Optimal assignment of cells in PCS networks - Springer Link

OptJmat Assignment of CeRs in PCS Networks Jie Li, Hisao Kameda and Hideto Itoh Institute of Information Sciences & Electronics, University of Tsukuba, Tsukuba Science City, Ibaraki, Japan Abstract: This paper deals with the assignment problem of cells to switches in a personal communication service network. Three types of

costs in a PCS network are considered in detail: the cost of handoffs, the cost of cabling, and the cost of switching. The optimal assignment problem is formulated as an integer-programming problem. A heuristic algorithm is proposed to obtain an assignment of cells in a PCS network. The proposed algorithm is compared with an existing heuristic cell assignment algorithm. By numerical examination, it is shown that the switching cost has a large effect on the solution of the cell assignment problem. The proposed algorithm obtains much better cell assignment in which the load of each switch is balanced and the total cost of a PCS network is much lower than what is obtained by the existing algorithm that does not take account of the switching cost. If the switching cost is taken into account, it has also been shown that our proposed algorithm achieves substantially the same results as the existing algorithm while requiring much less computation time.

1. Introduction Over the last decade, deployment of mobile communications has been phenomenal. The integration of mobile communications and computing results in a new distributed network, referred to as a mobile computing system. Personal communication service (PCS) networks [1-4] are new mobile communication systems which will enable users to economically transfer any form of information among any desired locations. In a PCS network, a given geographically serviced area is divided into cells. In each cell, there is a base station which is used to communicate with subscribers over some preassigned radio frequencies. Groups of several cells are connected to a Mobile Switching Center (MSC) through which the calls are then routed to the telephone networks. A group of cells which are connected to a common MSC is called an MSC district. Note that in a real PCS network there may be hundreds of MSCs and thousands of cells. It is an important problem to determine how to assign the cells to the MSCs in an optimal manner in the design of a PCS network. This paper studies the optimal assignment problem. A similar model of PCS networks is considered in Merchant and Sengupta [5]. In Merchant and Sengupta [5], however, there are only two types of costs: the cost of handoffs between neighbouring cells and the cost of cabling between cells and their

9 Springer-Verlag London L:d Personal Technologies ( 1997) 1:127 - 134

associated MSCs. Handoff involves location updates in the databases, which requires the execution of a complicated protocol between the two cells when a subscriber moves from one cell to another. In Merchant and Sengupta [5], an interesting enumeration-type heuristic algorithm is also provided to solve the assignment problem. It is known, however, that the implementation of the enumeration type algorithm may require a lot of CPU time for a large-scale problem. Furthermore, since there is no consideration of the cost of switching, the algorithm in Merchant and Sengupta [5] may obtain a cell assignment that would lead to an imbalance of load among the MSCs. In a real PCS network, unbalanced cell assignment should be avoided since it may cause network failure. In this paper, three types of costs in a PCS network are considered: the cost of handoffs, the cost of cabling, and the cost of switching, since the MSCs are also involved in the personal communication service. Another important reason for the introduction of the cost of switching is to balance the load among MSCs in a PCS network. If the load of an MSC reaches its capacity, the cost of switching will become extremely high since the MSC will not work at all under the very heavy load. The optimal solution to the assignment problem shall balance the load among MSCs so as to minimise the total cost of a PCS network. Furthermore, we use a more realistic form for the

============================= ......... .

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Location of

Location of switch 2

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. ..~176

Handoff

...

switch m-1

,/

Location of switch 3

&.

"~ Handoff

" ~ Cell

..........

switch m

*"

Fig. 2. An assignment of cells. % Cell %*. ...... ~ o~176176176176 ~176176176

9 MSC distdct "* . . . . . . . . . . . . . . . . . . . . . . .

..-"~

Fig. 1. The structure of a PCS network.

