A novel load balancing scheme for the tele-traffic hot spot problem in cellular networks

Wireless Networks 4 1998 325 340 325 A novel load balancing scheme for the tele-traffic hot spot problem in cellular networks Sajal K. Das Sanjoy K. Sen and Rajeev Jayaram Center for Research in Wireless Computing Department of Computer Sciences University of North Texas P.O. Box 311366 Denton TX 76203-1366 USA We propose a dynamic load balancing scheme for the tele-traffic hot spot problem in cellular networks. A tele-traffic hot spot is a region of adjacent hot cells where the channel demand has exceeded a certain threshold. A hot spot is depicted as a stack of hexagonal Rings of cells and is classified as complete if all cells within it are hot. Otherwise it is termed incomplete. The rings containing all cold cells outside the hot spot are called Peripheral Rings. Our load balancing scheme migrates channels through a structured borrowing mechanism from the cold cells within the Rings or Peripheral Rings to the hot cells constituting the hot spot. A hot cell in Ring i can only borrow a certain fixed number of channels from adjacent cells in Ring i + 1. We first propose a load balancing algorithm for a complete hot spot which is then extended to the more general case of an incomplete hot spot. In the latter case by further classifying a cell as cold safe cold semi-safe or cold unsafe a demand graph is constructed which describes the channel demand of each cell within the hot spot or its Peripheral Rings from its adjacent cells in the next outer ring. The channel borrowing algorithm works on the demand graph in a bottom up fashion satisfying the demands of the cells in each subsequent inner ring until Ring 0 is reached. A Markov chain model is first developed for a cell within a hot spot the results of which are used to develop a similar model which captures the evolution of the entire hot spot region. Detailed simulation experiments are conducted to evaluate the performance of our load balancing scheme. Comparison with another well known load balancing strategy known as CBWL shows that under moderate and heavy tele-traffic conditions a performance improvement as high as 12% in terms of call blockade is acheived by our load balancing scheme. 1. Introduction In view of the remarkable growth of the mobile communication users and the still very limited radio frequency spectrum allocated to this service by the FCC Federal Communications Commission the efficient management and sharing of the spectrum among numerous users become an important issue. This limitation means that the frequency channels have to be reused as much as possible in order to support the many thousands of simultaneous calls that may arise in any typical mobile communication environment. Thus evolved the concept of cellular architecture [15] which is conceived as a collection of geometric areas called cells typically hexagonal-shaped each serviced by a base station BS located at its center. A number of cells or BS s are again linked to a mobile switching center MSC which acts as a gateway of the cellular network to the existing wire-line networks like PSTN ISDN any LAN-WAN based networks or even the Internet. A base station communicates with the mobile stations or users through wireless links and with the MSC s through wired links. The model of such a system is shown in figure 1. Since frequency channels are a scarce resource in cellular mobile networks many schemes have been proposed to assign frequencies to the cells with a goal to maximize the frequency reuse. These can be broadly classified as fixed This work is partially supported by a grant from Nortel Wireless Technology Richardson TX and Texas Advanced Research Program grant TARP-003594-031. Figure 1. System model of a cellular mobile architecture. [8152123] dynamic [2161723] and flexible [1823] assignment strategies. In a fixed assignment FA scheme a set of channels is permanently allocated to each cell which can be reused in another cell sufficiently distant such that the co-channel interference is tolerable. Such a pair of cells is called cochannel cells. In one type of FA scheme clusters of cells called compact patterns are formed by finding the shortest distance between two co-channel cells [15]. Each cell within a compact pattern is assigned a different set of frequencies. The advantage of an FA scheme is its simplicity but the disadvantage is that if the number of calls exceeds the number of channels assigned to a cell the excess calls are blocked. Variants of an FA scheme generally use channel borrowing techniques [81119] in which a channel is borrowed from one of the neighboring cells in case of J.C. Baltzer AG Science Publishers

326 S.K. Das et al. / Tele-traffic load balancing blocked calls provided that it does not interfere with the existing calls. In a dynamic assignment DA scheme there is a global pool of channels wherefrom channels are allocated on demand. A channel assignment cost function is computed and the channel with the minimum cost is assigned. For example a channel can be selected for allocation from the global pool if it allows the minimum number of cells in which that channel will be locked. Flexible or hybrid channel assignment schemes combine the concepts of both fixed and dynamic schemes whereby there is a fixed set of channels for each cell but channels are also allocated from a global pool in case of shortage. While the motivation behind all of the preceding channel assignment strategies is the better utilization of the available frequency spectrum with the consequent reduction of the call blocking probability in each cell relatively few of them [371113] deal with the