An efficient load-balancing algorithm based on a two-threshold cell ...

Computer Communications 23 (2000) 452–461 www.elsevier.com/locate/comcom

An efficient load-balancing algorithm based on a two-threshold cell selection scheme in mobile cellular networks Yongbing Zhang a,*, Sajal K. Das b a

Institute of Policy and Planning Sciences, University of Tsukuba, Ibaraki 305-8573, Japan Department of Computer Science and Engineering, University of Texas at Arlington, Arlington, TX 76019-0015, USA

b

Abstract In this paper, we design and analyze an efficient load-balancing algorithm for channel assignment in mobile cellular networks. Based on a two-threshold cell selection scheme, our algorithm employs a fixed channel assignment as an underlying strategy and attempts, at run-time, to balance dynamically the available channels between the cells. Two thresholds, light and heavy, are introduced to classify the cells in the system into three categories—light, moderate, and heavy cells—according to the number of available channels in a cell. Whenever there exist any heavy cells, the load-balancing algorithm is activated by the MSC in order to borrow free channels from light cells to satisfy the demands of the heavy cells. When making a borrowing decision, the algorithm takes into account not only the state of a potential lender cell but also the states of the co-channel cells of the lender. The performance of our algorithm is evaluated and compared with other existing schemes. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Mobile cellular networks; Channel assignment; Channel borrowing; Load balancing; Performance evaluation

1. Introduction The rapid growth in demand for mobile communications has led the industries into intense research and development efforts towards a new generation of wireless cellular systems. One of the major challenges in such networks is the utilization of limited resources (radio channels) effectively in order to provide high availability of service. The basic idea in this problem is to divide the geographical area into cells and reuse the channels in the system in noninterfering cells. Many schemes have been proposed to assign channels to cells such that the available channels are efficiently used and thus channel reuse is maximized [1–8]. The performance index used for measuring the efficiency of a channel assignment scheme is the call blocking probability, that is, the sum of the probabilities of new call blocking and forced termination. There are three basic types of channel assignment algorithms such as fixed [9], flexible [10] and dynamic [1,7,11]. A fixed assignment (FA) strategy is to assign a fixed set of channels to each cell permanently. The same set of channels is reused by another cell at some distance away. The minimum distance at which the channel can be reused with no interference is called co-channel reuse distance. The advantage of the FA strategy is its simplicity. Its dis* Corresponding author.

advantage, however, is that if the number of calls exceeds the number of channels assigned to a cell the excess calls have to be blocked. A dynamic assignment (DA) strategy is to assign the channels in the system dynamically. Each cell has no channel for itself but requests for free channels if necessary. The system keeps a pool of free channels and any cell can use any channel that does not violate the channel reuse constraint. The DA strategies tend to be more efficient than the FA strategies in conditions of light, non-homogeneous and time-varying traffic but accompany with high implementation overhead. A flexible assignment strategy combines aspects of both the fixed and dynamic strategies. Each cell is assigned a fixed set of channels, but a pool of channels is reserved for dynamic assignment. In this paper, variations of a FA strategy are taken into account. That is, the FA strategy is used as the underlying scheme for channel assignment and, at run-time, channel borrowing techniques are used to reduce the call blockade [12–16]. Eklundh [14] proposed a directed retry strategy (DR) wherein a new user in a cell, where there is no free channel, tries to find a free channel from its neighboring cells. The user, however, should be in the overlap region between the two cells. Karlsson and Eklundh [16] proposed an extension of the DR strategy. Jiang and Rappaport [15] proposed a channel borrowing without locking strategy wherein a cell where free channel is exhausted tries to borrow some free channels from its neighboring cells. To

0140-3664/00/$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S01 40-3664(99)0020 0-5

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load-balancing algorithm for channel borrowing. Section 4 shows the performance evaluation of the algorithm in comparison with the LBSB and D-LBSB algorithms proposed by Das et al. [12,13]. The final section presents the conclusion of the paper.

