Wireless Personal Communications 12: 209–224, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.
A Simple Heuristic for Assignment of Cells to Switches in a PCS Network DEBASHIS SAHA Department of Computer Science and Engineering, Jadavpur University, Calcutta 700 032, India E-mail:
[email protected]
AMITAVA MUKHERJEE PricewaterhouseCoopers Ltd., Salt Lake City, Calcutta 700 091, India E-mail:
[email protected]
PARTHA SARATHI BHATTACHARYA Department of Telecommunications, 2 Judges Court Road, Calcutta 700 027, India
Abstract. This work deals with a design problem for a network of Personal Communication Services (PCS). The goal is to assign cells to switches in a PCS Network (PCSN) in an optimum manner so as to minimize the total cost which includes two types of cost, namely handoff cost between two adjacent cells, and cable cost between cells and switches. The design is to be optimized subject to the constraint that the call volume of each switch must not exceed its call handling capacity. In the literature, this problem has been conventionally formulated as an integer programming problem. However, because of the time complexity of the problem, the solution procedures are usually heuristic when the number of cells and switches are more. In this paper, we have proposed an assignment heuristic which is faster and much simpler than the existing algorithms. Despite its simplicity, experimental results show that it performs equally well in terms of solution quality, and, at the same time, it is faster than its predecessors. We present the algorithm as well as comparative results to justify our claim. Keywords: PCS, cell, switch, handoff, clustering, optimization, heuristic.
1. Introduction This paper describes a natural clustering algorithm for assigning cells to switches [1] in a Personal Communication Services Network (PCSN) [2, 3] in such a way that the total design cost is minimized under the constraint of call handling capacity of switches. The algorithm always produces an optimal/near-optimal solution, and it is faster than the existing techniques. Above all, it is very simple to execute. So the present work provides a better method for network planning where the computational resources are limited. The problem of assigning cells to switches [1] is often faced by the designer of mobile communication services [2, 3], and, in near future, it is likely to be faced by the designers of Personal Communication Services (PCS) too. 1.1. N ETWORK M ODEL In our cellular network model (Figure 1), we assume that, for each cell, there is a base station (BS) whose function is to provide radio interface to mobile subscribers within the cell. Cells are grouped into clusters [10, 11], and, for each cluster, a switch (SW) is allocated. Switches are co-located with signaling points (SPs) which interface with signaling network (Figure 1).
210 Debashis Saha et al. The BSs within a cluster talk with each other through the switch assigned for that cluster. This can be referred to as intra-cluster communication [10]. On the other hand, inter-cluster communication [10] involves two cells which are in two different clusters and, hence, are connected to different switches. Routing of calls between the switches is usually done through the telephone network (this is beyond the purview of our present work) [3]. Both intra- and inter-cluster movements involve handoff [4] as the subscriber moves from one cell to another. However, intra-cluster handoff is simpler than inter-cluster handoff because the latter requires more network resources [4]. For example, inter-cluster handoff involves a fairly complicated inter-switch protocol and location update in register databases [5]. Hence, it is costlier also. While clustering the cells, an important criterion should be reducing the frequency of handoff between the cells. Obviously, the cells, among which the handoff frequency is high, should come under one cluster so that handoff cost is minimized, provided the switch meant for the cluster can handle all the calls from the cells. The total cost involves not only the handoff cost but also the cabling cost (cells must be connected to the switch via trunk cables). The corresponding optimization problem is known as cell-to-switch assignment (CSA) problem [1] in the literature. The CSA problem can be of two types: single homing and dual homing. In single (dual) homing, each cell belongs to one (two) cluster(s). Thus, a single- (dual-) homed cell is connected to one (two) switch(es). In this paper, we restrict ourselves to single homing CSA problem only i.e., the clusters are non-overlapped. 