Unified Approach of GOS Optimization for Fixed Wireless Access Ran Giladi and Nathan Blaunstein Department of Communication Systems Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Avraham Freedman and Moshe Levin InnoWave- ECI Telecom, Petach-Tikva, Israel Hexagon, System Engineering, Tel-Aviv, Israel Tel: +972-7-647 2591 +972-3-5449471 Fax: +972-7- 647 2883 +972-3-6022447 E-mail: blaun@eesrv.bgu.ac.il avif@hexagonltd.com September 2000Unified Approach of GOS Optimization for Fixed Wireless Access Ran Giladi and Nathan Blaunstein Department of Communication Systems Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Avraham Freedman and Moshe Levin InnoWave – ECI Telecom and Hexagon System Engineering Ltd. Tel-Aviv, Israel Abstract Analytical and simulation models that obtain the minimum number of required communication channels for subscriber units and their allocations so as to increase the level of the grade of service (GOS) per user of Fixed Wireless Access Communication Systems (FWA or WACS) are presented in this article. On the basis of the proposed analytical and simulation models, a new methodology of using user radio-ports (RP) is introduced for evaluating the optimal number of RP-systems, their allocation policy and their spatial distribution for different configurations. This methodology enables us to increase the efficiency of FWA services on the basis of GOS maximization. 1. Introduction FWA (Fixed Wireless Access communication systems) are considered to be part of the PCS (Personal Communication Services), as a replacement for and enhancement of traditional wireline services. FWA are also often called WACS (Wireless Access Communication Systems), RLL or WLL (Radio/Wireless in the Local Loop). Standard organizations are currently involved in establishing and investigating various technical solutions for FWA [1-9]. These systems are useful in places where wireline telephone access solutions are impractical, expensive or temporary. More and more manufacturers and operators are considering FWA in order to enhance regular services (mainly bypass operators) and to provide better, cheaper and more flexible services. The main principle of these systems is to connect end-users to the telephone network via wireless radio links, which are allocated to them when required. This is in contrast to regular wireline end-users, where the reserved wireline to the end-user guarantees a free channel to each end-user. FWA therefore performs a kind of resource-switching by allocating channels (lines) to end-users per request, rather than by providing all users with resources to access the network. FWA systems, however, create problems in establishing reliable, accessible links between the end-users and the public switched telephone network (PSTN), in contrast to common wireline systems. The design of these systems must take into account user traffic intensity and the probability that the system cannot establish a call-for interconnection, which relates to the Grade of Service (GOS), i.e., the probability of a blocked system which causes unavailability of telephone lines to customers. Until recently a FWA infrastructure was designed by using the Erlang loss function [10] - the number of required channels was calculated as a function of user traffic intensity and desired GOS (generally smaller than 1%). The use of Erlang's formula is justified only in cases where all users have access to all resources of the system (i.e., lines), a situation referred to as a full availability system (see Fig. 1a), and where there is a large number of end-users using the available lines. In the case of a wireless system, various obstacles, as well as wave attenuation, require that the system components be deployed in a proper geographical distribution, so that users can access resources in a distributed way. This is referred to as a limited availability system (see Fig. 1b). In other words, not all the lines are available to all the users (some users have access to some lines, while others can access another set of lines). The limited availability problem has been encountered in the design of telephone exchanges [10, chapter 6], [11, chapter 4], where a source has access to only a part of the channels provided. The actual switching problem is solved by grading (see [10] and [11]). However, the limited availability problem we describe in this paper is different. The resource availability is determined by the propagation conditions and it cannot always be as well balanced as is achieved by grading within the exchange. In order to estimate the average probability of blocking the carried