2. Cellular system model

Fig. 1. Departing user in a cell.

avoid interference with the other co-channels of the lender, the borrowed channels are used with reduced power. Das et al. [13] recently proposed a novel load balancing strategy with selective borrowing strategy (LBSB) wherein a cell attempts to selectively borrow channels before the available channel in a cell is exhausted. The LBSB algorithm borrows channels for a so called ‘hot’ cell needing free channels from not only its neighboring cells but also others in its compact pattern. It has been shown in Ref. [13] that the LBSB algorithm outperforms the other strategies with respect to the call blocking probability. A distributed version of the LBSB algorithm, called D-LBSB algorithm, was also proposed by Das et al. [12] who have shown that the running time of the D-LBSB algorithm is smaller than the LBSB algorithm but the number of messages exchanged between the cells is higher. In this paper, we extend the scheme of Das et al. [13] and propose a new efficient load-balancing algorithm for channel assignment, based on a two-threshold cell selection scheme. The algorithm employs a heavy and a light threshold to classify the cells in the system into three categories according to their states (defined by the number of the available channels): light, moderate and heavy cells. This scheme is used for preventing a cell from the ping-pong state changes by being switched back and forth between light and heavy states. The values of the light and heavy thresholds are determined adaptively corresponding to the system state (i.e. the average number of available channels at a cell in the system) and therefore the algorithm is expected to be efficient even when the incoming call rate fluctuates. The proposed scheme runs the channel-borrowing algorithm on-demand whenever there exists any heavy cell needing free channels. Furthermore, when making a borrowing decision for a heavy cell, we take into account not only the states of the potential lender cells but also the states of their co-channel cells since their co-channel cells may also be heavy. This scheme is used to limit borrowing a channel from a cell with any heavy co-channel cells. The rest of this paper is organized as follows. Section 2 describes the cellular system model. Section 3 presents the

The cellular system model we consider is as follows. A given geographical area is divided into a number of hexagonal cells, each served by a base station (BS). A BS and the mobile users communicate through wireless links using radio channels. A number of cells (or BSs) are linked to a mobile switching center (MSC) through dedicated wire-line links. Each MSC is linked to the fixed telephone network again through a wire-line link and acts as a gateway of the cellular network to the fixed backbone network [17]. Each cell is allocated a fixed set of channels, C, which is reused in the other cells sufficiently far away without interference. The minimum distance at which the same channel can be reused without co-channel interference is called co-channel reuse distance [18–20]. The co-channel cells in a hexagonal cellular system can be determined by using two shift parameters, si and sj [19]. For example, in a system with a 7-cell group, the co-channel cells can be determined by the shift parameters 2 and 1. Cells in the system can be divided into groups in which there are no cells using the same set of channels. Such a group of cells using distinct channels is called a compact pattern. The number of cells in a compact pattern is given by ncp ˆ s2i 1 si sj 1 s2j : Let the number of channels available in cell i at any given instant be denoted by ci : To classify the cells into different classes, a light threshold, Tl, and a heavy threshold, Th, are used and it is assumed that C $ Tl . Th $ 0: If ci $ Tl ; i.e. the number of the available channels in cell i is equal to or greater than the light threshold, then cell i will be classified as a light cell. On the other hand, if ci # Th ; then cell i is classified as a heavy cell. Otherwise, cell i is classified as a moderate cell. The average numberP of the available channels in a cell is denoted by c ˆ Niˆ1 ci =N; where N denotes the number of cells in the system. It is assumed that each cell knows its own state (i.e. the number of available channels in the cell) at any given time. It is also assumed that each cell sends the state information message to the MSC independently and the MSC can process the messages sent from the cells concurrently. The MSC keeps the state information of all the cells in the system and makes channel assignment decisions based on the state information. When the state of a cell changes, the cell sends an update message to the MSC. Furthermore, when the number of available channels in a cell changes largely even though the state category of the cell remains unchanged; that is, the changes of the number of available channels in a cell becomes larger than threshold, D, the cell also need to send an update message to the MSC. The

