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Radio Planning of Wireless Local Area Networks Sandro Bosio, Antonio Capone, Matteo Cesana

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Radio Planning of Wireless Local Area Networks Sandro Bosio, Antonio Capone, Senior Member IEEE, Matteo Cesana, Member IEEE

Abstract— In this paper we propose mathematical models to tackle the WLAN planning problem. Our approach aims at maximizing the network efficiency by taking into account the inter APs domain interference and the access mechanism. Both the single channel and the multiple channels WLAN planning problems are considered. We give different formulations which capture at different levels of detail the effect of interference on the network efficiency. In order to evaluate the quality of the proposed models, we obtain the optimal solutions for synthetic network instances, and propose heuristics to get suboptimal solutions in a reasonable computation time. We show that the networks planned according to our approach feature higher efficiency than the one planned using classical models, like the minimum cardinality Set Covering Problem (SCP), by privileging network solutions with low power APs installed. The achieved gain reaches 167% in particular network scenarios. Moreover, we test the obtained solutions through simulation and real life testbed implementation; both analyses show that the networks planned with the proposed approaches are the ones with the highest saturation throughput with respect to those configurations obtained with SCP.

I. I NTRODUCTION The recent success of wireless technologies has boosted the development of Wireless Local Area Networks (WLANs). Among the WLAN standards, the IEEE 802.11 [1] is the most popular one. The IEEE 802.11 standard defines both the physical (PHY) and medium access control (MAC) layers of the network. The basic network building block defined by the standard is the infrastructure Basic Service Set (BSS) which is composed of a single Access Point (AP) connected to a wired backbone network providing wireless connectivity to a bunch of mobile users. The AP implements bridging functionalities between the wired realm and the wireless one. The BSS replicates in a wireless scenario a single segment of an Ethernet Local Area Network (LAN). Collisions are limited by the Carrier Sense Multiple Access (CSMA) scheme, like in a LAN segment. Such a network configuration is very well suited to provide coverage in hot spot-like scenarios, where the area to be covered is small sized (cafes, restaurants, conference rooms, etc.), and interference on the ongoing communications is limited by the carrier sensing mechanism [2], [3]. On the other hand, large facilities, such as office complexes, apartment buildings, hospitals [4], factories [5], university campuses, or warehouses can hardly be covered by a single BSS WLAN. In a wired LAN, as the dimensions of the area increase, the network can be expanded by deploying bridging and switching devices. Similarly, if the area to be covered by Paper submitted to IEEE Trans. on Networking on May 13, 2005. Preliminary results within the same framework of this work have been recently published in [9], [10] and [11]. The authors are with Politecnico di Milano, piazza Leonardo da Vinci 32, 20133, Milano, Italy, Email: {bosio, capone, cesana}@elet.polimi.it

a wireless LAN gets bigger and the deployment of a single AP is not enough to provide the required wireless coverage, multiple APs connected through a wired distribution system have to be used, giving rise to an Extended Service Set (ESS). However, in a LAN a complete separation among different segments can be achieved, and the transmissions on different segments do not collide. As a matter of fact, if source and destination of a transmission belong to the same segment, the corresponding physical signal is confined within the boundaries of the segment itself, and does not produce interference on other segments. On the contrary, in a wireless ESS different APs may share the same radio resources, and inter-domain interference can affect the performance of the overall network [6]. Intuitively, the ESS efficiency degradation due to the inter-domain interference increases with the degree of overlap among different APs’ coverage ranges. On the other side, a certain degree of overlap among contiguous covered domains should be maintained in order to ensure the continuity of service to nomadic users. In this scenario, a wise planning strategy for the positions and the radio resources (channels) assigned to each AP is crucial to limit the impact of inter-domain interference, thus determining the efficiency of the overall network. Radio planning schemes proposed in the literature are usually extensions of those adopted for cellular networks and aim at minimizing installation costs while providing a good signal level for the whole served area [7], [8]. In previous works [9], [10], we introduced the problem of planning a maximum efficiency WLAN by focusing on the single frequency channel only, proving also that the problems we are considering are NP-Hard, even if we limit the scope to instances defined on the Euclidean plane [11]. In this work we provide an optimization framework based on mathematical programming formulations to tackle the specific WLAN planning problem both for the single and for the multiple channel case. Moreover, in order to evaluate the quality of the proposed models, we obtain and comment on the optimal solutions for synthetic network instances, and propose heuristics to get suboptimal solutions in a reasonable computing time. The rationale behind our planning approach is the maximization of the network efficiency by taking into account the inter-domain interference and the peculiar access mechanism implemented in the WLANs. Within such general framework of maximum efficiency WLAN planning, the proposed formulations differ for the way they model the effect of inter-domain interference. In a first simple model we propose to measure the interference by the degree of overlap among APs’ coverage areas, and thus the model aims at minimizing such overlap. On the other

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S. BOSIO, A. CAPONE, M. CESANA: RADIO PLANNING OF WIRELESS LOCAL AREA NETWORKS

hand, the other models we introduce try to estimate directly the network efficiency. The definition and calculation of the WLAN network efficiency in a planning model can be cumbersome, since, as a matter of fact, the efficiency of a IEEE 802.11-based network depends not only on inter-domain interference, but also on many other parameters, such as traffic type and distribution, wireless terminal mobility, dynamic resource sharing, adaptive modulation schemes, etc., which can hardly be inserted in a planning optimization model. Since the primary goal of this paper is to define a planning framework that accounts for the effect of interference among APs, we adopt a simplified estimation of the network efficiency as a WLAN performance figure. The robustness of the adopted performance figure is then tested through simulation using the Network Simulator 2 (ns2) [12] and through the implementation of a real life testbed of a multi-APs WLAN. The paper is organized as follows: Section II gives a brief overview of the IEEE 802.11 technology and clarifies the novelty of our approach by reviewing the related work on the WLAN planning topic. In Section III we discuss on network efficiency and present the mathematical models for the case where a single frequency channel can be assigned through the WLAN. Section IV discusses the extensions to the case where multiple frequencies can be assigned to the APs. Section V presents enhanced formulations of the planning problems. Finally, Section VI gives some concluding remarks. II. WLAN D ESIGN I SSUES The task of the wireless network planner is basically to choose the positions of the APs and to assign to each of them a channel. The quality of the planned network can be constrained by multiple requirements on the signal level received by the wireless users or on the installation costs. A common approach to the radio planning problem is to consider feasible positions of traffic concentration points in the service area (Test Points, TPs) and feasible positions where APs can be installed (Candidate Sites, CSs) [13]. The placement of TPs and CSs depends on the traffic distribution and on the characteristics of the area to be covered. Although the concept of “test point” is distinguished from “end user” (formally, the end user is the traffic generation agent that is placed in a TP), we will use the two terms as synonyms throughout the paper. A simple model of the problem aims at selecting a subset of CSs where the APs should be installed so that the signal level is high enough in all the considered TPs. The problem of selecting a subset of CS positions able to cover all the TPs with the minimum total installation cost is a well known NP-hard combinatorial optimization problem, namely the Set Covering Problem (SCP) [14], for which effective heuristics and fast exact algorithms are known. This model has been adopted in most of the previous papers appeared in the literature on the design of general wireless networks [15] and specifically of cellular systems [13]. On the other side, the problem of planning a WLAN is quite different from the traditional approach adopted for cellular

