A novel cross-layer mesh router placement scheme for wireless mesh networks

In the near future, broadband wireless mesh networks (WMNs) [1, 2] are expected to be widely deployed for providing Internet connectivity to users in residential areas and offices and supplementing existing wired infrastructure. WMNs are characterized by self-organizing and self-configuring capabilities, and hence are easy to be deployed. In 3G and Wi-Fi networks, each access point (AP) is connected through extensive wired infrastructure to access the backhaul network, which is often expensive and time consuming to be built. On the other hand, WMNs only use a subset of APs, called internet gateways (IGWs), to access the wired network, while the rest of the APs, called mesh routers (MRs), are connected to the IGWs in multi-hop wireless fashion. Thus, they are easy to be built and can provide an economical alternative to broadband wireless Internet connectivity.

Although WMN products are available in the market [3–6], their deployment has faced tremendous challenges [1, 2] because of some inherent problems, such as interference and high bit error rate (BER). One of the biggest challenge in deploying a WMN is to meet users' requirements with minimal cost. Usually, we have only a limited number of selected places that may have ac power and many locations may not be appropriate for MR deployment. Thus, the problem is to choose some of the locations for MR deployment so as to achieve the best cost performance ratio (CPR). A good location of MRs not only can provide high network throughput but also can lead to minimum number of MRs for meeting users' demand in the WMN design.

In the past, various schemes in various layers [7–17] have been used in placing MRs and IGWs so as to enhance the performance. Intricate relationships among antenna type used, wireless links with adaptive modulation and coding (AMC) scheme, MAC scheduling, routing, and equipment cost render the problem of optimal WMN planning extremely complex to address. Similar to the IEEE 802.16j scheme, some researchers [7, 8] develop schemes to place MRs so as to improve the WMN throughput, while others discuss the problem of MR placement (MRP) either without considering the wireless backbone network specifications for users' demand [9] or just focusing on the user coverage while ignoring users' demand [10, 12–16], not to mention the wireless backbone network support. The authors in [11] present an MRP algorithm without considering the cost for various antenna types. To simplify the problem, these works only consider a part of the design parameters associated with the MRP. Therefore, a more sophisticated MRP scheme is desirable to design a cost effective WMN that can meet users' demand both at the local level and at the backbone, with various technical options, such as antenna types, MAC scheduling, and routing.

This article proposes a cross-layer MRP (CMRP) scheme for a comprehensive MRP problem. Many researchers have proved the cross-layer approach [18–21] to be an effective scheme in improving the network performance. Our new CMRP iteratively adjusts the user coverage by each MR while new MRs are being added. As the residual backbone capacity is being evaluated with respect to the incurred interference, additional demand can be satisfied by each newly added MR. Based on a minimal interference routing scheme and the concept of bottleneck collision domain (BCD) [22], the backbone capacity is also evaluated to see if it can really meet the users' demand. To design a WMN with minimal cost, CMRP deploys a pair of directional antennas whenever it is observed to be cost effective. Therefore, CMRP offers a powerful MR deployment scheme in planning a WMN.

In CMRP, the cross-layer metrics are encapsulated into three attributes: Local Coverage (LC), Backbone Residual Capacity (BRC), and Deployment Cost (DC), which are evaluated throughout the MR addition process. LC specifies the contribution of a new MR in offering additional access capacity to the local users, BRC indicates the contribution of MRs to the backhaul capacity, and DC represents the ratio of the cost of deploying an MR using directional antennas to the cost of deploying an MR using an omni-directional antenna. DC enables selection of antenna types, such as omni-directional or directional antenna, based on the CPR while a WMN is being planned. CPR is taken as the ratio of the total deployment cost to the aggregate throughput in IGW. To maximize the objective function (LC × BRC/DC), CMRP selects MRs one by one among all the MR candidate locations. In this way, the objective function picks MR candidate locations that largely adds to the backbone capacity, more users' demand coverage, and lower deployment cost.

