A Randomized Algorithm for Minimizing User Disturbance Due to Changes in Cellular Technology

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1 A Randomized Algorithm for Minimizing User Disturbance Due to Changes in Cellular Technology Carlos A. S. OLIVEIRA CAO Lab, Dept. of ISE, University of Florida Gainesville, FL 32611, USA David PAOLINI Cingular Wireless Corporation Panos M. PARDALOS CAO Lab., Dept. of ISE, University of Florida Gainesville, FL 32611, USA ABSTRACT We present a computational approach to a practical problem occurring in the mobile telecommunications industry. Due to changes in cellular technology, there is sometimes the necessity of moving users from one network to another (in this case from TDMA to GSM). In this paper we address the problem of minimizing the number of users affected by changes in cellular technology, subject to the constraint that the selected users must be uniformly spread along the covered area. The resulting problem is shown to be NP-hard and a Lagrangian relaxation method is used to find approximate results. We also propose a heuristic algorithm, which is shown to give good results for some instances. Keywords: Telecommunications, Cellular Telephony, Heuristics, Simulated Annealing, Massive Data Sets. 1. INTRODUCTION In the world of mobile telecommunications, many competing technologies are available to be used by cellular carriers. Eventually there is a necessity of changing from one technology to another, and at the same time maintaining the quality of service. In this paper we discuss the problem involving a change from TDMA to GSM technology. The change is planned to be done progressively, and at first just a certain number of users should be selected to change 303 Weil Hall, University of Florida, Gainesville, USA, Phone: 1 (352) Fax: 1 (352) addresses: {oliveira,pardalos}@ise.ufl.edu from one network to another. The number of users moved from the TDMA network must be proportional to the amount of channels assigned to the new GSM network. Moreover, the selected users must be spread evenly all over the network because the number of new GSM channels in each base station will be approximately equal. In the formulation of the problem we consider some characteristics of user access as occurring in practice. For example, in each call more than one cell is usually involved: the (possibly distinct) originating and destination cells along with the cells for which hand-offs are directed. The problem input is the data generated by usage of the cellular network during a period of days. To avoid unnecessary optimization, only the information about peak hours of usage (say, during two hours each day) is used as a baseline. It must be considered that there is a relatively large variability in usage data from day to day, since users have different calling patterns. For example, some users make calls in specific days only (e.g., each Tuesday and Thursday). The established goal for the problem we are modeling is to select which users to change from TDMA to GSM. In this process, we wish to minimize the total number of users affected by the change and at the same time guarantee that they will have a suitable service with the new available GSM capacity. It is also interesting to keep a minimum usage of this new GSM capacity. For example, we must ensure that each cell has at least some of the selected GSM users. It must be clear that the methods used in this paper are general enough to be applied to other technologies other than GSM and TDMA. This is true, since we consider a general selection problem, with the required constraints added. The available data con-

2 sists of daily listings of calls in a specified region, with source and destination cells, duration, and number of hand-offs made during each call. Some real instances of the problem can generate a large amount of data, with some examples having around 100 MB of size per day. Considering all variation and massive size of the data, it seams clear that its very difficult to give an exact solution to this scenario. We will show later that the resulting binary problem is NP-hard. That is why in this paper we are interested in giving only approximate good (near optimal) solutions to the problem. This paper is divided as follows. In the next section we present a mathematical formulation of the problem, which we call the Minimum User Disturbance Problem (MUDP). In Section 3 we propose a randomized algorithm to solve it. In Section 4 we give computational results of our approach. Finally, in Section 5 we make some concluding remarks about the work, as well as future directions of research. 