problem of non-uniformity of teletraffic demand in different cells which may lead to a gross imbalance in the system performance. An example might be the downtown area of a city consisting of a cluster of adjacent cells where the tele-traffic load can be very high for most part of the day. It is desirable that the system should be able to cope up with such traffic overloads in certain cells. We will designate those cells as hot where the tele-traffic demand exceeds a certain threshold value. A region consisting of multiple adjacent hot cells will be called a hot spot. A cell which is not hot will be denoted as cold. Fixed assignment schemes by themselves are unable to handle the hot spot problem [11] as the number of channels assigned to each cell cannot be changed although they usually perform better under heavy traffic conditions than dynamic schemes. On the other hand dynamic assignment schemes are expected to cope better with tele-traffic overloads to a certain extent but on high demands the computational overheads deceive the purpose of the scheme. Flexible schemes will face the same problem as they are basically reduced to dynamic schemes on high channel demand presumably after the fixed channel sets get exhausted. 1.1. Our contribution In this paper we propose a dynamic load balancing scheme employing channel borrowing technique to cope up with the problem of tele-traffic overloads in hot spots. We denote a particular cell within a hot spot as the center cell. Then the hot spot can be conceived as a stack of hexagonal Rings around the center cell such that each Ring consists of at least one hot cell among other classes of cells like cold safe cold unsafe and cold semi-safe. A hot spot with only hot cells is called complete otherwise it is incomplete. In our load balancing approach a hot cell in Ring i borrows channels from its adjacent cells in Ring i+1 to ease out its high channel demand. This structured lending mechanism decreases excessive co-channel interference and borrowing conflicts which are prevented through channel locking in other schemes. Also the number of channels to be borrowed by each cell will be predetermined by its class and its position in the hot spot. With the help of a simple and efficient construction of a demand graph unused channels are migrated from the cold cells within or in the periphery of the hot spot to the hot cells constituting the hot spot. Assuming a fixed channel assignment scheme is available to start with we have proposed a channel migration scheme through borrowing between cells of adjacent rings such that all the hot cells are provided with the required number of channels to cope up with their tele-traffic overloads. A discrete Markov model for a cell in our system incorporating load balancing is also proposed and another similar model is developed to capture the evolution of a hot spot region. Extensive simulation experiments are carried out to evaluate the performance of our scheme. A comparison with the CBWL scheme reveals that under moderate and severe tele-traffic demand the total number of blocked calls is reduced by as much as 12% in our scheme. A preliminary version of this paper appeared in [4]. The rest of the paper is organized as follows. Section 2 describes previous work related to load balancing in the context of the channel assignment problem. Section 3 classifies the cells and regions. The proposed channel borrowing algorithms for complete and incomplete hot spots are described in sections 4 and 5 respectively. Detailed performance analysis for a complete hot spot is laid down in section 6 where discrete Markov models for a cell and a hot spot are also developed. Section 7 gives the simulation results and Section 8 concludes the paper. 2. Channel assignment schemes with load balancing In this section we outline three methods [31113] proposed in the literature to include load balancing as one of the major criteria in the channel assignment strategy. 2.1. Directed retry The directed retry scheme due to Eklundh [7] assumes that the neighboring cells overlap and some of the users in the overlapping region are able to hear transmitters from the neighboring cells almost as well as in their own cell. If there is a channel request from a user and there is no free channel then the subscriber is requested to check for the signal strength of the transmitters in the neighboring cells. If a channel from a neighboring cell with adequate signal strength is found the call is set up using that channel. Otherwise the call attempt fails. To improve this scheme Karlsson and Eklundh [13] proposed to incorporate load sharing by treating subscribers differently based on whether they are able to hear more than one transmitter. Whenever the base station finds more than a certain number of voice channels occupied it requests the users to check for the quality of the channels in the neighboring cells. If some of the users report that they are able