1.2. P REVIOUS W ORKS The CSA problem is NP-complete [6] because it is mathematically equivalent to the binpacking problem [1, 8]. Therefore, heuristic solutions are necessary for the CSA problem. The CSA problem, or variants of it, has been considered earlier in [1, 7, 8]. Samadi and Wong [7] have considered an assignment based on other criteria (not cost) such as minimizing location updates and paging messages. Merchant and Sengupta [1, 8] first posed the CSA problem with respect to cost, and, in that sense, our paper can be described as an extension to their work reported in [1]. Related work on graph partitioning applied to CSA problem is considered in [8]. 1.3. O UR C ONTRIBUTION In [1] and [8], the CSA problem for single homing has been formulated as an integer program (IP) [9] and then solved by the integer programming method as well as by a heuristic. Expectedly, the integer programming method could not find any solution for large problem sizes, whereas the heuristic fared consistently well. Even then, we are unable to find any solution within a few seconds in our PC/AT 486 by the heuristic when the problem size grows around 100 cells and 20 switches. This motivates us to look for another heuristic which will be simpler and faster than the earlier one so that bigger problems can be tackled more effectively. For the sake of brevity, in the following sections of the paper, the CSA heuristic due to Merchant and Sengupta [1] is referred to as Heuristic-I and the heuristic proposed by us is called Heuristic-II. 1.4. O RGANIZATION
OF THE
PAPER
The paper is organized into five sections. Following Introduction in Section 1, a mathematical programming formulation of the problem is presented in Section 2. Heuristic-II is the theme
A Simple Heuristic for Assignment of Cells to Switches in a PCS Network 211
Figure 1. PCS network architecture.
of Section 3 that also contains a brief description of Heuristic-I at the beginning for the sake of comparative study. Section 4 contains experimental results with comparisons, and, finally, Section 5 concludes the paper. 2. Problem Formulation The CSA problem, considered in this work, can be stated as: given a group of cells and a group of switches (whose locations are known), the problem is to assign cells to switches in an optimum manner, such that the assignment attempts to minimize the cost comprising of two components, namely cost of handoff (which involves two switches) and cost of cabling (or trunking). A mapping representation of the problem is given below followed by its mathematical formulation. 2.1. M APPING R EPRESENTATION We assume that there are n cells and m switches. The number of n cells are divided into m groups of switches. Each group, i (1 ≤ i ≤ n) consists of one or more cells connected to a kth switch (1 ≤ k ≤ m). The function mapping is many to one, from cell domain to switch domain, as shown in Figure 2.
212 Debashis Saha et al.
Figure 2. Mapping representation of the CSA problem.
2.2. M ATHEMATICAL P ROGRAMMING F ORMULATION We assume that the location of n cells and m switches are fixed and known. If cells i and j are assigned to different switches then a cost is incurred every time a handoff occurs between cell i and j (i, j = 1,.., n). The following notations are used throughout the rest of the paper. Cik = fixed (initial) cost of connecting cell i (1 ≤ i ≤ n) to switch k (1 ≤ k ≤ n) (proportional to distance) hij = handoff cost per unit time between cell i and cell j (i, j = 1,.., n) λi = call volume that ith cell handles per unit time Mk = traffic handling capacity of switch k xik = an assignment variable = 1, if cell i (1 ≤ i ≤ n) is connected switch k, (1 ≤ k ≤ m) = 0, otherwise Total cost includes fixed cable cost plus the cost of handoff. Handoff cost, hij , is directly proportional to the frequency of handoff that occurs between cells i and j , which are assumed to be known. Since each cell is assigned to only one switch, we have the following constraint X xik = 1 for i = 1,.., n and k = 1, . . . m . (1) k
The optimization is carried out in such a way that the call handling capacity of a switch is not violated. The constraint on the call handling capacity is as follows X λi · xik ≤ Mk . (2) i
We consider a new variable zij k defined as zij k = xik · xj k for i, j = 1,.., n and k = 1,.. m .
(3)
A Simple Heuristic for Assignment of Cells to Switches in a PCS Network 213 If both cell i and cell j are connected to switch k, then zij k = 1, otherwise zero. We then consider another new variable yij defined below yij =
m X
zij k for i, j = 1,.. n.
(4)
k=1
Thus, yij = 1, if cell i and cell j are connected via a common switch = 0, if cell i and cell j are connected via different switches. Hence, handoff will occur for zero values of uij . Since the total cost = (cable cost + handoff cost), we will minimize the objective function given by XX XX Cik · xik + hij (1 − yij ) . (5) We now have a mathematical programming problem defined by the objective function (5) subject to the constraint (1) through (4). However, we also need an additional constraint xik = 1 or 0, for i = 1,.. n and k = 1,.. m .
