**4. Simulation results**

The performance evaluation of the HSRA algorithm was performed by means of simulation using the simulator proposed in [49]. The simulated network is an IEEE 802.16 mesh network with distributed scheduling under the configuration shown in Table 1. The number of nodes was specified as a simulation parameter. The nodes were uniformly distributed in a square area such that the node density was always kept at 15 nodes per unit area. The maximum transmission range of the nodes was set at 0.3 (i.e., max *r* = 0.3 ). The connectivity of the network under bidirectional links and with the nodes' transmission ranges set at max *r* was confirmed before executing the shortest-path routing algorithm. The number of flows was specified as a simulation parameter. The source and destination nodes of every flow were uniformly distributed among all the nodes in the network. The shortest-path routing algorithm calculated the flow paths under the MaxPower setup which is the power assignment when all the nodes' transmission ranges are set at the maximum (i.e., max *r* ). Once the paths were calculated, the transmission ranges of the nodes were found using the HSRA algorithm. Also, the optimal transmission ranges (i.e., the solution to the SRA-TM problem (Equation 13)), which we call OptPower, were found in [34] using the formulation of the SRA-TM problem as a mixed integer program with non-linear constraints (MIP-NLC). The MIP-NLC was solved using the Branch And Reduce Optimization Navigator (BARON) Solver [50], which is a system for solving non-convex optimization problems to global optimality. Finally, the network was simulated under the MaxPower, MinPower, OptPower, and HSRA setups.


**Table 1.** IEEE 808.16 mesh network configuration

[ † ] This is a parameter of the election algorithm used t specify the frequency that nodes transmit scheduling packets.

Figure 6 shows the average output-queue length for three networks with 20 nodes each and increasing number of flows (i.e., 10 flows in Figure 6a, 15 flows in Figure 6b, and 20 flows in Figure 6c). The input-queues have been omitted because they are guaranteed to always be stable [8, 11]. The flow rates in each network were all set at the same value. These are 8, 6, and 5 packets per frame for the networks in Figures 6a, 6b, and 6c respectively. These values were set so that their corresponding HSRA network became unstable if they were increased by at least one point. In this way, the network operates at a point inside the stability region and close to its boundary. Therefore, when any of the rates is increased by at least one point, the network operates outside the stability region, and therefore, it is unstable.

40 Wireless Mesh Networks – Efficient Link Scheduling, Channel Assignment and Network Planning Strategies

throughput (lines 14 to 23 of the HSRA Algorithm).

**4. Simulation results** 

and HSRA setups.

**Table 1.** IEEE 808.16 mesh network configuration

transmit scheduling packets.

TP. In the algorithm, this tradeoff is achieved by testing the improvement on the total

The performance evaluation of the HSRA algorithm was performed by means of simulation using the simulator proposed in [49]. The simulated network is an IEEE 802.16 mesh network with distributed scheduling under the configuration shown in Table 1. The number of nodes was specified as a simulation parameter. The nodes were uniformly distributed in a square area such that the node density was always kept at 15 nodes per unit area. The maximum transmission range of the nodes was set at 0.3 (i.e., max *r* = 0.3 ). The connectivity of the network under bidirectional links and with the nodes' transmission ranges set at max *r* was confirmed before executing the shortest-path routing algorithm. The number of flows was specified as a simulation parameter. The source and destination nodes of every flow were uniformly distributed among all the nodes in the network. The shortest-path routing algorithm calculated the flow paths under the MaxPower setup which is the power assignment when all the nodes' transmission ranges are set at the maximum (i.e., max *r* ). Once the paths were calculated, the transmission ranges of the nodes were found using the HSRA algorithm. Also, the optimal transmission ranges (i.e., the solution to the SRA-TM problem (Equation 13)), which we call OptPower, were found in [34] using the formulation of the SRA-TM problem as a mixed integer program with non-linear constraints (MIP-NLC). The MIP-NLC was solved using the Branch And Reduce Optimization Navigator (BARON) Solver [50], which is a system for solving non-convex optimization problems to global optimality. Finally, the network was simulated under the MaxPower, MinPower, OptPower,

> **Parameter Value**  Frame length 10 ms Control-time-slot length 63 µs

NextXmtMx † 7 XmtHoldoffExponent † 6

Link scheduling GM-RBDS Routing shortest-path

Number of control-time slots per frame 4 Number of data-time slots per frame 256

[ † ] This is a parameter of the election algorithm used t specify the frequency that nodes

Figure 6 shows the average output-queue length for three networks with 20 nodes each and increasing number of flows (i.e., 10 flows in Figure 6a, 15 flows in Figure 6b, and 20 flows in

**Figure 6.** Average output-queue comparison for the HSRA, MinPower, MaxPower, and OptPower configurations

Close to the end of the simulation time, when the transient behavior of the queues is over, the average queue lengths of the different power setups (i.e., MaxPower, MinPower, OptPower, and HSRA) can be compared. In Figure 6a, the MaxPower setup has the worst performance (i.e., the largest average queue length), and the MinPower, OptPower, and HSRA have similar performance. Therefore, when the number of flows is low (i.e., 10 flows), the MinPower and the HSRA algorithms are able to achieve queue lengths that are close to the lengths achieved by the optimal solution (i.e., OptPower). On the other hand, when the number of flows increases, the MinPower algorithm does not achieve a performance close to the optimal one while the HSRA algorithm does. This is shown in Figures 6b and 6c. The MaxPower and MinPower algorithms have similar performance which is worse when compared with the HSRA algorithm. The HSRA algorithm achieves average queue lengths that are close to the lengths achieved by the optimal solution. Therefore, the HSRA algorithm enables the flows to carry more traffic while guaranteeing stability than the MaxPower and

MinPower algorithms do. Also, it is confirmed that the technique of only maximizing the spatial reuse by reducing the transmission ranges (i.e., MinPower) does not perform well when the flow density increases (i.e., when the number of flows increases and the number of nodes is kept constant.). On the other hand, the technique of adapting the stability region to the given set of flows by means of TP control (i.e., HSRA) does perform well when the flow density increases.

Stability-Based Topology Control in Wireless Mesh Networks 43

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