traffic approximate formulas have been developed (e.g. the Palm Jacobaeus (PJ) and modified PJ formulas). However, because the model we are using is a simple one, we chose to use the exact solutions of the state equations in order to estimate the probability of blocking. The formulas described can, in some cases, be used for more complex problems. (a) (b) Figure 1: (a) Full and (b) limited availability The problem of limited availability has an effect on any cellular wireless system, be it mobile or fixed access and it should be taken into account in the cellular network design. There is a large variety of algorithms of resource allocations (or channel allocation, as the spectrum is the scarcest resource in a wireless system) among the various users. [12] provides a very thorough and extensive description of fixed channel allocation and dynamic channel allocation algorithms. None of the algorithms, however, takes limited availability explicitly into account. Generally, a balanced load assignment (as in fixed channel allocations) or an optimal approach (like e.g. [13]) may overcome this problem. In this paper we demonstrate the limited availability problem for relatively simple cases. The paper is designed as follows: In Section 2, we build a model that enables us to calculate the minimum number of required channels and their allocation in order to achieve a predetermined GOS per user. Analytical evaluation of the model is described in Section 3, and in Section 4 a methodology for carrying out the numerical simulations of the model is presented. Section 5 describes the simulation results, followed by an analysis of the simulations. The results obtained in section 5 are very similar to the results of load balancing algorithms, as described in [14] and [15]. Section 6 discusses the similarities and differences of the two problems. Section 7 presents a brief summary and some conclusions. 2. Model definition A fixed wireless system is composed of shared telephone lines that are potentially offered to a fixed population of end-users. This system is composed of antennas (Radio Ports, noted in Fig. 1 as RPs). These devices link the fixed public switched telephone network (PSTN) to the end-users’ devices (Subscriber Units, noted in Fig. 1 as SUs). RPs are “intelligent” devices that can allocate telephone lines to a large number of users (typically tens of SUs). The number of channels that the RP can allocate depends on the frequency bandwidth of the air transmission (each channel requires a certain frequency bandwidth). Given the users’ placement in a certain area, the objective is to deploy the minimum number of RPs in order to save costs while simultaneously achieving a predefined GOS per user. It is important to note that the optimal criterion here is the GOS that each user receives, and not, for instance, some other optimal criterion like the average GOS of all the users. In fact, the system must provide a sufficient level of service for each connected user. In other words, we face a minmax problem. The deployment of the RPs must also take into account geographical and topological constraints. Some zones are forbidden completely (military or private areas), while others are not convenient for installation (terrain accidents). For every RP deployment, the link loss between each RP and each SU is computed, resulting in numbers between 0 and 1 that define the quality of the connections. These numbers are rounded to 0 when they are smaller than a certain threshold and to 1 when they are larger than this threshold. They are stored in a matrix called the link matrix [L]. Rows in this link matrix represent the SUs, and columns represent the RPs. Thus, a "1" in row i and column j indicates a physically possible connection between the ith SU and the jth RP, while an "0" means that no physical connection is possible. Note that when a physical connection is possible, one can always cancel this connection in order to achieve better results in the optimization of the GOS (as we will see further on). Each user is supposed to have a fixed calling rate (l) and holding time (m). In fixed wireless systems, one must give each user an allocation policy for his connection to the system. The different policies that we considered are: ‭ ‬Random search‭ (‬denoted as the R method‭)‬:‭ ‬each user chooses randomly from among his available RPs‭ (‬with‭ "‬1‭" ‬in the corresponding row of the link matrix‭) ‬for his first attempt to get a connection.‭ ‬If the chosen RP has at least one free channel,‭ ‬the user will receive and use it,‭ ‬and will speak for an exponential time‭ (‬with parameter‭ ‬m‭)‬.