S.K. Das et al. / Tele-traffic load balancing 327 to receive transmission from neighboring cells adequately well a search for free channels begins in those cells and an attempt is made to move as many subscribers to those cells as possible. There is no concept of borrowing channels from neighboring cells but subscribers are simply moved from one cell to another by the process of hand-off. Ifno subscriber finds an adequate channel to setup or switch a call to the base station tries to find a free channel in the original cell or let the call proceed as usual. Although this load sharing scheme increases the number of potential channels to a certain extent the main disadvantages are the increased number of hand-offs and the co-channel interference. Also since a user has to be in the bordering regions of neighboring cells in order to be a potential candidate for a hand-off it puts a severe constraint on the efficacy of the algorithm to share load. The bordering region of two cells can be very small which reduces the probability that a sufficient number of users can be found in those regions to carry the load over to the neighboring cells in case of a drastic increase of the channel demand in a cell as might happen in the so called hot cells defined in section 3.1. 2.2. Channel Borrowing Without Locking CBWL In the CBWL scheme Jiang and Rappaport [11] proposed to use channel borrowing when the set of channels in a cell gets exhausted but to use them under reduced transmission power. This is done to avoid interference with the other co-channel cells of the lending cell using the same frequency. Channels can be borrowed only from adjacent cells in an orderly fashion. The set of channels in a particular cell is divided into seven groups. One group is exclusively for the users in that cell while each of the six other groups caters for channel requests from one of the neighboring cells. If the number of channels in a channel-group is exhausted a subscriber using one of the channels can be switched to an idle channel in another group thereby freeing up one in the occupied group. Since borrowed channels are transmitted at low power not all users within range are capable of receiving them. If such a user finds all the channels occupied an ordinary user occupying a regular channel can handover its channel to the former while itself switching over to a borrowed channel if available. This particular variant involving reassignment is called CBWL-R. The CBWL scheme has some advantages over the fixed and flexible assignment schemes since channel utilization is increased without locking. But one serious drawback of the reduced power transmission strategy is that not all users are in the right zone all the time for borrowing channels if the need arises. Also since only a fraction of the channels in all the neighboring cells is available for borrowing this coupled with the previous drawback can seriously degrade the system performance in case of high tele-traffic demand. For example if there is a hot spot where a hot cell is surrounded by six other hot cells whose channel sets are exhausted then the CBWL scheme performs very poorly for the hot cell in the center because no channel will be available for borrowing. The limitation on the number of channels available for borrowing places severe restriction on the system performance if some of the neighboring cells are hot. 2.3. Load Balancing with Selective Borrowing LBSB The problems mentioned in the directed retry with load balancing or the CBWL scheme are magnified when the channel demand is very high as in a hot cell. The LBSB scheme recently proposed by Das et al. [3] attempts to alleviate these problems by selectively borrowing channels by a cell before the available channel set is exhausted. A cell is classified as hot if its degree of coldness defined as the ratio of the number of available channels to the total number of channels allocated to that cell is less than or equal to some chosen threshold value. Otherwise the cell is cold. The LBSB scheme proposes to migrate a fixed number of channels from certain cold cells to a hot one according to a channel borrowing algorithm which can be implemented either as a centralized scheme [3] or a distributed scheme [5]. A channel assignment strategy is used in each cell by classifying the users into three broad types new departing and others and forming different priority classes of channel demands from these three types of users. Local and borrowed channels are assigned according to the priority classes. The main disadvantage of the directed retry scheme is that it makes load balancing a function of the number of users in the overlap region between two cells. If the number of such users is low proper load balancing is not achieved. However the LBSB approach achieves almost perfect load balancing in the sense that not only the overloaded cells get the neccessary number of channels but also the increase in load in the form of decreasing number of channels is shared evenly by multiple underloaded cells. The CBWL scheme on the other hand performs poorly for hot spots whereas LBSB performs equally well for all types of hot cell distribution. The disadvantage of LBSB is its computation-intensive nature as it performs user classification in a cell and computes the optimal lender cell at every stage of the load balancing algorithm. Our objective in this paper is to propose a new load balancing scheme based on structured channel borrowing mechanism between cells of adjacent rings which is not as computation-intensive as in [3] yet achieves almost perfect load balancing and eliminates the drawbacks of both the directed retry and the CBWL schemes i.e. performs equally well under all types of traffic conditions and load distribution in the cells within a given region. 3. Classification of cells and regions The high level architecture of our load balancing scheme is as in figure 1. To recall a given geographical area con-

