c British Computer Society 2002
User Tracking and Mobility Management Algorithm for Wireless Networks I RINA KOZATCHOK
AND
S AMUEL P IERRE
Mobile Computing and Networking Research Laboratory (LARIM), Department of Computer ´ Engineering, Ecole Polytechnique de Montr´eal, C.P. 6079, succ. Centre-ville, Montr´eal, Qu´ebec, Canada H3C 3A7 Email:
[email protected] This paper presents an algorithm for minimizing the cost associated with the management of users’ mobility in mobile communications networks. This algorithm allows one to determine the optimal size of a location area and to find the optimal cellular grouping model in polling regions. It guarantees the global minimum of the total cost function, while respecting the pre-established delay constraints. It takes into account the average probabilities of received calls, movements, updates and paging costs. This algorithm also allows one to avoid the problem of evaluating the cost function in a large number of points, which makes it usable on machines with limited computing power. Received 4 April 2001; revised 5 February 2002
1. INTRODUCTION When a mobile unit receives a call, the network must find its location in a reasonable time. To meet this requirement, the network must use efficient strategies to track the callers. To minimize the tracking process costs, several cells are grouped to form a location area [1]. In this context, the tracking process consists of two main procedures: the location update and the paging of the mobile unit [2, 3, 4, 5, 6]. The update process is carried out every time a mobile unit changes location area. It allows a network to track the approximate location of each unit at any time. In order to establish communications, when a call is being placed, the network begins paging, to locate the base station serving the mobile unit. To support communications among users, mobile networks use a very narrow bandwidth, with very high associated cost [7, 8]. The management of users’ mobility requires the use of the same bandwidth. The growth in mobile users causes network overload. In order to optimize resources and procedure costs, one has to find efficient management strategies [9, 10, 11, 12], such as: (1) determining the size of the location areas; (2) elaborating a model allowing for the grouping of cells into location areas; and (3) choosing a paging strategy. There are several models for grouping cells into location areas [1, 13]. In order to minimize the search cost, the location area could be divided into several regions. This procedure allows one to sequentially search the mobile unit, while respecting a pre-established delay threshold [13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Using the optimal size of a location area and by optimizing dividing T HE C OMPUTER J OURNAL,
the location area into several search regions, we limit our investigation to the elaboration and implementation of an algorithm which allows the minimization of the total cost of searching and updating. The total cost function is defined on the set of positive integers. Thus, this function must have several local minima, which makes its optimization even more difficult. Furthermore, there are several other strategies for subdividing a location area into several search regions. Traditional optimization methods, such as the gradient method, cannot be used to solve this problem, as they require multiple evaluations of the total cost function. In our perspective, a good method must be able to guarantee a satisfactory solution after a reasonable number of evaluations of the function. The algorithm we propose evaluates the total cost function at a limited number of points for which certain desired properties are guaranteed, albeit with some rounding errors. Moreover, this algorithm must respect certain efficiency criteria for calculating cost. For example, it should be able to use machines with limited computing power. In this paper, we analyze the principles of tracking mobile users. We identify the components constituting the total cost associated with mobility management. Finally, we propose an algorithm to find the optimal cell grouping and to optimize the size of a location area so as to minimize the total mobility management cost while respecting the established delay constraints. The rest of this paper is organized as follows. Section 2 describes the tracking principles as well as the most current updating strategies. Section 3 presents a summary of the Vol. 45, No. 5, 2002
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tracking model proposed by Ho and Akyildiz [3]; this model serves as a background to previous work and improvements referred to in this paper. Section 4 proposes a new algorithm for managing users’ mobility; this algorithm is based on the search for an optimal grouping of cells. Section 5 analyzes the simulation results of the proposed management model. 2. BACKGROUND AND RELATED WORK In mobile communications systems, mobile units are free to move within the service areas and therefore their point of attachment to the mobile network continuously changes [1]. When a user places a call, the mobile network must undertake numerous tasks including the identification of the caller, the search for the unit called and the routing of the call to the base station which serves the geographic area where the call is destined. When a user A calls a user B to initiate communication between A and B, the location problem consists of searching the location of B within a reasonable delay time. This procedure is known as user tracking. 2.1. Tracking principles There are different strategies for tracking mobile units in a wireless network [2, 3, 9, 23]. These strategies are divided into two groups: passive strategies and active strategies. Passive strategies register a mobile unit’s location in the network during the update operation, which is achieved by taking into account the last record of the unit’s location. Active strategies predict the movement of a mobile unit based on patterns of movements drawn from the user’s profile. Each type of strategy could be divided into two groups: strategies based on home location servers and those independent of the home location server. The latter assigns to a user a unique and lifetime number, regardless of the location of the user’s subscription in the network. In systems such as GSM and IS-41 [14], each user has a database called the Home Location Register (HLR) stored in his home location server, which keeps his current position in a user profile. When user A calls user B, the mobile network sends a request to user B’s HLR to determine the location and initiate the call. The drawback of this system is the slowness of the exchange procedure through which a remote database is looked up. To overcome the delay in searching for the user who is meant to receive the call, the network installs the Visitor’s Location Register (VLR) in each area. The VLR keeps copies of users’ profiles, located outside of the home location area. The search for a user in this HLR/VLR scheme consists of verifying the VLR registers before consulting the HLR. 2.2. Update strategies Location update strategies could be divided into two groups: static and dynamic [9]. According to static strategies, the network decides when a mobile unit must carry out the update. According to dynamic strategies, the mobile unit decides when to update. T HE C OMPUTER J OURNAL,