128

cost of cabling by considering the distance between the cells and their associated MSCs and the number of calls that a cell handles per time unit. The optimal assignment problem is formulated as an integer-programming problem. A heuristic algorithm is proposed to obtain a load-balancing assignment of cells in a PCS network. The proposed algorithm is compared with an existing heuristic cell assignment algorithm proposed in Merchant and Sengupta [5]. It is observed that the proposed algorithm obtains m u c h better cell assignment in which the load of each M S C is balanced and the total cost of a PCS network is much lower than what is obtained by the existing algorithm without taking account of the switching cost. In the case where the switching cost is taken into account, both algorithms achieve substantially the same results while the proposed algorithm requires much less computation time than the existing algorithm.

2. Model and Problem Formulation Consider a PCS network with n cells and m MSCs ( l < < m < < n ) . Assume that the locations of the cells and MSCs are fixed and different. A cell is connected to one and only one MSC. A group of cells that are connected to a c o m m o n M S C

J. Li et al.

is called an MSC district. There is a base station in each cell that is used to communicate with subscribers within the cell. A subscriber may move from one cell to a neighbouring cell, which will cause handoff processing. A neighbouring cell to cell i is the cell that has a common boundary with cell i. The structure of a PCS network is shown in Fig. 1. Figure 2 shows an example of an assignment of cells to MSCs. For a PCS network, there are three types of costs taken into account: the cost of handoffs between cells, the cost of cabling between cells and their associated MSCs, and the cost of switching. These types of costs are each formulated in turn.

2.1. Cost of handoffs We consider two types of handoffs [6-9], one which involves only one MSC district, and another which involves two M S C districts. T h e handoffs that occur between cells that belong to the same M S C district consume much less network resources (therefore, are much less costly) than what occur between ceils that belong to two different M S C districts. Let hij be the cost of handoffs incurred between two cells i and j. Note that hij c a n be estimated by using call statistics on existing systems from vehicular traffic measurements, or from analytic and simulation models [10]. Like [5], we assume that the cost of handoffs hij is not incurred if cells i and j belong to the same M S C district (i.e. ceils i andj are connected to a c o m m o n MSC). So the cost of handoffs in a PCS network is the sum of the cost of handoffs between all two cells i and j i, j = 1, 2 . . . . . n) minus the cost of handoffs that occur in the same MSC district.

We can express Cij(.~i)as a function of the number of calls that cell i handles by

This is given by 7l

~q

EE

hij(1-Oij)

(1)

i=1 j =1 where 0.._.[ 1 if cells i and j belong to the same MSC district, ~J- 1 0 otherwise One may find that it is convenient to have the above definition of function Oij in calculating the function below. We define the assignment function fiij (i = 1, ..., n, j = 1 . . . . , m) as follows: 6.._.[ 1 if cell i is assigned to MSCj, ~J- 1 0 otherwise Since each cell can be assigned to only one MSC, we have the constraint

• Sij

Cij(~i) = Aij + Bij~,i where Aij and Bij are constants and can be obtained through the price table of cables. Therefore, the cost of cabling in a PCS network is given as follows: m

i=1 j =1

2.3. Cost of switching T h e cost of s w i t c h i n g in a P C S n e t w o r k is expressed by

~3iFi(~i)

(3)

i=1 = 1,

for i = 1..... n

j=l Let

~ijk=fiijk6ijk for i, j = 1. . . . , n, and k = 1..... m

where j~i is the total number of calls that M S C i handles per unit time and Fi(fli) is the cost function of switching a call in M S C i. We refer to the i load of M S C i as/3i, which is given by &

Note that ~ijk equals 1 if both cells i and j belong to M S C district k, and 0 if they belong to different M S C districts. T h e n Oij is expressed as m

k=l

2.2. Cost of cabling We formulate the cost of cabling as a function of distances between base stations and MSCs, and the number of calls that a cell handles per unit time. Our formulation of the cost of cabling is more accurate t h a n that in M e r c h a n t and Sengupta [5]. Let dij be the distance between cell i and MSC j. Let ~i be the number of calls that cell i handles per unit time. N o t e that each cell is assigned to only one MSC. T h a t is, for a base station there is only one cable c o n n e c t e d to only one MSC. T h e n the cost of cabling for one cell is