328 S.K. Das et al. / Tele-traffic load balancing sists of a number of hexagonal cells each served by a base station communicating with the mobile users via wireless links. A group of base stations are served by a mobile switching center. The MSC s are connected through fixed wire-line networks and also act as gateways between the wire-line and wireless networks. Assuming that each cell is allocated a fixed set of C channels let us now classify the cells and regions as follows. Broadly a cell can be classified as either hot or cold. A cold cell can again be classified as cold safe cold semisafe or cold unsafe as defined in section 5. 3.1. Definition of a hot cell A cell is classified as hot or cold according to its degree of coldness defined as number of available channels d c = total number of channels number of available channels =. C If d c h whereh>0 is a fixed threshold parameter that particular cell is hot otherwise it is cold. Typical values of h are 0.2 0.25 etc. and determined by the average call arrival and termination rates and also by the channel borrowing rates from other cells. The usefulness of the parameter h is to keep a subset of channels available so that even when a cell reaches the hot state an originating call need not be blocked. When a cell reaches a hot state it merely serves as a warning that the available resource i.e. the channel set in that cell has reached a critical point and migration of resources is neccessary to mitigate the pressure arising due to a sudden traffic explosion. With this distinction between hot and cold cells let us now define a hot spot region. 3.2. Definition of a hot spot Two cells are said to be adjacent if they have a common edge. A set S of hot cells marked as H is said to form a hot spot if any cell in S is adjacent to at least another cell in S. Figure 2 shows an example of a hot spot in a cellular network. Let us introduce the concept of a Ring. We first select a center cell for the hot spot. A preferable way to make the selection might be to compute the diameter δ of the hot spot which can be defined as the maximum cell distance between any pair of cells in the hot spot. The cell lying at a distance of δ/2 from either end of the two farthest cells yielding the diameter is selected as the center cell. Now consider the center cell and the hexagonal rings of cells around it. If the center cell itself is hot we call it Ring 0. Otherwise the ring of cells nearest to the center cell containing at least one hot cell is denoted as Ring 0. We define Ring i i>0 as the ring of cells containing at least one hot cell at a cell distance i from Ring 0 and further away from the center cell. Note that if a hot spot consists of n Rings then all the Rings have to be contiguous by definition. In the example of figure 2 there are four Rings numbered as 0 1 2 and 3 consisting of 1 6 11 and 4 hot cells respectively. The first ring of cells outside the hot spot containing all cold cells is called the First Peripheral Ring the next such ring is called the Second Peripheral Ring and so on. So we distinguish between two types of hexagonal structures one type called Rings containing at least one hot cell and the other type called Peripheral Rings containing no hot cell at all. While referring to both of them we use the generic term ring. Henceforth the center cell will be called ring 0 and a Ring or a Peripheral Ring at a cell distance i from the center will be called ring i. If ν i denotes the number of cells in ring i then for the hexagonal geometry { 1 i = 0 ν i = 6i i>0. Let H n denote a hot spot whose outermost Ring is ring n. Then H 0 is equivalent to a single hot cell. The total number of cells in H n is given by n N n = ν i + 1 = 3nn + 1 + 1. 1 i=0 A hot spot H n is said to be complete if it contains N n hot cells. Otherwise it is defined as incomplete. Clearly for a complete hot spot Ring 0 always happens to be the center cell. Also the number of cells in the First Peripheral Ring of a complete hot spot H n is ν n+1 = 6n + 1. 4. A new load balancing scheme The underlying idea behind our load balancing strategy is the migration of channels between cells through channel borrowing mechanism in order to satisfy the channel demand of overloaded hot cells. Channel migration takes place between a borrower and a lender cell. Unlike the LBSB scheme [3] the borrower hot cell does not have the opportunity to select its lender from among all the cold cells in its compact pattern. A hot cell in Ring i ofacomplete hot spot can borrow channels only from its adjacent cells in Ring i + 1 or the First Peripheral Ring if Ring i isthe outermost Ring of the hot spot. This structured borrowing mechanism reduces the amount of co-channel interference between the borrower cell and the co-channel cells of the lender using the borrowed channel. Thus the number of cells is at most two in which the borrowed channel needs to be locked. Also a certain number X of channels is needed by each hot cell to relieve the excess traffic demand. The estimation of X is detailed in section 4.2 and is similar to the method used in our LBSB scheme. A hot cell in Ring i should borrow sufficient number of channels so that not only can it satisfy its own demand X but also cater for the channel demand from adjacent

S.K. Das et al. / Tele-traffic load balancing 329 Figure 2. A hot spot region shown as a collection of hexagonal rings. hot cells in Ring i 1. Note that only local channels of a cell are lended on demand to adjacent cells in the next inner ring. After borrowing channels from adjacent cells a ring i cell reassigns the borrowed channels by intra-cellular handoff to some of the users to release sufficient number of local channels to meet the demand of ring i 1 cells. In this way each Ring of cells caters for the channel demand of overloaded cells in the next inner Ring as well as within itself borrowing channels from the immediate outer Ring. The hot cells in the last Ring of the hot spot borrow channels from the cold cells in the First Peripheral Ring. 4.1. Assumptions The base station transmitter of each cell has the capability of transmitting any of the frequencies