2. MATHEMATICAL PROGRAMMING MODEL We start with some definitions. Let U = {u 1,..., u n} be the set of users in the system, and V = {v 1,..., v k } be the set of available cells. We need to select a subset A U such that the size of A is minimum and the selected users have some characteristics that ensure spreading over the network. We define next what we mean by spreading, by defining the constraints that need to be satisfied by a feasible solution. We assume that there is enough information about the network usage for each user in the system. This information is given by the baseline, collected during a period of m days. The information is given as the following set of functions, defined for each user u U, for each period i, of the m periods considered, during peak time. The peak time is considered to be a daily interval of two hours. C i : U N, the number of calls made by user u, for each user u U, i = 1,..., m; T i : U N, the total time of calls made by user u U, i = 1,..., m; H i : U N, the number of hand-offs make by user u U, i = 1,..., m; S i : U V N, the demand of user u U in cell j {v 1,..., v k }, i = 1,..., m. Based on the known data, we can consider some features of the desired users in our problem. In general, it would be beneficial to select users which make frequent calls during peak time, make long calls, make frequent hand-offs, are in the most crowded areas, maintain a predictable behavior during some days. With these features in mind, given the binary variables { 1 if user u is selected x u =, 0 otherwise we can define the Minimum User Disturbance Problem (MUDP) as: subject to min S i(u, j)x u s j d i j V, (1) where x u i = 1,..., m C i(u)x u c i i = 1,..., m (2) T i (u)x u t i i = 1,..., m (3) H i (u)x u h i i = 1,..., m (4) x u {0, 1} u = 1,..., n s j = capacity available for GSM in cell j, d i = maximum usage deviation on day i, and c i, t i, h i are minimal thresholds computed for each day i = 1,..., m. The first constraint means that the total GSM usage in cell j must be at most the allocated capacity s j, with maximum deviation d, for j = 1,..., m. The second through the forth constraints guarantee that the selected users are relevant to the system, i.e., they have usage parameters at least equal to the minimum parameters c i, t i and h i, for i = 1,..., m. This is an integer 0-1 programming problem with a large number of constraints. Thus, it is not surprising that this is a NP-hard problem, as shown bellow: Theorem 1 The Minimum User Disturbance problem is NP-hard. Proof: We need to make a reduction in polynomial time from a problem known to be NP-hard to the MUDP. The problem chosen (from [2]) is: Set Cover: Given a set S = {1,..., k} and a collection C = {S 1,..., S n }, with S i S for i = 1,..., m, we want to find the minimum cardinality subset T of

3 C such that the union of all S i T is equal to S. We say that the subset T is a cover for the set S. We can formulate the above problem using integer 0-1 programming as follows. Let us define the variable y j, for j = 1,..., n as and let { 1 if set Sj is selected x j = 0 otherwise a ij = { 1 if i Sj 0 otherwise for i = 1,..., k and j = 1,..., n. Then, Set Cover is equivalent to min subject to j=1 a ij y j 1 j=1 y j y {0, 1} n i S Now, suppose that we interpret S as the set of cells and C as the set of users in the cellular system. It is easy to see that, given an instance I of Set Cover we can construct an equivalent instance of the MUDP. To do this, let us use the following parameters: s 1 = 1/2, d 1 = 1/2, m = 1; { 1 if i Sj S 1(u, j) = 0 otherwise; C i (u) = c i = 0, T i (u) = t i = 0, H i (u) = h i = 0, for all u = 1,..., n, i = 1,..., m. The resulting instance of the MUDP has an optimal solution equal to the optimal solution of I. Thus, Set Cover can be reduced to MUDP in polynomial time and therefore MUDP is NP-hard. One of the biggest challenges in solving this problem exactly is the large number of variables. For example, a typical set of data can have as much as 300,000 variables. This is very hard to solve exactly using traditional methods for integer programming. Moreover, from the proof of Theorem 1 we see that even for a extremely simplified case, the problem is still NP-hard. The problem above can also be seen as a constraint satisfaction problem (CSP) [3]. In such problems, we are given a set Φ of discrete variables with constraints Γ i, i = 1,..., k, defined over Φ. The objective is finding an assignment of values for Φ such that each Γ i is satisfied. As often happens with CSP s, it is hard to find solutions satisfying all restrictions. This is shown in the next theorem. Theorem 2 Finding a feasible solution to the MUD problem is NP-hard. Proof: Again we must try to reduce a NP-hard problem to MUDP. This time we choose Set Cover Decision (which is also NP-complete): Set Cover Decision: Given an instance I of Set Cover and an integer B, determine if there is a solution x {0, 1} n to I such that x i < B (5) i=1 for a fixed integer number B. The reduction is similar, but this time we set m = 2, and, in addition, s 2 = B/2, d 2 = B/2 and { 1 if j = 1 S 2 (u, j) = 0 otherwise. With these values, it is easy to see that we create an additional restriction equal to (5). Thus, any feasible solution to the problem constructed as above is a solution to the NP-complete Set Cover Decision problem, and therefore it is NP-hard to find a feasible solution to MUDP. Lagrangian Relaxation to the MUDP A general method for handling a large number of restrictions is trying to incorporate some of them to the objective function. This is done while, at the same time, the corresponding restrictions are relaxed. This technique is the basis of the so-called Lagrangian relaxation (LR) method. To simplify notation, let us denote F 1,i(u) = N i(u), F 2,i (u) = T i (u), and F 3,i (u) = H i (u). Applying LR to our problem, and removing the constant factors c i, h i and t i, we are left with the following formulation subject to min x i 3 m k=1 i=1 w k,i n F k,i (u)x u S i(u, j)x u s j d j V, i = 1,..., m x u {0, 1} u = 1,..., n for some suitable weights w j,i, j = 1,..., 3, i = 1,..., m. The problem becomes finding a feasible vector x for some fixed w j,i such that the problem is minimized. It is clear that feasible solutions to this formulations represent lower bounds for the original problem. Note that, in general, the values w j,i can assume any positive value. Thus, the standard Lagrangian relaxation we need to find a vector w which maximizes the lower bound to the original problem. However, we can just say that the weights w j,i are defined according to our relative interest in solving each of the