Among static strategies one can distinguish the following: strategies based on the notion of location area; those based on the notion of overlapping location areas; and others based on the notion of reporting centers. Thus, the network decides when and where mobile units must carry out an update. Consequently, all mobile units carry out the update of their location in the same cells. A location area-based strategy is the simplest static update strategy. However, when a mobile unit moves close to the border among the location areas, this strategy exhibits a major problem: the risk that a mobile unit overloads the network with update messages every time the border among the location areas is crossed. The strategy based on overlapping location areas allows this problem to be overcome. When this strategy is adopted, the mobile unit could be part of several location areas, which necessitates an efficient strategy for allocating location areas to mobile units. The strategy based on the concept of the reporting center is very similar to the strategy based on overlapping location areas. One could consider this strategy as a particular case where the cells forming the location area’s border become reporting centers. Dynamic strategies can be distinguished in terms of whether they are time-based, movement-based or distancebased updates. Within these schemes, the location update is performed based on the time passed, the number of movements carried out and the distance covered since the latest update, respectively. According to these dynamic strategies, the mobile unit must decide itself where and when to carry out an update of its location on the network. The time-based update scheme has two major disadvantages. (1) The mobile unit must report its location in the network, even if the user has not moved since the latest update, the effect of which is a waste of network resources. (2) Considering the probability that the speed of a mobile unit’s movement could vary in time, it is difficult to identify the set of cells that could be reached in a given time. Therefore, there is a need for an algorithm capable of predicting the scope of cells that a mobile could reach in a given time interval. This has the effect of increasing the complexity of the search procedure. The movement-based update scheme could also generate a network overload in the case of a repetitive movement between two neighboring areas or in the case of a circular movement. In both cases, the mobile unit does not change network location, because it always returns to its starting point. However, it must report its location, thereby creating a waste of resources. Among all dynamic schemes, the most cost-effective and easy to manage is the distance-based scheme. According to this scheme, the mobile unit reports back only after having covered a distance D since its latest update. If, after having accomplished consecutive movements, it remains at a distance lower than D, it does not need network resources to update its location. Thus, the search procedure is easy to manage, considering the certainty of finding the mobile Vol. 45, No. 5, 2002
U SER T RACKING AND M OBILITY M ANAGEMENT A LGORITHM unit (since its latest update) in a scope of cells within radius (distance?) D. 3. TRACKING MODEL UNDER DELAY CONSTRAINTS Ho and Akyildiz [3] have proposed a management strategy using a distance-based update scheme in association with a search scheme that guarantees a limited maximum search delay m. According to this scheme, rings of cells surround a particular network cell, denoted as the 0 cell. Cells number 1 form the first ring, cells number 2 form the second ring, etc. The mobility of users corresponds to the discrete random movement model according to which, at each given instant t, the user moves in one of the neighboring cells with probability q or remains in the current cell with the probability 1 − q. Entering calls could arrive at each given instant t with probability c. The Markov chain model is used to represent the mobility of the unit and the arrival of calls. d denotes the updating distance. Transitions from state to state describe the movements of a unit outside of the current cell. The transition from a given state towards state 0 describes the arrival of a call or the update of a location by the unit after having covered a distance d. At the arrival of a call, the network determines the location of the unit through paging, which restores the central cell in the cell hosting the searched unit. U and V represent respectively the costs associated with the update and paging. The location area is divided into regions made of one or several rings, so that each region must be searched for the mobile unit. Thus, the total update and paging cost is CT (d, m) = Cu (d) + Cv (d, m)
(3.1)
where Cu and Cv are the total update and paging costs, respectively [3]. Ho and Akyildiz [3] have considered two mobility models: a one-dimensional (1-D) model and a twodimensional (2-D) model. In the 1-D model, the geographic space of a network is divided into cells of the same size, each having two neighboring cells. In the 2-D model, the space is divided into hexagonal cells of the same size, each having six neighboring cells. The balance equations issued from the Markov chain model allow exact computation of the steady-state probabilities for the 1-D model. To obtain closed-form expressions similar to those of the 1-D model, Ho and Akyildiz [3] have used approximate state transition probabilities.
the closed analytical equations allow approximate results to be obtained, while recursive methods allow exact results to be achieved. Using the recursive computation method in order to compute precise probabilities of all states pi,d of the Markov chain model introduced in [3], we define the non-normalized probability pd,d to be equal to 1 (100%). This probability serves to recursively compute the other non-normalized in a reverse order, 0 ≤ i ≤ d − 1, using the probabilities pi,d balance equation for the Markov chain model in [3]. Once all the probabilities i,d are achieved, we divide each of them p d in order to normalize and compute by the amount i=0 pi,d the precise probabilities pi,d . If we want to use closed analytical formulae to compute the approximate probabilities pi,d , we have to proceed by regularization and asymptotic expressions. Simulation results have shown that, for large enough values of d (d 1), the problem of computing pi,d is unstable, i.e. there is a correspondence between the finite growth of an argument and the infinitely large variation of the function due to computational errors and the level of precision granted by the computer. To achieve good results, one has to use appropriate regularization methods. In the context of this paper we propose the following procedure: (1) reduction of the common multipliers in both the numerator and denominator to eliminate singularities of types 1/0 and 0/0; (2) substitution of values of type Ri /Rj and Si in the steady-state equations [3] by a polynomial of the form i = 1 + ε + ε2 + · · · + εi ,
To compute the probability of each ring, we propose to use closed analytical equations and a recursive method derived from the balance equations of the Markov chain model proposed by Ho and Akyildiz [3]. For the 1-D model, both methods give identical results. For the 2-D model, T HE C OMPUTER J OURNAL,
i≥0
(3.2)
with the reduction and grouping of similar terms; (3) substitution of values of type Ri /Rj and Si (i, j 1) with their asymptotics such as ∞ = 1 + ε + ε2 + · · · + εi + · · · = (1 − ε)−1 , ε → 0, d → ∞.
(3.3)
Thus, we could obtain the following final expressions (see Appendix A for more details): A0 pd,d W A1 pd,d = W A2 pd,d = W = K1 W/(K2 B1 + K3 B2 + K4 B3 + 3W )
p0,d =
(3.4)
p1,d
(3.5)
p2,d pd,d
3.1. The proposed algorithm for calculating steady-state probabilities
527
(3.6) (3.7)
pi,d = (Si−2 A2 − Si−3 A1 )pd,d /W = e1i−2 (i−2 A2 − e1−1 i−3 A1 )pd,d W,
(3.8)
for i = 2, 3, . . . , d − 1. These expressions serve as a basis to compute pi,d (i = 0, 1, . . . , d). For sufficiently huge values of d (d ≫ 1), we can use the asymptotic expressions (see Appendix B for Vol. 45, No. 5, 2002
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FIGURE 1. The probability distribution using precise and approximated computation. 2-D model for q = 5%, U = 900, V = 10 and: (a) c = 0.1%; (b) c = 1%; (c) c = 5%.
more details) and transform expressions (3.7) and (3.8) as follows: pi,d ≈ pd,d ≈
e2i−2 A2 pd,d /W, 3 K1 (1 − ε)2 e2d−3
≤i ≤d −1
K2 + K3 /e1 + K4 /e12
.
