p-Epidemic forwarding method for heterogeneous delay-tolerant networks

Abstract

Delay-tolerant network (DTN) is a kind of wireless network that is specified by its discontinuous connectivity among the nodes. Due to the increasing use of wireless communications and infrastructure-less networks, DTNs should be considered accurate. epidemic routing, as a replication-based routing protocol is defined to overcome the intermittent connectivity issue in these networks. In this routing, all nodes maintain their buffer messages index which is called the summary vectors. The nodes exchange their summary vectors when they meet their neighbors. All carried messages by a node that are not presented in its neighbors are transmitted. So, a node sends multiple copies of a message to other nodes that do not have that message. The replication process consumes a high amount of network resources such as node energy. To address the resource problem, this paper intends to propose an effective mechanism for message delivering into those networks. This work formulated an energy-efficient probabilistic forwarding method of heterogeneous sets of nodes having two different transmission radii as well as two different amounts of available energies. We propose a static policy-based message forwarding method of two different forwarding probabilities for the heterogeneous sets of nodes. Our analytical result is supported by the simulation outcomes in terms of message delivery probabilities and the number of transmissions of the network. The results of this work can be considered in new applications such as delay-tolerant Internet of things.

Appendix

Coefficients of (18) are calculated as follows:

$$\begin{aligned}&T=\alpha \lambda _{22}N-\alpha \lambda _{11}N+\alpha \lambda _{11}\beta -\alpha \lambda _{22}\beta \\&Q=\alpha \lambda _{11}-\alpha \lambda _{22} \\&P=\alpha \lambda _{11}\beta -\alpha \lambda _{22}\beta \\&H=\alpha \lambda _{22}\beta -\alpha \lambda _{11}\beta \\&I=\lambda _{11}\beta -\lambda _{22}\beta \\&J=\alpha \lambda _{22}\beta -\alpha \lambda _{11}\beta \\&O=\lambda _{11}\beta -\lambda _{22}\beta \\&M=\lambda _{22}^{2}(2\beta \alpha -2N\alpha -\alpha ^{2})+\lambda _{22}\lambda _{11}(2N\alpha -\beta \alpha +\alpha ^{2}) \\&R=\alpha \lambda _{22}^{2}-\alpha \lambda _{22}\lambda _{11} \\&W=\lambda _{22}^{2}(2N\alpha ^{2}\beta -N^{2}\alpha ^{2}-\beta ^{2}\alpha ^{2})+\lambda _{22}\lambda _{11}(2\beta ^{2}\alpha ^{2}+N^{2}\alpha ^{2}-2N\alpha ^{2})-\beta ^{2}\alpha ^{2}\lambda _{11}^{2} \\&K=2\beta \alpha \lambda _{22}^{2}-2\beta \alpha \lambda _{22}\lambda _{11} \\&G=\lambda _{22}^{2}(2\beta ^{2}\alpha -2\alpha ^{2}\beta -2N\beta \alpha )+\lambda _{22}\lambda _{11}(2\alpha ^{2}\beta -4\beta ^{2}\alpha +2N\beta \alpha )+2\beta ^{2}\alpha \lambda _{11}^{2} \\&X=\lambda _{22}^{2}(2N\alpha ^{2}\beta -2\beta ^{2}\alpha ^{2})-2\beta ^{2}\alpha ^{2}\lambda _{11}^{2}+\lambda _{22}\lambda _{11}(4\beta ^{2}\alpha ^{2}-2N\alpha ^{2}\beta ) \\&S=\beta ^{2}\alpha \lambda _{11}^{2}+\beta ^{2}\alpha \lambda _{22}^{2}-2\beta ^{2}\alpha \lambda _{22}\lambda _{11} \\&F=2\beta ^{2}\alpha ^{2}\lambda _{22}\lambda _{11}-\beta ^{2}\alpha ^{2}\lambda _{11}^{2}-\beta ^{2}\alpha ^{2}\lambda _{22}^{2} \\&L=\lambda _{22}^{2}(N^{2}\alpha +2N\alpha ^{2}-2N\beta \alpha -\alpha ^{2}\beta +\beta ^{2}\alpha ) \\&\qquad +\,\lambda _{22}\lambda _{11}(2N\beta \alpha -\beta \alpha +2\alpha ^{2}\beta -2\beta ^{2}\alpha -N^{2}\alpha -2N\alpha ^{2})+\beta ^{2}\alpha \lambda _{11}^{2} \end{aligned}$$

(41)

Coefficients of (26) are:

$$\begin{aligned}&A=N^{2}\alpha \beta (\lambda _{11}-\lambda _{22})-N^{3}\alpha (\lambda _{11}-\lambda _{22}) \\&B=2N\beta \alpha \lambda _{22}+\alpha \beta \lambda _{11}-\beta \alpha \lambda _{22}-N^{3}\lambda _{22} \\&\qquad +\,3N^{2}\alpha \lambda _{11}-3N^{2}\alpha \lambda _{22}-N^{2}\beta (\lambda _{11}-\lambda _{22})-2N\beta \alpha \lambda _{11} \\&C=\alpha \beta \lambda _{11}-\alpha \beta \lambda _{22}+2N\beta (\lambda _{11}-\lambda _{22})-\beta (\lambda _{11}-\lambda _{22}) \\&\qquad +\,3N^{2}\lambda _{22}-3N\alpha \lambda _{11}+3N\alpha \lambda _{22} \\&D=\alpha (\lambda _{11}-\lambda _{22})-\beta (\lambda _{11}-\lambda _{22})-3N\lambda _{22} \\&E=\lambda _{22} \end{aligned}$$

(42)

To solve (36), if:

$$\begin{aligned}&a=\,-E \\&b=4NE \\&c=3DN+C \\&d=2CN+2B \\&e=BN+3A \end{aligned},$$

(43)

then from (36) and (43)

$$\begin{aligned} e+dy+cy^{2}+by^{3}+ay^{4}=0 \end{aligned}$$

(44)

The four roots $y_{1},y_{2},y_{3},y_{4}$ for the general quartic equation $e+dy+cy^{2}+by^{3}+ay^{4}=0$ are given in the following formula:

$$\begin{aligned} y_{1,2}=\, & {} \frac{-b}{4a}-S\pm \frac{1}{2}\sqrt{-4S^{2}-2p+\frac{q}{s}} \\ y_{3,4}=\, & {} \frac{-b}{4a}{+}S\pm \frac{1}{2}\sqrt{-4S^{2}-2p+\frac{q}{s}} \end{aligned}$$

(45)

where p and q are:

$$\begin{aligned} p=\, & {} \frac{8ac-3b^{2}}{8a^{2}} \\ q=\, & {} \frac{b^{3}-4abc+8a^{2}d}{8a^{3}} \end{aligned}$$

(46)

and

$$\begin{aligned} S=\, & {} \frac{1}{2}\sqrt{\frac{1}{3a}\left( Q+\frac{{{\varDelta }}_{0}}{Q}\right) -\frac{2p}{3}} \\ Q=\, & {} \root 3 \of {\frac{{{\varDelta }}_{1}+\sqrt{{{\varDelta }}_{1}^{2}-4{{\varDelta }}_{0}^{3}}}{2}} \end{aligned}$$

(47)

with

$$\begin{aligned} \varDelta _{0}=\, & {} c^{2}-3bd+12ae \\ \varDelta _{1}=\, & {} 2c^{3}-9bcd+27b^{2}e+27ad^{2}-72ace \end{aligned}$$

(48)