Location update optimization in personal communication systems

Transcription

1 Wireless Networks Location update optimization in personal communication systems Ahmed Abutaleb and Victor O.K. Li Communication Sciences Institute, Department of Electrical Engineering, University of Southern California, Los Angeles, CA , USA Mobility tracking is concerned with finding a mobile subscriber MS within the area serviced by the wireless network. The two basic operations for tracking an MS, location updating and paging, constitute additional load on the wireless network. The total cost of updating and paging can be minimized by optimally dividing the service area into location registration LR areas. There are various factors affecting this cost, including the mobility and call patterns of the individual MS, the shape, size and orientation of the LR area, and the method of searching for the MS within the LR area. Based on various mobility patterns of users and network architecture, the design of the LR area is formulated as a combinatorial optimization problem. The objective is to minimize the location update cost subject to a constraint on the size of the LR area. 1. Introduction A wireless personal communication system PCS consists of a group of fixed base stations BS covering the service area and interconnected by a fixed backbone network. The coverage area of one BS, determined by the terrain and the radio propagation characteristics, is referred to as a cell. Each cell contains a fixed base station BS and a number of mobile subscribers MS [4,11]. The tasks necessary to manage individual communication sessions in a cellular network can be divided into three categories [6], call processing, mobility management and radio resource management. The main task of mobility management is location tracking. In the current cellular systems, the networks are partitioned into location registration LR areas to facilitate the tracking of an MS within the network. Each LR area may contain one or more cells. The two fundamental operations associated with an LR area are: 1. Location update LU: When an MS enters a new LR area, it performs a location updating procedure; one of the BSs in the newly visited LR area is informed and the home directory of the MS is updated with the current MS location. Any incoming calls to that MS can thus be routed to the correct LR area. 2. Paging: Paging is used by the network to alert the MS of an incoming call. Paging messages are broadcast in thelrareawherethemsislocated. Paging and LU signaling take up valuable bandwidth in the wireless network. There is an inherent tradeoff between the costs of LU and paging. To illustrate, consider two simple strategies for tracking MSs, always-update and never-update [2]. In the always-update strategy, each MS transmits an LU upon entry into a new cell. Clearly the overhead is high for LU, but is minimal for paging. On the other hand, in the never-update strategy, the MS never updates its location. Consequently, network-wide paging is required for an incoming call. The current cellular radio network uses a combination of these two strategies. The service area is partitioned into LR areas, and within an LR area the MS does not perform LUs. When the MS enters a new LR area, it updates its location. The effectiveness of this strategy depends on how the partitions are done. The purpose of LR area design is to optimize the partitions. Recently, some approaches have been proposed to partition the cellular network into LR areas to minimize the total cost of LU and paging signaling. A static approach is considered in [2], where a subset of all the cells in the cellular network is selected and designated as reporting cells. An MS will transmit update messages only upon entering a reporting cell and a search for any user will always be restricted to the vicinity of the reporting cell to which the user last reported. Associated with each cell in the network is a weight that reflects the traffic intensity of MSs into this cell. Graph theoretic techniques are used to select the reporting centers in such a way that both the size of the largest vicinity and the total weight of the reporting centers are minimized. Since these are contradicting goals, the approach is to bound one goal and minimize the other. The strategy is global in the sense that all mobile users transmit their update messages in the same set of cells, and thus is a static LU strategy. Highly mobile users will update more frequently, thus using more channel resources. It may also happen that a mobile user will not transmit update messages for long periods of time, even if the MS is highly mobile, since its path may not take it into a reporting cell. We believe a better approach is to use a dynamic LU strategy that is specific to an MS or to a particular class of MSs, dependent on various MS parameters, such as mobility pattern and incoming call rate. In [13] the rate of updates for an MS is derived for arbitrary LR area sizes and shapes. The movements of MSs are based on the fluid model of mobility, where each MS s direction of movements is uniformly distributed over [0, 2π]. This formulation is used in [14] to determine the rate of location updates per terminal and the size of a square LR area that would minimize the total cost of signaling. This model does not exploit the indi- J.C. Baltzer AG, Science Publishers