4. THE PROPOSED MOBILITY MANAGEMENT ALGORITHM
(3.9)
If we fix the parameters c, q, U and V of the function, we obtain the following problem:
(3.10)
Min CT (d) by choice of d ∈ N
The approximate solution applied in [3] for the 2-D model provides an error in the computation of the probability pi,d , thereby distorting the interpretation of all the values of c and q, as illustrated in Figure 1. For the values of c q, the error is greater (see Figure 1(a)). This error decreases when c increases, as illustrated in Figures 1(b) and 1(c). The computational error of the total cost could reach 42% of the precise solution. T HE C OMPUTER J OURNAL,
(4.1)
where d denotes the updating distance. Ho and Akyildiz [3] have considered this problem and have proposed the method of grouping of cells in fixed paging regions. Simulation results have shown that such a cell grouping method is not optimal. Here, we propose a more appropriate formulation to the problem of optimizing the cost function. The proposed algorithm is used to search for an optimal cell grouping. It allows a global minimum for each given value of d and m to be achieved. The problem Vol. 45, No. 5, 2002
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FIGURE 2. Total cost curves for m = 3, q = 5%, V = 10 and: (a) c = 5%, U = 1000, 1-D; (b) c = 0.5%, U = 500, 2-D; (c) c = 1%, U = 100, 2-D.
of minimizing the cost function according to the new formulation is transformed into a new minimization problem of the function CT related to the scalar d and vector r: Min CT 0 (d, r(d, m)) by choice of d ∈ N and r ∈ R (4.2) where d denotes an updating distance, m the maximal paging delay and r a combination of grouping of cells. Thus, the iteration of each external cycle performing the search for the optimal distance carries out an internal search for an optimal grouping in order to achieve a monotonous convergence towards the case of unbounded delay. In so doing, the search for the optimal distance d is related to the search for the optimal grouping of cells in paging regions, T HE C OMPUTER J OURNAL,
because in the case of each distance d there exists an optimal grouping minimizing the total cost of managing mobility. The search for the optimal grouping for d ∈ N allows a cost function that only has a single minimum to be obtained, avoiding the problem of local minima encountered by Ho and Akyildiz [3]. The case of unbounded delay corresponds to the lowest costs of mobility management, which must take into account certain delay constraints. The solution is to find an efficient algorithm that insures the monotonous convergence of the cost function towards the unbounded case, when the paging delay increases. Numerical simulations show that the proposed algorithm satisfies this condition, integrating Vol. 45, No. 5, 2002
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FIGURE 3. Average total cost as a function of call probability for the 1-D model with q = 5%, U = 100 and V = 1.
external and internal cycles. The external cycle searches for the optimal d corresponding to the minimal cost. For the iteration of each external cycle, the algorithm optimizes the set of vectors r ∈ R. Since the total cost function is defined on natural numbers, this problem cannot be solved by gradient methods. The most appropriate method is selection. It is, therefore, important to limit the number of iterations of the internal cycle. One of the methods of reducing the number of iterations consists of choosing only suitable combinations of cell grouping methods. Based on numerical results of simulation, we have selected a limited number of suitable combinations of groupings of cells. The idea consists of selecting among the different grouping of the last five components of the vector r, taking into account that the first components do not affect the result of optimization. For a maximal paging delay m higher than 5, each of the first paging regions includes one ring. For T HE C OMPUTER J OURNAL,
example, for m = 7 and for d = 20, the initial vector takes the form [0 1 2 3 4 5 20]. Each of the first six paging regions contains one ring, while the rings from 6 to 20 constitute the last region. The algorithm reaches the optimum by modifying the configuration of the last five paging regions and by leaving the first two unchanged. Thus, the vector r could take the forms: [0
1 2
3 4
6 20]
[0
1 2
3 5
6 20]
[0
1 2
4 5
6 20]
··· . A special procedure has been designed to generate the matrix, which serves the basis to test different groupings. Vol. 45, No. 5, 2002
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FIGURE 4. Average total cost as a function of call probability for the 2-D model with q = 5%, U = 100 and V = 1.
The size of the four-column matrix is around 1800 lines. The three-, two-, and one-column matrices have 455, 91 and 13 lines, respectively. During the search for the optimum, the maximal paging delay m should be fixed. In certain cases, the optimal d could be lower than the paging delay m. Consequently, each optimization iteration could request a suitable matrix to calculate the optimal grouping. The reduction of the number of iterations in the case of both models is achieved through the application of an interval division method, which has the following steps. (1) Searching the optimal distance for the unbounded case d ∗ . When the cost function of the unbounded case reaches its maximum at the interval of the unchanged part, we take for d ∗ the minimal value of d in this interval. (2) If the optimal distance d ∗ is such that 0 ≤ d ∗ ≤ m − 1, this distance is the searched optimal distance; otherwise, T HE C OMPUTER J OURNAL,
divide the interval [m − 1, d ∗ ] to find two points situated near the middle between d1 and d2 , (ii) evaluate the value of the total cost function between these two points, (iii) if CT (d1 ) < CT (d2 ), then apply the same algorithm on the interval [m − 1, d1 ], otherwise apply it on the interval [d2 , d ∗ ].