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i1 ; i2 ; …; i6 are determined by the shift parameters, si and sj as shown in Fig. 2. When a channel of cell i is lent to another cell, some of its co-channels in the nearest co-channel cells have to be locked in order to avoid co-channel interference with the borrowing cell. All of the six co-channels can be locked in a conservative way, but it is apparently unnecessary. A co-channel locking scheme described in the next section shows that locking of at most three co-channels is sufficient to avoid interference. 3. Load-balancing algorithm

Fig. 2. Co-channel cells (Shift parameters si ˆ 4 and sj ˆ 1†:

determination of D can be based on the requirements of the system design. For example, it is preferable to set a relatively small D when the cost of the message exchanges between the BSs and the MSC is not high, and a relatively large D when the system load at a cell oscillates largely. As in Ref. [13], the mobile users in a cell are classified as either local or departing. To classify the users in a cell as local or departing, a shaded area between two adjunct cells is considered as shown in Fig. 1. • A departing user is designed as follows. A shaded area between two adjunct cells, i and j is considered (areas marked (2) and (3) in Fig. 1). The shaded area is denoted by parameter ps showing the percentage of the shaded region to the whole cell area. Typical values of ps are 20%, 10%, etc. When a user enters the shaded area the BS starts a timer to trace the user. That is, the BS triggers a timer for a user if the signal strength received from the user becomes less than a predefined level. A departing user in a cell is one who is within the shaded area, receiving a steadily diminishing signal strength from the BS of the cell for the last time units of a . If the signal strength stops diminishing within this time period, the departing user will be reset to be local. • A user who is not departing is local. Note that a local user can also be located in the shaded area, or not (areas marked (1) and (2) in Fig. 1). As in Ref. [13], each cell has six neighboring cells numbered from 1 to 6. An array, NumDepart[n], n ˆ 1; 2; …; 6 is used to store the number of departing users in a cell heading towards its neighboring cells. A heavy cell sends this array to the MSC along with the other state information when its state changes. For each cell i, the six nearest co-channel cells denoted by

The proposed load-balancing algorithm attempts to balance the number of available channels on-demand between the cells in the system. Whenever there exists any heavy cells needing free channels, the load-balancing algorithm is activated by the MSC to search for the potential channel lenders. Channel borrowing and channel lending, however, should obey the following rule. A heavy cell can borrow channels only from light cells, but a light cell is not allowed to borrow any channels from any other cells. A moderate cell, on the other hand, is neither allowed to borrow channels from any other cells nor lend any channels to any other cells. 3.1. Determination of the thresholds The basic idea of the channel assignment algorithm is to increase the number of available channels in the heavy cells to the average number of channels by channel borrowing and, therefore, the determination of the values of the thresholds is very important. The values of Tl and Th are determined corresponding to the average number of avail and a parameter, Dlh. That is, let able channels at a cell, c;  and Th ˆ T1 2 Dlh where Dlh . 0 and Th $ cmin : Tl ˆ bcc Here, Dlh denotes the difference of the values of Tl and Th and cmin denotes the minimum value that Th takes. The value of Tl is therefore adjusted adaptively based on c and bounded by the value of Dlh 1 cmin : This scheme is efficient even when the call arrival rate fluctuates. The average number of available channels in a cell c is computed at the MSC based on the information collected from the cells. Note that the MSC only listens to the reports of the state changes from the cells but never make any requests for them. 3.2. Channel borrowing scheme For each heavy cell, the MSC determines its compact pattern so that it is located as the central cell. It then divides the cells in the compact pattern into three categories as follows. 1. The neighboring cells towards which departing users are heading. 2. The neighboring cells but towards which no users are heading. 3. The other cells in the compact pattern.

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Fig. 3. The new channel borrowing algorithm.