Extensive simulations have been performed to examine the performance and feasibility of our approach. We also compare CMRP with existing WMN planning schemes that consider only either coverage, or connectivity, or a combination of both. The result illustrates that CMRP outperforms existing schemes both in terms of CPR and its deployment feasibility. In addition, CMRP can help determine the users' demand and the size of a WMN that can achieve the best CPR. This information can help in deciding how many IGWs are needed when a large WMN is being planned.

The remainder of this article is organized as follows. Section 2 describes the related work. Section 3 presents the network model and problem formulation. Section 4 describes our heuristic algorithm. Section 5 summarizes the simulation results. Finally, Section 6 concludes the article and discusses our future work.

The inherent drawbacks of WMNs, such as interference, power limitation, and high BER, significantly limit the performance of WMNs. In the past, many researches [18–21, 23, 24] have presented algorithms to improve the throughput in channel utilization, radio power setting, and time slots allocation of WMNs. However, these works do not consider the service point placement problem, which has been experimentally shown to have a great impact on the performance by Bicket et al. [25].

The service point can be divided into two types of placement: IGW and MRs. The IGW placement [26–33] focuses on the wide area WMN planning, wherein many service points are clustered, and an IGW is assigned to each cluster. The MRP [7–17] deploys MRs to cover all users' demand. MRs may interfere with one another. Thus, if one of the MR wants to improve its throughput or service range using power control, then the nearby MRs may adversely suffer serious interference. So, how to optimize the MRP is an important problem that dictates the overall performance in a WMN system. In this article, we only consider the MRP, while the IGW placement is left as our future work.

So and Liang [7] place a fixed number of tetherless relay points (TRPs) to improve the throughput of a wireless LAN. They present a rate adaptation scheme to estimate the link rate and analyze how various parameters, such as path loss exponent, power ratio of AP and TRP over the power of mobile host (MH), and the number of TRPs, affect the performance and TRP placement. Lin et al. [8] analyze the placement of a single relay node (RN) in the IEEE 802.16j point-to-multi-point (PMP) networks so as to extend the coverage and improve the throughput/capacity of the network. They use a cooperative relay strategy to improve spatial diversity. Wang et al. [9] use a distributed clustering scheme to place a minimum number of MRs on candidate locations. Although they ensure connectivity between MRs, users' demand and users' coverage are met. But, they do not consider the link scheduling at the WMN backbone, and thus they cannot guarantee users' demand to be supported by the wireless backbone. San and Raman [10] define a complex objective function to minimize the total cost of MR deployment. Their design considers the number of antennas, the type of antenna, and the height of the IGW which do affect the line-of-sight transmission. Although they have considered the user coverage and the interference problem, they do not take users' demand into account. Moreover, they limit their design to only two-hop networks.

To cover users' need, So and Liang [11] address the MRP problem by constructing a fixed power of local and backbone links. However, they do not consider the cost of different types of antennas. Robinson and Kinghtly [12] analyze the throughput of WMNs with various types of topology, such as triangle, rectangle, hexagon, and random, and then compare the coverage performance. But, they only consider the users' coverage, not the users' demand. Franklin and Murphy [13] consider both the network backbone connectivity and the local coverage problem and use signal strength to represent the connectivity. But, they do not incorporate users' demand, which limits the usefulness of their approach. Xhafa et al. [14–16] use several search methods to solve the MRP problem. They take both the network connectivity and the user coverage. However, they do not consider users' demand, antenna cost, link capacity, or interference issues. Wang et al. [17] minimize the number of MRs deployed, with the objectives of the network connectivity, the users coverage, and users' demand. They consider MRs with multiple rates, which influence both the transmission range and the link capacity. However, they consider only fixed number of antennas on an MR and thus fixed cost per MR.

