Energy optimal channel attempt rate and packet size for ALOHA based underwater acoustic sensor networks

Abstract

In this paper, we investigate the energy efficiency and throughput of ALOHA based underwater acoustic sensor networks (UASNs). We derive closed form expressions for the channel attempt rate and packet length at which energy efficiency and throughput are maximized separately. Based on our results, we observe that the packet length that maximizes energy efficiency leads to deterioration of throughput and vice versa. Motivated by these observations, we consider a cross layer optimization problem with the objective of maximizing the energy efficiency of the network, while meeting throughput criteria at the MAC layer and an SNR criteria at the PHY layer. With the aid of Karush–Kuhn–Tucker conditions, we derive closed form solutions for the optimal channel attempt rate and packet length that satisfy the desired objectives. For the analysis, we consider underwater acoustic channel specific parameters such as spreading losses and distance dependent bandwidth. Extensive performance evaluation study of our approach proves that, judicious selection of the packet lengths as well as channel attempt rates by the sensor nodes can ameliorate the energy efficiency of UASN remarkably, while satisfying the throughput criterion.

Appendix

1.1 1. Derivation of (7)

Notice that, (7) is obtained by differentiating (5) with respect to $\lambda$ and equating to zero, i. e., $\frac{dS}{d\lambda }_{|\lambda _S^*} = 0$. Accordingly, we get

$$\begin{aligned}&\delta B(l) e^{2 (n-1) \lambda T} (n-1)L\nonumber \\&\quad = (n-1) \lambda L \delta B(l) e^{2 (n-1) \lambda T} 2(n-1)T \end{aligned}$$

(26)

Now from (26), we get $\lambda _S^*$ which is given by (7)

1.2 2. Derivation of (8)

The throughput maximizng packet length $L_S^*$ given by (8) is obtained by differentiating (5) with respect to L and equating to zero, i.e., $\frac{dS}{dL}_{|L_S^*} = 0$. This gives the following Eq.:

$$\begin{aligned}&\delta B(l) e^{2 (n-1) \lambda T} (n-1) \lambda \nonumber \\&\quad = (n-1) \lambda L \delta B(l) e^{2 (n-1) \lambda T} \frac{2(n-1)\lambda }{\delta B(l)} \end{aligned}$$

(27)

Now, $L_S^*$ given by (8) can be obtained from (27).

1.3 3. Derivation of (9)

Energy efficiency maximizing channel attempt rate $\lambda _\eta ^*$ is obtained by differentiating (6) with respect to $\lambda$ and equating to zero. Accordingly, we get

$$\begin{aligned} \Big ( \frac{L}{\delta B(l)}+\frac{K}{\lambda } \Big ) \Big (\frac{2(n-1)L^2}{(\delta B(l))^2} e^{\frac{-2(n-1)\lambda L}{\delta B(l)}} \Big ) = \frac{KLe^{\frac{-2(n-1)\lambda L}{\delta B(l)}}}{\delta B(l) \lambda ^2} \nonumber \\ \end{aligned}$$

(28)

where $K=\frac{P_{sl}}{P_{t,min}}$. Simplifying (28), we get the following eqn:

$$\begin{aligned} \frac{2L^2 (n-1)}{(\delta B(l))^2} \lambda ^2 + \frac{2(n-1)LK}{\delta B(l)} \lambda - K =0 \end{aligned}$$

(29)

Now (9) is obtained by solving the quadratic equation given by (29).

1.4 4. Derivation of (10)

Energy efficiency maximizing packet size $L_\eta ^*$ is obtained by differentiating (6) with respect to L and equating to zero. Accordingly, we get the following equation:

$$\begin{aligned} \Big ( \frac{L}{\delta B(l) } + \frac{K}{\lambda } \Big ) \Big ( \frac{1}{\delta B(l)} - \frac{2(n-1)\lambda L}{(\delta B(l))^2}\Big ) = \frac{L}{(\delta B(l))^2} \end{aligned}$$

(30)

Rearranging and simplifying (30), we get

$$\begin{aligned} \frac{2(n-1)\lambda }{(\delta B(l))^2} L^2 + \frac{2(n-1)K}{\delta B(l)} L -\frac{K}{\lambda } =0 \end{aligned}$$

(31)

Now (10) is obtained by solving the quadratic equation given by (31).

1.5 5. Derivation of (20g)

Substituting the value for $\mu _1 = 0$ in (20c) gives, the following equation:

$$\begin{aligned} \frac{P_{sl}}{\lambda ^2} = \frac{2(n-1)L}{\delta B(l)} \Big (\frac{P_{t,min}L}{\delta B(l)} + \frac{P_{sl}}{\lambda }\Big ) \end{aligned}$$

(32)

Rearranging (32), we get the following quadratic equation:

$$\begin{aligned} \frac{2L^2 (n-1)}{(\delta B(l))^2} \lambda ^2 + \frac{2(n-1)LK}{\delta B(l)} \lambda - K =0 \end{aligned}$$

(33)

where $K=\frac{P_{sl}}{P_{t,min}}$. Now (20g) is obtained as the solution of (33).

1.6 6. Derivation of (23g)

Substituting the value for $\mu _1=0$ in (23c) gives the following equation:

$$\begin{aligned}&\Big ( \frac{P_{t,min}L}{\delta B(l)} + \frac{P_{sl}}{\lambda }\Big ) \Big (\frac{P_{t,min}}{\delta B(l)} + \frac{-2(n-1)\lambda P_{t,min}L}{(\delta B(l))^2}\Big )\nonumber \\&\quad = \frac{P_{t,min}^2L}{(\delta B(l))^2} \end{aligned}$$

(34)

Simplifying (34) gives the following:

$$\begin{aligned} \frac{P_{sl}}{\lambda } = \frac{2(n-1)\lambda P_{t,min}}{(\delta B(l)^2)} L^2 + \frac{2(n-1)P_{sl}}{\delta B(l)}L \end{aligned}$$

(35)

Dividing with $P_{t,min}$ and letting $K = \frac{P_{sl}}{P_{t,min}}$, we get

$$\begin{aligned} \frac{2(n-1)\lambda }{(\delta B(l))^2} L^2+ \frac{2(n-1)K}{\delta B(l)} L - \frac{K}{\lambda } = 0 \end{aligned}$$

(36)

Now (23g) is obtained as the solution of quadratic equation given by (36).

1.7 7. Tutorial example for Lingo

LINGO is a software tool designed to efficiently build and solve linear, nonlinear, and integer optimization models [40]. Given below is an example in our scenario, i.e., based on optimization problem (24). Here we are defining the nodes as a set, which have some associated characteristics called attributes (distance, bandwidth, transmit power, attempt rate). The set members are initialized in the data section (i.e., attributes are defined). For example, attribute ’DISTANCE’ has values ’d1–d9’ for the nine nodes considered for the problem. Then we start writing the objective function followed by the constraints. Here the objective is to maximize the energy efficiency (which is given in the LINGO syntax). Two constraints are defined; (i)throughput is less than or equal to the threshold value, (ii)attempt rate is greater than zero.

Once the LINGO model has been entered into the LINGO model window, the model can be solved, after which the solver status box describes the model classification (Linear Programming, Quadratic Programming, Integer Linear Programming, Non Linear Programming etc), state of the current solution (i.e., local or global optimum, feasible or infeasible), the value of the objective function and the number of iterations required to solve the model.