The MSC searches the appropriate light cells as the potential lenders following the above order for the heavy cell and then borrows the required channels and locks their related co-channels. The lender candidates in each category are treated equally; that is, the MSC chooses a lender at random or sequentially from the lender candidates. Furthermore, the algorithm can borrow multiple channels from a lender each time until the state of the lender alters. To determine a lender cell and how many channels the lender can lend for a heavy cell, the new algorithm takes into account the state of the lender cell as well as the states of the co-channel cells of the potential lender. Since a co-channel cell of a light cell may be a heavy cell, locking co-channels in such a co-channel cell may cause the situation in the co-channel cell to become even worse. This may result in extra channel borrowing in the heavy co-channel cell and, in extreme cases, a channel borrowing loop may occur, degrading the system performance seriously. In order to prevent this situation from happening, the algorithm

prohibits, after channel borrowing, any co-channel cell of a lender cell from becoming heavier than the borrower cell. The channel borrowing mechanism works as follows. For a heavy cell k, channel borrowing is performed according to the procedures listed below until either the borrowing request from cell k is satisfied, or the light cells are exhausted. The number of required channels that cell k needs to borrow from light cells is denoted by bk and determined by bk ˆ …Tl 2 1† 2 ck : The flowchart for the new algorithm is shown in Fig. 3. 1. Borrow channels from the light neighboring cells towards which departing users are heading. 1.1. Find the light neighboring cells for which there are non-zero NumDepart entries. Select those light cells as the lender candidates. 1.2. Borrow channels from such a light cell where NumDepart[n] is non-zero until either the light cell state alters, or any co-channel cell of the lender cell

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becomes heavier than the borrower cell, or the number of borrowed channels equals NumDepart[n]. 2. Borrow channels from the light neighboring cells towards which no departing users are heading. 2.1. Find the light neighboring cells and select them as the lender candidates. 2.2. Borrow channels from such a light cell until either the lender cell state alters, or any co-channel cell of the lender cell becomes heavier than the borrower cell, or the borrowing request is satisfied. 3. Borrow channels from other light cells in the compact pattern. 3.1. Find the light cells excluding the neighboring cells in the compact pattern such that cell k is located as the center cell and select them as the lender candidates. 3.2. Borrow channels from such a light cell until either the lender cell state alters, or any co-channel cell of the lender cell becomes heavier than the borrower cell, or the borrowing request is satisfied. After making the channel borrowing decision as described above, the MSC sends messages (including the borrowed channel id’s) to the lender cells and the co-channel cells of the lenders to lock the lent channels in order to avoid interference. It also sends a message to the borrower cell to inform about the borrowed channels (including the lender id’s and the borrowed channel id’s). 3.3. Borrowed channel returning scheme As mentioned, a light cell is not allowed to borrow any channels from any other cells. Therefore, if the number of available channels in cell k exceeds the light threshold, Tl, while cell k still has any borrowed channels, then cell k needs to return the borrowed channels until ck ˆ Tl 2 1: The channel returning is processed in an inverse order of the cells compared to that for channel borrowing. That is, cell k needs to return its borrowed channels to the lender cells according to the following order. 1. Return the channels that belong to the cells other than its neighboring cells in the compact pattern. 2. Return the channels that belong to its neighboring cells but towards which no users are heading. 3. Return the channels that belong to its neighboring cells towards which departing users are heading. 3.4. Co-channel locking scheme In order to determine which nearest co-channel cells should lock the co-channels of a lender cell i to avoid interference with the borrower cell, the following co-channel locking scheme is used. The cells in the system are divided into six sets as shown in Fig. 2 using six lines, l1 ; l2 ; …; l6 so that cell i is located as the central cell. The co-channel cells of cell i are determined anti-clockwise (or clockwise) and denoted by i1 ; i2 ; …; i6 : In this paper, it is only shown for the