2 206 A. Abutaleb, V.O.K. Li / Location update optimization in PCS vidual mobility patterns of different MS or MS classes, but rather averages the mobility pattern of all MSs and assumes a fluid model in determining LR area boundary crossings. For a specific user the direction of movement in the LR area is not uniformly distributed from [0, 2π], but rather exhibits directional preference. In [7] individual mobility patterns are considered and a dynamic iterative algorithm is proposed to find the optimal LR area size. However, the model considered is a simple one dimensional random walk model and the LR area is not two-dimensional but rather one-dimensional, i.e., a line. 2. Location update strategies Three dynamic location update methods exist [3]: timebased, movement-based, anddistance-based, in which updates are generated based on time, number of cell boundary crossings and distance thresholds. In [3], the three dynamic strategies are compared and it is concluded that the distance-based method is the most efficient in minimizing the number of LU messages transmitted by an MS, for the same paging cost. We define a new LU strategy based on a modified distance-based update strategy. The MS transmits an update message whenever the distance between the current cell, which the MS just entered, and the cell in which the MS last performed an LU exceeds a threshold D x, y, as shown in figure 1. The distance threshold D is a function of x and y, the relative positions of the current cell and the cell where the last LU was performed, the mobility model of the MS inside the LR area, and the topology of the cellular architecture. Our objective in this paper is to design an algorithm for determining the optimal LR area contour, D x, y, such that the total cost of signaling on the control channels due to LUs and paging is minimized. We begin by defining the total cost of signaling. Figure 1. Example of modified distance-based location updates and N k calculation. 3. Signaling cost The total signaling cost for LUs and paging per MS per unit time depends on the following parameters [7,14]: k, the number of cells in the LR area; µ a, the arrival rate of incoming calls for the MS calls/unit time; u k, the update rate for MS in LR containing k cells upds/unit time, which depends on the mobility pattern, speed profile and the design of the LR area; C p, the paging cost per cell BW/cell, where BW is the bandwidth required; C u,thecostfora location update BW/upd. The total signaling cost per MS is given by Ck, µ a = kµ a C p + u k C u. 1 In equation 1, it is assumed that all the cells in the LR area broadcast the paging message. In [1], we propose a selective-paging algorithm that exploits information about the mobility and call patterns of the MSs. On the average the number of cells paged is much less than k. Minimizing the total signaling cost by minimizing the LU cost and the paging cost constitutes contradicting goals. Increasing the LR area size decreases the update rate for an MS, but increases the number of cells that have to be paged in the LR area to alert the user of an incoming call. On the other hand, decreasing the LR area size decreases the number of cells paged per incoming call, but the duration of time the MS spends in the LR area decreases, which in turn increases the update rate per MS. The approach to minimize the total cost of LU and paging is to bound one cost and minimize the other. 4. Mobility model As the processing capability of the system increases, it would be possible to design MS specific LR areas which depend on the LU and incoming call rate for that specific user in the current geographical position in the system. However, for current systems MSs of similar incoming call rates and mobility patterns may be grouped together and the LR area is designed per group. The mobility pattern is related to the geographical area and the time of day, e.g., MSs may move from residential areas to business areas in the morning and in the opposite direction in the evening. MSs in a specific geographical area may be subdivided into groups using the incoming call rate of the MSs. This information may be accumulated in the handset of the MS and presented to the system at the time of LU. We shall illustrate the optimization of LR areas per MS which is representative of its group. To design per-user LR areas, we refer to a single MS. The MS moves in a two dimensional plane. The cellular topology used to illustrate the LR area design procedure is a grid architecture. In the grid architecture cell i, j has neighbors i, j + 1, i, j 1, i 1, j andi+1, j, as shown in figure 1. Even though the grid architecture is used for illustration, the same analysis applies to any 2-dimensional architecture.