‭ ‬If the chosen RP is occupied‭ (‬all its channels are currently busy‭)‬,‭ ‬the user randomly chooses another RP amongst the other RPs available to him,‭ ‬and so on.‭ ‬At the end of the process,‭ ‬the user that did not receive a connection gets a blocked call and tries again after an exponential time‭ (‬with parameter‭ ‬l‭)‬. ‭ ‬Predefined order search‭ (‬denoted as the P method‭)‬:‭ ‬each user chooses the order of the successive connection trials to his corresponding RPs from a predefined order given in advance and stored in a data base.‭ ‬The predefined order search is defined by a matrix‭ [‬P‭] ‬of size‭ [‬L‭]‬.‭ ‬It provides the order of the successive connection trials for each user when attempting to connect to its RP.‭ ‬The‭ ‬ith‭ ‬row of‭ [‬P‭] ‬will be completed with zeros if the‭ ‬ith‭ ‬user does not have access to all the deployed RPs in the area.‭ ‬For instance,‭ ‬and‭ ‬means that the first user will try only RP1‭ (‬from‭ [‬L‭] ‬we see that it is in fact the only possibility‭)‬,‭ ‬the second user will try only RP2,‭ ‬while the third user will try RP2,‭ ‬and,‭ ‬in case of a failed attempt,‭ ‬will try RP1.‭ ‬For each given link matrix,‭ ‬there are many possibilities of a predefined order search,‭ ‬but in each case the results will be presented with‭ [‬P‭] ‬optimized‭ (‬noted as Popt‭)‬,‭ ‬which means achieving the best minmax‭ (‬GOS‭) ‬for a given link matrix. ‭ ‬Controlled resources search‭ (‬denoted as the F method‭)‬:‭ ‬each user will know at any moment the availability of all its allowed RPs.‭ ‬He will use the RP with the highest number of free channels. We define a user-RP-balanced link matrix as a matrix representing a situation where each user can get connected to and use the same number p of RPs, and each RP can get connected to and be used by the same number n of SUs. We specify a user-RP-balanced link matrix with N customers and P RPs by an NxP matrix (n,p), where ntotal number of users‭ (=‬N‭) ‬and‭ ‬ptotal number of RPs‭ (=‬P‭) ‬in the area. ,‭ ‬for instance,‭ ‬is a balanced‭ ‬2x2‭ (‬1,1‭) ‬matrix,‭ ‬while‭ ‬is not balanced because the second user can get connected to and use two RPs while the first user can only get connected to and use the first one.‭ ‬Note that a full‭ "‬1‭" ‬matrix is always balanced‭ (‬and is an‭ (‬N,P‭) ‬matrix‭)‬,‭ ‬being in fact the full availability case‭ (‬presented in Fig.‭ ‬1a‭)‬,‭ ‬where the Erlang B formula is applicable‭ [‬6‭]‬.‭ ‬This formula can be used instead of the Engset distribution‭ [‬6‭] ‬for the case when the number of SUs,‭ ‬N,‭ ‬is infinite,‭ ‬whereas the number of channels per RP,‭ ‬S‭ ‬is finite. We further define an RP-balanced link matrix as a matrix where each RP can get connected to and be used by the same number n of SUs. Note that every user-RP-balanced matrix is also an RP-balanced matrix. ,‭ ‬for instance,‭ ‬is an‭ ‬RP-balanced‭ ‬link matrix,‭ ‬where each RP is connected to two users‭ (‬RP1‭ ‬to user‭ ‬1‭ ‬and‭ ‬3,‭ ‬and RP2‭ ‬to user‭ ‬2‭ ‬and‭ ‬3‭)‬. 3. Analytical evaluation In the analytical model and subsequently, in the numerical simulations, we assume that N= and use the Erlang B formula [6] since in our approach the parameter l is independent of the number of customers or servers in use. Below we analyze a two server system, first by using a predefined order search and then by using a random search. We use two user groups with a link matrix that describes their connections to the servers as ‭ (‬a non-balanced link matrix‭)‬.‭ ‬The analysis is based on two typical situations of SU-RP connections presented in Fig.‭ ‬1b.‭ ‬The state diagrams of these two servers in the two serving policies,‭ ‬i.e.,‭ ‬the predefined order and the random searches are shown in Fig.‭ ‬2a and Fig‭ ‬2b.‭ ‬Each state is marked‭ (‬n,‭ ‬m‭); ‬n‭ ‬specifies that the first server is occupied by a user either from the first group‭ (‬1‭)‬,‭ ‬or the second group‭ (‬2‭)‬,‭ ‬or is free‭ (‬0‭); ‬m‭ ‬specifies how the second server is occupied in the same manner. In the case of a predefined order search (state diagram in Fig. 2a) a user from the first group can use server RP1, and if it is occupied he can use server RP2 (if it is free; if not he leaves the system without service, which is a “blocked” situation). A user from the second group can use only RP2, if it is free. If it is not, he is blocked (he leaves the system without service). The predefined search matrix is therefore . Fig 2a: Predefined order search state diagram The probability of the system being in state (i,j) is Pi,j. The state transition equilibrium equations are given by: ‎‮ ‭(‬1a‭) ‎‮ ‭(‬1b‭) ‎‮ ‭(‬1c‭) ‎‮ ‭(‬1d‭) ‎‮ ‭(‬1e‭) ‎‮ ‭(‬1f‭) ‎‮ ‭(‬1g‭) After deriving system (1a)-(1g), we can obtain the probability of the event that a user of the first group gets an occupied system by the following equation: ‭ (‬2‭) The probability that a user of the second group gets an occupied system can also be obtained from system (1a)-(1g) as: ‎‮ ‭(‬3‭) Fig 2b: Random search state diagram The case of a random search is slightly different. The state diagram of this service is presented in Fig. 2b. For this case the state transition equilibrium is given by the following system: ‎‮ ‭(‬4a‭) ‎‮ ‭(‬4b‭) ‎‮ ‭(‬4c‭) ‎‮ ‭(‬4d‭) ‎‮ ‭(‬4e‭) ‎‮ ‭(‬4f‭) ‎‮ ‭(‬4g‭) From system (4a)-(4g) one can obtain the probability that a user of the first group gets an occupied system: ‎‮ ‭(‬5‭) and the probability that a user of the second group gets an occupied system: ‭ (‬6‭) It is worthwhile to compare the two search methods in this case. It turns out that the predefined order search GOS1 is slightly larger than the random search GOS1, which means that in terms of the first group of users, it is advisable to adopt the random search. The difference, GOS1(predefined)-GOS1(random) from (2) and (5), depends on the value of utilization, r=l/m, as follows from the graph, Fig. 3a, for values r in the range [0...1]. As can be seen from the illustration, the difference is limited to a value of about 0.015 for r>0.6. Fig 3a: The difference in GOS for users of group 1 In the case of the second group of users, however, the situation is the opposite. The random search GOS2 is larger than the predefined order search GOS2, which means that in terms of the second group of users it is advisable to adopt the predefined order search. The difference, GOS2(random)-GOS2(predefined) from (3) and (6), is presented in Fig. 3b as a function of the value of utilization r in the range [0...1]. As can be seen, there are large differences in GOS in the range 0.2<r<0.7. Fig 3b: The difference in GOS for users of group 2 The better search policy, in terms of maximal GOS for all the users, is obviously the predefined order search (users of group 2 show much higher differences in GOS). The better search policy, in terms of average minimal GOS for all the users, could be evaluated by calculating: N1/N[GOS1(predefined)-GOS1(random)]+N2/N[GOS2(predefined)-GOS2(random)] (7) where N is the total number of users, N1 is the number of users in the first group, and N2 is the number of users in the second group (N1+N2=N). The random search policy is better when (7) is positive, etc. The precise calculated difference is: This leaves us with the problem of comparing these search policies to the controlled resource search, and the problem of comparing policies with bigger link matrices. As can be seen from the graph depicted in Fig. 3c, the distance between the two differences presented in Figs. 3a and 3b for r in the range [0...1], which is equation (7) for case N1=N2, suggests that the predefined order search is still the best search policy. Fig 3c: Average random GOS - average predefined GOS 4. Methodology Because an analytical description for the general case is impossible, we used simulations of the model presented above. We assumed that there are two groups of users connected to two RPs, as in Fig. 1b, and used the structures of matrices defined above. To check how the structure of the matrix and the allocation policy affect the GOS, we ran simulations using Nx2 matrices in the general form: where there are ‭ ‬rows of‭ (‬1,0‭)‬,‭ ‬rows of‭ (‬0,1‭) ‬and‭ ‬rows of‭ (‬1,1‭)‬,‭ ‬where n1+n2+n3‭ =‬N.‭ ‬This L-matrix defines three classes of users:‭ ‬the‭ ‬first class‭ ‬is composed of n1‭ ‬users that can only use RP1‭; ‬the‭ ‬second class‭ ‬is composed of n2‭ ‬users that can only use RP2‭; ‬and the third class is composed of n3‭ ‬users that can use both RP1‭ ‬and RP2,‭ ‬as in Fig.‭ ‬1b. The general form of this presentation enables us to check a variety of situations, such as a user-RP-balanced matrix, where n1=n2=N/2 and n3=0 (so we get an (N/2,1) balanced matrix) or n1=n2=0 and n3=N (so we get an (N,2) balanced matrix). We can also check an RP-balanced matrix, when n1=n2<N/2, varying from a slightly user-RP-unbalanced link matrix (when n1=n2; n3<<n1 or n3n1), to a severely user-RP-unbalanced link matrix (when n1=n2; n3>n1 or n3n1). In order to optimize the number of RPs and their location in the area, we ran simulations with different configurations of the link matrix. We began with the balanced configuration, and then gradually added links to degrade this balanced situation. Then we compared different allocation policies for all cases. The user load is assumed constant, conforming to data that we obtained from experimental situations composed of incoming and outgoing calls symmetrically distributed. The main parameters assumed in the simulations are as follows: User load r=0.0625, size of the population N=40, and number of channels per RP (available lines from each RP) S=4. 