Si,,

for i = 1 , 2 ..... n

j=l where

i = 1 ..... m

j=l To determine Fi(bi) is difficult. It involves both the cost of switching a call in a normal condition and the cost of maintaining an MSC. Denote the call switching capacity of M S C i by pi. It is i m p o r t a n t to n o t e t h a t in a real e n v i r o n m e n t an M S C with a value of bi/J.li x 100% > 80% will n o t work. T h a t is, a heavy load causes an M S C failure, and it results in a large loss in switching service. We use the delay f u n c t i o n of the M/M/1 q u e u i n g m o d e l [11] multiplied by a weight (~z) as the cost function as follows:

Si(~i)-l.li_~ " _ . _ _ _aK _ ,~i < J.li,

i=1

..... m

To sum up, we express the total cost TC in a PCS network as the sum of the above three types of COSTS:

;q

;q

n

rfl

we = E E hij(1-Oij) + E E cij(~.i)dijSij +

m

s

~i=Zl~,j~ji

Cij(l~,i) is the cost of cabling per kilometre.

i=lj=l

i=ij=1

m

E•iFi(3i)

(4)

i=1

Optimal Assignment of Cells in PCS Networks

T h e goal of the optimal assignment of cells is to minimise the total cost. T h a t is

n

Minimise

m

(7)

s163 i=1 j =1

n

Minirnise

n

T C = E ~ hi,(1-Oij) +

(5)

i=1 j =t

with respect to the variables 6 0 (i = 1..... n, j = 1, .... m), subject to n

i=1 j = l

~__~2~i6ij=13i~1 M

X2+X5+X6< ~2

+ ),.4 < p l

Fig. 3.

Finding

an

Note that cell 5 is assigned to MSC 2 but not MSC 1 although M S C 1 is the nearest one to cell 5, since)~l + X4 + X5 > t-t1 and X2 + X5 + X6 < bt2. M S C 2 is the second nearest one to cell 5. Similarly, cells 2 and 6 are assigned to MSC 2, cells 3 and 7 are assigned to MSC 3. With an initial feasible solution, we improve the solution by taking the cost of handoffs and cost of switching into account.

initial feasible assignment. MSC should be assigned to the MSC in the algorithm with practical parameter settings. It is impossible to have such a case that set PC is empty. Also note that it should not have the empty set PC from the practical point of view. For the sake of completeness, we consider the case that set PC is empty in the algorithm.) .

3.2. The proposedalgorithm .

Initialisation. Obtain an initial feasible assignment to problem (5) by using the above algorithm.

Finding a set of possible changes for the assignment. Let PC be the set of all cell pairs (i, j) where cells i andj are neighbours and belong to different MSC districts. If the set PC is empty (i.e. PC = 0), then STOP. (The empty set PC means that only one MSC is necessary in a PCS network. In this paper, we consider a real PCS network that includes a number of MSCs (m >> 1) which locate in different places. Notice that the cell that contains an

Let hi*j* = max hij. Let a l be the assignment (~,j)~ PC that assigns cell i* to the MSC district that j* belongs to. Let A2 be the assignment that assigns cell j* to the M S C district that i* belongs to. If assignment A~ is feasible, set TC1 to be the total cost of assignment A1; else set TC~ to be a large enough value M (e.g. M = T C * + 105). Also if assignment A2 is feasible, set TC2 to be the total cost of assignment A2; else set TC2 to be the large enough value M.

Let T C * be the total cost of the assignment. .

~.,3+X7 T C * and TCz > T C * then PC = PC - {(i, j)}. If the set PC is empty (i.e. PC = O) then STOP; else go to step 3. If TC1 < T C z then 6iki = O, 6ikj = 1 (i.e. c h a n g e t h e a s s i g n m e n t of cell i to the MSC that cellj is connected to), T C * = TC1;

Optimal Assignment of Cells in PCS Networks

131

Boundary

MSC district

MSiCtJ

Possible boundarychanges

and the M&S algorithm in terms of the computation time requirements and the solution obtained by these two algorithms. We have programmed the M&S algorithm and the proposed algorithm in the C language and have run these two algorithms on a SPARC-20 workstation. In our numerical examination, we assume that the base station for each cell is at the center of the cell. We considered a PCS network that consists of six MSCs. The numbers of cells in the PCS network considered here are 50, 100, 150 and 200. The number of calls that cell i handles per unit time, Zi, is generated according to an exponential distribution with mean

Fig. 4. Improving the solution by changing the boundary.