3 A. Abutaleb, V.O.K. Li / Location update optimization in PCS 207 In other architectures the number of neighboring cells and the mobility patterns differ. Bounding the cost of paging, by keeping the number of cells in an LR area constant, we minimize the cost of LU. The LU cost depends on the update rate of the MS. Assuming that the durations different MSs spend in the cells are i.i.d independent identically distributed random variables, then the average update rate is u k = u 1, 2 N k where u 1 = rate of traversing one cell, and N k = expected number of cells traversed in the LR area with k cells before exiting the LR area. To show the calculation of N k for an arbitrary mobility model, consider the LR area A shown in figure 1. The cellular topology is a grid architecture and an MS in cell i, j can only move to cells i + 1, j, i 1, j, i, j + 1 or i, j 1. We assume without loss of generality that the LU cell has coordinates 0,0. Let the probability of visiting cell i, j on the boundary of the LR area after n movements n i + j inside the LR area, starting at the LU cell, be denoted by Pr vi, j n and let Prei, j n be the probability that cell i, j isthe last cell the MS visits before exiting the LR area after n movements, starting at the LU cell. Then, for the example shown in figure 1, Pr ei, j n = Pr vi, j n p 1 + Pr vi, j n p 2, where p ι, ι =1, 2, 3, 4, denotes the conditional probability of moving from cell i, j to cells i, j+1, i+1, j, i, j 1 and i 1, j, respectively, given the MS moves 1 Therefore N k = i,j RA p 1 + p 2 + p 3 + p 4 = 1. n=i+j n + 1 Pr ei, j n, 3 where RA is a set containing all cells on the boundary of A. So far in the formulation of the LR area optimization problem the i.i.d model of mobility of MS is used [3,7]. However, the movement of vehicular MSs is not independent of the previous movements. Vehicular MSs have directional preference which depends on the location of the source and destination of the MS. We would like to concentrate on vehicular MSs because they are highly mobile compared to pedestrian users, thus generating more LU traffic, due to their high speeds, and more paging traffic per incoming call, due to the larger LR area sizes associated with vehicular MSs. 1 In the example given in section 8, we will show how to handle the situation where the MS has arrived at its destination and therefore stops moving for a relatively long time Shortest distance model Consider again the grid architecture. Within the LR area up to the exit from the LR area we assume that the MS follows the shortest path, measured in number of cells traversed, from source to destination. This assumption is reasonable within the LR area borders particularly if the path length traversed within the LR area is small compared to the total path length between source and destination. The grid cellular architecture is exemplified by a Manhattan City streets model with the BSs located at the intersections of the streets [10]. More complex traffic patterns such as one way streets and other strange traffic patterns that may occur in urban areas are not considered by this model and have to be studied on a case-by-case basis. At each intersection the MS makes a decision to proceed to any of the neighboring cells such that the shortest distance assumption is maintained. On entry to the LU cell, we assume that the only information available to the system is the location of the previous cell visited. So in figure 2, on entry to the LU cell, the MS can visit any cell in the LR area to the right of the dotted line in addition to any cell on the dotted line. To maintain the shortest distance assumption within the LR area, the movements of the MS at each cell will depend on the previous movements within the LR area. To demonstrate such movements, we define the following: S, moving Straight ahead at an intersection; L, moving Left at an intersection; R, moving Right at an intersection; Λ, a random variable representing the last turn direction performed inside the LR area, Λ can take values {L, R}. Let the unconditional probabilities of the MS going straight, left and right at the intersections be given by PrS =, PrL = P l, PrR = P r, 4 where + P l + P r = 1. 5 If the last turn executed by the MS within the LR area is L then these probabilities become PrS Λ = L =, PrL Λ = L = 0, + P r 6 PrR Λ = L = P r. + P r Figure 2. Example of MS mobility pattern.