5. Simulation results Using simulations we examined the behavior of the maximal value of the Grade of Service, i.e., max(GOS), in terms of two main parameters: the structure of the connections between RP and SU; the allocation policy, i.e., the method used by each user to establish his connection to the system. In the following section we analyze these results for the case of two different link matrices. 5.1.The user-RP-balanced link matrix We start with balanced situation (N/2,1). The simulation uses the matrix [L(n1,n2,n3)] where n1=n2=20 and n3=0, and the predefined search order is (which is the only possibility in this case). The maximum GOS of all the users, for each of the three methods (R, P and F, which are the same in this particular case), are 2.98% maxgosR, 2.94% maxgosP and 2.98% maxgosF. In fact, in this case the Erlang B formula [6] is applicable because the first n1 users and the last n2 users form two independent user teams (they have their own independent servers (RP1 and RP2, as follows from Fig. 1b). The Erlang B formula yields GOS=2.94% and the Engset formula yields GOS=3.01%. These results are in accordance with our results. On this basis we assume that the results of the simulation should be closer to those obtained by the Engset formula; however, in our case with N=40, the Erlang B and the Engset approaches are almost the same. Again, in this particular case, we find that the expected theoretical GOS result is identical for every user and does not depend on the policy given to users. Now, we diverge from the user-RP-balanced link matrix [L(n1,n2,n3)] by gradually adding n3-type users to the configuration: n1=20, n2=20-n3, n3=0...20. The predefined search order will now be: (Note that this is the optimal order and is intuitively determined; RP1 serves more possible users than RP2 (Fig. 1b), so it is sensible to permit the third class of users to try RP2 first.) The results of simulation for the three methods (P, R and F) are presented in Table 1 and graphically in Fig. 4. Note that when n3 is 0, we are dealing with two independent groups - half the users are in the first group using RP1 and the other half are in the second group, using RP2. When n3 increases up to 20, we transfer users from the second group (that uses RP2) to a new group, a third one, that allows the use of both RP1 and RP2. When n3 is 20, we are in a situation where half the users are in the first group, using RP1, and the other half are in the third group, using both RP1 and RP2. N3 0 5 10 15 20 MaxgosP (%) 2.94 3.04 3.12 3.18 3.25 MaxgosF (%) 2.98 3.78 4.44 4.93 5.36 MaxgosR (%) 2.98 4.09 5.40 6.79 8.33 Table 1: GOS results of a diverted user-RP-balanced link matrix In this case the predefined order method is the best of the three methods, and adding links (until half of the users (n3=20) get connected to RP1 and RP2) makes matters worse; disconnecting links to establish a balanced situation (n1=n2=20; n3=0) might even be advisable. This is, of course, true in respect to the maximal GOS of all the users. This kind of configuration makes life easier to users in the new group (n3) in terms of their GOS. Fig. 4: GOS results of a diverted user-RP-balanced link matrix In the next cases, we check what happens if we shift the balanced matrix [L] with even more supplementary connections, getting closer to the full availability case (N, 2): n1=40-n3, n2=0, n3=21...39. The predefined search order is: (Note that this is the optimal order, intuitively determined; RP1 is granted first to the first team who can access only one RP, while the second team tries RP2 first.) The results of simulation for the three methods are presented in Table 2 and graphically in Fig. 5. n3 21 22 24 28 34 39 maxgosP (%) 2.91 2.57 1.99 1.21 0.68 0.55 maxgosF (%) 5.05 4.74 4.21 3.30 2.26 1.68 maxgosR (%) 8.07 7.69 7.14 6.02 4.51 3.45 Table 2: GOS results of diverting to a full availability link matrix Note again that we are essentially continuing the previous case. After the second group was emptied, we started to move users from the first group to the third one. In this situation, when n3 = 21, we are approximately in the same situation as the last phase in the previous example, i.e., half the users are in the first group, using RP1, and the other half are in the third group, using both RP1 