(6)
This cannot be considered as an integer programming because constraint (3) is non linear. However, addition of a few more constraints can convert the above formulation to a linear integer program as discussed in [1]. We replace constraint (3) with the following group of constraints (for i = 1,.. n and k = 1,.. m) zij k ≤ xik
(7)
zij k ≤ xj k
(8)
zij k ≥ xik + xj k − 1
(9)
zij k ≥ 0 .
(10)
Now the objective function (5) along with the constraints (1), (2), (4), (6) through (10) constitutes a valid linear integer program, and it can be solved by a combination of linear programming and branch and bound [8] algorithm. Any standard integer programming (IP) package can be used to implement it. 3. Assignment Heuristics In this section, we present Heuristics-I (briefly reproduced from [1] for the sake of continuity in discussion) and Heuristics-II and their comparative complexity analysis. 3.1. H EURISTICS -I We present the heuristic succinctly. For a detailed description, we refer to [1] and [8]. In this heuristic, the cells are ordered in decreasing call volume. For m switches, there will be m stages in this algorithm. At an intermediate stage j , the assignment of first (j − 1) cells has
214 Debashis Saha et al. already been done and cannot be changed. For j = 1,.., n we assign cell j in a feasible manner to that switch which minimizes the total cost for first j cells. Using this greedy heuristic [1], it is required to find best b solutions instead of just one. To do so, let us say that we have already obtained the best b solutions at the (j − 1) stage of the problem. Now, in the j th stage, we have m choices of assigning the j th cell to various switches. Thus, a total of b(m − 1) choices. Of this, we first throw out all the possibilities that are infeasible. Then, from the remaining, we again retain best b possibilities that have minimum cost. We continue in this fashion for j = 1,.., n. The pseudocode of the algorithm is presented next. ALGORITHM Heuristic-I Begin with a single empty assignment. STEP 0: INITIAL ASSIGNMENT (0.1) Order the cells in decreasing order of call volume. (0.2) Cells are assigned one at a time in n stages. For j = 1,2, . . . n extend each partial assignments under consideration by adding all possible assignment of the jth cell. Discard all assignment that violate the call handling capacity constraints. If no assignment remains, algorithm fails and go to endstep. If b or less partial assignments remain, keep them all, else keep the b best assignments based on the total cost of the j assigned cells only. (0.3) Keep the best assignments found. STEP l(l > 0): REFINEMENT (l.1) Mark all cells as unlocked. (l.2) Find the best feasible move: identify a cell j and a switch k, such that, of all feasible moves, moving cell j to switch k reduces the value of the objective function by the greatest amount. (l.3) Assign cell j to switch k. Mark cell j as temporarily locked. Note the current cell assignment pattern. (l.4) Repeat (l.2) and (l.3) until no move is found. In the sequence of cell assignment pattern generated in this pass, select the one with lowest objective value, and reset the current cell assignment to it. If the value of the objective function is not reduced, then go to endstep. ENDSTEP: Stop. Next, we discuss Heuristic-II and its implementation as worked out by us. This subsection represents the crux of the paper. 3.2. H EURISTIC II The idea of this algorithm follows from our previous work on clustering [10]. Natural clustering of cells around a switch assumes that a cluster should be generated incrementally with the switch at the center. This implies that a cluster around a switch must begin with the cell which
A Simple Heuristic for Assignment of Cells to Switches in a PCS Network 215 houses the switch. This cell is called the seed cell. Initially, every seed cell is a cluster. Since the number of clusters must be equal to the number of switches (because one switch is assigned per cluster), clusters should be formed around every seed cell. As the seed cell will be joined by its neighboring cells, the cluster corresponding to the seed cell will grow. The heuristic will expand all the clusters concurrently, until the set of cells is exhausted. While expanding each cluster, we will pick up one of the neighboring cells in an optimized way. At the same time, we will also verify that addition of the selected cell to the cluster under consideration will result in a feasible move satisfying the call handling capacity of the switch. It is worth noting that we consider only the neighboring cells of