**7.1. Constraints**

The constraints on the optimization as presented in (6) provide the boundaries for the selection of optimal resource parameters for both sensor and mesh nodes. The first constraint, *Pvx*, *fx* · *Pvx*+1, *fx* = 0, ∀*vx*, *vx*+<sup>1</sup> ∈ *pathvi*,*vj* , provides a power allocation restriction that a cluster node *vx*+<sup>1</sup> in an end-to-end path cannot receive and transmit on the same channel *fx*, that is *fx* �= *fx*+1. This restricts node *vx* and *vx*+1, which are successive nodes in a path, from allocating power on the same transmission channel. The second constraint, *Pvz*, *fz* · *Pvz*+1, *fz* = 0, ∀*vz*, *vz*+<sup>1</sup> ∈ *pathvi*,*vj* , is the corresponding power allocation restriction for the mesh network.

The third constraint, *Pvx*, *fx* , *Pvz*, *fz* ≥ *max*(*PMIRS*, *PSNR*, *Pmin*), ∀*vx*, *vz* ∈ *pathvi*,*vj* , restricts the minimum power that the receiver can receive. *PMIRS* corresponds to the minimum input signal power at the receiver, or the minimum input receiver sensitivity (MIRS), defined in Table 1. The *PSNR* constraint is the minimum power required to reach the signal-to-noise ratio (SNR) threshold at the receiver. The third term, *Pmin*, represents the lower bound on the transmit power that keeps *Lvx*,*vy*, *<sup>f</sup>*(*Rvx*,*vy*, *<sup>f</sup>* , *t*) positive. There is a fourth factor that is the lower bound on the available capacity at the transmitter, which is zero and is thus ignored.


**Table 1.** Operating Parameters for Zigbee, UWB and WiMax for Constraint Modeling [12–15]

The fourth constraint, *Pvx*, *fx* , *Pvz*, *fz* ≤ *min*(*Ptech*, *Pcap*, *Pmax*), ∀*vx*, *vz* ∈ *pathvi*,*vj* , restricts the maximum power of the optimality range. *Ptech* corresponds to the maximum allowable transmit power on a transmission channel for the technology being used. From Table 1, *Ptech* is calculated as *Ptech* (dBm) = Maximum EIRP (dBm) - *Gt* (dBi), where the Effective Isotropic Radiated Power (EIRP) is the maximum allowable power that can be put on the transmission channel and *Gt* is the transmit antenna gain in dBi. *Pcap* represents the power that corresponds to the available outgoing capacity on the channel at the transmitter and ensures that the channel capacity is not exceeded. While the lower bound on the outgoing capacity is zero in the previous constraint, the upper bound *Pcap* is not zero unless the full channel capacity is being used by the transmitter. Meanwhile, *Pmax* restricts the upper bound on transmit power as that which keeps *Lvx*,*vy*, *<sup>f</sup>*(*Rvx*,*vy*, *<sup>f</sup>* , *t*) positive.

The final constraint limits a path to at most *K*� hops to reduce power dissipation due to routing.

### **7.2. Steps in resource allocation**

12 Wireless Sensor Networks / Book 1

computed via the optimized link utility function *L*∗ across all nodes along a candidate path inside the cluster only; *g* represents the number of hops along the path within the cluster before reaching cluster-head *vm* and is computed as *g* = *find*(*pathvi*,*vj* == *vm*) − 1. The

We divide the utility sum by the hop count of the path to calculate the average link utility in the path. By doing so, we are able to more closely analyze the difference between a *k*-hop path and (*k+*)-hop path in choosing the optimal route. If we were to use total rather than average link utility, the algorithm would favor the (*k+*)-hop path as the summation of more

limited resources. Hence, by evaluating the average link utility, we are in fact reducing latency and conserving power. It should be noted, however, that, if a path with more hops has the

the best tradeoff between performance and energy consumption across all candidate paths.

The constraints on the optimization as presented in (6) provide the boundaries for the selection of optimal resource parameters for both sensor and mesh nodes. The first constraint,

node *vx*+<sup>1</sup> in an end-to-end path cannot receive and transmit on the same channel *fx*, that is *fx* �= *fx*+1. This restricts node *vx* and *vx*+1, which are successive nodes in a path, from allocating power on the same transmission channel. The second constraint, *Pvz*, *fz* · *Pvz*+1, *fz* =

minimum power that the receiver can receive. *PMIRS* corresponds to the minimum input signal power at the receiver, or the minimum input receiver sensitivity (MIRS), defined in Table 1. The *PSNR* constraint is the minimum power required to reach the signal-to-noise ratio (SNR) threshold at the receiver. The third term, *Pmin*, represents the lower bound on the transmit power that keeps *Lvx*,*vy*, *<sup>f</sup>*(*Rvx*,*vy*, *<sup>f</sup>* , *t*) positive. There is a fourth factor that is the lower

> Standard MIRS Minimum SNR Maximum EIRP Zigbee -85 dBm -1.2 dB 0 dBm UWB -85 dBm -1.59 dB -14.3 dBm WiMax -83.2 dBm 9.8 dB 24 dBm

maximum power of the optimality range. *Ptech* corresponds to the maximum allowable transmit power on a transmission channel for the technology being used. From Table 1, *Ptech* is calculated as *Ptech* (dBm) = Maximum EIRP (dBm) - *Gt* (dBi), where the Effective Isotropic Radiated Power (EIRP) is the maximum allowable power that can be put on the transmission channel and *Gt* is the transmit antenna gain in dBi. *Pcap* represents the power that corresponds to the available outgoing capacity on the channel at the transmitter and ensures that the channel capacity is not exceeded. While the lower bound on the outgoing capacity is zero in the previous constraint, the upper bound *Pcap* is not zero unless the full channel capacity is

The third constraint, *Pvx*, *fx* , *Pvz*, *fz* ≥ *max*(*PMIRS*, *PSNR*, *Pmin*), ∀*vx*, *vz* ∈ *pathvi*,*vj*

bound on the available capacity at the transmitter, which is zero and is thus ignored.

**Table 1.** Operating Parameters for Zigbee, UWB and WiMax for Constraint Modeling [12–15]

The fourth constraint, *Pvx*, *fx* , *Pvz*, *fz* ≤ *min*(*Ptech*, *Pcap*, *Pmax*), ∀*vx*, *vz* ∈ *pathvi*,*vj*

. In a power-constrained network, this over-utilizes already

200 Wireless Sensor Networks – Technology and Protocols Cross-Layer Design for Smart Routing in

, it will be chosen by the optimization policy as the path with

, provides a power allocation restriction that a cluster

, restricts the

, restricts the

, is the corresponding power allocation restriction for the mesh network.

second summation represents the sum of link utilities for a candidate path.

*vi*,*vj*

*vi*,*vj*

link utilities leads to a higher *U<sup>a</sup>*

highest average link utility *Ua*<sup>∗</sup>

*Pvx*, *fx* · *Pvx*+1, *fx* = 0, ∀*vx*, *vx*+<sup>1</sup> ∈ *pathvi*,*vj*

**7.1. Constraints**

0, ∀*vz*, *vz*+<sup>1</sup> ∈ *pathvi*,*vj*

The cluster-head executes the following steps to perform resource allocation for a sensor *vi*:


For node *vx* at hop *k*, the 4-tuple (*vy*, *f* ∗ *<sup>k</sup>* , *<sup>P</sup><sup>t</sup>* ∗ *vx*, *fk* , *r*∗ *vx*,*vy*, *fk* ) solves the smart routing problem.