(i)
The search for the optimal distance for CT 0 is carried out in a few iterations by starting from d ∗ . Such a modification reduces the optimization problem to a few iterations of the external cycle and considerably reduces the number of iterations of the internal cycle. Figure 2 illustrates the behavior of the total cost curve achieved after the application of this method. When the curve reaches its minimum at the starting point, it gradually increases. The total cost curves have been drawn for the typical values of the parameters c, q, U , V and m. The comparison of these curves shows that the total cost Vol. 45, No. 5, 2002
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FIGURE 5. Average total cost as a function of motion probability for the 1-D model with c = 1%, U = 100 and V = 1.
curve achieved by applying the proposed method is closer to the unbounded case curve, while insuring a lesser cost than the curve achieved by applying the Ho and Akyildiz scheme [3]. Simulation results have shown that:
5. ANALYSIS OF NUMERICAL RESULTS
(1) the optimization of the vector at each iteration allows a cost function that gradually decreases towards the minimum to be achieved; (2) after the achievement of the minimum the function grows more rapidly, but stays near the total cost of the unbounded case; (3) at each delay m, the algorithm could guarantee a minimal cost because the decrease is gradual without local minima; and finally
We first present the numerical results for the 1-D model by assuming that c = 3%, q = 5%, U = 1000 and V = 10. We considered three maximum paging delay bounds m (three, five and seven paging cycles). For the unbounded case, the optimal distance du∗ = 31 was found after 33 evaluations of the function CT (du∗ ). The value of the function at this point was CT (du∗ ) = 0.780. For m = 3, the function CT 0 was evaluated at the point du∗ = 31 and was equal to 1.3561. After the evaluation of the function
T HE C OMPUTER J OURNAL,
(4) the algorithm guarantees the solution of the minimization problem in a reasonable time.
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FIGURE 6. Average total cost as a function of motion probability for the 2-D model with c = 1%, U = 100 and V = 1.
CT (du∗ ) = 1.1279, we achieved the minimum of the function at the point d ∗ = 6:CT (du∗ ) = 1.1129. The results were obtained after eight internal cycles, each of 91 iterations. For m = 5, the function CT 0 was evaluated at the point du∗ = 31 and was equal to 0.8796. The minimum of the function CT (du∗ ) = 0.8211 was found at the point d ∗ = 8. The results were obtained after eight cycles, each of 1800 iterations. For m = 7, the function CT 0 was evaluated at the point du∗ = 31 and was equal to 0.7922. The minimum of the function CT 0 (du∗ ) = 0.7854 was found at the point d ∗ = 10. The results were obtained after eight internal cycles, each of 1800 iterations. In presenting the numerical results for the 2-D model, it is assumed that c = 0.5%, q = 10%, U = 1000 and V = 10. We considered two maximum paging delay bounds m (three and five paging cycles). For the unbounded case, the optimal T HE C OMPUTER J OURNAL,
distance du∗ = 8 is found after 10 evaluations of the function CT (du∗ ). The value at this point was CT (du∗ ) = 3.2791. For m = 3, the function CT 0 was evaluated at the point du∗ = 8 and was equal to 4.5067. The minimum of the function CT 0 (d ∗ ) = 4.1261 was found at the point d ∗ = 6. The results were obtained after four internal cycles, each of 91 iterations. For m = 5, the function CT 0 was evaluated at the point du∗ = 8 and was equal to 3.7531. The minimum of the function CT 0 (d ∗ ) = 3.6428 was found at the point d ∗ = 7. The results were achieved after three internal cycles, each of 1800 iterations. Figure 3 shows the average total cost achieved for the 1D model, when the probability c of receiving calls varies from 0.1 to 10%, the probability of movement q is fixed to 5%, the update cost U is 100 and the paging cost V is 1. For the 2-D model, we have used the probabilities obtained Vol. 45, No. 5, 2002
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I. KOZATCHOK AND S. P IERRE TABLE 1. Optimal distance and average total cost for the 1-D model. m=1 U
d∗
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000
0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 4 4 5 5 5 5 6 6
m=2
CT
d∗
0.150 0.200 0.250 0.300 0.350 0.368 0.380 0.391 0.402 0.414 0.527 0.630 0.673 0.716 0.760 0.803 0.846 0.878 0.897 1.095 1.193 1.290 1.351 1.401 1.451 1.501 1.537 1.563
0 0 1 1 1 1 1 1 1 1 1 2 3 3 3 3 3 4 4 5 6 7 7 7 8 8 8 8
m=3
CT
d∗
0.150 0.200 0.225 0.236 0.248 0.259 0.270 0.282 0.293 0.305 0.418 0.464 0.486 0.506 0.526 0.545 0.565 0.579 0.589 0.676 0.724 0.746 0.760 0.774 0.786 0.794 0.801 0.809
0 0 1 1 1 1 1 1 2 2 2 2 3 3 3 3 3 4 4 6 7 7 8 8 8 9 9 10
m=5
CT
d∗
0.150 0.200 0.225 0.236 0.248 0.259 0.270 0.282 0.291 0.296 0.339 0.382 0.415 0.435 0.454 0.474 0.494 0.505 0.514 0.548 0.565 0.579 0.587 0.594 0.602 0.606 0.610 0.613
0 0 1 1 1 1 1 1 2 2 3 3 4 4 4 4 5 5 5 7 8 8 9 9 9 11 11 12