a cluster (not all cells) when a cell is to be selected for the cluster. This follows intuitively from the fact that when a mobile subscriber moves out of a cell it enters into one of its neighboring cells. This type of natural, continuous mobility pattern (hence, handoff pattern) implies that natural clustering should resemble the mobility pattern as close as possible, subject to the call handling capacity of the switch. An algorithmic version of the heuristic is presented below. We assume that cj represents cell j and setlk represents the cluster around switch k formed in step l (l = 0, 1, 2, 3, . . .). ALGORITHM Heuristic-II Let setk denote the set of cells assigned to switch k. STEP 0: INITIAL ASSIGNMENT Find the cell cj that houses switch k and initialize setk to cj i.e., setk0 = {cj }. Correspondingly, reduce Mk by λj : Mk0 = Mk − λj STEP l(> 0): ITERATION (l.1) Identify the neighboring cells of setkl−1 and arrange them in descending order of ρ, where ρ = (handoff cost + cable cost) with respect to setkl−1 . (l.2) Choose the one (say cj ) with maximum ρ. If there is a tie, select the one with more hand-off cost. If, still, there is a tie, select one randomly. Do it for all switch i.e. {setkl−1 }, k = 0,1,2, . . . , m. (l.3) If cj is neighbor to more than one setkl−1 , then assign it to the setkl−1 for which cable cost is minimum provided its call volume can be supported by the remaining call handling capacity of the set. If there is a tie in cable cost, break it randomly. If cj can not be assigned to the nearest switch due to call handling limitation, try for the second nearest switch, if any, and so on. If cj cannot be accepted by any switch, leave it unassigned. (l.4) If cj is assigned to switch k, then reduce the call handling capacity of setk by the call volume of cell cj . (l.5) Go to step 2 until the call handling capacity of all the switches have become insufficient or all the cells have been assigned. Otherwise, go to ENDSTEP. ENDSTEP: If no cell remains unassigned, report SUCCESS else report FAILURE. Stop. The algorithm is best understood when applied to a concrete problem, say the problem of 15 cells and 2 switches (Figure 3) taken from [1]. In Figure 3, cells are hexagonal in shape with their base stations at the center. Two switches are housed at cell 5 and cell 8. Call volume of each cell is written beside the cell number. The handoff cost for each pair of adjacent cells
216 Debashis Saha et al.
Figure 3. An example network of 15 cells and 2 switches: partitions produced by Heuristic-I and Heuristic-II.
is marked to the edge connecting the centers of the cells. Heuristic-I produces the optimum solution for this test problem [1]. Later on, we will see that Heuristic-II also produces the optimum solution but in less time. Before that, we explain our algorithm with this test case. Since the two switches are located in cells 5 (switch 1) and 8 (switch 2), these two cells are taken as seed cells. For the seed cell 5, according to our algorithm, set01 = {5}. Similarly, for switch 2, set01 = {8}. In general, setik represents the cluster generated thus far upto the iteration i around the switch k. In Figure 3, for cell 5, these are 4 neighboring cells, namely cells 1, 2, 6 and 9, all of which happen to be at equal distance from cell 5. Out of these 4 equidistant (i.e., cable cost is the same) cells, cell 1 has highest handoff cost (4.7) with respect to set01 . So, in the next iteration, cell 5 is joined by cell 1, and the new cluster around switch 1 becomes, set11 = {5, 1}. Similarly, set12 becomes {8, 7}. Next, we have to select one of the 3 cells (namely 2, 6, 9) which are neighbor of the set11 (i.e., the combination of cells 5 and 1) and, at the same time, are equidistant from switch 1 (cell 5). Among the three cells, cell 6 has the highest handoff cost(1.7) with respect to set11 and so set21 becomes {5, 1, 2}. In this way, both set1 and set2 grow (by one cell) with every iteration. When a cell is added, its call volume is subtracted from the remaining call handling capacity of the switch. For this example problem, both switches have a capacity of 33.11. So, to begin with, set1 and set2 have capacity value 33.11. When set1 is initialized as set01 = {5}, capacity decreases by the call volume of cell 5 to become (33.11 − 4.7) = 28.4. For set11 = {5, 1}, capacity further decreases by 4.1 (call volume of cell 1) to remain (28.4 − 4.1) = 24.3. The algorithm stops, if either all the cells are exhausted (i.e., clusterized) or no switch can accommodate any more cell due to limitation in call handling capacity.