4 208 A. Abutaleb, V.O.K. Li / Location update optimization in PCS Similarly if the last turn executed by the MS within the LR area is R then PrS Λ = R =, PrR Λ = R = 0, + P l 7 PrL Λ = R = P l. + P l For example, the probability of visiting cell D 2 using the path denoted in figure 2 by 1, 3 and 6 9 is PrSPrRPrL Λ=RPrR Λ=L PrL Λ = RPrS Λ=L P l P r P l = P r. +P l +P r +P l +P r The vehicular traffic flow rates out of and into a cell can be used as a measure of the probabilities, P r and P l. Higher flow conditions represent higher probability of using that direction. Thus, P r and P l depend on the LR area and not on the MS. For small LR areas an average value of, P r and P l may be used for the whole area. To demonstrate the LR area optimization problem we will use an average value of, P r and P l to define the mobility pattern for an MS within the LR area. The analysis can easily be extended to the more general model. 5. Prvi, j and N k for the shortest distance model In this section we show how to calculate Prvi, j, the probability of visiting cell i, j, and N k for any arbitrary LR area, for the shortest distance model of mobility. We will then use N k as the criteria for searching for the optimum LR area shape subject to a paging constraint Calculation of Prvi, j Having defined the mobility model for vehicular MS in this section we describe an iterative algorithm to calculate the probability of visiting any cell i, j in the LR area from the W west, Nnorth, Eeast and Ssouth, where: W i, j, Ni, j, Ei, j, Si, j the unconditional probability of visiting cell i, j from the W, N, E, S directions, as shown in figure 3, using shortest distance paths, within the LR area, starting at the LU cell, and Pr vi, j = W i, j + Ei, j + Ni, j + Si, j. In the shortest distance model cell i, j can be visited starting at the LU cell only after traversing i α + j β cells in the LR area A, whereα,β denotes the coordinates of the LU cell. Let the LU cell be denoted α, β and let the binary m m matrix X represent the cell assignments to the LR area for a particular MS mobility pattern. Typically an LR area will include about 60 cells, but an LR area may be as Figure 3. Possible entry directions to cell i, j. large as the whole service, i.e., for a 400 cell system, m is 20. Thus the matrix X can be written as x 1m x 2m... x mm X =.... x αβ x 11 x x m1 and the elements of the matrix are given by { { } 1 assigned x ij = if cell i, j is 0 not assigned to the LR area. 8 The matrix X defines the shape of the LR area. Given matrix X we proceed to calculate the probability of visiting each cell in the LR area, then using this information we can calculate N k, the average number of cells traversed in the LR area before exiting. Let the movement of the MS at the intersections be given by equations 4, 6 and 7. The following is a description of the iterative algorithm for calculating the probability of visiting a cell from the W, N, E and S directions given the matrix X and the LU cell α, β is entered from the W direction as shown in figure 2. Similar algorithms can be used for an MS entering the LU cell from other directions. Algorithm Initialize all the probabilities to zero, i, j W i, j = Si, j = Ni, j = Ei, j = Initialize the LU cell, the entry direction of the MS into LR area is from W direction of cell α, β, W α, β = 1, Sα, β = Nα, β = Eα, β = Let n = This step calculates the probability of visiting all cells with the same y-coordinate as the LU cell. Shortest distance paths pass only through the W side of cells α + n, β figure 4a, therefore W α + n, β = W α + n 1, β x α+n 1,β. 5. This step calculates the probability of visiting all cells with the same x-coordinate as the LU cell. Shortest distance paths pass only through the S side of cells

5 A. Abutaleb, V.O.K. Li / Location update optimization in PCS 209 P r + Si 1, j + P r Si, j = Si, j 1 + P r if j<β 1 figure 4d, P l + W i, j 1 + P l W i, j = W i 1, j + P r P l + Ni 1, j + P l Ni, j = Ni, j P l if j = β + 1 figure 4e, P r + W i, j P r x i 1,j, x i,j 1 ; x i 1,j, x i,j+1 ; W i, j = W i 1, j + P l P r + Si 1, j x i 1,j, + P r Si, j = W i, j 1P l x i,j 1 ; Figure 4. The iterative algorithm. α, β + n and through the N side of cells α, β n figure 4b, therefore, Sα, β + n = W α, β + n 1P l + Sα, β + n 1 x α,β+n 1, + P r Nα, β n = W α, β n + 1P r + Nα, β n + 1 x α,β n+1. + P l 6. n = n Go to step 4 until the probabilities of visiting all cells with the same x- andy-coordinates as the LU cell have been calculated. 