networks. Firstly, the installation costs of WLAN APs are definitely lower than those of cellular networks base stations, and secondly, as already pointed out, WLANs weren’t devised to provide cellular coverage, and this should be taken into account in the planning phase. For the reasons above, up to now the development of coverage planning tools for WLANs has been looked over as too expensive with respect to the price of access points, and rules of thumb, rather than quantitative approaches, have been adopted. Nowadays the continuous increase of WLAN systems and of the services they can provide drives the focus on finding effective methods to determine high capacity and cost efficient solutions to the coverage planning problem. To this end, very few works have appeared on the specific issue. The general issues of WLAN planning are widely discussed and the impact of different planning choices on the quality of the planned network are qualitatively commented in terms of coverage degree and network capacity [7], [16]. Rodrigues et al. [17], [18] propose an Integer Linear Programming (ILP) formulation of the problem based on signal quality maximization in the TPs. Constraints on the distance of reuse of a single frequency are also inserted in the formulation, which is solved using a commercial tool. The proposed model does not require to cover all the TPs, and comprises a budget constraint on the number of installable APs. Kamenetsky et al. [19], [20] propose a coverage problem where the objective function is a convex combination of two terms: the first aims at minimizing the average path loss throughout the network, while the second one minimizes the maximum path loss among the users. Some heuristics based on pruning, local search and simulated annealing are proposed. Lee et al. [21] propose an extension of the models based on coverage maximization by changing the objective function. The basic idea is to achieve a balanced load by minimizing the maximum utilization among the installed APs. The resulting ILP formulation, which is proved to be NP-Complete by Leung et al. [22], is a particular case of the capacitated facility location problem, and is solved using commercial tools. Recently, Wang et al. [8] propose to apply dynamic resource allocation strategies to a planned WLAN to achieve network efficiency under dynamic traffic situations. In general, all the works cited above focus on the problem of achieving a high coverage level in terms of received signal quality or balanced traffic load among installed APs. Very few of them consider network efficiency as an optimization parameter, and, to the best of our knowledge, no one takes into account during the planning phase the peculiar channel access mechanism, which can dramatically affect the efficiency of the deployed WLAN. Prommak et al. [23] propose an interesting maximum capacity planning approach based on a constraint satisfaction model, with the aim of assuring the required local data rate to the end users. Their solution approach is based on a brute force technique divided into two distinct phases: in the first one the number of APs to be installed is obtained through qualitative considerations on the network efficiency, while in the second one the radio resources are optimally assigned to the APs according to the constraints on the received power,

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the perceived interference and the achieved data rate. The main differences between the aforementioned approach and ours is in the interference modeling. Prommak et al. define a constraint for which the overall interference perceived by each user cannot exceed a fixed threshold. This is very well suited for interference based wireless systems like cellular ones, where the main performance parameter is the Signal to Interference Ratio (SIR). On the other hand, it might not be the optimum one in the WLAN case, where a carrier sense access is implemented and the effect of interference is no longer to impair the quality of a communication but rather to block it. In fact, the basic access method for the IEEE 802.11 is the Distributed Coordination Function (DCF) [24] which uses CSMA with Collision Avoidance (CSMA/CA). This requires each station to listen for other users’ transmissions. If the channel is idle, the station may transmit. However, if it is busy, each station waits until the ongoing transmissions stop, and then enters a random back off procedure. A further channel sensing mechanism, often referred to as Virtual Carrier Sensing, is introduced to combat the hidden terminal problem [25]. The Virtual Carrier Sense enables a station to reserve the medium for a specified period of time through the use of RTS/CTS control frames. A sender transmits to the designated receiver an RTS frame, which contains a duration/ID field that specifies the period of time for which the medium is reserved for the subsequent transmission. Upon reception of the RTS, the receiving station responds with a CTS frame, which also contains a duration/ID field specifying the period of time for which the medium is reserved. The reservation information is stored in the Network Allocation Vector (NAV) of all the stations detecting the RTS and/or the CTS frame. Therefore, all the stations overhearing the RTS and/or the CTS packets refrain from accessing the medium for the whole period specified in the control packets. Collisions are thus avoided even though some stations are hidden from each other due to the limited transmission ranges. The carrier sense mechanisms described above has been devised explicitly to get rid of any interference during ongoing transmissions. However, this limits parallel transmissions in the network. In Figure 1(a) the mobile node A is covered by both AP1 and AP2 , using the same radio channel. If user A is engaged in a communication with AP1 , every other mobile user in the union of the coverage ranges of AP1 and AP2 is forbidden to start any new communication. The simple example above shows that the overlap of different APs’ coverage ranges can have a great impact on the WLAN capacity. Intuitively, if a WLAN is composed of two APs with non-overlapping coverage ranges (Figure 1(b)), the network capacity is the sum capacity of the two APs. On the other hand, if the two APs are within the sensing range one another (Figure 1(c)), then the capacity of the WLAN is the capacity of a single AP, since any transmission within the domain of each AP blocks the transmissions within the domain of the other AP. Thus, network efficiency is highly dependent on the degree of overlap among the coverage ranges of the APs, even if, on the other hand, a minimum overlap is required to guarantee service continuity. In the above intuitive comments on the effect of the overlap

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among coverage ranges on the network efficiency we neglected user-to-user direct interference that blocks transmissions if two or more users are within their respective transmission ranges even if the coverage ranges of their APs do not overlap. We comment more on this issue in Section V. The problem of inter-domain interference can be attenuated by using multiple radio channels within a WLAN. The actual availability of radio channels for WLANs depends on bandwidth regulation authorities of the different countries. For the IEEE 802.11b and 802.11g versions of the standard, the NorthAmerican and European bandwidth regulation institutions have allowed the use of three separated frequency channels when deploying a WLAN, while only one channel is available for use in Japan. Up to 12 disjoint channels can be used in the IEEE 802.11a version of the standard. The peculiar characteristics of IEEE 802.11 access mechanism affects the performances of the resulting network, and the planning procedure should take into account the incidence of overlapping regions besides all the other optimization parameters. That is what we propose in the next sections. III. S INGLE C HANNEL WLAN P LANNING Although using multiple channels has obvious advantages in the planning process, on the other hand it can create some problems in the network management. As a matter of fact, handover between APs on different frequency channels can be tricky and affects the service offered to the user. Furthermore, many of the IEEE 802.11b Network Interface Connectors (NICs) available on the market can’t handle fast frequency channel switching, i.e., when switching from one frequency to another the connection must be torn down and then must be re-established with high consequent delays. It is therefore reasonable to think that many multi-access point WLANs will use one single radio channel shared among all the APs. For these reasons, it is worth studying the general WLAN planning problem when only one radio channel is available within the network. The extension to the multiple channels case is reported in Section IV. In order to describe the problem formally we introduce the following notation: • I = {i1 , . . . , in } denotes the set of TPs. • J = { j1 , . . . , jm } denotes the set of CSs. • I j ⊆ I denotes the set of TPs covered by an AP installed in a given CS j ∈ J. • Ji ⊆ J denotes the set of CSs that (once an AP has been installed in) can cover a given TP i ∈ I. The subsets I j can be obtained through an accurate site survey by evaluating the power received in each TP from all the CSs through propagation prediction tools, while subsets Ji can then be derived from the simple rule i ∈ I j ↔ j ∈ Ji . However, propagation aspects of the problem are out of the scope of this paper, and we assume the coverage subsets to be provided as a given input to our planning tool. A solution of the planning problem is a subset S ⊆ J of CSs where APs have to be installed. By I(S) ⊆ I we denote the subset of TPs that can be covered by installing an AP in each CS j ∈ S: [ I(S) = Ij (1) j∈S

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(a) Fig. 1.

(b)

(c)

Spatial Reuse Impairment in WLAN and WLAN capacity.

The fundamental decision variables are, as in any covering problem, those selecting which subsets are part of the solution: ( 1 if an AP is installed in CS j xj = (2) 0 otherwise The information contained in I j or Ji sets can be equivalently given with the usual test points-candidate sites incidence matrix defined by the coefficients: ( 1 if TP i belongs to subset I j (3) ai j = 0 otherwise

overall interference in the WLAN. In order to limit the impact of interference on the WLAN efficiency, one should therefore try to minimize such overlap degree. A simplified way to minimize the overlap is to minimize the average number of installed APs which cover a given TP. Thus, the first model we propose to get higher efficiency WLANs is the Minimum Overlap Problem (MOP), that can be formulated as follows: min ∑ ∑ ai j x j (MOP) s.t.

As previously specified, a common approach to model the planning problem is to minimize the installation costs, in terms of weighted number of installed APs, while providing full coverage to all the TPs. The formulation coming from this approach depicts a typical minimum Set Covering Problem (SCP) and can be written as: (4)

j∈J

(SCP) s.t.