case where both si and sj are greater than zero. It is easy to extend this scheme, although omitted here, to the case of either si or sj being zero. From this figure, it can be seen that there is only one nearest co-channel cell in each set of cells. Before describing the co-channel locking algorithm, an example is used to explain the basic idea, where si ˆ 4 and sj ˆ 1 as shown in Fig. 2. It can be seen that when cell i lends a channel u to cell k which is between two lines, l2 and l3, there is no need to lock the co-channels of u in cells i 4, i 5 and i 6, since channel u will be used in borrower cell k that is farther than the lender cell i from those co-channel cells. Note also that cell i 1 has no need to lock the co-channel of u in this example. It is therefore sufficient to lock the co-channels of u in cells i 1, i 2 and i 3. When cell i lends a channel v to cell j which is on line l 5, the situation becomes a little complicated. There is however apparently no need to lock the co-channels of v in cells i 1 and i 2 for the same reason. Furthermore, at least one of cells i 3 and i 6 has no need to lock the co-channel of v, and the cell can be determined by the values of si and sj : When si . sj as in Fig. 2, cell i 3 has no need to lock the co-channel of v. Otherwise, cell i 6 has no need to lock the co-channel of v. Therefore, locking the co-channels in cells i 4, i 5 and i 6 is sufficient to avoid the co-channel interference. Let us outline the co-channel locking algorithm for the case of si ; sj . 0 for a lender cell i and borrower cell k. 1. Split the cells in the system centered at cell i into six sets by the line lp ; p ˆ 1; 2; …; 6 as shown in Fig. 2. 2. If cell k is not on a line but located in between lp and lp 1 1; then lock the co-channels in cells ip21 ; ip and ip11 : Here, the number p is treated as a ring from 1 to 6 so that if p ˆ 1 then p 2 1 ˆ 6 and if p ˆ 6 then p 1 1 ˆ 1: 3. If cell k is on a line, lp ; then lock the co-channels in cells ip21 and ip : Furthermore, if si $ sj then lock the co-channels in cell ip11 ; otherwise lock the co-channels in cell ip22 : 3.5. Channel assignment for users in a cell To assign a channel to an incoming call, either a new arrival call or a hand-off call, the same scheme as that proposed by Das et al. [13] can be used. That is, the channel requests are processed according to the priorities listed below. 1. Hand-off requests from the neighboring cells. 2. Local original calls. 3. Local user channel reassignment requests from a borrowed channel to a local channel. 4. Departing user channel reassignment requests from a local channel to a borrowed channel. When a departing user goes towards a light cell, the borrowed channel borrowed from the destination cell is unlocked as soon as the user enters that cell and does only a soft hand-off.

Y. Zhang, S.K. Das / Computer Communications 23 (2000) 452–461 Table 1 Implementation cost comparison of the algorithms Scheme

Number of messages

Running time

LBSB D-LBSB New

3…N 1 5X 1 1† 2…N 2 1† 1 2…uCPu 1 6†X N 1 6X 1 1

…3 1 …3 1 5X†Nh †d 4d…1 1 6X† 2d

4. Performance evaluation The performance of our load balancing algorithm is evaluated with respect to two parameters: the implementation cost and the call blocking probability. For the implementation cost, the number of messages exchanged between the BSs of the cells (also simply called BSs or cells) and the MSC during the process of channel borrowing for a heavy cell, as well as the running time of the algorithm are taken into account. The LBSB and D-LBSB algorithms are chosen for comparison since they outperformed the other existing algorithms [12,13]. 4.1. Implementation cost comparison It is assumed that the message delay between a BS and the MSC is fixed. It is also assumed that each BS sends messages to the MSC autonomously and the MSC is powerful enough to process concurrently the messages sent from the BSs. Since the computation time of the algorithm at the MSC is much shorter than the message delay between a BS and the MSC, it is not taken into account. The case of locking six co-channels for each lender cell is considered. The total number of cells in the system is denoted by N, and the number of light cells and heavy cells are denoted by Nl and Nh, respectively. In order not to bring any bias to the comparison results for the algorithms, the worst case scenario for the new algorithm is considered. It is assumed