4 constraints, and is why we consider the values w i,j to be fixed. Thus, we can define some weights w i,j and solve the problem for these specific values. In the next session we discuss a heuristic algorithm to solve this relaxed version of the MUD problem. 3. AN ALGORITHM TO THE MUD PROBLEM In this section, we present a heuristic algorithm for the MUD problem. Since the MUDP is NP-hard, we need a practical way to solve large instances arising in the real application. Although many advances have being made in enumerative algorithms in the last years, the only computationally efficient approach for such problems is to look for reasonably good (heuristic) solutions, instead of the global optimum. In our case, there is the added requirement that the algorithm must be applied for large data sets, which cannot in general fit the computer s main memory. As an example, the size of the data for a medium-size instance amounts to around 200MB per day. If we require that the problem be solved for an exted period of time, an algorithm with low complexity must be found. The algorithm we propose is a variation of the well known Simulated Annealing (SA) meta-heuristic. The SA meta-heuristic was initially proposed by [6] and has since being used successfully in many applications [1, 4, 5, 7]. In SA, the solution method is an extension of the steepest descent algorithm for combinatorial optimization. At each iteration, a new solution y is generated in the neighborhood of the current solution x. To be accepted, y must either be better than x, or be included randomly with exponentially distributed probability, with mean equal to α. The parameter α is used to control the amount of randomness acceptable at each stage of optimization. In the beginning, many non-improving solutions can be accepted. However, when α decreases, there are less opportunities for accepting non-improving solutions. The idea is that the system will converge to the optimum solution, as α converges to zero. The main difference between SA and the proposed algorithm is the need of dividing operations in more than one phase. The data representing one day of operation in the cellular network is called a data-frame. As we are processing data for m days, there are m data-frames. In our algorithm, each phase represents the processing of one data-frame. With the data for the first day we compute an initial solution (the construction phase). Then we use the next data-frames to compute improvements to the objective function. Due to the large amount of memory required to store all data-frames, we cannot expect to have the exact cost of the objective function every time we make a local change. For example, if we set x u = 1 for user u U in iteration j of the algorithm (corresponding to the j-th data-frame), we cannot calculate the exact change in the objective function since other dataframes are out of memory. Thus, we need to content ourselves with an approximation of the objective function, updated every time we read additional data. To compute an approximate objective function, we define a function P (x, p) as the penalty incurred by solution x until day p. In other words, we define P (x, p) = p 1 3 k=1 j=1 w k,j n F k,j (u)x u. This is exactly the part of the objective function that cannot be computed without knowledge of previous data. Thus, the objective function that we are really using at step j is f(x, j) = x u 3 k=1 w k,j n The Randomized Algorithm F k,j (u)x u P (x, j) There are two important steps in the operation of a meta-heuristic. In the construction phase a new solution is assembled from basic elements of the problem. In the improvement phase, the current solution(s) is (are) modified in order to find a local optimum, hopefully close to the the global optimum of the problem. In our approach, the initial phase, where the first data-frame is processed, corresponds to the construction phase. The remaining computation is divided in sub-phases, each corresponding to the remaining dataframes. Algorithm 1 gives a formal description of the general method used. Algorithm 1: Randomized algorithm for MUDP. Process Data1 for j 1 to m do Process-Data(j) while not terminate flag do r random(m) Process-Data(r) The description for the initial phase is shown in Algorithm 2. Initially, the users are sorted by increasing value of their contribution to the objective function. We define the contribution g(u) of user u to the objective function as g i(u) = w i,1n(u) + w i,2t (u) + w i,3h(u).

