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Symptotics: a framework for estimating the scalability of real-world wireless networks

Abstract

We present a framework for non-asymptotic analysis of real-world multi-hop wireless networks that captures protocol overhead, congestion bottlenecks, traffic heterogeneity and other real-world concerns. The framework introduces the concept of symptotic scalability to determine the number of nodes to which a network scales, and a metric called change impact value for comparing the impact of underlying system parameters on network scalability. A key idea is to divide analysis into generic and specific parts connected via a signature—a set of governing parameters of a network scenario—such that analyzing a new network scenario reduces mainly to identifying its signature. Using this framework, we present the first closed-form symptotic scalability expressions for line, grid, clique, randomized grid and mobile topologies. We model both TDMA and 802.11, as well as unicast and broadcast traffic. We compare the analysis with discrete event simulations and show that the model provides sufficiently accurate estimates of scalability. We show how our impact analysis methodology can be used to progressively tune network features to meet a scaling requirement. We uncover several new insights, for instance, on the limited impact of reducing routing overhead, the differential nature of flooding traffic, and the effect real-world mobility on scalability. Our work is applicable to the design and deployment of real-world multi-hop wireless networks including community mesh networks, military networks, disaster relief networks and sensor networks.

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Notes

  1. 1.

    For example a multi-hop wireless network with directional antennas—is asymptotically unscalable  [4], but theoretically scales to 14000 nodes with a \(30^{\circ }\) beamwidth [7].

  2. 2.

    The typical multi-hop wireless network is asymptotically unscalable [2]. While our framework can accommodate asymptotically scalable networks as well, these are largely uninteresting from a symptotic viewpoint as the scalability is infinite nodes.

  3. 3.

    An alternate model/assumption would be multiple unicast transmissions at the link layer, and can also be easily analyzed with our framework if necessary.

  4. 4.

    This is assuming the carrier sense range is same as transmission range. In reality, the carrier sense range depends on the radio. The assumptions is not critical in the context of the framework, that is, should the carrier sense range be two hops, one would merely replace the signature component to 4.

  5. 5.

    More precisely, \(N \times \lambda \times O\big (\sqrt{\frac{N}{\rho }}\big ) \le N/\rho \implies N \le O\big ( \frac{1}{\lambda ^2\rho }\big ).\)

  6. 6.

    Technology choices offer a range of factor-of improvements, and rather than pick different \(\alpha\)’s for each, we have simply picked the smallest integer factor, namely 2, for simplicity. The relative CIVs for \(\alpha = 2\) should adequately capture the relative CIVs with other \(\alpha\)’s for our purposes.

  7. 7.

    Although our impact analysis has not considered mobile networks, we have used an LSU source rate of 0.2 LSUs per second which captures mobilities with link dynamics of up to once every 5 s.

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Acknowledgments

Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-09-2-0053.

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Correspondence to Ram Ramanathan.

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The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding copyright notation here on.

Appendix: Shortest paths through degree-4 grid center

Appendix: Shortest paths through degree-4 grid center

We derive an approximate formula for the expected number of shortest paths between uniformly randomly selected nodes that go through the center of an m by m degree-4 grid containing \(N = m^2\) nodes.

Consider the route from source s to a destination d. For the path from s to d to go through the center, they must clearly lie in the opposite quadrant inclusive of the nodes parallel and perpendicular to the center. Let \(P_c(s,d)\) denote the probability that the path from s to d goes through the center, and Q(d) denote the probability that d is in the opposite quadrant. Then the expected number of paths through the center is given by

$$\begin{aligned} E[P_c] = Q(d_q)\cdot P_c(s,d)\cdot (N-1) \end{aligned}$$
(17)

Since we don’t count the center node c and s,

$$\begin{aligned} Q(d) = \frac{\left( \frac{m+1}{2}\right) ^2 - 1}{m^2 - 2} \approx \frac{1}{4} + \frac{1}{2m} \end{aligned}$$
(18)

\(P_c(s,d)\) depends upon the exact location of s and d. Instead of iterating over all values of s and d (which gets very complicated very quickly), we approximate it by assuming the average to be equal to when s and d are in the centers of their respective quadrants, and denote these nodes by \(s_q\) and \(d_q\). Let n(xy) denote the number of shortest paths from x to y. Then

$$\begin{aligned} P_c(s,d) = P_c(s_q,d_q) = \frac{n(s_q,c)\cdot n(c,d_q)}{n(s_q,d_q)} \end{aligned}$$
(19)
Fig. 12
figure12

Reference figure for derivation

Given a square z by z grid, the number of shortest paths from one corner to another is \({2z \atopwithdelims ()z}\)[37]. Let \(b = m/4\). Then, from Fig. 12 and the above, we have

$$\begin{aligned} P_c(s,d) = \frac{{2b \atopwithdelims ()b}^2}{{4b \atopwithdelims ()2b}} \end{aligned}$$
(20)

The above is related to the concept of Catalan numbers [37]. The \(n{\hbox {th}}\) Catalan number and an approximate formula thereof [38] are given by

$$\begin{aligned} C_n = \frac{1}{n+1}\cdot {2n \atopwithdelims ()n} \approx \frac{4^n}{n^{1.5}\sqrt{\pi }} \end{aligned}$$
(21)

Using Eq. 21 in 20, we have

$$\begin{aligned} P_c(s,d) = \frac{((b+1)C_b)^2}{(2b+1)C_{2b}} \approx \frac{0.8(b+1)}{b^{1.5}} \end{aligned}$$
(22)

Re-substituting \(b = m/4\) in the above, and using this and Eq. 18 in 17, we have

$$\begin{aligned} E[P_c] &\approx \left( 0.25 + \frac{1}{2m}\right) \cdot m^2\cdot \frac{1.6(m+4)}{m^{1.5}} \\ &\approx 0.4\left( 1 + \frac{2}{m}\right) (m^{1.5} + 4m^{0.5})\\ &\approx 0.4\left( 1+ \frac{2}{\sqrt{N}}\right) (N^{\frac{3}{4}} + 4N^{\frac{1}{4}}) \end{aligned}$$

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Ramanathan, R., Ciftcioglu, E., Samanta, A. et al. Symptotics: a framework for estimating the scalability of real-world wireless networks. Wireless Netw 23, 1063–1083 (2017). https://doi.org/10.1007/s11276-016-1204-4

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Keywords

  • Multi-hop wireless network
  • Network design
  • Scalability
  • Performance model