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Assignment of Cells to Switches in PCS Networks Arif Merchant and Bhaskar Sengupta
Abstract-In this paper, we consider a problem of network design of personal communication services (PCS). The problem is to assign cells to the switches of a PCS network in an optimum manner. We consider two types of costs. One is the cost of handoffs between cells. The other is the cost of cabling (or trunking) between a cell site and its associated switch. The problem is constrained by the call volume that each switch can handle. We formulate the problem exactly as an integer programming problem. We also propose a heuristic solution for this problem and show that it performs extremely well.
I SwPchI
I. INTRODUCTION
D
URING the last decade, there has been a tremendous growth in the deployment of mobile communication systems and in the future, this technology is likely to lead to universal personal communications [2], [4], [ 101. Although there are many differences between the two, they do make use of some concepts which are very similar. For example, both divide a given geographical territory into cells and have to solve problems such as handoffs and locates. In this paper, we address a problem which is currently faced by designers of mobile communication service and in the future, it is likely to be faced by designers of personal communication service (PCS). In PCS networks, each cell has an antenna which is used to communicate with subscribers over some preassigned radio frequencies. Groups of several cells are connected to a switch through which the calls are then routed to the telephone company network. The signal strength from a moving subscriber is monitored by the cells that are close enough to the subscriber, (i.e., the signal strength received by a cell must be higher than some threshold in order for it to be deemed significant). Note that significant levels of signal strength could be received by cells that are connected to different switches. This is illustrated in Fig. 1. Cells A and B are connected to switch 1 and cells C and D are connected to switch 2. Imagine that a subscriber is currently talking to someone and the call is being routed through cell B and switch 1. Assume also that the signal strengths at cells A and C are significant. Since cell C is adjacent to cell B , the signal strength information from cell C is fed to switch 1 through a signaling network (not shown in the figure). Thus, switch 1 receives all the signal strength information about this call and is able to figure out the cell which has the highest signal strength. If the subscriber moves from cell B to cell A, switch 1 will perform a handoff for this Manuscript received May 17, 1994; revised June 2, 1995; approved by IEEWACM TRANSACTIONS ON NETWORKING Editor K. Sabnani. This paper was presented in part was presented at IEEE INFOCOM’94, Toronto, Ont., Canada, June 14-16, 1994. The authors are with C&C Research Laboratories, NEC USA, Princeton, NJ 08540 USA (e-mail:
[email protected] and
[email protected]). IEEE Log Number 9414828.
(21 Fig. 1. Handoff from B to C is more expensive than from B to A.
call. This handoff is relatively simple and does not involve any location updates in the databases that record the position of the subscriber. The handoff also does not involve any network entity other than switch 1. Now imagine that the subscriber moves from cell B to cell C. Then the handoff involves the execution of a fairly complicated protocol between switches 1 and 2. In addition, the location of the subscriber in the databases has to be updated. There is actually one more fact that makes this type of handoff difficult. If switch 1 is responsible for keeping the billing information about the call, then switch 1 cannot simply remove itself from the connection as a result of the handoff. In fact, the call continues to be routed through switch 1 (for billing purposes). The connection, in this case, is from cell C to switch 2, then to switch 1 and finally to the telephone network. The brief description of the two types of handoffs given above is by no means complete. Its intention is to bring home to the reader one simple fact. There are two types of handoffs, one which involves only one switch and another which involves two switches. The handoffs that occur between cells that are connected to different switches consume much more network resources (therefore, are much more costly) than the handoffs that occur between cells that are connected to the same switch. For a more complete description of handoffs, the reader is referred to Yacoub [15]. Referring to Fig. 1, suppose that we knew that the frequency of handoffs between cells A and B was very high and that between B and C was not so high. Then it would make sense to connect cells A and B to the same switch (if possible) in order to reduce the cost of handoffs. This is the primary motivation for the cell assignment problem. We are given a group of cells and a group of switches (whose locations are known). The problem is to assign the cells to switches in an optimum manner. We would like to do the assignment in an attempt to minimize a cost criterion. The cost has two
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components. One is the cost of handoffs which involve two switches and the other is the cost of cabling (or trunking). (Assignment based on other criteria, such as minimizing paging messages and location updates, is considered in [ 111.) Based on the discussion given in the previous paragraph, we assume that the cost of handoffs involving only one switch is negligible; the case when such costs are significant can be shown to be mathematically equivalent, as we show later. Since the call handling capacity of each switch is limited, this constraint must be taken into account. If this were not done, one would obtain trivially incorrect assignments (like assign all cells to a single switch). Finally, we allow the cells to be single homed or dual homed. This means that we allow each cell to be connected to one or two switches, depending on needs. We allow dual homing because calling patterns at different times of the day could be different. As a result, one cell may be advantageously connected to one switch in the morning and a different one in the evening. This may allow one to reduce the cost of handoffs (and also the cost of switches) by increasing the cost of cables. The problem definition with dual homing allows one to study such tradeoffs in a natural manner. This paper is divided into four more sections. In Section 11, we formulate the cell assignment problem as an Integer Programming problem. In Section 111, we describe a heuristic method to solve this problem. In Section IV, we present some numerical results, and in Section V, the conclusions and some possible extensions.