A Simple Heuristic for Assignment of Cells to Switches in a PCS Network 217 3.3. T IME C OMPLEXITY C OMPARISON
OF
T WO A LGORITHMS
HEURISTIC-I The loop body consists of 1. Assigning a cell to a possible orientation, which requires time O(mn). 2. Sorting the list of size mb applying Quicksort, which requires O(mb(log(mb))) time. 3. Calculating cost of mb patterns of instant cell size i, which requires time of the order of τ , where i=n X τ = mb i2 i=1
= mbn(n + 1)(2n + 1)/6 As a whole, the loop is executed n times. Therefore, the net time complexity = O(n(mb+ mblog(mb))) + mbn(2n + 1)(n + 1)/6)) = O(n4 mb). HEURISTIC-II At each iteration, one cell is assigned to every switch. So the average number of iterations, needed to assign all n cells, is ceil(n/m). Again, in each iteration, we have to consider the following: 1. Assigning cells to m switches, which requires O(m) time. 2. Calculation of neighbor having the maximum hand off cost, which requires O(T ) time, where n X T = (1/n) neighbor i i=1
neighbori = number of neighboring cells to ith cell. Therefore, T represents the average number of neighbors of the cells. For example, T = 4.13 for the test network of Figure 3. 3. Calculation of handoff cost, which requires O(mn + nT ) time. 4. Calculation of cable cost, which requires O(n) time. Therefore, time complexity for cost calculation = O(mn + nT + n). One important characteristic of this heuristic is that time complexity depends not only upon the value of m and n, but also upon the structure of the network. The time complexity will be large for a network having large T , even if m and n remain unchanged. Now let us consider the worst case analysis. The worst case occurs when one switch has a large call handling capacity and rest of the switches has very small call handling capacity. The consequence is that the switches having less call handling capacity will be exhausted earlier than the switch having large call handling capacity. So the loop will be executed (mn) times instead of m(ceil(n/m)) times. Therefore, time complexity in the worst case, becomes O(mnT + nT + mn) = O(mnT). The above analysis clearly indicates that Heuristic-II is better than Heuristic-I in terms of handling more cells (i.e., larger input sizes). Since both heuristics depend on the number of switches linearly, increase in the number of switches will not result into considerable decrease in execution speed of either.
218 Debashis Saha et al. Table 1. Comparison between Heuristic I and II (– indicates insufficient data). # Cell (n)
# Switch (m)
Time taken by Heuristic-I (t1 ms)
Time taken by Heuristic-II (t2 ms)
t1/t2
Heuristic-II Obj/ Optimimum (IP)
2 5 15 3 8 5 15 20 25 70 90 100
2 2 2 4 4 5 10 15 15 10 15 20
10 16 1087 5 32 21 2052 7079 13095 61560 70800 1026000
3 5 10 3 10 5 10 10 10 16 21 32
3.33 3.2 108.7 2.67 3.2 4.21 205.2 707.9 1309.5 3847.5 3371.2 32062.5
1.00 1.00 1.00 1.00 1.00 1.00 1.01 1.00 1.00 1.02 –
4. Experimental Results In order to test the performance of Heuristic-II and compare it with IP and Heuristic-I, we have coded all the three algorithms in the C language to run a PC/AT-486 DX under Borland C environment. Several problems with varying number of cells, switches and call volume patterns have been generated by the method described in [1], and then the algorithms are experimented with these test problems. The results are described in Table 1 and Figures 4–9. Figures 4–6 have been drawn for a fixed number of switches (cells varying between 1 and 15). Heuristic-II is always the best. The time gap between Heuristic-I and Heuristic-II increases as the number of cells increases. Keeping the number of cells fixed, if we vary the number of switches, then also Heuristic II is better than other two, as shown in Figures 7 and 8. A comprehensive result, for different input sizes, is compiled in Table 1. A close scrutiny on the performance of Heuristic-I indicates that, since its time complexity is O(n4 mb), the shape of the curve cover depends mainly on the factor n4 . For a constant number of switches (i.e., fixed m), if we increase n, execution time will increase by a factor n4 . It can be easily shown that execution time of IP = (execution time of Heuristic-I)0.25n/logm n . Despite high execution time, Heuristic-I’s curve shows a linear behavior at the beginning, a mild increase in gradient in the middle, and a large sweep at the end. For varying switches (Figures 7 and 8), shape of the curve for Heuristic-I is generally characterized by the term mblog(mb). Finally, let us concentrate on the behavior of Heuristic-II whose time complexity is O(mnT). Figures 4–6, show that for a given number of switches, if the number of cells is increased, then curve-cover depends upon nT , which is a straight line. For the example problem of 15 cells and 2 switches (shown in Figure 3), we have run both the heuristics and ended up with a significant performance enhancement (in terms of absolute execution time) in case of Heuristic-II. In our PC/AT-486 (66 MHz), Heuristic-I took around 1 second, whereas Heuristic-II took only 10 ms. Optimum solution is found by
A Simple Heuristic for Assignment of Cells to Switches in a PCS Network 219
Figure 4. Execution time comparison of three algorithms when number of switches is 2.