using the exact computing methods described in the previous section. Figure 4 presents the cost results. We note that the average total cost increases as the probability of receiving calls c increases. On the other hand, the total cost decreases as the maximum paging delay m increases. It should be noted that, for all the values of probability c from 0.1 to 10%, the maximum paging delay m = 7 provides results similar or close to the unbounded case for both models. This means that the maximal delay m = 7 is sufficient to achieve the optimal or close to optimal total cost. Figure 5 shows the behavior of the total cost according to the 1-D model. The probability of movement q varies from 0.1 to 50%. The probability of receiving calls c is set to 10% and the update and paging costs are 100 and 1, respectively. We present results for m = 1, 2, 3, 5 and 7. It should also be noted that the increase in the maximal paging to seven cycles gives results near the optimal total cost achieved in the unbounded case. Figure 6 illustrates the same results for the 2-D model. In effect, when m = 7, as for the 1-D case, the total cost is near the optimal cost achieved in the unbounded case. However, the maximal delay m = 5 already gives results that are very close to the optimal for 1% ≤ q ≤ 5%. In both cases, we note that the average cost increases with an increase in the probability of movement q. The highest cost T HE C OMPUTER J OURNAL,
m=7
CT
d∗
0.150 0.200 0.225 0.236 0.248 0.259 0.270 0.282 0.291 0.296 0.338 0.357 0.371 0.381 0.391 0.400 0.406 0.411 0.416 0.443 0.455 0.462 0.467 0.471 0.475 0.477 0.478 0.479
0 0 1 1 1 1 1 1 2 2 3 3 4 4 5 6 6 6 6 9 9 10 11 11 11 12 13 13
m=9
Unbounded
CT
d∗
CT
d∗
CT
0.150 0.200 0.225 0.236 0.248 0.259 0.270 0.282 0.291 0.296 0.338 0.357 0.371 0.381 0.386 0.391 0.394 0.396 0.399 0.412 0.416 0.418 0.420 0.421 0.422 0.423 0.423 0.424
0 0 1 1 1 1 1 1 2 2 3 3 4 4 5 6 6 7 7 9 11 12 12 13 13 14 15 15
0.150 0.200 0.225 0.236 0.248 0.259 0.270 0.282 0.291 0.296 0.338 0.357 0.371 0.381 0.386 0.391 0.394 0.396 0.397 0.403 0.405 0.406 0.407 0.407 0.407 0.408 0.408 0.408
0 0 1 1 1 1 1 1 2 2 3 3 4 4 5 6 6 7 7 12 17 23 30 30 30 30 30 30
0.150 0.200 0.225 0.236 0.248 0.259 0.270 0.282 0.291 0.296 0.338 0.357 0.371 0.381 0.386 0.391 0.394 0.396 0.397 0.401 0.402 0.402 0.402 0.402 0.402 0.402 0.402 0.402
is achieved when the maximal delay is equal to one paging cycle. The cost decreases rapidly when the maximal paging delay increases. Table 1 presents the value of the optimal distance and the cost associated with the 1-D model when the paging cost V is fixed to 10 and the update cost U varies from 1 to 1000. The rate of calls received (c) and the rate of movement (q) are fixed to 1 and 5%, respectively. These results confirm that the total cost and the optimal distance increase, when the update cost increases. Thus, when the update cost is low, it is more profitable to frequently update in order to avoid paying for the cost of excess paging. However, when the paging cost is lower then the update cost, it becomes more profitable to update less frequently. Table 2 presents results that correspond to the 2-D model. As Tables 1 and 2 show, the maximal paging delay (m = 7 cycles) for the 1-D model induces costs that are very close to the optimal costs of the unbounded case. With a light worsening of results, we can choose the maximal delay of five cycles. The difference between the optimal cost and the cost closer to the optimum for m = 7 does not exceed 5.5%, while for m = 5 the difference does not exceed 19% for this model. The maximal paging delay m = 9 cycles allows results to be obtained that are still close to the optimum, but Vol. 45, No. 5, 2002
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TABLE 2. Optimal distance and average total cost for the 2-D model. m=1 U
d∗
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
m=2
CT
d∗
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.968 1.102 1.236 1.370 1.504 1.638 1.771 1.905 2.039 2.655 3.032 3.410 3.787 4.164 4.542 4.919 5.297 5.674
0 0 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4
m=3
CT
d∗
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.542 0.555 0.689 0.823 0.957 1.091 1.144 1.182 1.219 1.257 1.295 1.672 2.050 2.427 2.804 3.182 3.469 3.620 3.733 3.846
0 0 0 0 0 0 0 0 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4
m=5
CT
d∗
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.542 0.555 0.689 0.822 0.860 0.897 0.935 0.973 1.011 1.048 1.086 1.464 1.841 2.218 2.544 2.712 2.825 2.938 3.051 3.164
0 0 0 0 0 0 0 0 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 5 5 5
m=7
Unbounded
CT
d∗
CT
d∗
CT
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.542 0.555 0.689 0.822 0.860 0.897 0.935 0.973 1.011 1.048 1.086 1.464 1.841 2.025 2.138 2.251 2.364 2.447 2.503 2.559
0 0 0 0 0 0 0 0 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 5 5 5 6 6
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.542 0.555 0.689 0.822 0.860 0.897 0.935 0.973 1.011 1.048 1.086 1.464 1.841 2.025 2.138 2.204 2.260 2.315 2.346 2.374
0 0 0 0 0 0 0 0 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 5 5 5 6 6
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.542 0.555 0.689 0.822 0.860 0.897 0.935 0.973 1.011 1.048 1.086 1.464 1.841 2.025 2.138 2.204 2.260 2.315 2.346 2.374
FIGURE 7. Comparison with the Ho and Akyildiz scheme for their 2-D model with c = 1%, q = 5%, U = 1000, V = 10 and m = 3.