Since each cell can be assigned to only switch, we have the constraint m.
k=l
Further, the constraint on the call handling capacity is n
i=l
Also, the amortized cost of cabling is E:=l Er=lC i k x i k . The handoff costs are a little more difficult to handle. To this end, we define an additional variable
Thus, z ; j k equals 1 if both cells i and j are connected to a common switch IC, otherwise it is zero. Further, we define m
(4) k=l
Thus, yij takes a value of 1 if both cells i and j are connected to a common switch and 0 if they are connected to different switches. With this definition, it is easy to see that the cost of handoffs per unit time is given by hij(1 - y i j ) . This, together with our earlier statement about the amortized cabling cost, gives us the objective function:
cyZl
n
Minimize: 11. INTEGERPROGRAMMING FORMULATION Let there be n cells to be assigned to m switches. We assume that the location of the cells and switches are fixed and known. If cells and j are assigned to different switches, then a cost is incurred every time a handoff between cells i and j occurs; let h;; be the cost per unit time of the handoffs that occur between cells i and j ( i ,j = 1,.. . , n). Thus, haj is proportional to the frequency of handoffs that occur between cells i and j , which, we assume, is known. The rate of handoffs between cells can be estimated using call statistics on existing systems, when available, from vehicular traffic measurements, or from analytic and simulation models (see [9], [l], [14]). Let c;lc be the amortized cost of cabling per unit time between cell i and switch k (i = 1,.. . ,n;IC = 1 , . . . , m). Let A; denote the number of calls that cell i handles per unit time and let h f k denote the call handling capacity of switch k . Our objective is to assign each cell to a switch so as to minimize the total cost per unit time. The total cost per unit time has two components. One is the handoff cost per unit time due to handoffs between cells that are connected to different switches. The other is the amortized cost of cabling between the cells and the switches. The optimization has to be carried out in way such that the call handling capacity of each switch is not violated. To formulate this as an integer programming problem, let us define the following variables. Let
E
m
i=l k = l
n
CikXik
n
+
hi,(l - yij).
(5)
r=l j=1
We now have a mathematical programming problem defined by the objective function (5) and the constraints (1)-(4) and we also need the additional constraints xik
= 0 Or 1 for i = 11. . ' 1 n and k = 1,' ' . 1 m. (6)
-_I
xik
=
c
1 if cell i, is assigned to switch k 0 otherwise.
Note that we do not need any additional integrality or even nonnegativity constraints on the variables z i j k and yij since they are completely specified by (3) and (4) respectively. The difficulty with the formulation given above is that the constraint (3) is not a linear one. So this mathematical programming problem is not amenable to solution by standard techniques, such as a combination of Linear Programming and the Branch and Bound method [7]. However, it is possible to convert this problem to a linear integer program by adding more constraints. Specifically, it is possible to replace the constraint (3) by the group of constraints for i, j = 1, . . . ,n and k = l > . . . , m .
