Deploying Throwboxes to Enhance Fault-Tolerance Performance in Delay Tolerant Networks

Abstract

Delay Tolerant Networks (DTNs) have attracted various interests these days. Since DTNs are subject to high loss rate, large delay, intermittent connection, and even no end-to-end connectivity, relay nodes, such as throwboxes, are deployed to enhance network performance. Internet-based systems have contemporaneous connectivity between location-distributed nodes, and this does not apply to DTNs. Thus, the traditional relay node deployment strategies are no longer suitable for DTNs. In this paper, we propose a novel strategy, named Connection-2 ($CO_2$), to deploy throwboxes to enhance the fault tolerance of DTNs. $CO_2$ constructs a 2-connected DTN using an approximation algorithm. Every mobile node in the 2-connected DTN can reach another mobile node via two or more node-disjoint paths within its mobility range. While enhancing fault tolerance, the number of throwboxes that $CO_2$ requires is small. We conduct various experiments based on the simulation of the real Tuscaloosa bus transit system and compare its performance with two popular strategies. Experimental results show that $CO_2$ is effective.

Appendices

Appendix A

1.1 Proof of Theorem 1

Proof

We prove it with mathematical induction.

Firstly, we prove that the DTN $G_1(V_1,E_1,S_1,P_1)$ constructed in the first round by $CO_2$, e.g., $n=1$ round, is 2-connected.

According to $CO_2$, the vertex set $V_1$ of $G_1$ consists of the relay nodes in two node disjoint paths from $u_0$ to $u'$, and the mobile nodes covered by these relay nodes includes the following

$$ V_1 = \{u_0\}\cup Q_{u'} \cup C(q'), $$

(14)

where $q' \in \{u_0\}\cup Q_{u'}$. Since all the relay throwboxes are in the two paths from $u_0$ to $u'$, and the two paths are node disjoint, when one of the throwboxes fails, the others can still communicate with each other via one or multiple hops. Thus the throwbox set is 2-connected according to Definition 3. Now we only need to prove that each mobile node b in $C(q')$, where $q'$ in $\{u_0\}\cup Q_{u'}$ can directly communicate with at least two throwboxes, according to Definition 4. Since $u'$ is chosen according to Eq. (6), let’s define

$$ w = min(c(b,Q_{u}),2-label(b)), $$

(15)

where u is each node in U.

w only has three values: 0, 1 and 2.

• 1. When $c(b,Q_{u}) = 2-label(b) = 2$, $w =2$, which means that the mobile nodes are covered by $Q_{u'}$ twice and are not covered by $u_0$.

• 2. When $c(b,Q_{u}) = 1$ and $2-label(b) \ge 1$, or $c(b,Q_{u}) \ge 1$ and $2-label(b) = 1$, $w =1$, which means that the mobile nodes are covered at least once by both $u_0$ and $Q_{u'}$.

• 3. When $c(b,Q_{u}) = 0$ or $2-label(b) = 0$, $w =0$, which means that the mobile nodes are either not covered by $u_0$ or are covered by $Q_{u'}$ twice.

Since in Eq. (6), we choose the node with the largest value, all mobile nodes are covered twice by the 2-connected throwboxes.

Now define the constructed DTN as $G_1$. Add a virtual node $v'$ into $G_1$ and connect $v'$ with each node in R. The constructed throwbox set is 2-connected and the relay node $u''$ has two node disjoint paths to $v'$, which means that $u''$ can be connected to the throwbox set via two node-disjoint routes. Thus, the new throwbox set is also 2-connected.

We define the mobile nodes set covered by $V_1$ as $C(V_1)$ and the mobile nodes covered by $V_1 \cup u''$ as $C(V'_1)$.

• 1. If $C(V_1) = C(V'_1)$, each mobile node b can directly communicate with at least two throwboxes in $V_1 \cup u''$, because it can directly communicate with at least two throwboxes in $V_1$.

• 2. If $C(V_1) \not = C(V'_1)$, for each mobile node b in $C(V'_1) - C(V_1)$, it satisfies Eqs. (9) and (15). w can only have three values: 0, 1 and 2 according to Eq. (15). As proved above, b can be covered twice by the throwbox set if it satisfies Eq. (15) with the largest value of w.

Thus, each mobile node b covered by $V'_1$ can directly communicate with at least two throwboxes in $V'_1$.

Therefore when $n=1$, the DTN is 2-connected. Note that as the assumption, the possible solution is included in the unchosen location set U.