of the available spectrum. A channel borrow implies locking the same in the lender cell transmitter and unlocking it in the borrower cell transmitter. Due to the structured borrowing mechanism used in our load balancing scheme the borrowed channel needs to be locked in at most two cells. This will not significantly affect the system performance even under heavy load. All cells in the First Peripheral Ring are able to provide the required number of channels to the hot spot without exhausting their channel set or becoming hot themselves. If this is not true channels are borrowed from multiple Peripheral Rings and the algorithm adopted is the same as in the more general case of an incomplete hot spot see section 5. For the general case of an incomplete hot spot having cold cells along with hot ones a cold cell is further classified as cold safe cold unsafe and cold semi-safe. Cold unsafe cold semi-safe or hot cells in ring i need to borrow channels from adjacent cells in ring i + 1 the number of channels borrowed being different for different classes. A cold safe cell does not need to borrow any channel. A demand graph is constructed describing the number of channels required by the cells in a ring from their neighbors in the next outer ring. It can be shown that the channel borrowing algorithm for a complete hot spot described next is equivalent to the demand graph approach presented in section 5. 4.2. Number of channels retained by a hot cell Let us define a parameter d avg c called the average degree of coldness of a cell which is computed as the arithmetic mean of the d c s of all the cells over a certain period of time. In general d avg c is the degree of coldness which every hot cell tries to achieve through channel migration from cold cells. Therefore we can estimate the number X of channels a particular hot cell needs to retain assuming that the number of available channels in that cell is h C. Since retention of X channels by a hot cell leads to an increase in the number of its available channels by the same margin we get yielding d avg c = h C + X C X = davg C c h. 2 4.3. Channel borrowing for a complete hot spot Starting from the center cell all the cells along any of the six emanating chains of hexagons will be called corner cells. Each ring except ring 0 will then contain exactly six corner cells. The cells between two corner cells in a ring are called non-corner cells. For example ring 1 contains

330 S.K. Das et al. / Tele-traffic load balancing Figure 4. Channel lending in the complete hot spot H 4. Arrows indicate migration of channels from lender to borrower cells. Figure 3. Way to fix the coordinates of a cell within a hot spot. no non-corner cells ring 2 contains 6 ring 3 contains 12 andsoon. Every ring is a repetitive pattern of a corner cell and a fixed number of non-corner cells constituting what will be called a cell array. Ring 1 is composed of six such arrays where each array consists of only a single corner cell. Ring 2 is again composed of six arrays each array consisting of a corner and a non-corner cell. In general ring i is composed of six cell arrays each of which consists of a single corner cell and i 1 non-corner cells. Our load balancing scheme for a complete hot spot does not differentiate between corner or non-corner cells of different cell arrays within a ring. However for the more general case of an incomplete hot spot we will use a different convention of addressing the cells which will be described later. Referring to figure 3 all the corner cells in a particular ring i of a complete hot spot will have the same coordinate i 0. Moving in the anti-clockwise direction along ring i from one corner cell to the next the coordinate of the jth non-corner cell in a cell array will be i j. The coordinates repeat for the other cell arrays. Hence in a complete hot spot H n the coordinate of any cell except the center is given by i j where 1 i n and 0 j i 1. The coordinate 0 0 is assigned to the center cell. A cell with coordinate i j will be denoted by C ij. 4.3.1. Enumeration of channel borrow demand Consider a complete hot spot H n which is hexagonal region of cells containing n+1 Rings but containing no cold cell in any of the Rings within it. Figure 4 shows a complete H 4. The total channel demand of all the cells from Ring 0 to Ring i for 0 i n ofh n is {3ii + 1 + 1}X which according to our channel borrowing protocol must be provided by the 6i + 1 cells in ring i + 1. If each cell in ring i + 1 can lend l i+1 channels to adjacent hot cells in ring i then {3ii + 1 + 1}X l i+1 =. 3 6i + 1 Let us now derive the expressions for channel demand i.e. the number of channels to be borrowed by a cell in Ring i of a complete hot spot from its adjacent cells in ring i + 1. For this purpose let us consider the example of figure 4 and trace the channel bargain lending-borrowing between the cells starting from Ring 0 to Ring 3 and the First Peripheral Ring. Putting i = 0 in equation 3 we obtain l 1 = 1 6X implying that each of the six cells in Ring 1 lends 1 6X channels to the center cell thereby satisfying its demand of X channels. Since each of the Ring 1 cells require X channels themselves therefore it will borrow 1 6 X + X = 7 6X channels from adjacent cells in Ring 2. We make use of the following two facts. Fact 1. Any corner respectively non-corner cell in ring i has three respectively two adjacent cells in ring i + 1. Fact 2. Any corner respectively non-corner cell in ring i can lend channels to its only one respectively two adjacent corner cell in ring i 1. Since all the cells in Ring 1 are corner cells each of them can borrow channels from three adjacent cells in Ring 2. Putting i = 1 in equation 3 we obtain l 2 = 7 12X. Each cell in Ring 2 can lend 7 12X channels to adjacent cells in Ring 1. A borrower cell C 10 in Ring 1 requires 7 6 X channels and it has three adjacent lenders in Ring 2 one corner cell C 20 and two non-corner cells one belonging to the same cell array as C 20 and another to an adjacent