Figure 5. Execution time comparison of three algorithms when number of switches is 4.
both heuristics (objective value = 107.3). This huge disparity in execution times is due to the fact that Heuristic-I is more (of the order of three) sensitive than Heuristic II on the number of cells. Consequently, we have noticed that Heuristic-II has outperformed Heuristic-I by a large margin whenever the number of cells has been increased. This is predicted by our time complexity analysis also. Lastly, we summarize the inferences drawn from the presented work as: (i) Heuristic-II has always showed an excellent performance when run with large number of cells and/or switches. In Table 1, we have shown upto 100 cells and 20 switches. The
220 Debashis Saha et al.
Figure 6. Execution time comparison of three algorithms when number of switches is 10.
Figure 7. Execution time comparison of three algorithms when number of cells is 2.
execution time it takes is only 32 milliseconds on PC-AT/486 for the combination of 100 cells and 20 switches. (ii) The time complexity of Heuristic-I is O(mn4 ), where m is the number of switch, and n is number of cells. The time complexity of Heuristic-II is O(mnT), which is linear. So complexity of Heuristic-I is clearly more than that of Heuristic-II. (iii) Using Heuristic-II, a solution could be found very rapidly (faster than Heuristic-I), and, on the average, the solution had an objective value within 2–3% of that of the optimum
A Simple Heuristic for Assignment of Cells to Switches in a PCS Network 221
Figure 8. Execution time comparison of three algorithms when number of cells is 5.
Figure 9. Deviation of objective value from the optimum value.
solution (Figure 9). This small amount of deviation is not really significant, if we consider the speed-up obtained in Heuristic-II.
5. Conclusion During the last decade, there has been a tremendous growth in the deployment of mobile communication systems (universal personal communication) [2–4]. The basic concept behind this technology is that we divide a given geographical territory into cells, and groups of cells are connected to a switch.
222 Debashis Saha et al. In this paper, we have solved the problem of optimum assignment of cells to switches in a PCS network. A heuristic that is simpler and faster than the one reported in [1] has been presented in this paper. In fact, for all the nontrivial problems, it is faster among all existing CSA algorithms and it gives the optimal solution for 95% cases. The objective value may not be optimum in 5% cases deviating by only 2–3% from the optimum. As reported in [1] and verified by us, for problems of 15 switches and 25 cells (or more), IP is unable to produce any result within a few hours of CPU time, whereas our heuristic method is versatile with any size of the input. So IP is a good approach only for less number of cells and switches, whereas heuristic is a very good approach for large size of cells and switches. But we may have to sacrifice the optimality of solution with the gain in CPU time. So IP is applicable when CPU time is less important than optimality of objective function, whereas heuristic is a must when CPU time is more important than the optimality of the objective function. We are presently working on an extension of the algorithm that will take into account the “load balancing” of switches. Another important modification of the algorithm will be to incorporate the dual homing problem that is currently under investigation. Modification of the algorithm so as to solve the dual homing problem is currently under investigation and will be reported soon in our forthcoming paper.