the difference between m = 7 and m = 9 cannot justify an increase in this delay. As for the 2-D model, the maximum paging delay m = 7 also induces very good performances compared to the unbounded case. In the worst case, the worsening does not exceed 7.8%. T HE C OMPUTER J OURNAL,
Figure 7 compares the results from Ho and Akyildiz’s [3] algorithm with those of the proposed algorithm for values of m = 2 and 3 for the 1-D model. One notes the difference in the results obtained using both algorithms. (In the cost calculations for values of U ranging from 1 to 6, an error Vol. 45, No. 5, 2002
536
I. KOZATCHOK AND S. P IERRE TABLE 3. Comparative result analysis for the 1-D model. m = 2 H&Aa U
d∗
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000
0 0 0 0 0 0 1 1 1 1 1 2 3 3 3 3 3 4 4 4 6 6 6 6 6 6 8 8
m=2
CT
d∗
0.125 0.150 0.175 0.200 0.225 0.250 0.270 0.282 0.293 0.305 0.418 0.465 0.486 0.506 0.526 0.545 0.565 0.579 0.589 0.686 0.724 0.750 0.776 0.803 0.829 0.855 0.868 0.876
0 0 1 1 1 1 1 1 1 1 1 2 3 3 3 3 3 4 4 5 6 7 7 7 8 8 8 8
m = 3 H&Aa
CT
d∗
0.150 0.200 0.225 0.236 0.248 0.259 0.270 0.282 0.293 0.305 0.418 0.464 0.486 0.506 0.526 0.545 0.565 0.579 0.589 0.676 0.724 0.746 0.760 0.774 0.786 0.794 0.801 0.809
0 0 0 0 0 0 1 1 2 2 2 2 3 3 3 3 3 5 5 6 7 7 7 7 7 7 7 7
m=3
CT
d∗
0.125 0.150 0.175 0.200 0.225 0.250 0.270 0.282 0.291 0.296 0.339 0.382 0.415 0.435 0.454 0.474 0.494 0.510 0.515 0.548 0.565 0.579 0.593 0.607 0.621 0.635 0.649 0.663
0 0 1 1 1 1 1 1 2 2 2 2 3 3 3 3 3 4 4 6 7 7 8 8 8 9 9 10
m=5
CT
d∗
0.150 0.200 0.225 0.236 0.248 0.259 0.270 0.282 0.291 0.296 0.339 0.382 0.415 0.435 0.454 0.474 0.494 0.505 0.514 0.548 0.565 0.579 0.587 0.594 0.602 0.606 0.610 0.613
0 0 1 1 1 1 1 1 2 2 3 3 4 4 4 4 5 5 5 7 8 8 9 9 9 11 11 12
m=7
CT
d∗
CT
0.150 0.200 0.225 0.236 0.248 0.259 0.270 0.282 0.291 0.296 0.338 0.357 0.371 0.381 0.391 0.400 0.406 0.411 0.416 0.443 0.455 0.462 0.467 0.471 0.475 0.477 0.478 0.479
0 0 1 1 1 1 1 1 2 2 3 3 4 4 5 6 6 6 6 9 9 10 11 11 11 12 13 13
0.150 0.200 0.225 0.236 0.248 0.259 0.270 0.282 0.291 0.296 0.338 0.357 0.371 0.381 0.386 0.391 0.394 0.396 0.399 0.412 0.416 0.418 0.420 0.421 0.422 0.423 0.423 0.424
a Ho and Akyildiz’s scheme.
has been committed in [3], which is easy to verify by means of straightforward calculations for d = 0 and d = 1.) For values of U ranging from 7 to 80 inclusively and from 200 to 400 inclusively, with a paging delay of m = 3, our results are identical to the results obtained by Ho and Akyildiz. For values of U ranging from 90 to 100 inclusively and from 500 to 1000 inclusively, their algorithm did not allow them to obtain the real optimal distance d ∗ . This led them to achieve the highest average total costs. Similar gaps also exist for the paging delay of m = 2. In both cases, the difference between the optimal cost and the cost obtained with Ho and Akyildiz’s algorithm reached more than 8%. Table 3 presents the results of the 1-D model. For values of U ranging from 1 to 20 inclusively and from 600 to 1000 inclusively, with a paging delay of m = 3, our results are identical to those obtained by Ho and Akyildiz [3]. However, for values of U ranging from 30 to 500 inclusively, their algorithm did not allow the real optimal distance d ∗ to be obtained in all cases, which led them to obtain high average total costs. In certain cases, regardless of the matching between the value of the optimal distance d ∗ achieved by applying both T HE C OMPUTER J OURNAL,
algorithms, the cost obtained by Ho and Akyildiz remains too high. This could be partly explained by the nonoptimal grouping of cells in the paging regions. We also observe similar gaps for a paging delay of m = 2. In both cases, the difference between the optimal cost and the cost obtained by applying Ho’s and Akyildiz’s algorithm can reach 29%. An even more important gap emerges when we compare the costs obtained by Ho and Akyildiz for the maximal paging delay they suggest (m = 3) with the results obtained from the application of the proposed algorithm for m = 5 and m = 7. In both cases, the cost differences can reach 33% and 34%, respectively. In the case of the 2-D model, to find the distance near the optimum, Ho and Akyildiz [3] used approximate analytical expressions. We also present these results in Table 4. It should be noted that the near to optimal distance d ∗ obtained this way corresponds in most cases to the optimal distance d ∗ . In some cases, the value of d ∗ slightly differs from that of d ∗ . However, in these cases, we encounter situations where the approximate total cost CT at the distance d ∗ is greater than the total average cost CT at the distance d ∗ . Vol. 45, No. 5, 2002
U SER T RACKING AND M OBILITY M ANAGEMENT A LGORITHM
537
TABLE 4. Comparative result analysis for the 2-D model. m = 2 H&Aa U
d∗
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 3 3 5
m=2
m = 3 H&Aa
m = 3 H&Ab
CT
d ∗
CT
d∗
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.542 0.555 0.689 0.823 0.957 1.074 1.126 1.178 1.231 1.283 1.335 1.858 2.372 2.608 2.843 2.955 3.011 3.066 3.122 3.177
0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 3 3 3 5 5 5 5
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 1.100 1.600 2.100 1.074 1.126 1.178 1.231 1.283 1.335 1.858 2.381 2.608 2.843 3.079 3.011 3.066 3.122 3.177
0 0 0 0 0 0 0 0 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4
CT
d∗
CT
d∗
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.542 0.555 0.689 0.823 0.957 1.091 1.225 1.359 1.493 1.614 1.651 2.029 2.406 2.762 2.998 3.233 3.469 3.704 3.940 4.133
0 0 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.542 0.555 0.689 0.823 0.957 1.091 1.144 1.182 1.219 1.257 1.295 1.672 2.050 2.427 2.804 3.182 3.469 3.620 3.733 3.846
0 0 0 0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 5 5 5 5 5
m=3
m=5
CT
d∗
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.542 0.555 0.689 0.822 0.860 0.897 0.935 0.973 1.011 1.048 1.086 1.464 1.841 2.218 2.544 2.712 2.825 2.938 3.051 3.164
0 0 0 0 0 0 0 0 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 5 5 5
m=7
CT
d∗
CT
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.542 0.555 0.689 0.822 0.860 0.897 0.935 0.973 1.011 1.048 1.086 1.464 1.841 2.025 2.138 2.251 2.364 2.447 2.503 2.559
0 0 0 0 0 0 0 0 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 5 5 5 6 6
0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.542 0.555 0.689 0.822 0.860 0.897 0.935 0.973 1.011 1.048 1.086 1.464 1.841 2.025 2.138 2.204 2.260 2.315 2.346 2.374
a Ho and Akyildiz’s exact solution scheme. b Ho and Akyildiz’s approximate solution scheme.