Selecting network service points with the minimum cost is a challenging task. Although the above researches have worked on this issue, they do not consider comprehensive metrics, such as users' demand, signal interference, MR deployment cost, and antenna type. This article presents a CMRP scheme to minimize the cost of MR deployment by taking users' demand, MAC scheduling, routing, and costs of MR and antenna into consideration.

This article focuses on IEEE 802.16d-based WMNs, in which a set of MRs are connected with multi-hop wireless links to form a wireless backbone, which is then connected to the Internet through an IGW.

3.3 Cross-layer design

Our cross-layer design contains two major parts: the local network and the backbone network. In the local network, we try to satisfy all local users' demand with a minimal number of MRs. In the backbone network, we must ensure all the MRs have sufficient bandwidth to forward their traffic hop-by-hop to the IGW through a MAC scheduling algorithm and a good routing tree. This subsection first discusses the AMC model in the physical layer and a tree-based minimum cost routing (TMCR) in the network layer. Then, we do the MAC layer scheduling based on the AMC and TMCR.

• Physical layer

In the PHY layer, what we care about is the transmission quality and the link rate. In the measurement-based deployment, the received signal strength (RSS) is measured for each candidate MR using the path loss model [13] as given in Equation 8. The path loss model describes the attenuation experienced by a wireless signal as a function of distance. The signal power decays exponentially with the distance. Given a reference signal strength P _dBm (d ₀ ) at distance d ₀ , the RSS at distance d is given as

$P_{dBm}\left(d\right)=P_{dBm}\left(d_0\right)-10\gamma {log}_{10}\left(\frac{d}{d_0}\right)+ϵ,$

(8)

where γ is the path loss exponent, and ε is the shadowing term.

With a given transmission power, higher rate modulation requires a higher RSS or a shorter transmission distance between two nodes. In order to increase the link capacity while maintaining transmission quality, the AMC technique is used at the physical layer that improves the data transmission rate. To estimate the link rates of the local and the backbone networks, we apply the distance between two nodes using Equation 8 to obtain the RSS first. Then, the RSS is applied into the 802.16 AMC table given in [34] to select an appropriate modulation scheme and thus the corresponding raw bit rate.

• Network layer

A multi-hop wireless network must have a routing scheme that selects a path to relay packets between IGW and MRs. The shortest path routing and the minimum hop routing (e.g., AODV) are two popular routing schemes. However, different routing schemes are suitable for different networks, such as ad hoc networks, sensor networks, and stationary networks, such as WMNs. Routing has been primarily designed to maintain connectivity for ad hoc networks or sensor networks, whereas it is more important to maximize the network throughput for WMNs.

We define the distance between two links as the shortest distance between two opposite end nodes. As shown in Figure 1, four possible distances between two opposite end nodes of links l _jk and l _lm are d(j, l), d(j, m), d(k, l), and d(k, m). Then, in this case, the distance between links l _jk and l _lm equals to d(k, m), i.e., the shortest among the four. Interference occurs when the distance between two links is smaller than the transmission range of an MR. Next, we define the degree of interference for link l _jk , denoted as I _jk , as the number of links that are restrained from transmitting because of the interference caused by the transmitting of link l _jk . As shown in Figure 2, the degree of interference for link l_1,2 is 5, since there are five links, i.e., l_3,4, l_5,6, l_7,8, l_9,10, and l_11,12, are interfered by the transmission of link l_1,2 and thus are restrained from transmission. In Figure 2, the transparent and shaded circles show the transmission ranges of MR 1 and MR 2, respectively. Conversely, link l_1,2 is restrained from transmission when any one of the interfered links, i.e., l_3,4, l_5,6, l_7,8, l_9,10, or l_11,12, is transmitting.

We define the cost of the link between MR j and MR k as

$Cos{t}_{jk}=I_{jk}/r_{jk},$

(9)

where I _jk is the degree of interference that interfere link l _jk and r _jk is its link rate. Thus, Cost _jk represents the time duration of interference incurred when transmitting a unit of data over the link l _jk . The larger r _jk is, the shorter will be the transmission time for a data packet, and hence, the shorter the blocking time will be for other links in link l _jk 's collision domain. Also, the smaller I _jk is, the fewer number of links is interfered by the link l _jk , and the shorter will be the aggregate blocking time.

