'_n\) contains a disk of area \(\frac{\pi s^2(n)}{64}\), then for large n we have

$$ Pr \left\{\hbox{number\,of\,nodes\,in }V \geq \frac{f(n)s^2(n)} {32}\right\} \geq 1-\delta(n) $$
(87)

Next we show that the traffic to be served by each cell V is bounded from above with high probability. Towards this objective we let i and j denote two randomly located points on S 2, and L ij denote the line segment connecting these two points. We claim that for any cell \(V\in\mathcal{V}'_n\),

$$ Pr(L_{ij} \hbox{ intersects } V) \leq c_{24}s(n) $$
(88)

This is proved as followsFootnote 2. From property (A2), every cell \(V\in\mathcal{V}'_n\) is contained in a disk of radius s(n)/4. If i lies at a distance x from the disk, then the angle θ subtended at i by the disk is no more than c 18 s(n)/x. The area of the sector so formed is no more than \(\frac{c_{19}\theta} {2\pi}\). If j does not lie in this sector, Then the line L ij cannot intersect the disk containing the cell V. Hence for a point i at a distance x from the disk of radius s(n)/4 containing the cell V, the probability that the line L ij intersects the disk is no more than c 20 s(n)/x. Since i is uniformly distributed on S 2, the probability density that it is at a distance x from the disk is bounded above by 2c 21π(x + s(n)/4). Integrating, we obtain

$$ \begin{aligned} Pr(L_{ij} \hbox{ intersects } V) &\leq \int\limits_{s(n)/4}^{\sqrt\pi/2}\left( \frac{c_{20}s(n)}{x}\right)\\ & \cdot 2c_{21}\pi\left(x+\frac{s(n)}{4}\right)dx \\ &\leq c_{24}s(n) \end{aligned} $$
(89)

By a similar reasoning through Lemma 4.10–4.14 in [2] we have

$$ \begin{aligned} &Pr\{ \sup_{V \in {\mathcal{V}}'_m}(\hbox{traffic needing to be carried by cell V})\\ &\quad \leq c_{24} {\lambda'} n s(n) \} \geq 1-\delta_1(m) \end{aligned} $$
(90)

It can be implied by Eq. 87 that each cell \(V\in\mathcal{V}'_n\) is able to transmit at least \(\frac{f(n)s^2(n)}{32}W_2\) bits/sec. Then with high probability, the rate c 24 λ′ n s(n) can be accommodated by all cell if

$$ c_{5} {\lambda'} n s(n) \leq \frac{f(n)s^2(n)}{32}W_2 $$
(91)

Then the per session achievable throughput is given by

$$\lambda' = \frac{W_2f(n)s(n)}{32c_{24}n} $$
(92)

which proves Lemma 1. □

Appendix 2

Proof of Lemma 4

Clearly the number of type-1 RF λ2-traffic segments is (n − f). Let random variable \(\mathcal{B}\) denote the number of type-2 RF λ2-traffic segments. Then

$$ {\mathcal{N}}={\mathcal{B}}+n-f $$
(93)

For each source-destination pair, the destination node is a regular node with probability \(\frac{n-f}{n}\). Then \(\mathcal{B}\) follows binomial distribution \(B(n,\frac{n-f}{n})\). It follows from the law of large numbers that given any \(\epsilon>0\),

$$ \lim_{n\rightarrow\infty}Pr\left\{\left|\frac{{\mathcal{B}}} {n}-\frac{n-f}{n}\right|<\epsilon\right\} =1 $$
(94)

Let \(\epsilon=c_{22}(n-f)/n\), then for large n and f < n, there is \(\delta_2(n) \rightarrow 0\) such that

$$ Pr\left\{{\mathcal{B}}\leq (1+c_{22})(n-f)\right\} \geq 1-\delta_2(n) $$
(95)

Then Eq. 39 follows directly from Eqs. 93 and 95. □

Appendix 3

Proof of Lemma 5

This proof uses a similar technique introduced in the proof of Lemma 4.9 in [2].

According to the property (V2), there exists a disk of radius 2ρ that contains the Voronoi cell V. Let D denote such a disk. If i lies at a distance x from the disk D, then the angle \(\alpha(\leq\pi)\) of the sector subtended at node i by the disk is no more than \(\frac{c_{23}}{x}\sqrt{\frac{\log n} {n}}\). If the segment L i joining node i and node g(i) intersects the disk D, then node g(i) should lie inside the sector and \(|X_i-X_{g(i)}|\geq x\). Then the probability that L i intersects the disk D is no more than \(\frac{\alpha}{2\pi} Pr\{|X_i-X_{g(i)}|\geq x\}\). Note that \(|X_i-X_{g(i)}|\) follows the probability distribution of random variable T defined in Section IV, which is given by Eq. 23. Since node i is uniformly distributed on S 2, the probability density that it is at a distance x from the disk is bounded above by c 24(x + 2ρ). Let x = z u 1, then