∑ ai j x j ≥ 1

i∈I

x j ∈ {0, 1}

j∈J

∑ ai j x j ≥ 1

i∈I

(8)

x j ∈ {0, 1}

j∈J

(9)

j∈J

A. Set Covering and Minimum Overlap

min ∑ c j x j

(7)

i∈I j∈J

(5)

In the objective function we sum, for each TP i, the number of the installed APs that cover it. The minimization of this sum leads to solutions where the average number of installed APs covering a single TP is minimum. The MOP formulation can be easily reduced to an equivalent SCP formulation with appropriate costs c j . In fact, the MOP objective function can be rewritten according to the following: !

∑ ∑ ai j x j = ∑ ∑ ai j i∈I j∈J

j∈J

(6)

where c j is the cost associated to the installation of an AP in CS j. Constraints (5) force each TP to be covered by one installed AP at least, while constraints (6) are the integrality constraints for the binary decision variables x. The SCP is a well-known NP-Hard problem [26], [27], and a number of exact algorithms, typically based on tree-search procedures, have been proposed [28]. The installation cost is the central parameter to be optimized in the SCP. As already pointed out, installation costs may not be the main issue in WLAN planning, since the deployment of an AP is pretty cheap with respect, for example, to the deployment of a cellular network base station. Furthermore, the SCP approach completely neglects the optimization of the network capacity, and can thus provide solutions of installed WLAN with low efficiency. To this end, hereafter we model and change the objective function of the SCP, maintaining its basic structure, so as to derive a planning model that provides WLANs with higher efficiency. As already mentioned in the previous section, the more the coverage regions of different APs overlap the higher is the

j∈J

i∈I

x j = ∑ c jx j

(10)

j∈J

The term c j = ∑i∈I ai j gives the cardinality of the subset I j of TPs covered by CS j. MOP is therefore simply a SCP with the “installation costs” of each CS equal to the number of TPs covered by the CS itself (i.e., with c j = |I j |). The approach of minimizing the number of installed APs which cover a single TP, as we do in MOP, can provide better solutions than SCP. However, it does not really address the problem of maximizing the network efficiency, because it is not tailored to the peculiar access mechanism of WLANs. In the next section we define a simplified estimation of the network capacity to be used during the planning phase, which takes into account the peculiar access mechanism and includes the concept of overlap among coverage ranges. B. Network Efficiency Estimation The capacity of a WLAN strictly depends on the actual positions of the active users, on the traffic dynamics and on signal propagation issues. Therefore, it can be hardly defined a priori during the planning phase, and approximations are required. In the following we introduce the concept of balanced share, which we use as a simplified estimation


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of the network saturation throughput to be adopted in the optimization models as a network quality parameter. The balanced share for a given TP i in a given solution S can be defined by the following equation:

BS(S, i) =

 

1 Int(S, i) 0

if i ∈ I(S)

(11)

otherwise

where Int(S, i) is the number of users interfering with user i in the solution S, that is, user i’s competitors to gain access to the channel. We assume that each user is connected to a single AP whose capacity is shared by all the users within its coverage range. Without loss of generality, we assume that the overall capacity of an AP is equal to 1. However, due to the multiple access mechanism described in the previous section, a user that is in the interfering range of a set of APs blocks (and is blocked by) any transmission to/from any AP in this set. Therefore, we assume here that the fraction of the AP capacity available to a given user is equal to the reciprocal of the number of users in the interference range of the set of APs covering that user. In our assumption, this number is therefore given by Int(S, i) = |I(S ∩ Ji )| where S ∩ Ji is the set of CSs covering TP i in the solution S, and therefore |I(S ∩ Ji )| is the total number of TPs covered by the same APs covering i, i.e., the TPs competing with i for accessing the channel (counting i itself). Enhanced definitions of balanced share, and therefore of Int(S, i), will be provided in Section V. In the case of Figure 1(b), where two APs with disjoint coverage ranges provide coverage to three users, the balanced share of users A and B is 21 , while for user C is 1, since it does not have any interferer. On the other hand, the balanced share of user A in Figure 1(a) is 13 , since it is blocked by the transmission of either of the other two users in the network. The balanced share can be interpreted as the probability that each user has to access the shared resource under the assumption of uniform and maximal traffic, and clearly depends on the overlap among coverage ranges. In the definition of balanced share we have considered the transmission ranges of the APs only, while due to the RTS/CTS mechanism also those of the TPs themselves should be accounted for. In Section V we show how to integrate this refinement in our models, and we also show that it has a small impact on the quality of the solutions provided. Therefore, for the sake of simplicity, we present our models with the assumption described above. The efficiency of a given WLAN in terms of saturation throughput can be consequently estimated as the sum of the balanced share of all the users. In Section III-E we present simulation results and real-life measurements showing that if we maximize the balanced share we plan networks with higher saturation throughput with respect to other planning criteria (SCP and MOP).

C. Maximum Efficiency Planning The concept of balanced share can be used to give novel formulations for the WLAN planning problem, which can be now formalized as a network efficiency maximization problem. Besides the decision variables x (see Section III), we need to introduce a further set of variables to measure Int(S, i). To this end, we introduce the variables y, which actually depend on variables x as follows: ( 1 if TPs i and h appear together in some selected subset yih = 0 otherwise (12) Having defined variables y in this way it is easy to note that |I(S ∩ Ji )| = ∑ yih . The Maximum Efficiency Problem (MEP) h∈I

can then be formulated as follows: 1 max ∑ yih ∑ i∈I

(13)

h∈I

(MEP) s.t.

∑ ai j x j ≥ 1

i∈I

(14)

yih ≥ ai j ah j x j

j ∈ J, i, h ∈ I

(15)

x j ∈ {0, 1}, yih ∈ {0, 1}

j ∈ J, i, h ∈ I

(16)

j∈J

Constraints (14), exactly as in the SCP problem, impose the complete coverage. Constraints (15) express the interference relation between TP i and TP h: if TPs i and h have a common covering installed AP, then yih = 1. Otherwise, because of the objective function, in any optimal solution yih = 0. Thus, integrality constraints (16) for y variables holds implicitly. Note that yii = 1 for any TP i, since all the TPs are covered. In some situations, the WLAN service provider can drop the full coverage constraint and tolerate solutions where not all the users are covered, and the balanced share of some users is 0. The MEP formulation can be extended to match the aforementioned situation giving rise to an Uncovered Maximum Efficiency Problem (U-MEP). We skip here for the sake of brevity the details of the U-MEP formulation, since it is a trivial extension of the MEP one. A network planner may be required to specify a certain cost budget in the APs installation. To this end, both the MOP and the MEP formulations can be easily modified to account for cost limitations. With c j the cost, as in the SCP problem, and with B the budget, this can be done simply by adding the constraint: (17) ∑ c jx j ≤ B j∈J

Unfortunately, the MEP formulation is an hyperbolic 0-1 formulation, and medium-large sized instances can hardly be solved to optimality with generic solvers [29], [30]. Linearization techniques have been applied to the problem, while other promising solution approaches are currently under study [11]. D. WLAN Instance Generator In order to test the effectiveness of the above problem formulations, we have implemented an instance generator able to create synthetic instances representing WLANs. The software takes as input the following parameters:

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#CS=10

#CS=20

#CS=30

#CS=40

r 50 m 100 m 200 m 50 m 100 m 200 m 50 m 100 m 200 m 50 m 100 m 200 m

# APs 9.85 9.27 7.23 18.86 15.78 8.45 26.37 19.73 8.69 31.73 21.45 8.32

SCP MEP O.F. 9.3205 7.5884 4.1337 17.0289 11.4325 4.9542 23.5605 14.0345 5.1796 28.1535 15.5946 5.2408

# APs 9.85 9.27 7.23 18.86 15.79 8.50 26.38 19.76 8.80 31.85 21.51 8.52

MOP MEP O.F. 9.3221 7.5983 4.2210 17.0451 11.5467 5.3083 23.6296 14.4216 5.9243 28.4944 16.3021 6.1929

# APs 9.85 9.28 7.27 18.87 15.90 8.94 26.42 20.24 9.76 32.02 22.56 10.19

MEP MEP O.F. 9.3221 7.6006 4.2354 17.0462 11.5829 5.4496 23.6472 14.5593 6.1980 28.5581 16.6047 6.5957

TABLE I O PTIMUM SOLUTIONS OF THE

#CS=10 #CS=20 #CS=30 #CS=40

# APs 5.08 5.36 4.98 4.69

O PTIMUM SOLUTIONS OF THE

THREE APPROACHES SCP,

SCP MEP O.F. 3.749 3.194 2.724 2.333

# APs 5.09 5.85 6.20 6.82

MOP AND MEP ON

MOP MEP O.F. 3.774 3.745 4.021 4.788

# APs 5.13 6.24 7.75 9.57

TABLE II MOP AND MEP ON

THREE APPROACHES SCP,

the edge of the square area to be simulated (L), the number of Test Points (m), • the number of Candidate Sites (n), • the coverage range of each AP, expressed in meters (r). In a basic set of instances, each AP is assumed to have a circular coverage region with radius r. However, we underline that the proposed model formulations are general and can adopt any propagation model. In Section III-F we describe the behavior of our models both considering non circular coverage ranges obtained assuming shadowed propagation models and the real propagation conditions of a testbed. According to the above parameters, the generating tool randomly draws the positions for the n candidate sites and of the m test points. In order to generate only feasible instances (i.e., for which all the TP can be covered), the generator does the following: firstly, the positions of the CSs are randomly generated within the simulated area, and secondly, TPs are randomly located in the area covered by at least one CS. All the results reported in the remainder, unless differently specified, are the average values on 100 randomly generated instances. To make numerical results more easily readable we replace in the tables n with #CS and m with #TP. •

•

E. Experimental Results Using the tool described above we have generated two sets of WLAN instances: “uniform” instances, where all the APs to be installed have the same value of coverage radius, and “hierarchical” instances, where for 10% of the CSs the coverage radius is 400 meters, for 40% is 200 meters and for the remaining 50% is 100 meters. In all our instances the edge is L = 1km and the number of Test Points is m = 100. In Table I we compare on uniform instances the optimum solutions for the three planning approaches: the set covering

UNIFORM INSTANCES .

MEP MEP O.F. 3.798 3.908 4.746 6.221

HIERARCHICAL INSTANCES .

problem (SCP), the minimum overlap problem (MOP), and the maximum efficiency problem (MEP). Each formulation has been solved to optimality, and the obtained solution has been evaluated with the MEP objective function (MEP O.F. in the table), which represents an estimation of network efficiency. Since the optimum solutions have been computed by enumeration, the size of the instances is limited to n = 40. For these uniform instances the three approaches provide very similar results with short coverage range (r = 50 m), i.e., when the overlap degree is low. A non-negligible difference can be appreciated when the overlap degree and the number of CSs increase. For example, for n = 40 and r = 200 m the MOP and MEP efficiencies are respectively 18% and 26% higher than that of SCP. The different rationale behind the three approaches can be better appreciated in Table II, where we compare the three approaches on the set of hierarchical instances, for which the CSs to be installed have different coverage ranges. When the number of CSs is small, the three approaches provides almost the same result, since the coverage constraint limits the number of feasible solutions. However, by increasing the number of CSs we can observe a remarkable difference: MEP provides the best efficiency, SCP the poorest one, and MOP intermediate values, and the difference increases with the number of CSs. Obviously, higher efficiency solutions are usually characterized by a higher number of APs installed. For example, with n = 40, MEP provides an efficiency 167% higher than that of SCP with 104% more APs. The difference among the models can be clearly appreciated by comparing Figure 2(a) and Figure 2(b), which portrait the solutions provided respectively by the SCP and the MEP for a sample hierarchical instance (#CS =20). While the SCP tends to install few APs (6), using the highest coverage


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(a) Fig. 2.

(b)

WLAN planning solution provided by respectively SCP and MEP for a sample hierarchical instance is shown in Figure (a) and (b).

#CS=10 #CS=20 #CS=30 #CS=40

NETWORK THROUGHPUT (Mb/s) SCP MOP MEP 1.34 2.96 2.96 1.38 3.09 3.09 1.43 3.56 3.6 1.29 2.99 3.37

NETWORK EFFICIENCY (MEP O.F.) SCP MOP MEP 3.434 3.436 3.436 3.755 3.939 3.971 2.909 4.156 4.338 2.58 4.152 5.208

TABLE III C OMPARISON OF N ETWORK THROUGHPUT (M B / S ) AND NETWORK EFFICIENCY FOR THE

radius possible, the MEP does install more APs (11) with lower coverage radius and a much lower degree of overlap. In other words, while SCP does install the smallest number of APs regardless of the interference, MEP tends to favor network efficiency and consequently chooses those network configurations with limited inter-domain interference. F. Simulation and Testbed Analysis As already mentioned in previous sections, the focus of this work is on the definition of optimization approaches for the WLAN planning problem which take into account the peculiarities of the MAC algorithms adopted within such networks. To this end, we introduced in Section III-C the concept of balanced share as a rather simplified network performance figure with the general purpose to get a simplified but consistent measure of the actual network efficiency. Therefore, it is worth testing the quality of such a measure, i.e., analyzing the relation between the balanced share we get solving the proposed models and the actual network capacity. To this purpose, we evaluated through dynamic network simulations the performance of the networks obtained solving our models on the hierarchical instances. In particular, for each planned WLAN, we evaluate the corresponding maximum network throughput using the Network Simulator 2 (ns-2) version 2.29 [12]. In each simulation run, the TPs are assigned to the AP received with the strongest signal level, and generate traffic towards this AP according to a Constant Bit Rate (CBR) traffic source, with a packet size of 500 bytes. The User Datagram Protocol (UDP) is used at the transport layer, with a maximum segment size of 600 bytes. Similar results have

THREE APPROACHES ON HIERARCHICAL INSTANCES .

been obtained using TCP at the transport layer. As far as the MAC layer is concerned, the DCF operation mode of the IEEE 802.11b standard is adopted with the RTS/CTS control packet exchange. The physical layer is the one defined in the IEEE 802.11b standard, with a maximum data rate of 11 Mb/s. As to the propagation model, the received power Pr is related to the ε transmitting one Pt according by the formula: Pr = d −4 10 10 Pt , ε where d is the distance between sender and receiver and 10 10 accounts for the losses due to log-normal shadowing, where ε is a normal variate with zero mean and standard deviation σ = 6 dB. The overall network throughput, defined as the total number of delivered bits over the simulation duration time has been measured. Different values of the traffic intensity have been considered to get the maximum network throughput. Table III reports the network throughput in Mb/s provided by the three planning approaches for different values of the number of CSs, compared with the network efficiency estimated by the MEP objective function. Each throughput value in the table has been obtained averaging on the 100 hierarchical instances. The simulation time for each instance has been set to 50 s, with the first 10 s used as warm-up time. The output results on the 100 instances have been tested according to the t-student statistical test. For all the throughput measures reported in the following the confidence interval is under 5%, given a confidence level of 95%. For the three optimization approaches considered, the table shows that the MEP approach does provide network topologies with higher throughput with respect to the SCP and MOP approaches. Furthermore, the behavior of the network throughput follows that of the network efficiency, i.e., the WLANs planned using the MEP approach are always the ones with the

8


Test Points (TP)

Fig. 3.

High Power CS (Pt=20dBm)

Low Power CS (Pt=5dBm)

Testbed topology description.

(a)

and standard connectivity test tools (e.g. ping, traceroute, and others). Second, we formalized the instance according to the proposed models and we solved it applying the MEP and SCP approaches. Figure 4 reports the network topologies planned using the two approaches. As expected, while the SCP installs few APs with high power, the MEP tends to install more APs with lower power to avoid inter-domain interference. In order to test the actual quality of the planned networks, we have injected uplink traffic from the TPs. Namely, each TP has been manually assigned to the best received installed AP, and runs an iperf [31] client generating UDP/TCP 30 seconds long connections towards a server located on the wired network. UDP connections are configured with a 6 Mb/s data rate and both TCP and UDP connections adopt a maximum segment size of 500 bytes. Table IV reports the overall network throughput for the two networks when considering UDP and TCP connections. The value of the MEP objective function is also reported, for the sake of comparison. As it is clear from the table, the overall throughput provided by the network planned according to the MEP approach is considerably higher than the one obtained with the network planned with the SCP approach.