S.K. Das et al. / Tele-traffic load balancing 331 7 array. Cell C 20 lends all it can i.e. 12X channels to the only borrower cell C 10 while the remaining channel demand 7 6 X 7 12 X = 7 12 X of C 10 is satisfied by equal contributions from the other two non-corner lender cells. Since all the cells in Ring 1 are corner cells the same distribution is applicable to all of them. We propose the following convention of channel borrowing for the corner cells in every Ring. Proposition 1. A cell C i0 will borrow the whole amount offered by its adjacent corner cell C i+10 in ring i + 1and the remaining channel demand will be satisfied by equal contributions from its two other adjacent non-corner cells in ring i + 1. Now each of the Ring 2 cells requires X channels themselves and hence will borrow 7 12 X + X = 19 12 X channels from adjacent cells in Ring 3. Let us find out what each cell in Ring 3 has to offer. Putting i = 2 in equation 3 we obtain l 3 = 19 18X. Proceeding similarly to the case for i = 1 a corner cell C 20 in Ring 2 will borrow l 3 channels from its adjacent corner cell C 30 in Ring 3 and 1 4 l 3 channels each from its two adjacent non-corner cells in the same ring one of which is C 31 and belongs to the same cell array as C 30. Note that C 31 has another adjacent cell C 21 in Ring 2. It lends its remaining 3 4 l 3 borrowable channels to C 21 which in turn requires another 3 4 l 3 channels to fulfil its demand of 19 12 X = 3 2 l 3 channels and thus borrows from its other adjacent cell C 32 in Ring 3. Next Ring 3 has one corner and two non-corner cells. This is the last Ring of our hot spot and its cells borrow channels from the adjacent cold cells of the First Peripheral Ring. Equation 3 still applies to each cell in the First Peripheral Ring giving l 4 = 37 24X. Proceeding as above it can be shown that a corner cell C 30 in Ring 3 borrows l 4 channels from a corner cell in the First Peripheral Ring and 1 6 l 4 channels from two adjacent non-corner cells in the same ring. The non-corner cell C 31 borrows 5 6 l 4 and 3 6 l 4 channels from the adjacent cells C 41 and C 42 respectively in the First Peripheral Ring. The other non-corner cell C 32 borrows 3 6 l 4 and 5 6 l 4 channels from the adjacent cells C 42 and C 43 respectively in the First Peripheral Ring. Now it is easy to derive the generalized expressions for the number of channels borrowed by a hot cell in Ring i from adjacent cells in Ring i + 1. Recall that {3ii + 1 + 1}X l i+1 = 6i + 1 is the number of channels that each cell in Ring i + 1 can lend to the adjacent cells in Ring i. The following lemma holds for a complete hot spot. Lemma 1. a A corner cell C i0 in ring i will borrow l i+1 channels from its adjacent corner cell C i+10 and l i+1 /2i channels from each of its two adjacent non-corner cells in ring i + 1. b A non-corner cell C ij in ring i will borrow 1 2j 1 2j + 1 l i+1 and l i+1 2i 2i channels from its adjacent non-corner cells C i+1j and C i+1j+1 respectively. From the viewpoint of a lender cell the generalized expressions for the number of channels lended is as follows: Lemma 2. a A corner cell C i0 in ring i will lend all of its l i lendable channels to its adjacent corner cell C i 10 in ring i 1. b A non-corner cell C ij will lend 2j 1 2i 1 l i and 1 2j 1 l i 2i 1 channels to the adjacent cells C i 1j 1 and C i 1j in ring i 1. The channel borrowing algorithm is straightforward for a complete hot spot H n. Each cold cell in the First Peripheral Ring lends l n+1 channels to adjacent cells in Ring n each of which in turn retains X channels and lends l n channels to adjacent cells in Ring n 1 and so on until Ring 0 is reached. When the channel borrowing algorithm terminates the number of available channels in all the cells within the hot spot will be increased exactly by X. If all cells in the First Peripheral Ring are not capable of lending the required l n+1 channels then channels are borrowed from the other Peripheral Rings using the method described in the next section. 5. Channel borrowing for an incomplete hot spot In this section we propose to further classify a cold cell into three subclasses which will lead to a general channel borrowing algorithm for cells in an incomplete hot spot. Since a complete hot spot is a special case of an incomplete hot spot where all the cells are hot any channel borrowing algorithm for an incomplete hot spot can be used for the complete hot spot as well as for Peripheral Rings. Fixing any one of the six emanating chains of corner cells as the reference chain the coordinate of such a corner cell is i 0 if it belongs to ring i. Moving in the anti-clockwise direction along ring i from the corner cell C i0 the coordinate of the jth cell will be i j. Here we distinguish between the cell arrays since the same set of coordinates cannot be assigned to cells within different cell arrays.