and RP2. As n3 increases, and more users are moved from the first group to the third one, we end up with n3 = 39. This is the most efficient situation since almost all users are in the third group and are using both RP1 and RP2. From the histograms presented in Fig. 5 we see that in this case the predefined order method is again the best of the three methods. For n3=21 to n3=39, the maxgosP (2.91% to 0.54%, respectively) becomes smaller than the balanced situation ((N/2,1) with 2.94%). Note that for n3=40, we reach the full availability case (Fig. 1a), which we can predict by the Erlang B formula [6] (GOS=0.31%). This is the (N,2) balanced matrix case which achieves the best level of service. Fig. 5: GOS results of diverting to a full availability link matrix 5.2. The RP-balanced matrix We start by examining how the GOS behaves in the RP-balanced matrix case for the three methods (P, R and F). The link matrix [L] will have the following configuration: n1=n2, n3=2...38, and the predefined search order matrix: where the last two partitions of [P] belong to the third class of n3 users, and the number of (1,2) rows is equal to the number of (2,1) rows (and equal to (N-2xn1)/2). Note that [P] is intuitively optimized by equalizing the load on each RP. The results of simulation for the three methods are presented in Table 3 and graphically in Fig. 6. n3 2 4 8 12 16 20 24 28 32 36 38 maxgosP (%) 2.99 3.01 3.05 3.07 3.13 3.16 3.19 3.24 3.25 3.26 3.28 maxgosF (%) 2.87 2.75 2.45 2.20 1.99 1.80 1.58 1.43 1.32 1.16 1.11 maxgosR (%) 2.99 3.02 3.06 3.10 3.12 3.15 3.18 3.26 3.29 3.31 3.33 Table 3: GOS results of the RP-balanced link matrix In this situation, we are starting with two independent groups, each being served by a separate RP. Half the customers are in the first group, and the rest in the second. As n3 increases, we take users from the two groups and move them to a new third group being served by both RP1 and RP2. When n3 is 20, then a quarter of the users are in the first group, being served by RP1, another quarter are in the second group, being served by RP2, and half are in the third one, using both RP1 and RP2. When n3 approaches 38, almost all users are in the third group, using both RP1 and RP2. The histograms presented in Fig. 6 reveal a strange phenomenon: using the P and R methods and adding more links from the balanced situation (N/2,1) to the pseudo-balanced situation only increases the max(GOS) (in the P method we get a degradation of the max(GOS) from 2.99% to 3.28%, and for the R method from 2.99% to 3.33%). The F method behaves in the opposite way: the max(GOS) decreases from 2.87% to 1.11%. The GOS in the balanced situation (from the Erlang B formula [6]), as mentioned earlier, is equal to 3.01%. Thus, the P and R methods are never worthwhile, but using the F method always improves the GOS in comparison with the balanced situation. Fig. 6: GOS results of the RP-balanced link matrix 5.3.Analysis From the simulation results presented in Tables 1-3 and Figs. 4-6 it follows that for the case of a user-RP-balanced link matrix, the maximum GOS of all users for the three methods of the search is 3.0%, and it degrades as links are added (the maximum GOS increases) in non RP-balanced configurations. Moreover, the predefined search order method (P) is the best of the three methods for different non-RP-balanced configurations of link matrices. In the case of the RP-balanced link matrix for various kinds of configurations, only the controlled resources search (F) method improves the GOS level, decreasing it from 2.87% to 1.11% as we move from one user-RP-balanced situation (N/2,1) (two separate groups) to a second user-RP-balanced situation (N,2) (full availability case). The other two methods increase the max(GOS): The P-search increase is from 2.99% to 3.28% and the R-search from 2.99% to 3.33%. We also notice that going from one user-RP-balanced situation to another, ((N/2,1) to (N,2)), the link matrix can reach many configurations, including RP-balanced link matrix ones. When the configuration is not an RP-balanced link matrix case, it is better to use the P-search method in order to achieve the best (minimal) maximum GOS. However, if an RP-balanced situation does not exist, links can be added or deleted, so that it can be configured to the nearest balanced configuration, an effect which depends on the degree of non-balancing of the link matrix. After a certain point, when links are disconnected in order to return to the balanced situation, a better max(GOS) can be achieved using the F-search method. This point depends on the ratio of the links to the size