(b) Fig. 4.

Planned networks applying the MEP (a) and the SCP (b). SCP MEP

highest network throughput, followed by those planned using the MOP approach, while the SCP approach always provides WLAN configurations with lower network throughput. Therefore, in the considered network scenarios, the balanced share seems to be a consistent estimation of the network throughput. However, we have to observe that the differences predicted with the balanced share are higher than those obtained with the maximum throughput. In order to get the feeling on the quality of the proposed approaches in a real life scenario, we have applied the proposed models to the problem of covering with a multi APs WLAN the offices of the Telecommunications Network Group at our Department. To this end, we have built up a planning instance with 10 TPs and 11 CSs placed as described in Figure 3. In each CS we have placed a Linksys WRT54G Broadband Wireless Router, equipped with a MIPSEL (Broadcom 4712) 200 MHz processor, with 4 MB of flash memory and 16 MB of RAM with maximum data rate fixed to 11 Mb/s. Two of the CSs were configured to use a transmission power level of 20 dBm (triangles in Figure 3) while all the others transmit at 5 dBm (diamonds in Figure 3). All the wireless routers are placed on the same segment of fully switched LAN at 100 Mb/s. The end users placed in the TPs feature 10 laptops equipped with D-Link DWL-G650 Cardbus wireless interfaces controlled by MadWifi drivers which allow flexible card management via software. The installed OS is GNU/Linux Debian distribution with 2.6.8 kernel. First, we gathered the incidence matrix {ai j }i∈I, j∈J by evaluating the connectivity between each TP and all the CSs. To this end we used standard wireless signal capturing tools

UDP THROUGHPUT (Mb/s) 5.1 14.1

TCP THROUGHPUT (Mb/s) 4.7 12.7

NETWORK EFFICIENCY (MEP O.F.) 1.04 5.33

TABLE IV N ETWORK THROUGHPUT MEASURED THROUGH THE

TESTBED .

G. Heuristics to Solve MEP The computing time required to obtain an optimal solution for MEP might be very long for medium-to-large-size instances. As an example, the average processing time required to solve at optimum a uniform instances with coverage radius r = 200 m grows exponentially from 1 s for #CS=10 up to 74654 s for #CS=40, when run on a 3 GHz Pentium 4 with 512 Mb of RAM. We therefore propose hereafter a simple but effective heuristic approach providing near-optimal solutions in a reasonable amount of time. The proposed algorithm is composed of two phases: in the first one a greedy approach is used to build a feasible solution, while in the second one the resulting solution is improved through local search (Algorithm 1). The functions BuildUpSolution and LocalSearch both take as input parameters the incidence matrix A and the current solution set S. BuildUpSolution implements the greedy phase of the heuristics and iteratively converges to a feasible solution S, while LocalSearch refines the solution S through a local search maintaining feasibility. The greedy phase (function BuildUpSolution) keeps adding to the solution the CS j that maximizes a certain benefit function fb (S, j) among those increasing the coverage (function PickBestCS), until complete coverage is reached. If the current solution is empty, the benefit function f b (S, j) is 1 over the maximum overlap with the other CSs. Otherwise, it is the ratio


of the increase in the objective function (OF) over the increase in the cardinality of the coverage (that is by construction greater than zero). Given the current solution S (i.e., a set of installed APs) and given a CS j not in S, the benefit function fb (S, j) can be written as:  OF(S ∪ j) − OF(S)   if S 6= 0/    |I(S ∪ j) \ I(S)| (18) fb (S, j) = 1   otherwise    max I j ∩ I`

9

#CS=10

#CS=20

#CS=30

`∈J\{ j}

where we recall that I(S) is the set of TPs covered in the given solution S. Pseudo-code for the of the greedy phase is given by Algorithm 2 and Algorithm 3. Algorithm 1 HEURISTIC (A) 1: S = 0/ 2: BuildUpSolution (A, S) 3: LocalSearch (A, S) 4: return S

Algorithm 2 BuildUpSolution (A, S) 1: while (I(S) 6= I) do 2: Best_CS = PickBestCS (A, S) 3: S = S ∪ {Best_CS} 4: end while 5: return

Algorithm 3 PickBest (A, S) 1: MaxFunction = 0 2: for all j ∈ / S do 3: if ( fb (S, j) > MaxFunction) then 4: CS_To_Add = j 5: MaxFunction = f b (S, j) 6: end if 7: end for 8: return CS_To_Add

Algorithm 4 LocalSearch (A, S) 1: repeat 2: Enhanced = FALSE 3: MaxSol = S 4: for all R ⊆ S such that |R| ≤ 2 do 5: S’ = S \ R 6: BuildUpSolution (A, S’) 7: if (OF(S’) > OF(MaxSol)) then 8: MaxSol = S’ 9: Enhanced = TRUE 10: end if 11: end for 12: S = MaxSol 13: until ! Enhanced 14: return

#CS=40

r 50 m 100 m 200 m 50 m 100 m 200 m 50 m 100 m 200 m 50 m 100 m 200 m

MEP O.F. OPTIMUM HEURISTICS 9.322 9.322 7.600 7.600 4.235 4.235 17.046 17.046 11.583 11.405 5.449 5.330 23.647 23.451 14.559 14.329 6.198 6.018 28.558 27.992 16.604 16.208 6.596 6.203

TABLE V C OMPARISON BETWEEN THE

OPTIMAL AND THE HEURISTIC SOLUTIONS

ON UNIFORM INSTANCES FOR

MEP.

Since each iteration adds to the solution a CS which covers one TP at least, the procedure BuildUpSolution requires at most min(n, m) steps to converge. The local search phase takes in input the solution S provided by the greedy phase and tries to enhance it. The algorithm searches for the best solution in the neighborhood N(S) of the current solution S. If such a solution is better than S, it becomes the current solution and the procedure is repeated. Otherwise the algorithm stops. The neighborhood N(S) is defined by removing a set R of CSs from the S and then applying BuildUpSolution to obtain a feasible solution. The sets R are all the subsets of S of cardinality 1 or 2 (single CSs and couples of CSs). Pseudocode of the LocalSearch function is given by Algorithm 4. Table V reports the comparison between the optimum value of MEP and the value obtained through the heuristic on the same uniform instances used for Table I. In some cases the heuristics come out with the optimum values of the objective function, and in all the cases the computation time is below 30 seconds for all the tested instances. A slight discrepancy from the optimum occurs for high sized instances. In these cases the number of feasible solutions is greater and the heuristic approach presents slight differences with respect to the optimum. Similar results showing the effectiveness of the proposed heuristic have been derived for hierarchical instances. IV. M AXIMUM E FFICIENCY M ULTIPLE C HANNEL WLAN S A higher number of frequency channels can enhance the efficiency of the planned network, since the distance of interfering stations (stations working at the same frequency) is augmented, and the interference consequently reduced. In this section we extend the models presented above to the case of multiple channel WLANs. In Section IV-A we extend the MEP formulation for the single channel case to the case where multiple frequency

10


channels are available, by defining the multi frequency balanced share as the average of many single frequency partial balanced shares. This formulation is the natural extension of the previous one, and provides a quite simplified model of the problem of planning WLANs in the multi frequencies scenario. In Section IV-B we consider a more accurate model of the problem, considering that every user selects as his working frequency the frequency of the AP received with the strongest signal. Therefore, we add variables in order to assign TPs to APs. This complicates the formulations, but permits to describe the interference in a more accurate way. Finally, Section IV-C contains the results of the experiments we have run on realistic WLAN instances in order to test the quality of the proposed formulations. A. Simplified Multiple Frequencies WLAN Planning The formalism introduced in Section III needs some modifications in order to express this optimization problem. Let F denote the set of available frequency channels. For every CS we need to know not only if it is installed, but also which frequency it is assigned. Let S be a solution (subset of selected CSs, each one with an assigned frequency), and let S f ⊆ S be the set of selected CSs with assigned frequency f . In this scenario a given TP i can be potentially covered in a solution S with more than one frequency. For any frequency f we evaluate a partial balanced share that is equal to the balanced share BS(S f , i). Let ki be the number of frequencies with which user i is covered. We define the mean balanced share as the mean of the partial balanced shares over all the covering frequencies:  BS(S f , i)   f∑ ∈F if ki > 0 (19) MBS(S, i) = ki   0 otherwise As in the single frequency case, we define a first approximation of the multiple frequencies network efficiency as the sum over all users of their mean balanced share. In order to formulate this problem, let us introduce a new set of binary decision variables: ( 1 if an AP is installed in CS j with frequency f ¯xj f = 0 otherwise (20) We need to know whether a TP i is covered at frequency f . We thus introduce the set of binary decision variables: ( 1 if TP i is covered at frequency f ¯zi f = (21) 0 otherwise

To calculate the partial balanced share we introduce the following set of binary decision variables:    1 if TP i and TP h appear together in some subset selected with frequency f yih f =   0 otherwise (22) The Simplified Multiple Frequencies Maximum Efficiency Problem (S-MF-MEP) can then be formulated as:

max ∑ i∈I

(S-MF-MEP) s.t.