Note that it is easy to verify that (3) implies (7)-(10)and vice vemu, so we omit this proof for the sake of brevity. Now, the objective function (5) together with the constraints (l), ( 2 ) , (4),(6), (7)-(10) constitute a valid linear integer program and
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MERCHANT AND SENGUFTA: ASSIGNMENT OF CELLS TO SWITCHES IN PCS NETWORKS
it can be solved by a combination of linear programming and the branch-and-bound algorithm. We now discuss the dual homing problem. In this version of the problem, there are two different times of the day during which the call volume and the handoff patterns are different. Thus the definition of A, and h,, remain the same as before, except that they are applicable to one time of the day. In addition, at some other time during the day, the call volume per unit time on cell i is assumed to be A: and the cost of handoffs per unit time from cell z to cell 3 is h:,. The definition of b f k and c& remain the same as before. This time we allow upto two connections per cell. If it is economica! to have some of the cells to be dual homed, the integer programming solution should indicate this. To facilitate the discussion, we say that there are two assignment patterns that we are seeking, one corresponding to the cell connections for each of the two call volume and handoff patterns. The first assignment pattern is pretty much the same as the single homed problem. The second assignment pattern is also a single homed problem which uses the primed versions of the parameters of the problem. The only quirk is that if a cell is to be connected to the same switch in both patterns, its cable cost should not be doubled. For pattern 1, let x,?, y,? and Z t 3 k be defined as before and they need to satisfy the constraints (I), (2), (4), (6), (7)-(IO). For pattern 2, let us define the corresponding variables x i j , y,: and z : , k . These variables must also satisfy the constraints (l), (2), (4), (6),(7)-(10) except that A, is replaced by A: for all 2. In order to ensure that the amortized cable costs are not counted twice in the event that a cell is connected to the same switch in both patterns, we define W&
= ~ & v x : , for z = 1,... , n and k = I , . . . , m (11)
where the “V” symbol means the “or” operation. With this definition, we can now specify the objective function as Minimize: n
m
i = l k=l
n
n
n
is called feasible if the resulting assignment is feasible. Let us first find a method of generating a feasible cell assignment. To do so, we order the cells in the decreasing order of call volume (A,). There are m stages in this algorithm. At stage k , the assignment of the first k - 1 cells has already been done and cannot be changed. For k = 1, ’ . . , n, we assign cell Ic in a feasible manner to that switch which minimizes the total cost of handoffs and cabling for the first k cells. This is a greedy or myopic heuristic. We next extend the method given in the last paragraph, so that we find b feasible solutions, instead of just one. To do so, let us say that we already have b feasible solutions at the (k - 1)st stage of the problem. For k = 2, this is done trivially. Now in the kth stage, we have m choices of assigning the kth cell to the various switches. Thus, we have a total of b(m - 1) possibilities of assigning k cells to m switches. Of these, we first throw out all the possibilities which are infeasible. Then, from the remaining possibilities, we retain b possibilities which have the minimum cost. We continue in this fashion for IC = 1,. . . ,n. The heuristic now selects the best solution from the set of up to b solutions and uses the tuning mechanism to improve the result in a sequential manner. The tuning mechanism repeatedly applies the feasible move which gives the best improvement in the objective function, until a heuristic optimality criterion is met. In our numerical examples, we chose b = 10. Initial Cell Assignment: 1) Order the cells in decreasing order of call volume. We start with a single empty assignment. 2 ) Cells are assigned one at a time in m stages. For IC = 1 , 2 ; . . . , m . a)
n
i=l j=1
(12) Although this defines a valid mathematical program, it is still not a linear integer program on account of the constraint (1 1). As before, the “or” operation in (1 1) can be converted to a set of linear constraints as follows:
Again we omit the proof that (11) is equivalent to (13)-(16). This completes the integer programming formulation of the single and dual homed problems. 111. A HEURISTICMETHOD
In this section we describe a heuristic method for solving the single-homing problem. We call a cell assignment feasible if it satisfies the call handling capacity constraints ( 2 ) ;a change in the cell assignment
523
Extend each partial assignment under consideration by adding all possible assignments of the kth cell. Discard all assignments that violate the call handling capacity constraints. If no assignments remain, the algorithm fails. If b or less partial assignments remain, keep them all, else keep the b best assignments, based on the cable costs and handoff costs of the IC assigned cells only.
3 ) Finally, return the best of the assignments found. Using this method to find a good initial solution, the improvement algorithm reduces the objective function through feasible moves. A feasible move is a move that leads to a feasible cell assignment which satisfies the call handling capacity constraints. Single Homing Heuristic: 1) Find an initial feasible partition, using the method above. 2 ) Repeatedly perform improvement passes on the initial partition as follows until a pass does not reduce the value of the objective function, then exit. a) b)
Mark all cells as unlocked. Find the best feasible move: a cell i and a switch k , such that of all feasible moves, moving cell i to switch k reduces the objective by the greatest amount.
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c)
d)
Assign cell i to switch IC. Mark cell i as temporarily locked. Note the current cell assignment pattern. Repeat (b) and (c) until no move is found. In the sequence of cell assignment patterns generated in this pass, select the one with the lowest objective value, and reset the current cell assignments to it. This ends the pass.