For example, for the value of U = 40. the approximate optimal distance d ∗ is zero and the average total cost at this distance is equal to 2.100. while the optimal distance obtained through the exact formulae d ∗ is 1 and the average total cost at this distance is 0.957. This estimated error of the optimal distance leads to a difference in average total cost of more than 120%. If we compare this result with the result obtained by applying our algorithm, we obtain a difference in total average cost of more than 140%. These comparative results prevent us from using approximate analytical expressions. In effect, the examples presented show that the use of closed formulae could lead to a critical increase in costs, as recognized by Ho and Akyildiz, without any advantages for the search algorithm of the optimal distance d ∗ . The numerical results obtained have shown the following.
•
•
• •
A significant reduction of the average total cost CT could be obtained by increasing the maximum number of paging cycles. In some cases, this increase allows a minimum average total cost to be reached which corresponds to the unbounded case. T HE C OMPUTER J OURNAL,
The approximate equations for calculating probabilities give results different from those obtained by applying the exact calculations. The proposed algorithm uses recursion to calculate the probabilities; it is therefore more easy to use and more efficient than the algorithm based on the approximate equations. If the computation is significant or critical, our algorithm is more efficient and more precise than that based on approximate equations, because it is based on exact calculations. When the cellular system operator has no exact statistical data, it could take average statistical data from industry and calculate the optimum, according to the proposed algorithm. The gap between average industrial data and real data from the operator (according to the proposed algorithm) is less than the algorithm presented in [3], as shown in Figure 7. The analysis of the formulae presented in Section 3 allows us to conclude that the function of the total cost is homogeneous in relation to the parameters c, q, U and V . This means that we could expand the results presented in Figure 7 and Table 3 to a larger set of c, q, U and V values, by expressing the cost function CT as CT = cVf (c/q, U/V , d, r).
Vol. 45, No. 5, 2002
538
I. KOZATCHOK AND S. P IERRE
6. CONCLUSIONS In this paper, we have presented an algorithm to minimize the cost associated with the management of users’ mobility in mobile communications networks. This algorithm allows one to determine the optimal size of a location area and to achieve the optimal cellular grouping model in polling regions. It guarantees the global minimum of the total cost function, while respecting the pre-established delay constraints. It also takes into account the average probabilities of received calls, movements, updates and paging costs. Thanks to the integration of the internal optimization cycle algorithm, we have succeeded in avoiding the problem of local minima for the total cost function, which is discrete by nature. This algorithm allows us to avoid the problem of evaluating a cost function at a large number of points, which makes it usable on machines with limited computing power. This has been possible through both the application of the interval division method and the reduction in the number of cell groupings studied. The algorithm evaluates the cost function at a set of limited points. Nevertheless, the number of iterations of the internal cycle remains relatively high. The processing speed of current computers allows calculations to be carried out in a few milliseconds. However, one can improve this algorithm in order to use it in dynamic mobility management systems that are running on portable computers working in an autonomous mode. The study of the total cost function has shown that this function is homogeneous in relation to the parameters c, q, U and V . However, we have not generalized the results to all the values of these parameters. Future research could be oriented towards separate studies of the behavior of both update and paging cost functions. This is to further understand certain behavioral mechanisms which may allow for simplification and optimization of the minimization algorithm of the total cost function. ACKNOWLEDGEMENTS This work was supported in part by the Natural Science and Engineering Research Council (NSERC) of Canada under grants 140264-98 and 224132. REFERENCES [1] Markoulidakis, J. G., Lyberopoulos, G. L., Tsirkas, D. F. and Sykas, E. D. (1995) Evaluation of location area planning scenarios in future mobile telecommunication systems. Wireless Networks, 1, 17–29. [2] Bar-noy, A., Kesseler, I. and Sidi, M. (1995) Mobile users: to update or not to update? Wireless Networks, 1, 175–186. [3] Ho, J. S. M. and Akyildiz, I. F. (1995) Mobile user location update and paging under delay constraint. Wireless Networks, 1, 413–425. [4] Lin, Y. B. (1997) Reducing location update cost in a PCS network. IEEE/ACM Trans. Networking, 5, 26–33. [5] Lin, Y. B. (1994) Determining the user locations for personal communication service networks. IEEE Trans. Veh. Technol., 43, 466–473.