In this study, we use a routing scheme, called TMCR, for the backbone relay traffic. TMCR works similar to that in [35], but considers both the capacity and the degree of interference along the path. Thus, TMCR selects a path with the minimum interference capacity. The goal of TMCR is to minimize the aggregate cost along a routing path. We define the routing cost of an MR l as

$COS{T}_i=\sum _{l_{jk}\in P_i}Cos{t}_{jk}=\sum _{l_{jk}\in P_i}\left(I_{jk}/r_{jk}\right),$

(10)

where P _l represents the routing path from MR l all the way to IGW. Thus, COST _l represents the backbone capacity consumed when a unit of data is transmitted on P _l .

Algorithm 1 shows the TMCR algorithm. TMCR is a variant of the Prim's algorithm. It finds a minimum spanning tree using a greedy strategy based on COST _l . After TMCR terminates, a routing tree T is obtained.

• MAC layer

It is important to handle all users' demand evenly by nearby MRs. However, since all the internetwork traffic goes through the IGW, MRs closer to the IGW have shorter paths to the IGW and therefore consume less network resource than MRs farer away from the IGW. Thus, we shall give higher priority to MRs closer to the IGW when we assign users' demand to MRs. To achieve this goal, we define a weight-based link assignment (WLA) at the MAC layer. In WLA, we first sort MRs in an increasing order based on their routing cost, as defined in Equation 10. Then, we assign users' demand to MRs according to their order by the nearest neighborhood scheme, i.e., we assign user demand q _i to MR j whose r _ij is the largest while guaranteeing such an allocation is supported by the backbone. If the backbone cannot support such a user demand, then WLA terminates, which implies that the scheduling fails. Algorithm 2 shows the procedure for WLA.

As MR is deployed incrementally, the routing tree also changes accordingly. Thus, WLA must be repeated for every MR added. With such a dynamic allocation, we are able to achieve close-to-optimal assignment while ensuring the feasibility of the MRP.

3.4 Performance metrics

On the MRP problem, it is hard to solve Equation 1 while satisfying Equations 4 and 5 because of interference. In this subsection, based on the concept of collision domain, we first consider the upper bound for the capacity of a WMN. Then, we introduce two performance metrics: local coverage (LC) and backbone residual capacity (BRC). Using these two metrics, we can quantify the degree of their contribution when deploying an MR both in the users' demand coverage and in the backbone. Then, we present a novel heuristic algorithm based on the metrics for MRP.

• WMN capacity upper bound

Evaluating the upper bound C _wmn for the capacity in a WMN is important for the network planning. It indicates how well users' demand can be satisfied. To estimate C _wmn , this study utilizes the heuristic of [22] which utilizes the concept of collision domain (CD), and then the most congested CD, called BCD, is identified and used to compute C _wmn .

A CD covers a set of nodes which should not transmit or receive any data at the same time so as to avoid the mutual interference. To demonstrate how C _wmn of a WMN is computed, a chain topology of Figure 3, taken from [22], is used as an example. Here, every node has a demand of 1G to gateway. The CD centered at link 2-3 contains links 1-2, 2-3, 3-4, and 4-5. When the link 2-3 is activated, the links in the 2-3 CD cannot be active at the same time. With similar arguments, we can readily find out CDs of all the links, out of which the CD of link 2-3 contains the most link flows (4 + 5 + 6 + 7 + 8)G and hence is the BCD of the WMN. If each link in the collision domain of 2-3 cannot forward more than the nominal MAC layer capacity B, then the maximal throughput cannot exceed C _wmn = B/(4 + 5 + 6 + 7 + 8)G = B/30G.