$$ \begin{aligned} p_{V} = & Pr\{ L_{i} \,{\text{intersects}}V\} \\ \le & \int_{0}^{\infty } {{\frac{\alpha }{{2\pi }}}} Pr\{ |X_{i} - X_{{g(i)}} | \ge x\} c_{{24}} (x + 2\rho )dx \\ \le & \int_{0}^{\infty } {\min } \left\{ {\pi ,{\frac{{c_{{22}} }}{x}}\sqrt {{\frac{{\log n}}{n}}} } \right\}{\frac{{c_{{23}} }}{{2\pi }}}\cdot f^{{ - z^{2} }} (x + 2\rho )dx \\ = & \int_{0}^{{{\frac{{c_{{22}} }}{\pi }}\sqrt {{\frac{{\log n}}{n}}} }} \pi \cdot{\frac{{c_{{23}} }}{{2\pi }}}\cdot f^{{ - z^{2} }} (x + 2\rho )dx \\ & + \int_{{{\frac{{c_{{22}} }}{\pi }}\sqrt {{\frac{{\log n}}{n}}} }}^{\infty } {{\frac{{c_{{22}} }}{x}}} \sqrt {{\frac{{\log n}}{n}}} \cdot{\frac{{c_{{23}} }}{{2\pi }}}\cdot f^{{ - z^{2} }} (x + 2\rho )dx \\ \le & c_{{14}} \sqrt {{\frac{{\log n}}{{fn}}}} \\ \end{aligned} $$
(96)

Hence we have proved the lemma. □

Appendix 4

Proof of Lemma 6

Given \(\mathcal{N}=N\), the number of RF λ2-traffic segments \(\mathcal{M}\) follows the binomial distribution B(Np V ).

If N = ω(1), i.e. N is large, then it follows from the law of large numbers that given any \(\epsilon>0\),

$$ \lim_{n\rightarrow\infty}Pr\left\{\left|\frac{{\mathcal{M}}} {N}-p_V\right|<\epsilon\right\} =1 $$
(97)

Let \(\epsilon=p_V\), then according to Lemma 4, there is a \(\delta_4(n) \rightarrow 0\) such that

$$ Pr\left\{{\mathcal{M}} \leq 2c_{14} N \sqrt{\frac{\log n}{fn}}\right\} \geq 1-\delta_4(n) $$
(98)

If N = O(1), then for large n,

$$ E\{{\mathcal{M}}\} \leq c_{14}N\sqrt{\frac{\log n}{f n}} << 1 $$
(99)

Thus for large n,

$$ \begin{aligned} Pr\left\{{\mathcal{M}} \leq 2c_{14} N \sqrt{\frac{\log n}{f n}}\right\} &\geq Pr\left\{{\mathcal{M}} \leq E\{{\mathcal{M}}\}\right\} \\ &= Pr\left\{{\mathcal{M}}=0\right\} \\ &\geq \left(1-c_{14}\sqrt{\frac{\log n}{f n}}\right)^{N} \\ \rightarrow& 1 \end{aligned} $$
(100)

Note that Eqs. 98 and 23 indicate that for any \({N\in\mathbb{N}}\), there is a \(\delta_5(n) \rightarrow 0\) such that

$$ Pr\left\{\left.{\mathcal{M}} \leq 2c_{14} {\mathcal{N}} \sqrt{\frac{\log n} {f n}}\right|{\mathcal{N}}=N\right\} \geq 1-\delta_5(n) $$
(101)

According to Lemma 3, we have

$$ \begin{aligned} Pr & \left\{{\mathcal{M}} \leq (2+c_{13})(n-f)\cdot 2c_{14} \sqrt{ \frac{\log n}{f n}}\right\} &\geq Pr\left\{{\mathcal{M}} \leq 2c_{14} {\mathcal{N}} \sqrt{\frac{\log n}{f n}}, {\mathcal{N}}\leq(2+c_{13})(n-f)\right\} & = Pr\left\{{\mathcal{N}}\leq(2+c_{13})(n-f)\right\} &\quad \cdot Pr\left\{\left.{\mathcal{M}} \leq 2c_{14} {\mathcal{N}} \sqrt{\frac{\log n}{f n}}\right|{\mathcal{N}}\leq(2+c_{13})(n-f)\right\} &\geq (1-\delta_2(n))(1-\delta_5(n)) \end{aligned} $$
(102)

Hence we have proved the lemma. □

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Wang, D., Abouzeid, A.A. Throughput and delay analysis for hybrid radio-frequency and free-space-optical (RF/FSO) networks. Wireless Netw 17, 877–892 (2011). https://doi.org/10.1007/s11276-011-0321-3

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Keywords

  • Ad hoc networks
  • Hybrid networks
  • Throughput capacity
  • Delay
  • Free space optics