Secondly, assume that the DTN $G_k(V_k,E_k,S_k,P_k)$ constructed in the $n = k$ round by $CO_2$ is 2-connected, we prove that the DTN $G_{k+1}(V_{k+1},E_{k+1},S_{k+1},P_{k+1})$ constructed in the $n = k+1$ round by $CO_2$ is also 2-connected.

Add a virtual node $v'$ into $G_k$, and connect $v'$ with each node in R. Since

• 1. the throwbox set constructed in $n = k$ round is 2-connected, because the DTN $G_k(V_k,E_k,S_k,P_k)$ is 2-connected;

• 2. the relay node $u''$ found in $k+1$ round has two node disjoint paths to $v'$, which means that $u''$ can be connected to the throwbox set via two node-disjoint routes;

The throwbox set constructed in $k+1$ round is also 2-connected. Now we only need to prove that each mobile node b covered by $V_{k+1}$ can directly communicate with at least two throwboxes in $V_{k+1}$.

We define the mobile nodes set covered by $V_k$ as $C(V_k)$ and the mobile nodes covered by $V_{k+1}$ as $C(V_{k+1})$.

• 1. If $C(V_k) = C(V_{k+1})$, each mobile node b can directly communicate with at least two throwboxes in $V_{k+1}$, because it can directly communicate with at least two throwboxes in $V_k$.

• 2. If $C(V_k) \not = C(V_{k+1})$, for each mobile node b in $C(V_{k+1}) - C(V_k)$, it satisfies Eqs. (9) and (15). w can only have three values: 0, 1 and 2 according to Eq. (15). As proved above, b can be covered twice by the throwbox set if it satisfies Eq. (15) with the largest value of w.

Thus, each mobile node b covered by $V_{k+1}$ can directly communicate with at least two throwboxes in $V_{k+1}$.

Therefore, when the DTN $G_k(V_k,E_k,S_k,P_k)$ constructed in the $n = k$ round by $CO_2$ is 2-connected, the DTN $G_{k+1}(V_{k+1},E_{k+1},S_{k+1},P_{k+1})$ constructed in the $n = k+1$ round by $CO_2$ is also 2-connected.

Now, we have proved that the DTN G(V, E, S, P) constructed by $CO_2$ is 2-connected.$\square$

Appendix B

1.1 Proof of Theorem 2

Proof

As previously defined, $B = \{b_1,b_2,\ldots ,b_i,\ldots ,b_n \}$ is the set of mobile nodes. As shown in Fig. 13, $B^1$ and $B^2$ are 2 copies of B, where $B^j = \{b_1^j,b_2^j,\ldots ,b_i^j,\ldots ,b_n^j\}$, $i\in [1,n]$, and $j\in [1,2]$. U is the set of unchosen possible locations of relay throwboxes. $R_{min} = \{r^*_1, r^*_2, \ldots , r^*_i, \ldots , r^*_{min}\}$ is a minimal 2-connected throwbox set. $R_{CO_2} = \{r_1, r_2, \dots , r_i, \ldots , r_{CO_2}\}$ is the throwbox set chosen by the proposed $CO_2$ algorithm, and $|R_{CO_2}| \ge |R_{min}|$. Their edge sets are $E_{min}$ and $E_{CO_2}$, respectively.

$$ |R_{CO_2}| < \mathcal O(D\ln n) |R_{min}|. $$

(16)

Fig. 13

The 2-connected DTN graph constructed by $CO_2$ shown on the right side and the corresponding minimal 2-connected DTN shown on the left side

For each $b_i^j \in B^1 \cup B^2$, we define

$$ w(b^j_i) = |Q \cap U|/C(Q), $$

(17)

where Q is the throwbox set of two disjoint paths and C(Q) is the mobile node set covered by Q, as previously defined.

Assume that $CO_2$ can find a solution $R_{CO_2}$ after M steps. Then we have

$$ |R_{CO_2}| = \quad \underset{m=1}{\overset{M}{\sum }}|Q_m \cup U|. $$

(18)

Since $|Q \cap U| = w(b^j_i)C(Q)$, according to Eq. (17), we get

$$ |R_{CO_2}| = \quad \underset{m=1}{\overset{M}{\sum }}\quad \underset{b^j_i covered \, by \, Q_m}{\sum }w(b^j_i) C(Q_m). $$

(19)

From Step 1 to Step M, $CO_2$ has covered all the nodes in $U^1$ and $U^2$, thus

$$ \underset{m=1}{\overset{M}{\sum }}\quad \underset{b^j_i covered \, by \, Q_m}{\sum }w(b^j_i) C(Q_m) = \quad \underset{b^j_i\in B^1\cup B^2}{\sum }w(b^j_i). $$