8. Let n = This step calculates the probabilities of visiting all the remaining cells that are reachable from the LU cell using shortest distance paths, and the paths are in the LR area. i, j such that i α + j β = n, i>α,j β: if j>β+1 figure 4c, W i, j = W i 1, j + P l if j = β 1 figure 4f, 10. n = n + 1. W i, j = W i 1, j + P r P l + Ni 1, j x i 1,j, + P l Ni, j = W i, j + 1P r x i,j Go to step 9 until all cells that are reachable from the LU cell using shortest distance paths have been calculated. 12. i, j Prvi,j = Ei, j+si, j+ni, j+w i, j. The iterative algorithm above considers only the shortest distance paths inside any arbitrary LR area in the calculation of the probability of visiting a cell, Prvi, j, since paths inside the LR area pass only through cells i, j such that x ij = 1 along the entire path. The above algorithm evaluates Prvi, j for all cells inside the LR area and on the perimeter just outside the LR area. This is required for the calculation of N k. Figure 5 shows the different mobility patterns for an MS, as specified by, P r and P l, entering the LU cell 51, 51 from the W direction and following shortest path

6 210 A. Abutaleb, V.O.K. Li / Location update optimization in PCS Figure 5. Examples of MS mobility pattern. routes. The LR area assumed is the half-plane x 51. The contours shown are the equal probability contours of the MS visiting the cells on the contour given the MS enters the LR at cell 51, 51 from W. The values Prvi, j obtained using the iterative algorithm are indicated on the contours. The higher the probability of proceeding straight at the intersections,, the more likely the MS is to arrive at the destination with fewer turns Calculation of N k Consider the LR area shown in figure 1. Assume the shortest distance model for the MS mobility inside the LR area until the MS exits the LR area. The cell i, j onthe boundary of the LR area is reached after traversing i + j cells inside the LR area, where the LU cell in figure 1 is cell 0, 0. The probability that cell i, j is the last cell the MS visits before exiting the LR area A is the probability that cell i, j + 1 is visited from the S direction or cell i+1, j is visited from the W direction. Thus Prei, j = Si, j W i + 1, j fori,j, shown in figure 1. Prei, j for any cell i, j LR area A, i.e., x ij = 1, is the probability that the MS moves from inside the LR area to any cell outside the LR area, thus, Pr ei, j = Si, j + 1x i,j+1 + W i + 1, jx i+1,j + Ni, j 1x i,j 1 + Ei 1, jx i 1,j, 9 where { 0, if xij = 1, x ij = 1, if x ij = 0. Therefore, for any cell i, j Pr ei, j = x ij Si, j + 1xi,j+1 + W i + 1, jx i+1,j

7 A. Abutaleb, V.O.K. Li / Location update optimization in PCS Ni, j 1x i,j 1 + Ei 1, jx i 1,j. 10 When the MS exits the LR area from cell i, j, the number of cells traversed in the LR area is i α + j β + 1, where α, β are the LU cell coordinates. Therefore N k = i,j i α + j β + 1 Pr ei, j, 11 where Prei, j is given by equation LR area optimization problem We have defined the vehicular mobility model in section 4.1, and have formulated algorithm 1 to calculate the probability of visiting a cell i, j from the E, W, S and N directions. Our objective is to find the optimal LR area shape that would minimize the rate of updates equation 2, or, in other words, maximize the average number of cells traversed in the LR area, N k equation 11, for a particular MS, subject to a constraint on the paging cost. The problem can be formulated in the following form: Problem 1. Maximize N k X = subject to: m i=1 j=1 m i=1 j=1 m c ij X 12 m x ij = k, 13 where k is the number of cells in LR area, and N k is the expected number of cells traversed in an LR area of size k. N k is a function of X equations 10 and 11. Associated with every cell i, j isacost c ij X = i α + j β + 1 Pr ei, j, where Prei, j is a function of X itself, as shown in equation 10. Once an incoming call arrives to the MS, the cost of paging is dependent on the number of cells paged in the LR area. In the conventional scheme all the BSs in the LR area broadcast the paging call, resulting in the paging cost constraint shown in 13. Proposition 1. Problem 1 is NP-complete. Proof. The proof is by restriction. The problem formulated by equations 12 and 13 can be reduced to the conventional Knapsack problem by restricting c ij X = c ij, independent of X. Thus the Knapsack problem is a special case of the LR area optimization problem. Since the Knapsack problem is NP-complete [5,12], problem 1 is also NP-complete Method of solution Exact solution to the Knapsack optimization problem have very high time complexity. Approximate solutions which give up accuracy in exchange for efficiency can be used [5,12]; but the cost c ij has to be defined at the beginning of the problem for each cell i, j. However, problem 1 is more complex because the costs c ij X are not only a function of x ij, but of all other cells, p, q, that lie on the shortest paths from the LU cell to cell i, j and of their x pq