Because all the traffic must be forwarded toward/from the IGW, IGW is the most heavily loaded CD in the network and often becomes the BCD of a WMN [36]. By analyzing the capacity of BCD, we can compute C _wmn of a WMN, by which we can decide if the backbone capacity is sufficient to support all the users. The BCD concept holds true for single channel. For multiple channel case, it is easy to iterate for C _wmn in a WMN. If each channel has the same characteristics, then it is simply c × C _wmn for c subchannels. However, for simplicity, we assume a single channel case, thereby enabling the use of only C _wmn .

• Local coverage

The location of an MR is very important for serving MCs. A user demand is satisfied when both the local and the backbone networks have sufficient capacities to handle. As per PHY layer property, if the distance between two nodes is shorter, then the transmission rate becomes large with AMC. Thus, if we want to enhance the backbone link quality, then we must reduce the transmission distance between the MRs. On the contrary, if we want to serve more MCs, then we should place an MR close to as many uncovered MCs as possible, i.e., extending the transmission distance between MRs. However, the local coverage metric (LC) simply considers serving as many MCs as it can.

The users' demand allocation not only must meet the link and local network constraints, respectively, in Equations 3 and 4, but ought to be supported by the backbone network as well as follows:

$\Theta =Q_{IGW}+\sum _{j\in BCD}{Q}_j\le C_{wmn},$

(11)

where Q _IGW and Q _j are the local demands of IGW and MR j, respectively. Equation 11 computes the total throughput of the mesh network Θ, which must be smaller than or equal to the network capacity C _wmn .

When the backbone capacity is large enough to support more users' demand, every newly added MR can cover more users' demand and hence contributes additional throughput. To evaluate the value of a candidate MR, the Local Coverage (LC) metric is used to represent the contribution of an MR in enhancing the network throughput. We define R_{( n )} as an increment to the network throughput when n ^th MR is deployed:

$R_{\left(n\right)}={\Theta }_{\left(n\right)}-{\Theta }_{\left(n-1\right)},$

(12)

where Θ_(n)denotes the throughput of the WMN after the nth MR has been deployed. To determine the n th MR, the local coverage of every MR, denoted by Equation 4 as R _j for MR j, is calculated first. Apparently, it is beneficial to select the MR with the largest LC.

• Backbone residual capacity

Transmitting data in a wireless multi-hop network consumes substantial resources because of interference among the links. Thus, we must try to cover more users' demand while reducing resource consumption. Because we place MR one by one, it is necessary to compute how much residual resource is available for other unserved users. We define the Backbone Residual Capacity (BRC) metric that estimates the amount of backbone capacity available to serve un-assigned users' demand after placing an MR. BRC computes the residual capacity of all the links in the BCD, i.e., the CD of the IGW. Because all the data flows must be transmitted through the BCD, the resource in the BCD will be exhausted first. Thus, if BRC is larger, then more MCs far away from the IGW can be served.

Algorithm 3 presents the BRC computation algorithm. The residual capacity of link l _jk , denoted as ${Ĉ}_{jk}^r$, is its link capacity minus its current aggregated traffic load:

${Ĉ}_{jk}^r=r_{jk}-{\widehat{Q}}_j.$

(13)

The total residual capacity in the BCD, denoted as Ĉ ^r , is the sum of the residual capacity of each link in the BCD:

${Ĉ}^r=\sum _{jk\in BCD}{Ĉ}_{jk}^r=\sum _{jk\in BCD}\left(r_{jk}-{\widehat{Q}}_j\right),$

(14)

where l _jk means MR k is the uplink node of MR j. We denote ${Ĉ}_j^r$ as the total residual capacity if MR j is deployed.

When Ĉ ^r is zero while some users' demand are still un-assigned, the WMN design either fails or omni-directional antennas for some MRs must be replaced by directional antennas so as to reduce the interference, and thus the link rate could be increased.