(20)

Then we can obtain

$$ |R_{CO_2}| = \quad \underset{b^j_i\in B^1\cup B^2}{\sum }w(b^j_i). $$

(21)

Because all the nodes in $U^1$ and $U^2$ are also covered by previously assumed algorithm with a minimal 2-connected solution $R_{min}$, as shown on the left side of Fig. 13, then

$$\begin{aligned} {\begin{matrix} \quad \underset{b^j_i\in B^1\cup B^2}{\sum }w(b^j_i) &{}= \quad \underset{r^*_i \in R_{min}}{\sum } \quad \underset{b^j_i covered \, by \, r^*_i}{\sum }w(b^j_i)\\ &{} = |R_{min}| \quad \underset{b^j_i covered \, by \, r^*_i}{\sum }w(b^j_i). \end{matrix}} \end{aligned}$$

(22)

Now, $|R_{CO_2}|$ and $|R_{min}|$ have a relationship as follows:

$$ |R_{CO_2}| = |R_{min}| \! \quad \underset{b^j_i covered \, by \, r^*_i}{\sum }w(b^j_i). $$

(23)

Let define

$$ {\mathscr {W} }= \quad \underset{b^j_i covered \, by \, r^*_i}{\sum }w(b^j_i), $$

(24)

and discuss its range. Because of Eq. (17) and that the step number of getting $R_{min}$ is not larger than getting $R_{CO_2}$,

$$ {\mathscr {W}} \le \underset{m=1}{\overset{M}{\sum }}(|B_{m-1}|-|B_m|)\frac{|Q_m \cap U|}{C(Q_m)}, $$

(25)

where $|B_m|$ is the number of uncovered neighbor nodes of $Q_m$ after Step m, $B_m \in B^1 \cup B^2$. Note that they are only uncovered neighbor nodes, not all uncovered nodes in B.

Since $C(Q_m) \ge |B{m-1}|$ and $|Q_m \cap U|\le D$,

$$ \mathscr {W} \le D \underset{m=1}{\overset{M}{\sum }}\frac{|B_{m-1}|-|B_m|}{|B_{m-1}|}. $$

(26)

Actually $|B{m-1}|-|B_m|$ is the newly covered number of nodes in Step m. Thus

$$ 1 \le |B_{m-1}|-|B_m| \le |B_0|. $$

(27)

The best situation occurs when all nodes are covered in the first step, that is

$$ \underset{m=1}{\overset{M}{\sum }}\frac{|B_{m-1}|-|B_m|}{|B_{m-1}|} = 1. $$

(28)

The worst situation occurs when only one node is covered in each step. Thus

$$ \mathscr {W} \le D \left( \frac{1}{|B_0|} + \frac{1}{|B_0|-1} + \frac{1}{|B_0|-2} + \cdots + 1\right). $$

(29)

Let define

$$\begin{aligned} f= & {} \underset{n \rightarrow \infty }{\lim }(\underset{k=1}{\overset{n}{\sum }}{\frac{1}{k}} - \ln {(n)})\nonumber \\= & {} \underset{n \rightarrow \infty }{\lim }(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} - \ln {(n)})\nonumber \\= & {} \underset{n \rightarrow \infty }{\lim }(\underset{k=1}{\overset{n}{\sum }}\frac{1}{k} - \int ^n_1 \! \frac{1}{x} \, \mathrm {d} x)\nonumber \\= &\, {} \gamma \nonumber \\\approx\,&\, {} 0.577. \end{aligned}$$

(30)

Thus,

$$\begin{aligned} \underset{k=1}{\overset{n}{\sum }}{\frac{1}{k}}&\approx {} \int ^n_1 \! \frac{1}{x} \, \mathrm {d} x + \gamma \nonumber \\&= {} \gamma + \ln x\nonumber \\&< {} 1+\ln x. \end{aligned}$$

(31)

Therefore, according to Eqs. (29) (31), we have

$$ \mathscr {W} < \,D(1+\ln |B_0|). $$

(32)

Now put $\mathscr {W}$ into Eqs. (23) (24), we can obtain

$$ |R_{CO_2}| <\, D(1+\ln |B_0|)|R_{min}| $$

(33)

And since $|B_0| \le n$, where n is the number of mobile nodes, we get

$$ |R_{CO_2}| <\, D(1+\ln n)|R_{min}|. $$

(34)

Then Eq. (16) is established. Therefore, to construct a 2-connected DTN with $CO_2$ strategy, the time complexity is $\mathcal O(D\log n)$ times of constructing a minimal 2-connected DTN.$\square$