We note that Kemighan and Lin [5] and Sanchis [12], [13] have studied other heuristics for graph partitioning, particularly in the context of VLSI design. In [8], we have compared the above heuristic to theirs for the cell assignment problem and shown that our results are substantially better. A. Dual Homing Our heuristic for a dual homing cell assignment problem is based on solving a series of single homing problems using any of the methods we have described. The intuition behind this method is that we solve a single homing problem based on one of the patterns in the dual homing problem, then temporarily fix the cell assignments based on this. Now we may make a second set of cell assignments based on the second pattern. However, if a cell is assigned to the same switch in the second assignment as in the first, then no cable cost need be added for it, since it was already accounted for in the first assignment. Once we have the assignments based on the second pattern, we temporarily fix these assignments instead, and redo the first problem with a similar modification. This is repeated until the assignments give no further improvement in the objective function. Dual Homing Heuristic: Form the single homing cell assignment problem incorporating the first pattern using (A,, h,, , h f k , czk) for i , j = 1 , 2 , . . .,m, and a second single homing cell assignment problem using (A:, h:J,h f k , C & ) for i , j = 1 , 2 , . . . , m. Solve the two problems using one of the algorithms given earlier. Let Q be the problem whose solution A has the lower objective value; let Q’ denote the other problem. Let Q1 be the single homing problem identical to Q‘, with the modification that if A assigns cell i to switch IC, then the cable cost C t k is zero in & I . Solve Q l , and call its solution A’. Similarly, let Q2 be the single homing problem identical to Q, with the modification that if A’ assigns cell i to switch IC, then the cable cost C,k is zero in Q2. Solve Q2, and reset A to be the solution. Repeat steps (3) and (4) until the assignments give no further improvement in the objective function. The assignments in A and A’ combined form a solution to the dual homing problem.
IV. NUMERICALRESULTS We tested the integer programming formulation and the heuristic methods on several hundred single and dual homing
problems with varying numbers of cells, switches and different patterns of call volume. In this section we discuss how these problems were generated, and the results of the tests. A. Generation of Test Problems The test problems were generated by assuming that cells lie on a hexagonal grid of roughly equal dimensions in two axes (see Fig. 1). We assumed that the antenna for each cell was at the center of the cell. Switches were uniformly distributed randomly over all the cells, and were also assumed to be at the center of the cell; however, switch assignments in which two or more switches were very close were then rejected, and new assignments made. The cable cost between a switch and a cell antenna was taken to be proportional to the geometric distance between the two. The rate of calls originated in each cell was generated from a Gamma distribution with mean one and coefficient of variation 0.25. The holding time of a call within each cell was taken to be exponentially distributed with mean one. If a cell had IC neighbors, then we divided the range ( 0 , l ) into IC 1 intervals by selecting IC random numbers from a uniform distribution between 0 and 1. The length of the ith interval represented the probability of handoff from the cell to its ith neighbor for i = 1 , 2 , . . . , IC at the end of the holding time. With probability equal to the length of the IC 1st interval, the call ends. To find a consistent set of call volume and handoff rates, we formed a Jackson network (see Kleinrock [6]) by treating cells as infinite server nodes, call originations as customer arrivals, handoff probabilities as node transition probabilities, and call holding times in each cell as the average service requirement of a customer at the corresponding node. This was solved to give the queue lengths, which translated into call volume for the cells (Ai), and the rate of customers moving between cells, which gave handoff rates (proportional to htj). We assumed switch capacities to be equal, based on an overall excess capacity of between 10 and 50%.
+
+
B. Experimental Results
The heuristic methods for the single and dual homing problems were coded in the C language to run on a Sun Sparc2 machine. The integer programming forms of these problems were solved by using the CPLEX mixed integer programming software. The problem of finding a feasible cell assignment is mathematically equivalent to the bin-packing problem, which known to be NP-complete [3]. This implies that for any solution technique (integer programming or heuristic), we are not guaranteed a feasible solution in a reasonable time. In Table I, we compare the performance of the single homing heuristic method against the integer programming method, parametrized by the number of cells in the problem. The integer programming method failed to find any solution after the problem size goes beyond 35 (because the constraint set grew too large for the memory available). The heuristic method, on the other hand, found feasible solutions consistently for all the problems we tried. When the Integer programming method worked, the CPU time requirement was high, while
MERCHANT 4 Y D SENGUPTA: ASSICNMEN'T OF. CELLS TO SWITCttES I N PC'S NE1 WORKS
TABLE I IN7 EGER PROGRAMMING AND HEURISTIC METHODS i\i SIW;IEHOMIVG PROHLI:F*?S ( * Indicates inadequate data).