Instead of selecting MR with a maximal BRC or a maximal LC, we select MR with the maximum product of BRC and LC first. The reason we select BRC × LC is to maximize the backbone capacity while covering more users' demand. If only LC is used, then the MR deployment will always select an MR with maximal user demand coverage, which can cover substantial users' demand initially, but completely consume the backbone resource soon, resulting in a non-optimal placement. Thus, the product of BRC and LC can allow us to balance the effectiveness between the coverage of users' demand and improvement in the backbone capacity.

Based on OF₁, Algorithm 4 contains three main parts. The first part is to select an IGW location, the second part is to deploy an MR, and the last part is to deploy directional antennas for backbone links.

Step 1 initializes the routing tree T using TMRC as given in the Algorithm 1. Step 3 determines the IGW location. The IGW deployment problem is beyond the scope of this article. So, we use an existing approach given in [9] to select the IGW location. In Step 6, we calculate all the candidate MR locations to find a location with the maximum OF ₁ and deploy it. In Steps 10 and 11, we reconstruct the scheduling routing tree and re-allocate users' demand with TMRC and WLA, respectively. If we cannot find an MR location with OF ₁ > 0, then it implies that the backbone capacity is exhausted and cannot satisfy any more demand. However, there may be some links that could be split by another MR to enhance their link rates and hence increase the backbone throughput. Thus, we temporarily ignore LC and select an MR with a maximal average BRC, denoted as Max_R, larger than a threshold and deploy it. If no such MR exists, then Step 20 finds a link that interferes with other links for the longest time period and replaces it with a pair of directional antennas. This algorithm will be terminated either when all the users' demands are satisfied or when we cannot deploy an MR at a new location to increase the backbone throughput any further.

In CMRP-1, directional antennas are used only when the WMN topology cannot meet all users' demand and the MR locations are not changed when directional antennas are added. Because using directional antennas not only reduces the interference, but also enhances the transmission range, such a deployment scheme may not be optimal. Furthermore, CMRP-1 does not consider the deployment cost of an MR and an antenna and cannot optimize the cost of the WMN deployment.

If MR k is selected and its OF ₂ is larger than its OF ₁ , then MR k will use a pair of directional antennas on the link between itself and its parent node in the routing tree. We denote β/α in Equation 17 as the cost ratio (CR). Lower cost of directional antenna means smaller CR, i.e., DC is closer to unity and OF₁ ≈ OF ₂ , and thus CMRP-2 performs nearly as CMRP-1. The cost of deploying a new MR is always there. What we consider in CR is the difference between the cost of adopting directional antenna and omni-directional antenna. We show how CR affects CMRP-2 in Section 5.

The complexities of TMCR, WLA, and BRC are O (m²), O (m²) + O (m × n log n), and O (m²), respectively. The complexity of CMRP is thus O (m³) + O (m ² × n log n).

In this article, we present a CMRP scheme for IEEE 802.16d WMNs. CMRP integrates the AMC technology and the antenna type at the PHY layer, TMCR at the network layer, MAC scheduling and WLA at the data link layer to derive a cost effective WMN design. CMRP encapsulates the complex design metrics into three design attributes: the local coverage, the backbone residual capacity, and the deployment cost. Numeric results show that CMRP works well, and provides a good CPR in the WMN network planning. Simulation results also confirm that our novel TMCR and WLA schemes can effectively improve the performance of CMRP. Moreover, by incorporating the cost ratio of directional antenna to MR in the network planning, a WMN with a low CPR can be obtained.

From the simulation results, we also see that the CPR increases substantially as a WMN covers larger users' demand. Based on this observation, we plan to develop an IGW placement algorithm based on CMRP to achieve low CPR in the large-scale WMN planning.

Input: backbone topology G = (V',E), link rates r_jk|jk∈E, and IGW.

Output: a routing tree T.

               //M _c : the set of MR candidates that have not yet been

               //   included in a routing tree.

               //M _r : the set of MR candidates that have been

               //   included in a routing tree.