1 ~6 - ~ , # i x 70%

else 6jkj = O, &jki = 1 (i.e. change the assignment of cell j to the MSC that cell i is connected to), TC* = TC> 5.

v

132

Repeat steps 2, 3 and 4.

In the proposed algorithm, step 2 finds a set of possible changes for the assignment by finding all pairs of two neighbouring ceils i and j such that they belong to different MSC districts. Steps 3 and 4 try to find the best feasible change for the assignment. We say that a change in the cell assignment is feasible if the resulting assignment is feasible. Note that the proposed algorithm tries to minimise the total cost function including the cost of switching. That is, we can obtain a balanced assignment of cells by using the algorithm which we will show in the next section. Figure 4 illustrates the boundary between MSC districts i and j, and the possible boundary changes for the assignment. The algorithm tries to find the best change that minimises the total cost function.

4. Numerical Examination Merchant and Sengupta [5] have proposed another interesting heuristic algorithm (we will call it the M&S algorithm) to solve the cell assignment problem without considering the cost of switching. The M&S algorithm is an enumeration-type heuristic algorithm, i.e. it obtains a solution by using enumeration. In Merchant and Sengupta [5], the M&S algorithm was compared with the standard integer-programming algorithm. It is shown that M&S can be used to obtain a good solution within a shorter CPU computation time than the standard integer-programming algorithm. Here, we present a comparison of the proposed algorithm

J. Li et al.

"~" 7""7'=

where n is the number of cells. The call switching capacity of MSC i,/.zi, is generated from a uniform distribution between 50 and 150. The handoff cost of each pair of two neighbouring cells is taken to be a random number from a uniform distribution between 0 and 30. For the cost of cabling, we chose A i j = 3 0 , B i j = 2 ( i = 1 ..... n, j = l, ..., m). The distance between two neighbouring ceils was set to be 1. We first consider the impact of the switching cost function on the cell assignment. Note that in Merchant and Sengupta [5] there is no consideration of switching cost. We found that the proposed algorithm obtained much better cell assignment in which the load of each MSC is balanced and the total cost of a PCS network is much lower than in the assignment obtained by the M&S algorithm, which does not take account of the switching cost. We also note that we can apply the M&S algorithm to obtain a solution for the cell assignment problem (5) considering switching cost. For the sake of brevity, we only give the data that show the comparison of the M&S algorithm and the proposed algorithm for the case that the switching cost is taken into account. Figure 5 illustrates the utilisation of each MSC, pi> (= ~i/I..li), for the case of six MSCs and 150 cells. It shows that utilisation of MSCs tends to be uniform as the weight c~of the switching cost increases. Figure 6 gives the variance of utilisation of MSCs (s2) vs. weight of switching cost (00. 6

(s2 =

6

= s 0 /6) i=l

i =1

Both the two figures show that loads among MSCs become more balanced as the weight (x increases

switch1

[~

switch4

I@B s w i t c h 5

switch2

~

switch3

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0.8

0.7

03

0.6

0.4

0.2

Proposed

M&S

Proposed

o~=1

M&S

Proposed

o~=4

M&S

Proposed

a =16

M&S

ot =64

Fig. 5. Utilisation of MSCs vs. weight of switching cost (number of cells = I50, number of MSCs = 6).

M&S . . . . . Proposed - -

M&S -PropOsed . . . . .

004

8000

i &03

6000

~ g o.02

4000

.-!

2000

I

2

4

I

t

8 16 Weigh of SwitchingCost (~)

32

64

0 50

16D

100

600

Num 01 Cells

Fig. 6. Variance of utilisation of MSCs vs. weight of switching cost (nmnber of cells = 150, number of MSCs = 6).

Fig. 7. Comparison of c o m p u t a t i o n time requirements (a = 40, number of MSCs = 6).