PERFORMANCE OF
# cell-
# problems
15
i 4
10 25
74 74 i4 i4
30
35 40 45 50 55 60 65 70 75 80 85
90 95
100
% Feasible solutions IF Heuristic ____
100 100
100
CPU time (7e.I IP 65.32 9a 74
100
100 100
100
100
261 64 7RQ 96
loo
856 47
sa
29 9
0 0
9
n
9
0
9
0
9 9 9
0
'
100 100 100 100 1no
*
nU
* *
0 54 0 64
*
0
loo 100 100 ino 100 100
0
loa
* * * *
0
100
*
0
0 0 0
9 9
9 9 9
Heuristic 0 05 o 09 0 11 0 16 n 20 030
I
Beurlst,c Ob. O p t i m u m (IP) 1 on 102
101 1 02
0 43
*
Oil
*
088 1 OG
122 1.36 153 183 1.96
*
,
*
TABLE 11 PERFORMANCF OF IUTEGER PROGRAMM1IVG /!PI AND Hf:URISTIC METHODS IN DLiAI-HoMINc; k 0 8 L l : . \ l 5 ( * Indicates inadequate data).
# cells
# problems
30 35
72 72 2 72 72
40 45 50
63 63 63
55
63 63
15 20 25
60 65 i0 75
80 85 90 95
100
63 63 63
63 63 63 63 63
% Feasible soiutlons IP 96
-c
100
a
100
0
loll 100 100
0 0 0 0 0 0 U
0 0 0 0 0
CPC' time IP 49346 6
*
IOU
*
100 1 00 100 100
n n
100
0
100
134 145
I56 1 i,'j
IOU LOO 1 no 1U 0 100 100
(sec)
*
H~~~~~~~~ ob. Optimum (IP)
Heuristic
1951 2 27 2 47 2 95
* *
*
9 11
*
V. CONCI,USIONS
* *
3 34 3 66
4.71 5il 6 30 681 7 84
each cell is noted in the cell. Cable costs are assumed to be proportional to distance between the switch and cell center. The partition shown is optimum, and was produced by both the Integer Programming method and the single-homing heuristic method.
1.115
4 19
*
Fig. 2. Partition produced by single homing heuristic method: objective value = 107.3 (optiniuml.
* 1
*
9.52
the heuristic method required very little time. The solution found by the Integer programming method is guaranteed to be optimal in the objective value. however, the heuristic method came close-on the average, within 2% of the optimum. For the dual homing problem (Table II), the integer programming formulation could be used only for problems with up to 15 cells. since the memory requirement grows very quickly with the number of cells considered. As in the case of single homing, the CPU time required for solving the integer programming formulation was very high when the solution was possible. Using the heuristic method given above. a solution could be found very rapidly, and had an objective value within 5% of the optimum on the average, for the cases where an optimum could be computed using the lnteger Programming method. C. An Emmple
Fig. 2 shows the assignment of cells in one problem with IS cells and 2 switches. The handoff cost for each pair of adjacent cells is marked next to the edge connecting the centers of the cells, where the cell antennas are assumed to lie. Switch 1 , connected to the unshaded cells. is located in cell S , and switch 2. connected with the shaded cells. is in cell 8. Both switches have a capacity of 33.1 1 , and the call volume associated with
In this paper, we have solved a problem of optimum assignment of cells to switches in a PCS network. We considered an exact solution by integer programming and compared it to a heuristic method. For problems of size 30 and IS, respectively, or smaller, the integer programming method is often able to find the optimum solution within a few hours of CPU time, Since we know that the solution is optimal, this solution technique may be preferable for small problems. For all problems (small and large), our heuristic methods consistently found feasible solutionf with objective values close to the optin~umsolution. ReForc we end this section, we discuss two variants of the problem. In the first variant, we consider two types of handoff costs. Lct H,, denote the handoff costs incurred i f i and ,j are assigned to the same switch. This cost is incurred if ! j , / = 1. Let denote the handoff costs if i and ,j are assigned to different switches. This cost is incurred if y,./ = 0. For this problem. the objective function for the single homed problem (5) will be replaced by Mininiizi::
1=1 k - l
i = l /=I
r=1.;=1
Clearly. by letting hi, = H,, - H,.,, we can rewrite (17) as (5) plus ;1 constant. Therefore, this variant is mathematically equivalent to the earlier integer programming formulation (for both the single homed and dual homed versions of the problem 1. The second variant that we consider is concemed with load balancing. Note that in the problem considered so far, we have included 110 consideration of balancing the load between the switches. except that the call handling capacity constraint (2) applies to the switches. As a result of this, some switches may be close 10 fully loaded and others may be lightly loaded,
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leading to an uneven distribution of the load on the switches. For a practical problem, if such an assignment is undesirable, it can be improved easily, to give better load balancing. TO do SO, we first look at the solution of the original problem and if it out to have characteristics, we solve a new problem by reducing the capacity Of each switch by a factor a. For instance, Mk for each IC is multiplied by a = 0.9 and the problem is again‘ The either exhibit better load balancing characteristics (at a higher cost) or be infeasible. If the former, the same sensitivity analysis can be carried out by dropping the value of a further. If the latter. the value of cy must be increased to allow a feasible solution. By this rather simple technique, the user can generate a series of solutions and choose one which appears to give the best compromise between cable costs, handoff costs and load balancing.