values. Thus we resort to an iterative greedy heuristic to find the approximate solution. The heuristic starts with an LR area containing the LU cell only, then at each iteration, the cell on the perimeter of the current LR area providing the maximum increase in the number of cells traversed in the LR area is added to the LR area. The algorithm continues until the LR area contains k cells. Algorithm 2. Define the following: O an ordered list, elements are added to the end of the list; X = m m matrix, with elements x ij ; max N n X the value of the maximized N n X found by the algorithm, for LR area containing n cells; BX a set containing all the cells just outside the boundary of the LR area defined by X, i.e., the cells on the perimeter. 1. Initialize all the variables: n = 1; max N n X = 1; O = {α, β}, where α, β is the LU cell; X = 0; x α,β = n = n BX = {i, j x ij = 0andx i+1,j = 1orx i 1,j = 1orx i,j+1 =1orx i,j 1 =1}. 4. For cell i, j BX Let x ij = 1; if N n X > max N n X max N n X = N n X; x new, y new = i, j; x ij = Repeat step 4 i, j BX. 6. O x new, y new ; x xnew,y new = Go to step 2 until n = k. Note that the list O contains the cells in the LR area in the order that they were added by the algorithm. So the first n elements of the list O, n k, are the coordinates of the cells in the optimal LR area containing n cells. Figure 6 shows examples of the optimum LR areas obtained using algorithm 2 for an MS entering cell 51, 51 from the W direction. The contours shown are the boundaries of the LR area, and labeled on each contour is the size

8 212 A. Abutaleb, V.O.K. Li / Location update optimization in PCS Figure 6. Examples of optimum LR area shapes. of the area, in number of cells that it encompasses. As increases the LR area becomes more directional as shown. Mobility tracking of an MS requires the system and the MS to agree on the exact shape and dimensions of the LR area so that 1 when an incoming call for the MS arrives to the system, the system would page the cells in the LR area that the MS is currently registered in; 2 when the MS moves between cells, the MS makes a decision whether or not to update based on whether the new cell is inside the LR area or not. The LR area shapes shown in figure 6 depend on the MS mobility pattern and their sizes are a function of the incoming call rate. The larger the rate the smaller the LR area size required to keep the paging cost per unit time the same equation 1. The mobility pattern for a particular MS is not constant but depends on the traffic characteristics in that region section 4.1. Thus in one region the MS may have a mobility pattern characterized by = 0.6 and P r = P l = 0.2, and in another region the mobility characteristics may be characterized by = 0.9 andp r = P l =0.05, and as seen from figures 6a and 6d the LR area shapes for the same paging constraint are different. The MS and the system have to agree on the LR shape and size. One way to do this is for the system and the MS to store different numbered lists O which represent various mobility models. When the MS performs an LU the system specifies the number for the list O representing the mobility model in that area, and the size of the area. This completely specifies the LR area shape, as generated by algorithm 2. The overhead is the amount of storage required to store tables of sets O. Another solution is for the system to download the set O to the MS when the MS performs the LU; however, this is very wasteful of radio bandwidth. A third solution is to approximate the irregular LR area shapes obtained by algorithm 2, using regular shapes that are easier to specify at the time the

9 A. Abutaleb, V.O.K. Li / Location update optimization in PCS 213 Figure 7. Examples of optimum rectangular LR area shapes. MS performs the LU. In the next section we compare the performance of optimal rectangular shapes to the optimum irregular shapes obtained in this section. 7. Optimal rectangular shapes As seen from figure 6 the rectangular shapes might be a good approximation to the optimum LR area shapes obtained using the greedy heuristic algorithm. To find the optimal rectangular LR area shapes we state the optimization problem as follows: Problem 2. Maximize N k X = m i=1 j=1 m c ij X 14 subject to: x ij = 1 i {i,...,i +L 1}, i {1,...,m L+1}, j {j,...,j +W 1}, j {1,...,m W +1} and LW = k. 15 The number of solutions to the above optimization problem is not large, so the optimal solution can be found by complete enumeration. The length and width of the optimum rectangular shapes for the mobility patterns shown in figure 5 are shown in figure 7. The zigzagging in length and width is due to the integer optimization problem and constraints. We now compare the LR areas found using