(i.e., the switching cost increases). It was also found that there was not much difference among the load of each MSC in the results. Since the cell assignment problem (5) is NPcomplete, the computation time requirement is an important metric in comparing the performance of algorithms. In Fig. 7, we compare the performance of the proposed algorithm and the M&S algorithm, parameterised by the number of cells in the problem with six MSCs. The weight of switching cost was set to be 40 (a = 40). Figure 7 shows that the M&S algorithm requires much more CPU computation time (in sec. 5 times more) than the proposed algorithm for all numbers of cells in the problem. For the case considered here, Fig. 8 shows that there is not much difference between experiments were conducted to examine different parameter settings. They showed results similar to those of the above example.

Conclusion We have studied the optimal assignment of cells to MSCs in a PCS network. A detailed cost model of a PCS network for the assignment problem has been given. The factor of load balancing among MSCs is taken into account in the cost model. The assignment problem has been formulated as an integer-programming problem. A heuristic algorithm has been provided to solve the optimal assignment problem. The proposed algorithm is compared with the heuristic cell assignment algorithm proposed in Merchant and Sengupta [5]. By numerical examination, it has been shown that the switching cost has a significant effect on the solution of the cell assignment problem. It was found that the proposed algorithm obtained much better cell assignment in which the load of each MSC is balanced and the total cost of a PCS network is

Optimal Assignment of Ceils in PCS Networks

handoff cost

6000

cabling cost switching cost 5000

4000 O

o

I~ 3000

2000

1000

~m Proposed M&S

Proposed M&S

Num of Cell = 50

Num of Cell = 100

Proposed M&S Num of Cell = 150

Proposed M&S Num of Cell = 200

Fig. 8. Total cost vs. number of cells (number of MSCs = 6).

r

! 34

much lower than what obtained by the M&S algorithm, which does not take account of the switching cost. If the switching cost is taken into account, it has also been shown that the proposed achieves substantially the same results as the M&S algorithm while requiring much less computation time. Our solution is based on the given number, location and capacities of MSCs. For practical use, it may also be necessary to determine these parameters optimally. As a whole, it may be a significant problem to determine the number, location and capacities of MSCs as well as the cell assignment. This problem is also NP-complete. We are currently investigating solutions to this problem.

Acknowledgements The authors thank Professor Huizinga and the anonymous reviewers for their helpful suggestions that improved the paper.

References 1. Cox DC. Personal communications: a viewpoint. IEEE Commun Mag 1990; Nov: 8-20. 2. Homa J, Harris S. Intelligent network requirements for personal communications services. IEEE Commun Mag 1992; Feb: 70-76.

J. Li et al.

3. Grag VK, Wilkes JE. Wireless and personal communications systems. Prentice-Hall, Englewood Cliffs, NJ 1996. 4. Tuttlebee WHW. Cordless personal communications. IEEE Commun Mag i992; Dec: 42-53. 5. Merchant A, Sengupta B. Assignraent of cells to switches in PCS networks. IEEE/ACM Trans Networking 1995; 3(5): 521-526. 6. Gibson JD (ed in chief). The mobile communications handbook. CRC Press/IEEE Press, Boca Raton, FL, 1996. 7. Goodman DJ, Pollini GP, Meier-Hellstem KS. Network control for wireless communications. IEEE Commun Mag 1992; Dec: 116-124. 8. Lin Y. Mobility management for cellular telephony networks. IEEE Parallel Distrib Technol 1996; 4(4): 65 73. 9. Yacoub MD. Foundations of mobile radio engineering. CRC Press, Boca Raton, FL 1993. 10. Alonso E, Meier-Hellstern KS, Pollini GE Influence of cell geometry on handover and registration rates in cellular and universal personal telecommunications networks. In: 8th ITC Specialist Seminar in Universal Personal Communications, Genova, Italy, 1992. 11. Kleinrock L. Queueing systems. Vol 2: Computer applications. Wiley, New York 1976. 12. Garey MR, Johnson DS. Computers and intractability: a guide to the theory of NP-completeness. Freeman, New York 1979.

Correspondence and offprint requests to: Jie Li (JieLi@is. tsukuba.ac.jp.) Institute of Information Sciences and Electronics University of Tsukuba, Tsukuba Science City, Ibaraki 305, Japan.