1 ki

∑

f ∈F

zif ∑ yihf

(23)

h∈I

∑ ¯xjf ≤ 1

j∈J

(24)

∑ ai j ¯xjf ≥ zif

i ∈ I, f ∈ F

(25)

zif ≥ ai j ¯xjf

i ∈ I, j ∈ J, f ∈ F

(26)

i∈I

(27)

ki ≥ 1

i∈I

(28)

yihf ≥ ai j ah j ¯xjf

i, h ∈ I, j ∈ J, f ∈ F (29)

yiif = 1

i ∈ I, f ∈ F

(30)

¯xjf ∈ {0, 1}

j ∈ J, f ∈ F

(31)

zif ∈ {0, 1}

i ∈ I, f ∈ F

(32)

yihf ∈ {0, 1}

i, h ∈ I, f ∈ F

(33)

f ∈F

j∈J

ki =

∑ zif

f ∈F

Constraints (24) state that every CS can be either excluded from the solution, or included with a unique assigned frequency. Constraints (25) state that if there is no covering AP at a given frequency for a given TP, then the corresponding z variable must have 0 value, while constraints (26) state that if there is a covering AP, then the z variable must have value 1. Constraints (27) define for every user i the number of covering frequencies ki , while constraints (28) impose the complete coverage (at least one covering frequency). Constraints (29) define the variables yihf . We have to add constraints (30) because a given user may be uncovered at a particular frequency, and in this case we do not want the denominator to evaluate to zero. In order to extend this formulation to the uncovered case, we have simply to relax constraints (27) to: ki ≥

∑ zif

(34)

f ∈F

If a given TP i is not covered then it will result in ∑f ∈F zif = 0, and because of constraint (28) it will be ki = 1, thus giving in the objective function a mean balanced share of 0. If otherwise TP i is covered, then because of the objective function constraint, (34) will hold to equality. B. Multiple Frequencies WLAN Planning with Assignment The formulation of S-MF-MEP given above does not take into account the assignment of TPs to CSs, so we give here a different formulation that should better describe the problem. We consider the following assumptions: 1) a station works at the same frequency of the AP that hears with the strongest signal, 2) a station h interfere with a station i if and only if both stations work at the same frequency f and there is an installed AP covering both stations and working at the same frequency f . Assigning a frequency to the stations (TPs) changes radically the way interferers have to be counted.


11

Let us denote by f (i) the frequency assigned to a TP i in a given solution S, and by S f ⊆ S the partitioning of the APs in S in frequency classes. We define also the partitioning of the covered TPs in frequency classes by T f = {i ∈ I(S) : f (i) = f }. The balanced share of a TP i is then given by the reciprocal of the number of TPs working at frequency f (i) that share with i a selected covering AP working at frequency f (i) :  1  if i ∈ I(S) (35) BS(S, i) = Tf (i) ∩ I Sf (i) ∩ Ji  0 otherwise

As for the previous models, the assumption made for interference neglects direct interference. In Section V we show how to integrate this refinement in our models. In order to model the way a station selects its own working frequency, we need to add some variables and parameters. We reintroduce the previous variables x j and we link them to the new set ¯xjf . We also reintroduce variables yih , defining the interference with users i and h. In order to formalize the association of a TP to a particular installed CS we introduce variables: ( 1 if TP i is associated to an AP installed in CS j li j = 0 otherwise (36) We need a new parameter, that defines the ordering in which the APs have to be considered, i.e., from strongest to weakest signal. We define parameter pi j as the power received in a given TP i of a signal emitted by an AP installed in CS j. We assume that pi j > 0 ∀ j ∈ Ji and that pi j = 0 ∀ j ∈ / Ji . In fact we do not need the power estimate itself, but just the total ordering that it induces over the set Ji . We can then formalize the Multiple Frequency Maximum Efficiency Problem (MF-MEP) as follows: 1 max ∑ (37) i∈I ∑ yih h∈I

(MF-MEP) s.t.

∑ ¯xjf = x j

j∈J

(38)

f ∈F

∑ li j = 1

i∈I

(39)

li j ≤ a i j x j

i ∈ I, j ∈ J

(40)

∑

li` ≤ 1 i ∈ I, j ∈ Ji

(41)

zif = ∑ li j xjf

i ∈ I, f ∈ F

(42)

yih ≥ ai j ah j zif zhf xjf

i, h ∈ I, j ∈ J,

state that every TP must be assigned to a single AP. Constraints (40) state that we can assign a TP to a CS only if the CS is selected and covers the TP, while constraints (41) state that if a CS having preference order k in the preference list of a given TP is selected, then this TP cannot be assigned to any CS having a worse preference order. The combination of these three constraints makes sure that if a station i is covered by many APs, then it will be li j = 1 for the nearest and li j = 0 for all the others. The quadratic constraints (42) define the frequency of work for the TPs, basing on the assignment of TPs to APs and on the frequency of work of the APs. Finally the cubic constraints (43) define the interference among two TPs according to the assumption described before: both the interferers and a common covering AP have to work at the same frequency. The quality of the solutions provided by the two formulations are analyzed in Section IV-C on synthetic instances of WLAN topologies. C. Experimental Results Since the two above formulations are hard to tackle even for small instances, we extended the heuristics presented in III-G for single channel networks to the case of multiple available channels within the WLAN. The structure of the heuristics remains the same as before.

#CS=10

#CS=15

#CS=20

#CS=30

r 100 m 200 m 300 m 100 m 200 m 300 m 100 m 200 m 300 m 100 m 200 m 300 m

# AP 10.0 10.0 9.3 15.0 14.1 12.4 19.9 18.0 13.8 28.9 23.6 17.4

MF-MEP MF-MEP O.F. 10.000 9.554 7.928 14.942 13.072 10.011 19.522 16.015 11.499 27.861 19.996 12.882

# AP 9.6 7.9 6.0 14.1 9.9 7.0 18.1 11.2 7.8 24.4 12.7 9.2

TABLE VI H EURISTIC S OLUTIONS OF THE MF-MEP AND

S-MF-MEP MF-MEP O.F. 9.584 7.794 5.922 14.100 9.718 6.651 17.940 11.200 7.335 23.271 12.265 8.376

THE

S-MF-MEP

FORMULATIONS ON UNIFORM INSTANCES .

j∈J

xj +

` : pi j >pi`

j∈J

f ∈F

(43)

x j ∈ {0, 1}

j∈J

(44)

¯xjf ∈ {0, 1}

j ∈ J, f ∈ F

(45)

li j ∈ {0, 1}

j ∈ J, i ∈ I

(46)

zif ∈ {0, 1}

i ∈ I, f ∈ F

(47)

yih ∈ {0, 1}

i, h ∈ I

(48)

Constraints (38) link ¯xjf variables to x j ones. Constraints (39)