REFERENCES E. Alonso, K. S. Meier-Hellstem, and G. P. Pollini, “Influence of cell geometry on handover and registration rates in cellular and universal personal telecommunications networks,” in 8th ITC Specialist Sem. Universal Personal Commun., Genova, Italy, 1992. D. C. Cox, “Personal Communications-A viewpoint,” IEEE Cnmmim. Mag., pp. 8-20, Nov. 1990. M. R. Garey and D. S. Johnson, Computers and Intmcrczhilih; A Guide to the T h e o p of NP-Completeness. New York: W. H. Freeman, 1979. T. Hattori, “Personal communication-Concept and architecture,” in Proc. ICC’90, 1990, pp. 333.4.1-335.4.7. B. W. Kemighan and S. Lin, “An efficient heuristic procedure for partitioning graphs,” Bell Syst. Tech. J . , vol. 49, pp. 291-307, 1970. L. Kleinrock, Queueing Systems I: T h e o n . New York: Wiley, 1975. G. L. Nemhauser and L. A. Woolsey, Integer and Comhinarorial Optimization. New York: Wiley, 1988. A. Merchant and B. Sengupta, “Multiway graph partitioning with applications to PCS networks,” NEC USA, Tech. Rep. TR-93-CO024-5021-1, 1993. S. Nanda, “Teletraffic models for urban and suburban microcells: cell sizes and handoff rates,’’ in 8th ITC Specialist Sem. Unit~ersalPersonul Conzmun., Genova, Italy, 1992. R. Steele, “Deploying personal communication networks,” IEEE Commu17. Mag., Sept. 1990.
[ I 11 B. Samadi and W. S. Wong, “Optimization techniques for location area partitioning,” in 8th ITC Specialist Sem. Universal Personal Cornmun., Genova, Italy, 1992. 1121 L, A. Sanchis, “Multiple-way network partitioning,” IEEE Trans. Comput., vol. 38, pp. 62-81, 1989. [ 131 -, “Multiple-way network partitioning with different cost functions,’’ Colgate Univ., Tech. Rep. TR-1992-1. Feb. 1992. [14] I. Seskar, S. v. Maric, J. Holtzman, and J. Wasserman, “Rate of location area updates in cellular systems,” in Proc. IEEE Veh. Techno). Conf, Denver, CO, May 1992. [ 151 M. D. Yacoub, Foundations of Mobile Radio Glgineering. Boca Raton, FL: CRC Press. 1993.
Arif Merchant received the B.Tech. degree in computer science from the Indian Institute of Technology, Bombay, India, in 1984 and the Ph.D. degree in computer science from Stanford University in 1991. He was with the IBM Thomas J. Watson Research Center, Yorktown Heights, NY, from 1991 to 1992. He is currently a Research Staff Member at the NEC Computer and Communications Research Laboratories in Princeton, NJ. His research interests include multimedia server architectures, disk arrays, personal communications networks, computer architecture, and performance modeling.
Bhaskar Sengupta received the doctoral degree from the Operations Research Department of Columbia University in 1976. He was a faculty member in the Department of Applied Mathematics and Statistics in the State University of New York at Stony Brook from 1976 to 1981. Subsequently, he worked in AT&T Bell Laboratories and was a Distinguished Member of Technical Staff when he left in 1990. He has been an Adjunct Faculty Member at Columbia University and has consulted with many companies in New York. He is currently the Head of the Performance Analysis Department at the C&C Laboratorie s, NEC USA. His research interests are in queueing theory and its applical.ions to computer and communications problems.
