## Appendix B Symbols  
\begin{tabular}{l l} \(\alpha\) & the path loss exponent \\ \(\beta_{1}\) & the energy spent in the electronics circuitry for transmitting each bit data \\ [PERSON]beta_{2}\) & the coefficient affected by the transmit amplifier efficiency, antenna gains and other system parameters \\ \(\beta_{3}\) & the energy spent in the electronics circuitry for receiving each bit data \\ \(\beta_{4}\) & the energy spent for each bit message in the idle state \\ \(\tau\) & the optimal lifetime \\ \(\tau^{*}\) & the lifetime requirement of the WSN \\ \(\omega_{ack}\) & the expected number of transmission attempts of an ACK packet \\ \(\omega_{ack,n}\) & the probability of transmitting an ACK packet \(n\) times \\ \end{tabular}  
\begin{tabular}{l l} \(\omega_{data}\) & the expected number of transmission attempts of a DATA packet \\ \(\omega_{data,n}\) & the probability of transmitting a DATA packet \(n\) times \\ \(a\) & the number of packets sensed by each sensor in each data gathering cycle \\ \(A\) & the retransmission rate of a DATA packet \\ \(B\) & the retransmission rate of an ACK packet \\ \(C\) & the speed of light \\ \(d\) & the distance between the neighboring sensors \\ \(d_{c}(i)\) & the circumcircle radius of a sensor in layer \(i\) \\ \(E\) & the total energy consumed by a sensor in a data gathering cycle \\ \(E_{0}\) & the initial energy of a sensor \\ \(E_{i,j}\) & the energy consumption at location \((i,j)\) in a data gathering cycle \\ \(E^{\prime}_{i,j}\) & the optimal energy consumed by the sensor at location \((i,j)\) in a data gathering cycle \\ \(E_{id}\) & the energy consumption in the idle state \\ \(E_{non}\) & the energy consumption of a sensor without retransmission in a data gathering cycle \\ \(E_{r}\) & the energy consumption of reception \\ \(E_{re}\) & the energy consumption of a sensor with retransmission in a data gathering cycle \\ \(E_{t}\) & the energy consumption of transmission \\ \(f\) & the bandwidth \\ \(i\) & the layer number \\ \(j\) & the location number in a layer \\ \(l\) & the length of the packet preamble \\ \(L\) & the number of layers \\ \(l_{ack}\) & the preamble length of the ACK packet \\ \(l_{data}\) & the preamble length of the DATA packet \\ \(m\) & the length of a single packet \\ \(m_{ack}\) & the length of the ACK packet \\ \(m_{data}\) & the length of the DATA packet \\ \(m_{r}\) & the length of the message received \\ \(m_{t}\) & the length of the message transmitted \\ \(n\) & the transmission times \\ \(n_{i,j}\) & the number of sensors deployed at location \((i,j)\) \\ \(N\) & the optimal total number of sensors deployed in the target area \\ \(N^{*}\) & the maximum number of sensors allowed to be deployed in the target area \\ \(N_{t}(i,j)\) & the number of packets transmitted at location \((i,j)\) \\ \(N_{t,non}\) & the numbers of packets transmitted without retransmission \\ \(N_{r,non}\) & the numbers of packets received without retransmission \\ \(N_{t,re,ack}\) & the numbers of ACK packets transmitted with retransmission \\ \(N_{r,re,ack}\) & the numbers of ACK packets received with retransmission \\ \(N_{t,re,data}\) & the numbers of DATA packets transmitted with retransmission \\ \(P\) & the dealing rate of the message sensed by sensors \\ \(P_{t}\) & the transmission power \\ \(R_{a}\) & the radius of the circular target area \\ \end{tabular}