Table VI compares the solutions of the two approaches in terms of number of installed APs and network efficiency. The network efficiency is estimated considering the assignment of TPs to APs according to the objective function of MF-MEP. Also this table has two parts referring to the two different approaches, like the previous one. For each approach, the number of installed APs and the network efficiency are reported. The network efficiency for the simplified approach is computed in post processing from the solution of the S-MF-MEP, i.e., given the set of installed AP each TP is assigned to the closest one, and the network efficiency is computed consequently. On the other hand, the network efficiency for the MF-MEP is simply the value of the objective function of MF-MEP obtained solving MF-MEP formulation. The solutions of the S-MF-MEP have a lower number of installed APs and the actual network efficiency provided by the

12


#CS=10

#CS=15

#CS=20

#CS=30

r 100 m 200 m 300 m 100 m 200 m 300 m 100 m 200 m 300 m 100 m 200 m 300 m

MF-MEP MF-MEP O.F. MF-MEP * O.F. 10.000 8.927 9.554 9.413 7.928 7.802 14.942 14.791 13.072 12.892 10.011 9.737 19.522 19.361 16.015 15.631 11.499 11.010 27.861 27.415 19.996 19.491 12.882 12.507

S-MF-MEP S-MF-MEP O.F. S-MF-MEP * O.F. 7.957 7.873 5.140 4.996 3.324 3.186 11.398 11.228 6.026 5.824 3.738 3.589 13.517 13.237 6.747 6.453 4.083 3.849 16.963 16.620 7.493 7.198 4.368 4.038

TABLE VII H EURISTIC S OLUTIONS OF THE MF-MEP AND

THE

solution of MF-MEP is generally higher than the one provided by the solution of S-MF-MEP. The difference between the two values gets bigger as the degree of overlapping among APs’ coverage radii increases, in percentage, it goes from 4.16% in the case with #CS = 10, r = 100 m, to 24% in the case with #CS = 20, r = 200 m. Similar results have been obtained with hierarchical instances (see Section III-D) where the APs to be installed have different values of coverage radius. V. E NHANCED WLAN E FFICIENCY E STIMATION All the formulations presented in the previous sections are based on the definition of balanced share given in Section IIIC. The definition of balanced share is based on the concept of interferer, i.e., a WLAN device each user must compete with to gain the access to the channel. In the definition of balanced share, given a TP, its interferers are those TPs which fall within the transmission range of all the APs covering the TP itself. However, due to the CSMA/CA mechanism, the TPs that are in the transmission range of the considered TP are also interferers. In order to introduce this refinement in our models, we need some extra input. For any TP i we need to know the subset of users that are in direct interference with it, and this can be done with an incidence matrix defined by the coefficients: ( 1 if TP h is within the hearing range of TP i bih = 0 otherwise (49) For the MEP formulation we simply need to add the constraints: yih ≥ bih i, h ∈ I (50) In S-MF-MEP we have to repeat the additional constraints for any frequency: yihf ≥ bih zhf

i, h ∈ I, f ∈ F

(51)

In formulation MF-MEP we need to verify that both users are working at the same frequency, so we have to add the following quadratic constraints: yih ≥ bih zif zhf

i, h ∈ I, f ∈ F

(52)

S-MF-MEP FORMULATIONS ON

UNIFORM INSTANCES .

There is also another problem that may take place with a hierarchical instance. Due to the presence of different transmission powers it can happen that a user produces interference over an AP without being covered by it. It is the case of a user i covered by a distant large range AP j and that has nearby a low range AP ` that do not cover it. User i, in order to communicate with AP j, has to use a high power, therefore producing interference over AP `. In order to introduce also this refinement in our models, again we need some extra input. For any CS j we need to know not only the subset of users I j that are covered, but also the subset of users I 0j ⊇ I j that produce interference. Then we can define a new incidence matrix by the coefficients: ( 1 if TP i belongs to subset I 0j 0 (53) ai j = 0 otherwise Then we simply have to redefine all the constraints that in our models define variables y substituting the matrix a with this new matrix a0 . Note that again in this operation we are adding constraints to our models, because I 0j ⊇ I j for all CSs and therefore a0i j ≥ ai j for all CSs and for all TPs. This means that the optimal solution given by the formulations described in the previous sections always gives an upper bound for the optimal solution of the corresponding formulations where the enhanced capacity measure is considered. A. Experimental Results Table VII reports the results of the heuristics applied to uniform instances of different sizes (#CS = 10, 15, 20, 30, r= 100, 200, 300 m). The left hand part of the table refers to the solution of MF-MEP, while the right hand part refers to the solutions of the S-MF-MEP. In each part, the heuristic local optimum solution is evaluated according to the standard formulation (MF-MEP O.F. and S-MF-MEP O.F.)and to the formulation modified according to the new definition of balanced share introduced in this section (MF-MEP * O.F. and S-MF-MEP * O.F.). The value of the objective function of S-MF-MEP is actually a lower bound for the objective function of the MF-MEP, but its quality is quite poor and decreases when increasing the degree of radio coverage overlap, i.e., when increasing either the number of CS, or, mainly, the coverage radius of each CS.


13

A C AP 3 AP1 B AP 2

Fig. 5. Balanced share enhancement accounting for end user association to APs.

Furthermore, the values of network efficiency computed with the two definitions of the balanced share are very close for both the formulations. B. Further possible enhancements The problem formulations proposed in the paper differ for the way the interference and the multiple access scheme are accounted in the model. As already stated, the balanced share remains an approximate estimation of the network efficiency. We briefly mention here further possible enhancements that could be introduced to our models with a limited effort. The balanced share definition can be enhanced by taking into account the effect of the association of users to APs. Consider, for example, the case in Figure 5: according to the basic definition, all the users should have a balanced share of 1 3 , because all of them are covered by AP 1. However, suppose to have the association relationships described by the arrows in the figure (user A to AP 1, user B to AP 2, user C to AP 3). If users B and C are not within communication range, then users B and C do not have to be considered as interferers, since their communications are confined in separated domains. Anyways, both of them still interfere with user A, since they still produce interference on AP 1. In this case, the actual balanced share should be 21 for both users B and C, and 13 for user A. Moreover, some specific features of the carrier sensing mechanism. could be accounted for. For example, according to the carrier sense threshold setting, the carrier sense range in which transmissions are blocked may be bigger than the communication range in which transmissions are actually correctly received. The mathematical programming formulations can be easily modified to include this feature considering two TP sets for each AP, one for the balanced share calculation and one for the coverage constraint. Moreover, since WLAN physical layer can optionally adopt rate adaptation schemes to adjust the transmission rate according to channel conditions, we could extend our models by considering multiple TP sets corresponding to the areas covered with different rates. VI. C ONCLUSIONS As the dimensions of the WLANs increase the network planning phase becomes more and more important in determining the network efficiency and consequently the future success of the WLAN technology.

In this work we have proposed a novel framework for the WLAN access points positioning problem based on the maximization of the network efficiency. To this end, we defined simplified network efficiency estimations which endorse the peculiarities of the WLAN access mechanism and we tailored on them mathematical programming formulations of the maximum WLAN efficiency problem. All the proposed formulations have been commented in details and have been tested on synthetic WLAN instances. The results show that our planning approach does provide planned WLANs with higher efficiency with respect to the ones provided by classical planning approaches like the minimum cardinality set covering, by privileging the installation of low power APs, which limit the inter domain interference. The gain we get in terms of network efficiency can be up to 167% for particular instances. We have also tested the quality of the network efficiency estimation used in our optimization models through detailed dynamic network simulation and real-life testbed implementation, and we have shown that a good match exists between the estimated efficiency and the actual network saturation throughput. The general side result coming from our analysis is that the efficiency of a multi-APs WLAN can be highly impaired by the effect of the interference among APs’ domains, thus a wise planning strategy should aim at minimizing such effect. To this end, we envision that radio resource management techniques like transmission power control can lead to high throughput WLANs. ACKNOWLEDGMENTS The authors would like to thank Edoardo Amaldi and Federico Malucelli for the helpful support they provided in the definition of the mathematical programming formulations. R EFERENCES [1] IEEE 802.11 Working Group: http://grouper.ieee.org/groups/802/11/ [2] Starbucks News: http://www.starbucks.com/retail/wireless.asp [3] A. Doufexi, E. Tameh, A. Nix, S. Armour, A. Molina, Hotspot wireless LANs to enhance the performance of 3G and beyond cellular networks, IEEE Communications Magazine, July 2003, Volume: 41, Issue: 7, Page(s): 58–65 [4] Brian Senese, Implementing Wireless Communication in Hospital Environments with Bluetooth, 802.11b, and Other Technologies, Medical Device & Diagnostic Industry, July 2003 [5] Kit-Sang Tang, Kim-Fung Man, S. Kwong, Wireless communication network design in IC factory, IEEE Transactions on Industrial Electronics, April 2001, Volume: 48, Issue: 2, Page(s): 452–459 [6] Haitao Wu, Shiduan Cheng, Yong Peng, Keping Long, Jian Ma, Does the IEEE 802.11 MAC protocol work well in multihop wireless ad hoc networks?, IEEE Communication Magazine, June 2001, Volume: 39, Issue: 6, Page(s): 130–137 [7] A. Hills, Large-Scale Wireless LAN Design, IEEE Communications Magazine, November 2001, Volume: 39, Issue: 11, Page(s): 98–107 [8] Y. Wang, L.G. Cuthbert, J. Bigham, Intelligent radio resource management for IEEE 802.11 WLAN, in proceedings of the IEEE Wireless Communications and Networking Conference 2004 (WCNC 2004), 21– 25 March 2004, Volume: 3, Page(s): 1365–1370 [9] E. Amaldi, A. Capone, M. Cesana, F. Malucelli, Optimizing WLAN Radio Coverage, in proceedings of the IEEE International Conference on Communications 2004 (ICC 2004), 20–24 June 2004, Volume: 1, Page(s): 180–184