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that a heavy cell needs to borrow X channels and each light cell can lend only one channel to the heavy cell. It is assumed that Nl . X; and therefore each heavy cell needs X light cells for channel borrowing. The messages exchanged between the BSs and the MSC for each heavy cell can be listed as follows. (1) Each BS initially sends a message to the MSC to inform its state. (2) After determining the lender cells, the MSC sends a message to each of them to inform about the decision and which channels should be blocked. (3) The MSC also sends a message to the borrower cell to inform it from where and which channels are borrowed. (4) The MSC then informs the related co-channel cells of the lenders in order to lock the co-channels in order to avoid interference. Since the MSC can concurrently process messages (2)–(4) listed above, sending these messages experiences only one message delay of d . The running time of the new algorithm is therefore determined by 2d . Table 1 shows the total number of messages exchanged between the BSs and the MSC and the running time of the algorithms for each heavy cell for borrowing X channels. In this table, uCPu denotes the number of cells in a compact pattern. 4.2. Call blocking probability comparison Simulation was used to evaluate the call blocking probability of the new algorithm and compare it with the LBSB and D-LBSB algorithms proposed by Das et al. [12,13]. Since the D-LBSB algorithm yields the same call blocking probability as the LBSB algorithm, only the results of the LBSB algorithm were used here. The results shown in the figures were obtained with 90% confidence interval and within 5% of the sample mean. The simulated cellular system contains 100 hexagonal

Fig. 4. Simulated cellular system.

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Fig. 5. User position in a cell.

cells shown in Fig. 4. Two integer values x and y …1 # x; y # 10† are used to describe the locations of the cells. The shift parameters, si and sj ; are 3 and 2, respectively. The letters, a; b; c; …; on the cells in Fig. 4 denote distinct sets of channels and the cells with the same letter are assigned with the same set of channels. Incoming call arrival at each cell is assumed to follow a Poisson process with a mean l . The holding time of a call is assumed to be distributed based on an exponential distribution with a mean 1=m of 500 s. The other parameters used in the simulation runs are as follows: Dlh ˆ 2; D ˆ 2; ps ˆ 20% and cmin ˆ 0:05C (i.e. 5% of C). According to the results of Refs. [12,13], the call blocking probability is the lowest when the value of the threshold for the degree of coldness at a cell is 0.1. In order not to bring any bias to the LBSB algorithm in

comparison with the new algorithm, this value was used for the LBSB algorithm. Both the algorithms used the co-channel locking scheme proposed in this paper to lock the co-channels for a lender cell. The effects of the values of D and cmin are not shown here, since they depend on the requirements of the system design and are not very sensitive to the performance. The determination of the value of D is governed by the communication cost between the BSs and the MSC and the degree of fluctuation of the system load. The value of cmin is affected by the value of C, but a value within the range of several percents of C yields a reasonable performance. In order to determine the position of a user in a cell, a cell is modeled as a circle with a grid of size 100 × 360 as shown in Fig. 5. A user location is determined by using a radial method, that is, a pair of …g; u† to indicate the position of a user where g denotes the distance of the user from the center of the cell and u denotes the angle from a common reference line. The location of a new incoming user is given at random; that is, the newly arriving calls are dispersed uniformly in a cell. It is assumed that a user can move, with the same probability, to one of the four directions or remain in the same position. When a user moves to a position over 100 from the center, a hand-off occurs. If a user moves out of the service area of 100 hexagonal cells (see Fig. 4), the call is simply lost. The time period, a , used for determining a departing user is set to 10 time units. If a user keeps departing for at least 5 times in the shaded region for the last period, it is classified as a departing user. Otherwise, it is a local user. Fig. 6 shows the call blocking probability of the two algorithms for various values of C, the number of channels initially allocated to each cell under the fixed assignment scheme. The load level used here denotes the ratio of the call arrival rate, at one channel in a cell, to the call service rate at

Fig. 6. Call blocking probability: the new algorithm vs. the LBSB algorithm.

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Fig. 7. LBSB algorithm: the rate of borrowed channels.

a cell; that is, (load level ˆ call holding time/(call interarrival time × C)) It is assumed that an arriving call in a cell is served immediately if there are any channels available for use in that cell. From Fig. 6, it is shown that the value of C has strong effects on the call blocking probability for both algorithms. Similar to the LBSB algorithm, for a larger C, the new algorithm yields a lower call blocking probability. As expected, it is observed that, over a wide range of the load level, the new algorithm performs much better than the LBSB algorithm. The improvements are mainly due to the following reasons.