the greedy heuristic algorithm 2 for problem 1, the optimal rectangular shapes of problem 2, and square LR areas which use the fluid model of mobility, where each MS s direction of movements is uniformly distributed over [0, 2π]. Since the fluid model of mobility averages the mobility pattern of all MSs and assumes uniformly distributed movements over [0, 2π], the optimum position of the MS would be at the center of the square at the time of LU [8]. The criteria used for comparison is the number of cells traversed inside the LR area for the same paging constraint. As shown in figure 8 the performance of the LR areas obtained by solving problem 1 and problem 2 are comparable, while the performance of the square LR area with the MS at the center of the area at the time of LU is much worse. This shows the importance of considering the direction of travel and the mobility pattern for a specific area and MS. Thus

10 214 A. Abutaleb, V.O.K. Li / Location update optimization in PCS Figure 8. Performance comparison of irregular and rectangular shapes. the optimal rectangular LR areas are a good approximation to the irregular LR areas of problem 1. This greatly simplifies the LU procedure. So when an MS performs an LU the system can just inform the mobile terminal of the LR area boundaries by specifying the length, width, and the position of the LU cell in the rectangular area. This completely specifies the LR area for an MS with a known incoming call rate and mobility pattern. 8. Example In this section we introduce an example of the optimization of the total cost of signaling on the control channel for a given MS. Consider the following example of the Manhattan City model, with the BSs located at the intersections of the streets: The distance between the BSs on the grid is L = 300 m. The mean and standard deviation of the speed of vehicular users in inner city are given by v = m/s and σ v = 2.19 m/s, respectively. At each intersection, there is a traffic light with green cycle of duration g = 30 s and red cycle of duration r = 30 s. The average arrival rate for calls for that customer is µ a calls/hr or µ a /3600 calls/s. The mobility characteristics of that region are given by = 0.8 andp r =P l =0.1. µ a C p /C u = γ.

11 A. Abutaleb, V.O.K. Li / Location update optimization in PCS 215 All cells have equal probability of being the destination of the MS. For a network of size 400 cells, the probability of any cell being the destination cell is P dest = 1/400. The mean time spent in the destination cell is T dest = s. Using the equation of total cost equation 1 [14] Ck, µ a = k µ a 3600 C p + u k C u. Normalizing C u to1andck,µ a /C u = ck, µ a, then ck, µ a = kγ u k. The average time a user spends moving in a cell can be approximated by [9] T i L v + 2Lσ2 v 2v 3. The probability that an MS is delayed at an intersection is P d = r/r + g. The average delay for an MS delayed at an intersection is T D = r/2. Thus the average delay for any MS is T d = r 2 P r 2 d = 2r + g. Therefore the average time an MS resides in a cell is X i = T i + T d 1 Pdest + T dest P dest = 1 u 1. From equation 2, u k = u 1 1 =. N k X i N k Therefore, the normalized cost ck, µ a = γk X i N k Using the optimal rectangular LR shapes, the maximized values of N k for different constraint sizes, shown in figure 7c, are used to find the optimal values of N k and k that minimize the normalized cost ck, µ a. Figure 9 shows the normalized cost as a function of the number of cells in the LR area k for various values of γ. Table 1 gives the optimal values of k, W and L that minimize the normalized cost ck, µ a for various values of γ. As shown from table 1 the optimum area sizes decrease as γ or the incoming call rate increases. For typical values µ a = 0.5 call/hr and C p /C u = 0.1, the optimum LR area size is 63 cells, as the calling rate increases to µ a = 1 call/hr Table 1 Optimum LR area dimensions and minimum normalized cost. γ c mink, µ a k opt W opt L opt N k Figure 9. Normalized cost function versus constraint size. the optimum LR area size decreases to 13 cells. Thus the LR area sizes for vehicular MS in a city environment are small. Within a small LR area, the MS mobility pattern is more likely to follow the shortest distance path within the LR area. So the shortest distance model is a good model to use for vehicular MS in a microcellular urban environment. 