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[10] E. Amaldi, A. Capone, M. Cesana, F. Malucelli, F. Palazzo, WLAN Coverage Planning: Optimization Models and Algorithms, in proceedings of the IEEE Vehicular Technology Conference (VTC-Spring 2004), 17–19 May 2004, Volume: 4, Page(s): 2219–2223 [11] E. Amaldi, S. Bosio, F. Malucelli, D. Yuan, On a new class of set covering problems arising in WLAN design, in proceedings of the International Network Optimization Conference 2005 (INOC 2005), March 2005, Lisbon, Portugal [12] Network Simulator 2, http://www.isi.edu/nsnam/ns [13] K. Tutschku, Demand-based radio network planning of cellular mobile communication systems, in proceedings of the Joint Conference of the IEEE Computer and Communications Societies 1998 (INFOCOM 1998), 29 March – 2 April 1998, Volume: 3, Page(s): 1054–1061 [14] S. Ceria, P. Nobili, A. Sassano , Set Covering Problem, in Annotated bibliographies in Combinatorial Optimization, Dell’Amico M., F. Maffioli, S. Martello eds, John Wiley and Sons - Chichester [15] H. R. Anderson, J.P. McGeehan, Optimizing microcell base station locations using simulated annealing techniques, in proceedings of the IEEE Vehicular Technology Conference 1994 (VTC 1994 Spring), 8–10 June 1994, Volume: 2, Page(s): 858–862 [16] N.R. Prasad, IEEE 802.11 system design, in proceedings of the IEEE International Conference on Personal Wireless Communications 2000 (PWC 2000), 17-20 December 2000, Page(s): 490–494 [17] R.C. Rodrigues, G.R. Mateus, A.A.F. Loureiro, On the design and capacity planning of a wireless local area network, in proceedings of the IEEE/IFIP Network Operations and Management Symposium 2000 (NOMS 2000), 10-14 April 2000, Page(s): 335–348 [18] G. R. Mateus, A. A. F. Loureiro, R. C. Rodrigues, Optimal Network Design for Wireless Local Area Network, Annals of Operations Research, September 2001, Volume: 106, Number: 1–4, Page(s): 331–345 [19] M. Kamenetsky, M. Unbehaun, Coverage planning for outdoor wireless LAN systems, International Zurich Seminar on Broadband Communications Access, Transmission, Networking 2002, 19–21 February 2002, Page(s): 491–496 [20] M. Unbehaun, M. Kamenetky, On the deployment of picocellular wireless infrastructure, IEEE Wireless Communications, December 2003, Volume: 10, Issue: 6, Page(s): 70–80 [21] Y. Lee, K. Kim, Y. Choi, Optimization of AP placement and Channel Assignment in Wireless LANs, in proceedings of the IEEE Conference on Local Computer Networks 2002 (LCN 2002), 6–8 November 2002, Page(s): 831–836 [22] K.K. Leung, B. J. Kim, Frequency assignment for IEEE 802.11 wireless networks, in proceedings of the IEEE Vehicular Technology Conference 2003 (VTC 2003–Fall), 6–9 October 2003, Volume: 3, Page(s): 1422– 1426 [23] C. Prommak, J. Kabara, D. Tipper, C. Charnsripinyo, Next generation wireless LAN system design, in proceedings of the IEEE Military Conference 2002 (MILCOM 2002), 7–10 October 2002, Volume: 1, Page(s): 473–477 [24] IEEE Standard 802.11, Wireless LAN media access control (MAC) and physical layer (PHY) specifications, 1999 [25] F. A. Tobagi and L. Kleinrock, Packet switching in radio channels: Part II-The hidden terminal problem in carrier sense multiple-access and the busy-tone solution, IEEE Transactions on Communications, December 1975, Volume: COM-23, Page(s): 1417–1433 [26] R. M. Karp, Reducibility among Combinatorial Problems, in R. E. Miller and J. W. Thatcher, Eds., Complexity of Computer Computations, Plenum Press, New York, 1972 [27] E. Balas, A. Ho, Set Covering Algorithms Using Cutting Planes, Heuristics and Subgradient Optimization: A Computational Study, Mathematical Programming 1980, Issue: 12, Page(s): 37–60 [28] J. E. Beasley, A Lagrangian Heuristic for Set-Covering Problems, Naval Research Logistics 1990, Issue: 37, Page(s): 151–164 [29] P. Hansen, M. V. Poggi de Aragao, Boolean query optimization and the 0-1 hyperbolic sum problem, Annals of Mathematics and Artificial Intelligence 1990, Volume: 1, Page(s): 97–109 [30] P. Hansen, M. V. Poggi de Aragao, Hyperbolic 0-1 programming and query optimization in information retrieval, Mathematical Programming 1991, Volume: 52, Page(s): 255–263 [31] Iperf version 1.7.0, http://dast.nlanr.net/Projects/Iperf

Sandro Bosio received his M.S. in Computer Science from Università degli Studi di Milano in December 2002, and his Ph.D. in Mathematical Engineering from the Politecnico di Milano in April 2006. From April 2006 he has been working as a postdoc at the Electronics and Information Department of the Politecnico di Milano. His research activities are in the Operations Research field, and especially in Integer Programming, Computational Complexity and Approximability.

Antonio Capone (SM ’95) is an Associate Professor at the Dipartimento di Elettronica e Informazione of the Technical University of Milan (Politecnico di Milano). His expertise is in networking and main research activities include protocol design (MAC and routing) and performance evaluation of wireless access and multi-hop networks, traffic management and quality of service issues in IP networks, and network planning and optimization. He received the M.S. and Ph.D. degrees in electrical engineering from the Politecnico di Milano in 1994 and 1998, respectively. In 2000 he was a visiting professor at UCLA, Computer Science department. He currently serves as editor of Wireless Communications and Mobile Computing (Wiley) and Computer Networks (Elsevier). He served in the technical program committee of several international conferences and he is a regular reviewer of the main journals in the networking area. He is currently involved in the scientific and technical activities of several national and European research projects, and he leads several industrial projects.

Matteo Cesana (M ’00) received his M.S. in Telecommunication Engineering and his Ph.D. in Information Engineering from the Politecnico di Milano in July 2000 and in September 2004 respectively. From September 2002 to March 2003 he has been working as a visiting researcher at the Computer Science Department of the University of California in Los Angeles (UCLA). He is now an Assistant Professor of the Electronics and Information Department of the Politecnico di Milano. His research activities are in the field of cellular systems performance evaluation, ad-hoc networks protocol design and evaluation and wireless networks optimization. He is a member of IEEE Communication and Computer societies.