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[6] Rose, C. and Yates, R. (1995) Minimizing the average cost of paging under delay constraints. Wireless Networks, 1, 211– 219. [7] Jordan, S. and Schwabe, E. J. (1996) Worst-case performance of cellular channel assignment policies. Wireless Network, 2, 265–275. [8] Katzela, I. and Naghshineh, M. (1996) Channel assignment schemes for cellular mobile telecommunication systems: a comprehensive survey. IEEE Personal Commun., 3, 10–31. [9] Bar-noy, A., Kessler, I. and Nahgshineh, M. (1996) Topologybased tracking strategies for personal communication networks. Mobile Network Applic., 1, 49–56. [10] Jain, R. and Lin, Y. B. (1995) An auxiliary user location strategy employing forwarding pointers to reduce network impacts of PCS. ACM-Baltzer J. Wireless Networks, 1, 197– 210. [11] Jain, R., Lin, Y. B., Lo, C. and Mohan, S. (1994) A caching strategy to reduce network impacts of PCS. IEEE J. Selected Areas Commun, 12, 1434–1444. [12] Jannik, J., Lam, D., Shivakumar, N., Widom, J. and Cox, D. C. (1997) Efficient and flexible location management techniques for wireless communication systems. Wireless Networks 3, 361–374. [13] Gavish, B. and Sridhar, S. (1995) Economic aspects of configuring cellular networks. Wireless Networks, 1, 115– 128. [14] Gallagher, M. D. and Randall, S. A. (1997) Mobile Telecommunications Networking With IS-41. McGraw-Hill, New York. [15] Garg, V. K., Sneed, E. L. and Gooding, W. E. (1997) User data management in personal communications services networks. IEEE Personal Commun., 4, 33–39. [16] Khrisna, P., Vaidya, N. H. and Pradhan, D. K. (1994) Forwarding Pointers for Efficient Location Management in Distributed Mobile Environments. Computer Science Technical Report, 94-061, Texas A&M University, College Station. [17] Linnartz, J.-P. (1995) On the performance of packet-switched cellular networks for wireless data communications. Wireless Networks, 1, 129–138. [18] Markoulidakis, J. G., Lyberopoulos, G. L., Tsirkas, D. F. and Sykas, E. D. (1997) Mobility modeling in third-generation mobile telecommunications systems. IEEE Personal Commun., 4, 41–56. [19] Rajagopalan, S. and Badrinath, B. R. (1995) An adaptive location management strategy for mobile IP. In Proc. 1st ACM Mobicom 95, pp. 170–180. [20] Raju, J. V. C., Kumar, V. and Datta, A. (1997) An Adaptive Location Management Algorithm for Mobile Networking. IEEE 0742-1303, pp. 133–140. [21] Safa, H. , Pierre, S. and Conan, J. (1999) A new model for reducing location update cost in a wireless network. In ’Wireless 99’ the 11th Ann. Int. Conf. on Wireless Communications, Calgary, Canada, pp. 39–47. [22] Wang, J. Z. (1993) A fully distributed location registration strategy for universal personal communication systems. IEEE J. Selected Areas Commun., 11, 850–860. [23] Ho, J. S. M. and Akyildiz, I. F. (1997) Dynamic hierarchical database architecture for location management in PCS Networks. IEEE/ACM Trans. Networking, 5, 646–660.
Vol. 45, No. 5, 2002
U SER T RACKING AND M OBILITY M ANAGEMENT A LGORITHM APPENDIX A
Now, we introduce the following notation:
By using Equation (3.2) as well as the expressions 1−εi = (1−ε)(1+ε+ε2 +· · ·+εi−1 ) = (1−ε)i−1 (A.1) we can write for i ≥ 0 Ri = e1d−i − e2d−i = e1d−i (1 − εd−i ) = e1d−i (1 − ε)d−i−1
= e1
− e2d−i − e2d−i−1 1 − εd−i
(A.2)
e1d−i e1d−i−1
Ri = Ri+1
1 − εd−i−1
= e1
d−i−1 d−i−2
(A.3)
e1i+1 − e2i=1 1 − εi+1 = e1i e1 − e2 1−ε i 2 = e1 (1 + ε + ε + · · · + εi ) = e1i i
Si =
(A.4)
where ε = e2 /e1 . In particular, for i = 1, 2, 3 and d − 1, we obtain the following expression: R1 e1 d−2 = R2 d−3
(A.5)
e−1 d−4 R3 = 1 R2 d−3 Rd−1 = e1 (1 − ε).
(A.6)
R1 R3 −1 − 2α + A0 = 3 α 2 R2 R2 −1 d−4 2 = 3 α e1 d−2 /d−3 − 2α + e1 d−3 (A.11) (R1 α − R2 ) αe1 d−2 A1 = = −1 (A.12) R2 d−3 (R2 α − R3 ) d−4 A2 = =α− (A.13) R2 e1 d−3 R1 d−2 = (A.14) B1 = R2 Rd−1 (1 − ε)d−3 1 −1 (A.15) B2 = Rd−1 = e1 (1 − ε) R3 d−4 B3 = = 2 (A.16) R2 Rd−1 e1 (1 − ε)d−3 W =
R22
=
p1,d
(A.19)
=
pd,d
(d−3 + εd−2 )(d−3 2 d−3
−
εd−3 )
.
(A.8)
This implies that R22
(A.20) (A.21)
pi,d = (Si−2 A2 − Si−3 A1 )pd,d /W = e1i−2 (i−2 A2 − e1−1 i−3 A1 )pd,d /W,
(A.22)
for i = 2, 3, . . . , d − 1.
2 − R1 R3 = e12d−4(1 − ε)2 (d−3 − d−2 d−4 ) 2d−4 2 = e1 (1 − ε) 2 × [d−3 − (d−3 − +εd−2 )(d−3 − εd−3 )] = e12d−4(1 − ε)2 × [d−3 εd−3 − d−3 εd−2 + ε2d−5 ]
= e12d−4(1 − ε)2 εd−3 d−3 1 − ε +
and then
A0 Pd,d W A1 Pd,d = W A2 Pd,d = W = K1 W/(K2 B1 + K3 B2 + K4 B3 + 3W )
(A.18)
p2,d
2 d−3
(A.17)
p0,d =
(A.7)
d−2 d−4
(R22 − R1 R3 ) . (R2 Rd−1 )
By using equation (A.8), as well as the notation from (A.11)–(A.17) to calculate pi,d , we obtain the following final expressions:
By using equations (A.5)–(A.7), it is easy to obtain R1 R3
539
εd−2
d−3 (A.9)
(R22 − R1 R3 ) εd−2 = e1d−3 εd−3 1 − ε (R2 Rd−1 ) d−3 d−2 ε = e2d−3 1 − ε + . d−3
(A.10)
T HE C OMPUTER J OURNAL,
APPENDIX B For sufficiently huge values of d (d ≫ 1), we can use the following asymptotic expressions: R1 /R2 ≈ e1
(B.1)
R3 /R2 ≈ e1−1
(B.2)
W ≈
e2d−3 (1 − ε)(1 + 0(εd−2 )),
d → ∞, ε → 0 (B.3)
A1 ≈ αe1 − 1
(B.4)
A2 ≈ α − 1/e1
(B.5)
B1 ≈ (1 − ε)−1
(B.6)
B2 ≈ B3 ≈
e1−1 (1 − ε)−1 e1−2 (1 − ε)−1 (1 + O(εd−3 )).
Vol. 45, No. 5, 2002
(B.7) (B.8)