332 S.K. Das et al. / Tele-traffic load balancing Table 1 Cell classification in incomplete hot spot. Cell position Class Conditions Corner cold safe N avail d avg c C andn avail d 1 X>hC cold semi-safe does not exist cold unsafe N avail <d avg c C orn avail d 1 X hc hot N avail hc Non-corner cold safe N avail d avg c C andn avail d 2 + d 3 X>hC cold semi-safe N avail d avg c C andn avail {maxd 2 d 3 }X hc and N avail d 2 + d 3 X hc cold unsafe N avail <d avg c C orn avail {mind 2 d 3 }X hc hot N avail hc 5.1. Classification of cold cells We will classify a cold cell into three groups cold safe cold semi-safe and cold unsafe according to the demands of the adjacent cells of the next inner ring and the number of channels available within the cell denoted as N avail. The definitions will be different for each class depending on whether the cell is corner or non-corner. Consider first a cold corner cell C ij in ring i having a channel demand d 1 X where d 1 is given as w or w figures 5vii x depending on whether its adjacent cell in ring i 1 is hot or cold. From section 4.3 3ii 1 + 1 w =. 6i Let w represent the demand from a cold cell i.e. w will assume different values for different classes of cold cells. On the other hand let the channel demands for a noncorner cell in ring i be denoted as d 2 X and d 3 X.Thend 2 is represented by a or a depending on whether the demanding cell is hot or cold a will assume different values for the different classes of coldness. Similarly d 3 can assume the values b or b as in figures 5i vi. Here a = 2j 1{3ii 1 + 1} 12ii 1 and b = 1 2j 1 3ii 1 + 1. 2i 1 6i The detailed classification of the cells and the associated conditions are specified in table 1. The intuition behind the classification and the channel lending-borrowing protocol followed by each class of cells is described below. 1. A corner cell is termed cold unsafe if its channel availability falls below the average or it does not have enough channels to satisfy the demand d 1 X without itself becoming hot. Such a cell borrows d 1 X channels from its three adjacent neighbors in the next outer ring and lends them entirely to its neighboring cell in the next inner ring. 2. A corner cell is termed cold safe if it has sufficient number of channels to satisfy the demand of its adjacent cells without itself changing state. Hence a cold safe cell does not need to borrow any channel. 3. A non-corner cell is classified as cold unsafe if its channel availability is less than average or it does not have enough available channels to cater for the minimum of the two demands d 2 X and d 3 X without itself becoming hot. Such a cell borrows d 2 + d 3 X channels from its adjacent cells in the next outer ring and lends the whole of it without retaining any for itself. 4. A non-corner cell is in the cold semi-safe state if its channel availability is more than average and it has enough available channels to satisfy the maximum demand out of d 2 and d 3 but does not have enough to satisfy both. Such a cell will lend channels from its available channel set to the adjacent cell with the minimum channel demand {mind 2 d 3 }X. This kind of lending strategy guarantees that the available channel set will not be reduced to the threshold value hc in the worst case after channels are lended. To cater for the channel demand of the other adjacent cells a cold semi-safe cell acts like a cold unsafe cell i.e. it borrows the requisite channels from adjacent cells in the next outer ring only to lend them to the demanding cell. 5. A non-corner cell is termed cold safe if it has sufficient number of channels to satisfy the demand of its adjacent cells without itself changing state. A cold safe cell does not need to borrow any channel. 5.2. Modification of channel demands Figure 5 shows the channel demand of a ring i cold cell from adjacent cells in ring i+1 which is a function of its class and the demand from its adjacent cells in ring i 1. Figure 5i shows the case for a complete hot spot where both cells in ring i 1 are hot. Thus the coefficients a b c and d can be exactly computed as shown in section 4.2. For the other classes of cells in ring i our objective is to determine the coefficients c d etc. refer to figures 5ii vi in terms of the known coefficients and the modified channel demands a and b from ring i 1 cells whenever applicable.

S.K. Das et al. / Tele-traffic load balancing 333 Rewriting equation 4 as follows: 1 + mina b 1 + mina b c 1 X + d 1 X { maxa b } X = 0 8 Figure 5. Channel demand graphs for corner and non-corner cells. In the case of corner cells figure 5vii shows the situation for a complete hot spot where the ring i 1 cell is hot. The coefficients w e f and g are then determined as in section 4.2 and our objective is to determine the modified demands e f g etc. refer to figures 5viii x in terms of these known parameters and the modified demand w whenever applicable. 5.2.1. Non-corner cells of types hot cold unsafe and cold semi-safe With respect to figure 5i the channel demand equation for a hot non-corner cell in ring i whose adjacent cells in ring i 1 are also hot is given by cx + dx ax bx = X. 4 The channel demand equation for a cold unsafe non-corner cell in ring i whose adjacent cells in ring i 1 are hot figure 5ii is given by c X + d X ax bx = 0. 5 Rewriting equation 4 as follows: c 1 1 X + d 1 1 X ax bx = 0 6 we get c = c 1 1 and d = d 1 1. The channel demand equation for a cold semi-safe noncorner cell in ring i whose adjacent cells in ring i 1are hot figure 5v is given by c iv X + d iv X { maxa b } X = 0. 7 we get and c iv = c 1 d iv = d 1 1 + mina b 1 + mina b. Similarly it can be shown that c = c 1 a + b a + b c = c 1 1 + a + b a + b c v = c 1 a + b a + b d = d 1 a + b a + b d = d 1 1 + a + b a + b 1 + mina b d v = d 1 a + b a + b 1 + mina b 5.2.2. Corner cells of types hot and cold unsafe The channel demand equation for a hot corner cell in ring i whose adjacent cells in ring i 1 are also hot is given as ex + fx + gx hx = X. 9 Proceeding in a similar manner as in the case of non-corner cells it can be shown that refer to figures 5vii x e 1 = e 1 e = e 1 h h e = e 1 1 + h h f 1 = f 1 f = f 1 h h f = f 1 1 + h h g = g 1 1.

