Suppressive Fair Buffer Management Policy for Intermittently Connected Mobile Ad Hoc Networks

Abstract

We propose a suppressive fair buffer management policy for intermittently connected mobile ad-hoc networks. So far, several buffer management policies have been considered. These existing buffer management policies assume that all stored messages can be replaced when nodes encounter each other. Buffer management policies, however, can prioritize messages stored at the receiving side over those at the sending side. By doing this, message transmissions are suppressive, and thus energy consumption in terms of sending messages is reduced. Moreover, our proposed policy gives relay messages with a small number of message copies to high priority. Specifically, our proposed policy maintains a sharing of buffer spaces that is as fair as possible. In this paper, we reveal how the suppression of receiving messages affects the system performance compared with existing buffer management policies. Through simulation experiments, we show that the suppressive fair buffer management policy improves energy consumption without largely degrading the delivery failure probability and the mean delivery delay.

Appendices

Appendix 1: Minimization of the Delivery Failure Probability

Suppose that all message copies are randomly discarded and the life time of each message copy is independent and identically distributed according to an exponential distribution with parameter $\alpha$. Let K denote the number of distinct messages in the network. Copies of K messages occupy all of the buffer space in the network. Formally,

$$\begin{aligned} \sum ^{K}_{k=1} N_k = ( N + 1 ) B . \end{aligned}$$

(1)

Furthermore, we assume that new messages are not generated. We denote $L_k$ ($k=1,2,\ldots ,K$) as the delivery failure probability of the kth distinct message. Probability $L_k$ is given by

$$\begin{aligned} L_k= & {} \frac{ N_k \alpha }{ N_k (\lambda + \alpha ) } \cdot \frac{ (N_k -1) \alpha }{ (N_k-1) (\lambda + \alpha ) } \cdots \frac{ \alpha }{ \lambda + \alpha } \\= & {} \left( \frac{ \alpha }{ \lambda + \alpha } \right) ^{N_k}. \end{aligned}$$

Moreover, we denote $N_E$ as the total number of distinct messages that are vanished away from the network without being delivered to their destination nodes. The mean number $\mathrm {E}[N_E]$ is given by

$$\begin{aligned} E[N_E] = \left( \frac{ \alpha }{ \lambda + \alpha } \right) ^{N_1} + \left( \frac{ \alpha }{ \lambda + \alpha } \right) ^{N_2} + \cdots + \left( \frac{ \alpha }{ \lambda + \alpha } \right) ^{N_K}. \end{aligned}$$

Note that delivery failure probability L equals $\mathrm {E}[N_E] / K$. When K is fixed, minimizing the delivery failure probability is equivalent to minimizing $\mathrm {E}[N_E]$ over all integers $N_1,N_2, \ldots , N_K$ satisfying Eq. (1). We neglect the integer constraint on $N_k$. Hence, we can write the constrained minimization using Lagrange multipliers as the minimization of

$$\begin{aligned} J = \sum _{k=1}^{K} \left( \frac{ \alpha }{ \lambda + \alpha } \right) ^{N_k} - \gamma \left( \sum _{k=1}^{K} N_k - (N+1) B \right) . \end{aligned}$$

Differentiating with respect to $N_k$ ($k=1,2,\ldots ,K$), we obtain

$$\begin{aligned} \frac{\partial J}{\partial N_k} =\left( \frac{\alpha }{\lambda + \alpha } \right) ^{N_k} \log \frac{ \alpha }{ \lambda + \alpha } - \gamma . \end{aligned}$$

Setting the derivative to 0, we obtain

$$\begin{aligned} \gamma ^* = \left( \frac{ \alpha }{ \lambda + \alpha } \right) ^{N_k^*} \log \frac{ \alpha }{ \lambda + \alpha }, \qquad k = 1,2,\ldots ,K. \end{aligned}$$

Therefore, when $(\alpha / \lambda + \alpha )^{N_1^*} = (\alpha / \lambda + \alpha )^{N_2^*} = \cdots = (\alpha / \lambda + \alpha )^{N_K^*}$, i.e., $N_1^* = N_2^* = \cdots = N_K^*$, J reaches a minimum. When $N_1 = N_2 = \cdots = N_K$, we consider the distribution of message copies to be fair. Otherwise, we regard it to be unfair. This indicates that when the message copy distribution is fair, the delivery failure probability is minimized.

When $N_k^*$ ($k=1,2,\ldots ,K$) is not an integer, in order to achieve the minimum, buffer space is utilized with fair time sharing, i.e., the time that each distinct message occupies buffer space is equal. For example, we consider that there are two distinct messages and three nodes with $B=1$ in the network. In this case, $N_1^* = N_2^* = 3/2$. Figure 11 shows an example of fair time sharing. Buffers 1 and 2 are occupied by Messages 1 and 2 from time 0 to time $4 \tau$, respectively. In contrast, in Buffer 3, the buffer occupancy time of Message 1 and that of Message 2 equal $2 \tau$. Thus, the average number of copies of Message 1 and 2 is $1 \times (2 \tau / 4 \tau ) + 2 \times (2 \tau / 4 \tau ) = 3 / 2$ from time 0 to time $4 \tau$.

Fig. 11

Example of fair time sharing

Appendix 2: Minimization of the Mean Delivery Delay

To show the effectiveness of the fair sharing distribution in terms of the mean delivery delay, we consider a simple model, where there are K types of messages in the network and all nodes have B message copies. Moreover, we assume that, after time t, all messages and their copies are not generated and vanish away from the network. Under these assumptions, the mean delivery delay of the kth distinct message is represented by $1 / ( N_k \lambda ).$ The average message delivery delay W in the network is given by

$$\begin{aligned} W = \frac{1}{K} \sum ^{K}_{k=1} \frac{1}{N_k \lambda }. \end{aligned}$$

Minimizing W is equivalent to minimizing $\sum ^{K}_{k=1} 1 / N_k$. For the relationship between the arithmetic mean and harmonic mean of $N_k$ ($k= 1,2, \ldots ,K$), the following inequality is given by

$$\begin{aligned} \frac{K}{\sum ^{K}_{k=1} \frac{1}{N_k} } \le \frac{1}{K} \sum ^{K}_{k=1} N_k = \frac{1}{K} ( N + 1 ) B, \end{aligned}$$

where $\sum ^{K}_{k=1} N_k$ = $(N+1)B$ when the message copies occupy all of the buffer space. Thus, we obtain the following inequality.

$$\begin{aligned} \sum ^{K}_{k=1} \frac{1}{N_k} \ge \frac{K^2}{ (N+1) B }. \end{aligned}$$

Equality only holds when $N_1 = N_2 = \cdots = N_K$. When $N_1 = N_2 = \cdots = N_K$, W reaches a minimum. Thus, the mean delivery delay is minimized when each message in the network has the same number of message copies, i.e., the message copy distribution is fair.