On estimating the interest satisfaction ratio in IEEE 802.15.4-based named-data networks

Abstract

Named-Data Networking (NDN) over Low Power and Lossy Networks (LLNs), employing IEEE 802.15.4 communication technology, is projected to provide native support for mobility and efficient content delivery for the emerging Internet of Things (IoT). While many interest forwarding strategies have been proposed for NDNs over LLNs, most existing studies have relied on software simulations to evaluate their performance due to the lack of analytical modeling tools. This paper introduces the first analytical model for estimating the Interest Satisfaction Ratio (ISR) in NDN over LLNs, which is a crucial metric for assessing the effectiveness of interest forwarding strategies. We develop the analytical model specifically for the broadcast forwarding strategy, which has been extensively studied due to its simplicity and ease of implementation. Simulation results confirm that the proposed model predicts the ISR with reasonable accuracy. The model is then used to elucidate the strong interaction between the CSMA/CA parameters of the IEEE 802.15.4 standard and the achieved ISR.

Appendix. Computing the reachability probability, R(p)

This appendix briefly presents the methodology of [35, 36] for computing the reachability probability in 2-dimensional grids. Consider Fig. 6, which illustrates a 6 × 6 square with directed links, and suppose a packet is originated at node (0, 0) and destined for node (5, 5). The objective is to determine the reachability probability, R(p). It is the likelihood that the packet from the source node (0, 0) reaches the destination node (5, 5), given that each network node retransmits the packet to the next adjacent node with probability p. We assume that the probability p is uniform across all nodes. Moreover, each node independently decides whether or not to retransmit a given packet.

Despite the above assumptions, it is challenging to derive the reachability probability that the packet from node (0, 0) will reach node (5, 5). This is because to reach node (5, 5), the packet may pass through node (4, 5) or node (5, 4) at the last hop. Even if the reachability probability to node (4, 5) and node (5, 4) are known, it will still be challenging to determine the reachability probability to node (5, 5) since the paths from node (0, 0) to node (5, 5) are not independent.

A brute force approach would examine all possible paths between the source and destination. However, this approach’s complexity is hampered by a combinatorial explosion due to the exponentially increasing number of paths as the grid grows. Furthermore, most paths share common links, making them non-independent.

To address this challenge, the authors of [35, 36] have developed a technique for computing the reachability probability in a 2-dimensional grid. They first calculated the reachability probability for the two fundamental paths: serial and parallel. These probabilities were subsequently used to determine the reachability probability for the 2 × 2 square and triangular grid. After that, the authors employed serial and parallel paths, 2 × 2 squares, and triangular grids as building blocks to create larger grids hierarchically. They combined their respective reachability probabilities to compute the reachability probability in larger grids, such as 3 × 3, 4 × 4, etc.

Let us illustrate the approach for computing the reachability probabilities for “serial” and “parallel” paths. As depicted in Fig. 

Fig. 27

Two types of paths exist in the grid: serial and parallel paths

27, in a “serial” path where two directed links connect nodes A to C through a common node B, the reachability probability, ${R}_{0}\left(p\right)$, that a packet generated by node A reaches node C is given by.

$${R}_{0}\left(p\right)= p \cdot p = {p}^{2}$$

(21)

On the other hand, Fig. 27 b shows two “parallel” paths connecting node A to node C, each of which is serial. The reachability probability, ${R}_{1}\left(p\right)$, can be expressed, using the principle of inclusion and exclusion because the two parallel paths are not independent, as.

$${R}_{1}\left(p\right) = {R}_{0}\left(p\right) + {R}_{0}\left(p\right)- {R}_{0}\left(p\right)\cdot {R}_{0}\left(p\right)= {p}^{2}+{p}^{2}-{p}^{2}\cdot {p}^{2}= -1 \cdot {p}^{4} +2 \cdot {p}^{2}$$

(22)

The probability ${R}_{1}\left(p\right)$ can then be used to compute the reachability probability for the 2 × 2 square. Figure 

Fig. 28

The 2 × 2 square is composed of two parallel paths

28 reveals that the 2 × 2 square is composed of two parallel paths. Consequently, the reachability probability in the 2 × 2 square is ${R}_{1}\left(p\right)$, given by Eq. (A.2).

The authors in [35, 36] used the serial and parallel paths, and the 2 × 2 square, to compose larger grids hierarchically. In doing so, they encountered the triangular grid when dealing with larger grids. The triangular grid provides a means to summarize a path between two nodes at the opposite corner in a large grid. Figure 

Fig. 29

The triangular grid is composed of two parallel paths

29 indicates that the triangular grid consists of two parallel paths, each is a serial path. The first serial path consists of a single link; thus, its reachability probability is simply p. However, the second path is serial, and its reachability probability is.${R}_{0}\left(p\right)$. Consequently, we can write the reachability probability of the triangular grid, ${R}_{T}\left(p\right)$ using the principle of inclusion and exclusion principle as

$$R_T\left(p\right)=p+R_0\left(p\right)-p{\cdot R}_0\left(p\right)=p+p^2-p\cdot p^2=p+p^2-p^3.$$

(23)

The study of [35, 36] has utilized the reachability probabilities obtained from Eqs. (A.1) to (A.3) to calculate the reachability probability of larger grids, including 3 × 3 and 4 × 4 squares, and so forth. It is important to note that constructing larger grids from fundamental components, such as serial paths, parallel paths, 2 × 2 squares, and triangular grids, requires addressing various unique scenarios separately. Due to the lengthy nature of explaining these cases, we refer the interested reader to [35, 36] for further information.