9. Conclusions In this paper we focused on the design of the dynamic LR areas that minimize the rate of LU, subject to a constraint on the number of cells contained inside the LR area. The dynamic LR area design exploits information about the mobility pattern of the MS to maximize the number of cells traversed by the MS inside the LR area, before performing an LU. We introduced the shortest distance model for vehicular MS mobility patterns, which is more realistic than the i.i.d model usually used for pedestrian users. Using the shortest distance model we formulated the LR optimization problem and proved that it is NP-complete. Approximate solution to the LR area optimization problem was obtained using a heuristic greedy algorithm. The shapes obtained by the heuristic greedy algorithm were found to be irregular and therefore will be difficult to implement. So we solved the LR optimization problem for rectangular shapes and found that the optimum rectangular LR areas are very good approximation to the irregular shapes found by the greedy algorithm. The dimensions of the optimum rectangles vary with the constraint size and the mobility pattern. References [1] A. Abutaleb and V.O.K. Li, Paging strategy optimization in personal communication systems, Wireless Networks, this issue. [2] A. Bar-Noy and I. Kessler, Tracking mobile users in wireless communications networks, in: Proceedings IEEE INFOCOM 1993 pp

12 216 A. Abutaleb, V.O.K. Li / Location update optimization in PCS [3] A. Bar-Noy, I. Kessler and M. Sidi, Mobile users: To update or not to update?, in: Proceedings IEEE INFOCOM 1994 pp [4] S. Chia, The universal mobile telecommunication system, IEEE Communications Magazine [5] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness Freeman, San Francisco, CA, [6] D.J. Goodman, G.P. Pollini and K.S. Meier-Hellstern, Network control for wireless communications, IEEE Communications Magazine [7] U. Madhow, M.L. Honig and K. Steiglitz, Optimization of wireless resources for personal communications mobility tracking, in: Proceedings IEEE INFOCOM 1994 pp [8] S. Okasaka and S. Onoe, A new location updating method for digital cellular systems, in: Proceedings IEEE Vehicular Technology Conference 1991 pp [9] A. Papoulis, Probability and Statistics Prentice-Hall, Englewood Cliffs, NJ, [10] H. Persson, Microcellular structures and their performance, in: Proceedings IEEE Vehicular Technology Conference 1992 pp [11] R. Steele, Mobile Radio Communications Pentech Press, [12] K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity Prentice-Hall, Englewood Cliffs, NJ, [13] R. Thomas, M. Mouly and H. Gilbert, Performance evaluation of the channel organization of the european digital mobile communication system, in: Proceedings IEEE Vehicular Technology Conference 1988 pp [14] H. Xie, S. Tabbane and D.J. Goodman, Dynamic location area management and performance analysis, in: Proceedings IEEE Vehicular Technology Conference 1993 pp Ahmed Abutaleb graduated from the University of Southampton, England, with First Class Honors in He obtained his M.Sc. from the University of Southern California USC in 1992, and is pursuing a Ph.D. at USC. His research interests include traffic engineering and mobility management in wireless communication networks. He is a Member of Technical Staff at Lucent Bell Laboratories. Victor O.K. Li was born in Hong Kong in He received his SB, SM, and ScD degrees in electrical engineering and computer science from the Massachusetts Institute of Technology, Cambridge, Massachusetts, in 1977, 1979, and 1981, respectively. Since February 1981 he has been with the University of Southern California USC, Los Angeles, California, where he is Professor of Electrical Engineering and Director of the USC Communication Sciences Institute. His research interests include high speed communication networks, personal communication networks, distributed multimedia systems, distributed databases, queueing theory, graph theory, and applied probability. He has lectured and consulted extensively around the world. Dr. Li chaired the Computer Communications Technical Committee of the IEEE Communications Society , and the Los Angeles Chapter of the IEEE Information Theory Group He was the Co- Founder and Steering Committee Chair of the International Conference on Computer Communications and Networks IC 3 N, General Chair of the 1st Annual IC 3 N, June 1992, Technical Program Chair of the Institution of Electrical Engineers IEE Personal Communication Services Symposium, June 1995, and Chair of the 4th IEEE Workshop on Computer Communications, October Dr. Li has served as an editor of IEEE Network and of Telecommunication Systems, guest editor of IEEE JSAC and of Computer Networks and ISDN Systems, and is now serving as an editor of Wireless Networks. He serves on the International Advisory Board of IEEE TENCON 90, IEEE TENCON 94, IEEE SICON 91, IEEE SICON 93, IEEE SICON/ICIE 95, the International Conference on Microwaves and Communications 92, and the International Symposium on Communications 91. He is a member of ACM was elected an IEEE Fellow in 1992.