Acknowledgement We acknowledge the valuable comments from the reviewers who helped us in improving the quality of the paper. We thank the All India Council for Technical Education (AICTE) for supporting Debashis Saha through a Career Award grant.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
A. Merchant and B. Sengupta, “Assignment of Cells to Switches in PCS Networks”, IEEE/ACM Trans. on Networking, Vol. 3, No. 5, pp. 521–526, 1995. D.C. Cox, “Personal Communications – A Viewpoint”, IEEE Commun. Mag., pp 8–12, 1990. R. Steele, Mobile Radio Communications, Pentech Press, 1992. M.D. Yacoub, Foundations of Mobile Radio Engineering, CRC Press: Boca Raton, FL, 1993. R.H. Katz, “Adaptation and Mobility in Wireless Information Systems”, IEEE Personal Commun., Vol. 1, No. 1, pp. 6–17, 1994. M.R. Garey and D.S. Johnson, Computers and Intractability, a Guide to the Theory of NP-Completeness, Freeman: New York, 1979. B. Samadi and W.S. Wong, “Optimization Techniques for Location Area Partitioning”, 8th ITC Specialist Sem. UPC, Geneva, Italy, 1992. A. Merchant and B. Sengupta, “Multiway Graph Partitioning with Applications to PCS Networks”, Tech. Report, TR-93-C002-4-5021-1, NEC, U.S.A., 1993. G.L. Nemhauser and L.A. Woolly, Integer and Combinatorial Optimization, Wiley: New York, 1988. D. Saha and A. Mukherjee, “Design of Hierarchical Communication Networks under Node/Link Failure Constraints”, Computer Communication, Vol. 18, No. 5, pp. 378–383, 1995. D. Saha, A. Mukherjee and P.S. Bhattacharya, “Design of a Personal Communication Services Network (PCSN) for Optimum Location Area Size”, in Proc. IEEE, ICPWC ’97, Mumbai, India, Dec. 1997.
A Simple Heuristic for Assignment of Cells to Switches in a PCS Network 223
Dr. Debashis Saha received his Bachelor of Electronics and Telecommunication Engineering Degree from Jadavpur University , Calcutta and his M.Tech. and Ph.D. degrees in 1988 and 1996 respectively, from the Electronics and Electrical Communication Engineering Department of Indian Institute of Technology (IIT) at Kharagpur in India. He joined Jadavpur University as a faculty member in the Computer Science and Engineering Department in 1990 and is currently a Reader there. His research areas are: Formal modeling of network protocols, Topological design of computer networks, All-optical networks, and Mobile Communication. He has coauthored two books and a monograph. Dr. Saha is a senior Member of IEEE and a member of IEEE Computer Society. He is recipient of the prestigious Career Award for Young Teachers from the All India Council for Technical Education (AICTE).
Amitava Mukherjee received his Ph.D. degree in Computer Science from Jadavpur University, Calcutta, India. He was in the Department of Electronics and Telecommunication Engineering at Jadavpur University, Calcutta, India from 1982–1995. Since June 1995, he is a Principal Consultant in PricewaterhouseCoopers Ltd., Calcutta, India. His research interests are in the area of Personal Communication Services, Mobile Ad-hoc Networks, Optical Networks, Combinatorial Optimization and Distributed Systems. His interests also include the Mathematical Modeling and its applications in the fields of Societal Engineering and International Relations. He is author of over 45 technical papers, and few papers, one monograph and one book in Societal Engineering. He is a member of IEEE and IEEE Communication Society.
224 Debashis Saha et al.
Partha Sarathi Bhattacharjee was born in Calcutta in 1963. He received Bachelor of Electronics and Telecommunication Engineering and Master of Electronics and Telecommunication Engineering from Jadavpur University, Calcutta in 1986 and 1988 respectively. He joined the Department of Telecommunication , Government of India as Assistant Divisional Engineer in 1989. He worked on microstrip antennas and components from 1993 to 1997. His current research interest includes personal communication services network (PCSN), third generation cellular/personal communications. He has written several papers in national and international journals. Since 1988, he has been a member of IEEE.