of population and RP number. In the RP-balanced situation, the F-search method allows us to obtain the correct solution to the SU and RP optimization problems. When using the non-balanced link matrix, the P-search is best in terms of maximal GOS, but the F-search gives us a good alternative as well, in comparison with the R and P search methods. In practice, the main parameters might be different from those presented in the numerical simulation. In fact, a typical size of the population, N, might be in the hundreds, and the number of deployed RPs might be about ten. Fortunately, in the case of RP-balancing, the F-policy gives us the correct solution to the optimization problem. In other cases (where the L-matrix is severely non-balanced), the predefined search order (P) method is optimal, but might not be so easy to calculate. In these cases the F-search method still provides a good alternative. Comparison to other work The results of the analysis suggest that balancing the load on each RP gives the optimal solution of the problem. Load balancing has been treated extensively, mainly in the context of networks, however, it is not identical to the problem described above. On the other hand, the similar solutions might suggest that some of the methods and algorithms described in the context of load balancing can be applied for the case of limited availability. [14] introduces an algorithm for re-homing in order to balance the load of mobiles on Mobile Switching Centers (MSC). The home MSC handles call related network activities, such as mobile termination and call deliver, and inter Visitor Location Register (VLR) registration, activities which load the MSC, and can be balanced by moving a group of mobiles to another MSC. The algorithm suggested therein is doing the re-homing process according to the criterion of maximum benefit to the load standard deviation ratio, where benefit is expressed in the terms of maximal load reduction. This algorithm could be applied to the limited availability case if there is an overlap of coverage area of more than two RP. The measure of the benefits and costs would of course depend on the characteristics of the system in question. [15] formulates the load balancing problem as an optimal control problem and shows that variants of the “Least Loaded Routing” policy is asymptotically optimal for large call arrival rates. The problems descirbed in [15] are very similar to the problem described above, especially the “basic dynamic model” given in section 3. However, the cost function chosen in [15] does not reflect the real cost for a FWA system, which aims to minimize the probability of blocking of the the user located in the worst location. In [15], on the other hand, the cost function was a quadratic function of the load, which is not at all identical. The Least Loaded Routing algorithm studied in [15] isn’t but the F algorithm suggeted above, which was shown here to be the best choice in most cases. 7. Summary This work presents a new methodology that is based on three different search methods by the user/subscriber unit, whose purpose is to optimize the number of RPs and their spatial allocation: Random (R), predefined order (P) and controlled resources (F). For each of these methods we carried out simulations for two different configurations of link matrices: the user-RP-balanced and RP-balanced matrices. The grade of service (GOS) was evaluated for two variants of communication channels service policies. The simulated results resembled those of the traditional approach that is based on Erlang's canonical formula. Generally, the controlled resource approach was proven to be the optimal policy, in terms of GOS, as the simulations showed. In some cases, however, the predefined order approach was preferred. Both analytical and numerical analysis enabled us to predict the maximal GOS level and to obtain the optimal number of radio ports (RP) and their effective allocation with subscriber units (SU) for Wireless Access Communication Systems (WACS). Similar results were obtained in studies of the load balancing problems, although the problem by itself is not identical. Load balancing algorithms can thus be applied, at least on a first trial basis, for the limited availability problem as well. References Bellcore, Generic Framework Criteria for Version 1.0 Wireless Access Communications Systems (WACS), FA-NWT-001318, 1992. Bellcore, Comparison of the Economics of Radio and Conventional Distribution for Rural Areas, SR-NPL-000676, 1987. ETSI, ETR on Radio in the Local Loop, 1994. 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