• In the LBSB algorithm, the states of only the potential lenders are taken into account. Locking the co-channels in a heavy co-channel cell of a lender in order to avoid interference, however, may cause the heavy co-channel cell to become heavier and attempt to borrow further extra channels. Under an extreme situation, a borrowing loop may occur, thus degrading the channel utilization dramatically. To show how often a channel is borrowed from a cell with any heavy co-channel cells, the simulation was carried out and the result is shown in Fig. 7. On the other hand, under the new algorithm, such a situation will not occur.

Fig. 8. Call blocking probability: the new algorithm with various values of Dlh.

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• In the LBSB algorithm, only a fixed value of the threshold, h, is used to determine if a cell in the system is to be heavy or light. When the value of h is too large or the degree of coldness at most of the cells are less than h, there are fewer chances to find a light cell, resulting in frequent call blockade. In our algorithm, on the other hand, a two-threshold scheme is used and the values of the thresholds are adjusted adaptively corresponding to the fluctuation of the system load. Fig. 7 shows the rate of borrowed channels under the LBSB algorithm. The rate of borrowed channels, Rb, is defined by the ratio of the number of borrowed channels to the total number of calls in a simulation run. The rate of channels that borrowed from the cells with heavy co-channel cells, Rh, is defined by the ratio of the number of channels borrowed from cells with heavy co-channel cells to the total number of calls in a simulation run. It is observed that Rb increases if the load level increases, but decreases if the load level becomes higher than 0.9. This is because, from this point, the degrees of coldness at many cells become less than h, and as a result it becomes difficult to find a light cell. It is also observed in Fig. 7 that the ratio Rh =Rb continues to increase when the load level becomes high. This means that the rate of channel borrowing from a cell with heavy co-channel cells increases constantly. For example, the rate can reach a level close to 50% when the load level is 1.0 as shown in Fig. 7. This situation made the heavy co-channel cells attempt to borrow extra free channels, resulting in waste of the system resources and rapid performance degradation. Fig. 8 shows the call blocking probability of the new algorithm for various values of Dlh when C ˆ 30: The value of Dlh determines the dispersion between the light and the heavy thresholds. Therefore, when the value of Dlh is large, the state changes between the light and heavy cells rarely happen. When the value of Dlh is small, on the other hand, the state change may become easy. It is, therefore, preferable to use a large value of Dlh if the system load oscillates. Simulation results (including Fig. 8) show that a relatively small value of Dlh yields a reasonably smaller call blocking probability. 5. Conclusions In this paper, a new load-balancing algorithm for channel assignment based on a two-threshold cell selection scheme has been proposed and evaluated in mobile cellular networks. The proposed algorithm is shown to provide significant improvements on the implementation cost, both on message complexity and running time of the algorithm, over the LBSB algorithm [13]. Furthermore, it has been shown that the new algorithm performs much better than the LBSB algorithm with respect to the call blocking probability. The advantages of our scheme are summarized below.

1. The message exchanges between the BSs and the MSC can be carried out autonomously and in parallel, and therefore there is no need to wait for any messages for making borrowing decisions at the MSC. 2. It uses a two-threshold scheme that prevents a cell from the ping-pong state changes of being switched back and forth between light and heavy. 3. It also takes into account the states of the co-channel cells of a lender cell when making a borrowing decision so that potential wrong borrowing decisions can be avoided. 4. It adjusts, adaptively, the values of the light and heavy thresholds according to the system load-state so that better channel borrowing decisions can be made. We assume that a BS has more intelligence and is capable of monitoring its cell state and sending the update messages to the MSC autonomously. Such a mechanism, however, can be implemented with minor modification on the platform of the existing algorithms. All the work a BS has to do, in our new algorithm more than the other algorithms, is to determine its own cell state and send the update messages to the MSC autonomously. By distributing some basic functions from the MSC to the BSs, the workload at the MSC can be alleviated and channel borrowing can also be exploited more efficiently.

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