1:   M _c = V - {IGW};

2:   M _r = {IGW}; T = {IGW}; COST _IGw = 0;

3:   for j ∈ M _c

4:      if r _j,IGW ≠ 0

5:         Cost _j,IGW = I _{j,IGW/rj,IGW ;}

6:         uplink(j) = IGW;

7:      else

8:         Cost _j,IGW = ∞;

9:         uplink(j) = ϕ;

10:      endif

11:      COST _j = Cost _j,IGW ;

12:   endfor

13:   while M _c ≠ ϕ

14:       $k=\arg {\min }_{j\in M_c}$COST _j ; l = uplink(k);

15:      M _c - {k}; M _r ∪ {k}; T ∪ {l _kl };

16:      for j ∈ M _c

17:         if (Cost _jk + COST _k ) < COST _j

18:            COST _j = Cost _jk + COST _k ;

19:            uplink(j) = k;

20:         endif

21:      endfor

22:   endwhile

Input: a routing tree T, MR routing costs $COS{T}_{j|j\in M_r}$, users' demand q _i | _i∈V .

Output: the user's demand allocation A.

1:   S = M _r ; D = {q _i |i ∈ V};

2:   flag = 1;

3:   while S ≠ ϕ && flag == 1

4:      k = arg min_j∈SCOST _j ;

5:      sort q _i in D in a descending order of r _ik ;

6:      R _k = R _L ;

7:      while D ≠ ϕ and R _k is not exhausted

8:         assign the first q _i in D to A(k);

9:         D - {q _i }; R _k = R _k - q _i ;

10:         allocate backbone resource to q _i ;

11:          if backbone resource is exhausted

12:            flag = 0; break;

13:         endif

14:      endwhile

15:      S - {k};

16:   endwhile

Input: the set of scheduled backbone links L _B .

Output: the backbone residual capacity Ĉ ^r .

               //${Ĉ}_{jk}^r$: the residual capacity of link l _jk

1:   U = L _B ;

2:   Ĉ ^r = 0;

3:   while U ≠ ϕ

4:      select a link l _jk from U;

5:      U - {l _jk };

6:      if l _jk is in the BCD

7:         ${Ĉ}_{jk}^r=r_{jk}-{\widehat{Q}}_j;$

8:         ${Ĉ}^r={Ĉ}^r+{Ĉ}_{jk}^r;$

9:      endif

10:   endwhile

Input: user location V, demand q _{i | i ∈ V} , MR candidate locations M _c , and IGW location IGW.

Output: the number of selected MRs and their respective locations M _r .

1:   initialize the routing tree T by TMCR;

2:   M _r = {IGW};

3:   set up IGW location;

4:   M _c - {IGW};

5:   while $\Theta <{\sum }_{i=1}^n{q}_i$

6:      j = arg max _Mc {OF₁ > 0};

7:      if j ≠ - 1//current backbone capacity

                     //   is large enough

8:         M _c - {j};

9:         M _r ∪ {j};

10:         update the routing tree T by TMCR;

11:         reallocate users' demand by WLA;

12:      else   //j = -1

13:         find a location k ∈ M _c

                        with $Max\_R={\max }_{k\in M_r}\{BR{C}_k/|M_r\left|\right\}$;

14:         if(Max_R > ω)//ω is a threshold

15:            M_c - {k};

16:            M_r ∪ {k};

17:            update the routing tree T by TMCR;

18:            reallocate users' demand by WLA;

19:         else//   no MR can be selected from M _c

                              //with Max_R > ω

20:            find a link l _ij ∈ BCD

                                 with maximal $\frac{Q_i}{r_{ij}}\times I_{ij}$

21:            if no such a link is found

22:               break;

23:            endif

24:            replace l _ij by a pair of directional antennas;

25:            update the routing tree T by TMCR;

26:            reallocate users' demand by WLA;

27:         endif

28:      endif

29:   endwhile