5 A user u is said to be viable for a partial solution x if, after making x u = 1, constraint (1) in the integer formulation is still valid. During the first phase, we use viable users to create an initial solution to the problem. The algorithm analyzes all data for the first data-frame and sets x u = 1 for viable users u with probability P 1 = ρ (ρ is a parameter of the algorithm). After the of the algorithm for the first dataframe, the process continues with the remaining data. The technique for subsequent data-frames, shown in Algorithm 3, is similar, with exception of the construction phase, where we are now concerned in improving the existing solution. For each subsequent data-frame i, the file is read into the memory and viable users are changed with a probability P i = γp i 1. We repeat this step while improvements are possible. The general idea is that if user u is important, it should appear in most of the days, so the probability of assign users appearing in just some days is smaller. The steps in the algorithm are detailed bellow. Algorithm 2: Process-Data(1) for u U do Compute T (u, i), H(u, i), C(u, i). P 1 ρ. Sort(U, g(u)). for i = 1 to n do if Viable(u) then x u 1 w.p. P 1. ImprovementPhase() Algorithm 3: Process-Data(j). Data : j Result : A set of users initialization for u U do Compute T (u, i), H(u, i), C(u, i). P i γp i 1, for (0 < γ < 1). P 1 ρ Sort(U, g(u)) for i = 1 to n do if Viable(u) then if x u = 0 then x u 1 w.p. P 1 else if x u = 1 then x u 0 w.p. P 1 ImprovementPhase() An improvement phase is applied for each dataframe, as show in Algorithm 2 and 3. We change the solution by verifying if the status of a user can be changed (x u = 1 if x u was 0, or x u = 0 if x u was 1). If this improves the objective function, then we accept the change. Otherwise, we can still accept the change with probability P i. After a fixed number of iterations without change, probability p i is multiplied by ρ, such that 0 < ρ < 1. Thus, the probability of accepting a non-improving change goes to 0 as the number of iterations increase. The resulting improvement procedure is shown in Algorithm 4. Algorithm 4: ImprovementPhase() P ρ j i 0 while i < MAX IT do u random(n) if user u improves the objective then change x u w.p. P i 0 else i i + 1 j j + 1 if j = n then P γp ; j 0 In this algorithm there are some parameters that must be set. They are the Lagrangian multipliers w i,j, for j = 1,..., 3, i = 1,..., m, and the randomization parameters ρ and γ. The last two represent, respectively, the initial probability of selecting users and the multiplicative factor for the probability of changing values in the current solution. 4. COMPUTATIONAL EXPERIMENTS To evaluate the efficiency of the proposed algorithm we developed a simple local search (LS) and used it as a baseline for comparison. The instances used in the experiments are derived from some real data supplied by Cingular Wireless. Two data frames of real data were used as the initial description of the instances. From these data frames, other 15 instances were created, using random numbers distributed according to the values observed in the real files. The experiment was performed by first running LS on a random initial solution, for each instance. We then run Algorithm 1 in these instances, for the same number of iterations. Both algorithms were implemented using the C programming language. The random parameters used were ρ = 0.7, and γ = 0.8. The resulting code was compiled using the gcc optimized compiler,

6 Objective function cost Pure Local Search Algorith Instance number Figure 1: Results for Algorithm 1 applied to generated instances of the MUDP. Objective function cost Pure Local Search Algorithm time (seconds) Figure 2: Comparison of solution quality for local search and Algorithm 1, applied to an instance of MUDP. In this paper we formulate and propose an algorithmic solution to a real-life problem occurring in the telecommunications industry. We call it the Minimum User Disturbance Problem. The problem is modeled using mathematical formulation and solved by a heuristic procedure, based on randomized local search, and using ideas from the Simulated Annealing metaheuristic. Additional computational experiments with the proposed method are being performed. It remains as a future topic of research to determine new ways of improving this solution. For example, the algorithm seams to be very suitable for parallelization. Also, it would be beneficial to compare our approach with other heuristic methods that could exploit different characteristics of the problem. 6. REFERENCES [1] N. E. Collins, R. W. Eglese, and B. L. Golden. Simulated annealing an annotated bibliography. AJMMS, 8: , [2] M. R. Garey and D. S. Johnson. Computers and Intractability a Guide to the Theory of NP- Completeness. W. H. Freeman and Company, [3] Walter Hower. Constraint satisfaction - algorithms and complexity analysis. Information Processing Letters, 55: , [4] D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon. Optimization by simulated annealing: and experimental evaluation; part I, graph partitioning. Operations Research, [5] D. S. Jonhson, C. A. Aragon, L. A. Mcgeoch, and C. Schevon. Optimization by simulated annealing: an experimental evaluation Part II (graph coloring and number partition). Operations research, 31: , [6] S. Kirkpatrick, C. D. Gellat Jr., and M. P.Vicchi. Optimization by simulated annealing. Science, 220: , [7] P. J. M van Laarhoren and E. H. L. Aarts. Simulated Annealing: theory and applications. D. Reidel Publishing Co., Holanda, in a PC using Linux, with 256 MB of RAM and a 1.8 GHz processor. The results of the comparisons are shown in Figure 1. In Figure 2 we present a comparison of solution quality for the two methods when applied to a particular instance. The same pattern was also observed in other instances. 5. CONCLUDING REMARKS