334 S.K. Das et al. / Tele-traffic load balancing g = g 1 h h g = g 1 1 + h h 5.3. Channel demand graph Figure 6. Channel demand graph for an incomplete hot spot. Initially a channel demand graph is constructed based on the demand and class of each cell in the hot spot and Peripheral Rings. A demand graph is a layered graph with the uppermost layer representing the Ring 0 of the hot spot and each subsequent layer consists of nodes representing cells in the next outer ring until we reach a Peripheral Ring consisting only of cold safe cells which form the lowermost layer of the demand graph. Thus this graph spans all the Rings of the hot spot as well at least one of its Peripheral Rings. Let ring i correspond to the layer j of the demand graph. Then there is an edge between a node u in layer j and a node v in layer j + 1ifthereis channel demand from the ring i cell corresponding to node u to the ring i + 1 cell corresponding to node v. The edge weight is given by demandu v/x. Hence the lowermost layer of the demand graph consists only of cells in the cold safe state which do not have any channel demand. Section 5.2 discussed that given the channel demands towards a cell in ring i in an incomplete hot spot we can. find exactly how the channel demands of ring i cells from adjacent ring i + 1 cells are modified with respect to the same demands if it belonged to a complete hot spot. The construction of the demand graph involves computing these modified demands for the cells in a layer by layer fashion such that the uppermost layer consists of nodes representing the hot cells of Ring 0. Let us assume that the uppermost layer contains the center cell itself. The demands from each of its six adjacent cells in ring 1 is 1 6XwhereX is the demand in the center cell. Depending on this demand and the channel availability these six cells are classified as hot cold safe cold unsafe or cold semi-safe and thus form the second layer of nodes in the demand graph. The demands of the ring 1 cells from ring 2 cells can now be computed as laid down in section 5.2 which lead to the classification of the ring 2 cells. In this way the demand graph is constructed in a top down fashion. The construction of the demand graph terminates when a ring consisting only of cold safe cells is reached. An example of an incomplete hot spot and its demand graph is shown in figure 6. For simplicity the edge weights are not shown in this figure. Next consider the case when the center cell does not constitute the uppermost layer of the demand graph i.e. the center cell is not hot. Then the uppermost layer consists of nodes representing the hot cellss of the ring j say

S.K. Das et al. / Tele-traffic load balancing 335 nearest to the center cell and containing at least one hot cell. To compute the channel demand of the hot cells in ring j from the adjacent cells in ring j + 1 we proceed as earlier for non-corner hot cells in ring j and obtain c vi = c 1 a + b d vi = d 1 a + b. 10 Similarly for corner hot cells in ring j we obtain e iv w = e 1 f iv w = f 1 11 g iv w = g 1. Let us now sketch the channel borrowing algorithm. After the channel demand graph for the incomplete hot spot is constructed the borrowing algorithm works in a bottom up fashion starting from the lowermost layer of the demand graph. These cold safe cells lend the necessary amount of channels to adjacent nodes in the next higher layer to satisfy the edge demands. This brings all the cells in this layer to the cold safe state and they are now ready to lend channels to nodes in the next higher layer and so on. The algorithm continues till the uppermost layer of the demand graph is reached. At the end all the cells in the hot spot will have their channel demands fulfilled and thereby will be in the cold state. 6. Performance modeling We develop two discrete time Markov models one for a complete hot spot and the other for a cell within the hot spot. The Markov model for a cell is developed first and some of the analytical results obtained are then used to develop another model capturing the evolution of a complete hot spot. The model can be extended easily to the case of incomplete hot spot. 6.1. Markov chain model of a cell Our Markov chain model of a cell in a complete hot spot captures the channel availability pattern in that cell with respect to discrete time intervals see figure 7. Let the stochastic process describing this Markov chain take up the discrete set of values {n: 0 n C} where n denotes the number of available channels in the cell and C is the total number of channels allocated to that cell initially. The process is said to be in state S i if n = i. Assume that the call arrival process in the cell is Poisson with rate λ. Let the call termination process be also Poisson with parameter µ. These are valid assumptions so far as the normal telephone calls are concerned. If a cell is in any one of the states S i for0 i hc then it is a hot cell. From now on hc will be interpreted as hc. When the cell enters the state S hc+1 from S hc it becomes a cold cell from a hot one and remains in the cold state as long as it is in one of the states S i where hc + 1 i C. By definition a cell is in state S i if the number of available channels is i or the numberof channels in use is C i. Hence the cell will make a state transition from S i to S i+1 whenever any one of C i ongoing simultaneous calls terminate. This implies that the forward state transition probability from S i to S i+1 is given as C iµ. Someof these forward transition probabilities are shown in figure 7. The reverse transition will take place whenever there is a new call arrival implying that the probability is given as λ. Also there can be a discrete increase of X the number of channels to be retained by a hot cell in the number of available channels of the cell in hot state when the load balancing algorithm runs. We assume that this increase in the number of channels can take place with a constant probability µ from any of the states S i 0 i hc which accounts for the forward arcs from S i to S i+x in figure 7. In the next subsection we will describe a method to estimate the value of µ. Let us now compute the steady state probability Π i for state S i of the Markov chain. It is evident from figure 7 that the balance equations are not identical for all the states. We can actually partition the whole chain into four subchains and derive the limiting probabilities of the states within each subchain in terms of the limiting probability Π 0 of the state S 0. The subchains are defined as Subchain-1: consists of the states S i for 0 i hc Subchain-2: consists of S i forhc + 1 i X 1 Subchain-3: consists of S i forx i hc + X Subchain-4: consists of the remaining states. The recursive balance equation for a state S i Subchain-1 is given by Π i 1 C i + 1µ + Π i+1 λ = Π i C iµ + µ + λ 12 where the boundary cases are { Π0 i = 0 Π i = Cµ+µ 13 λ Π 0 i = 1. To solve for the closed form expression for Π i we apply a geometric transform G-transform to equation 12. The G-transform of Π i is given as GΠ i = Gz = Π i z i. i=0 By the shifting and scaling properties of G-transform we have r 1 GΠ i+r = z Gz r Π j z j and GiΠ i = z dgz dz j=0 = zg z in

336 S.K. Das et al. / Tele-traffic load balancing Figure 7. Markov model for a cell in a complete hot spot. respectively. Using the initial conditions of Π i given in equation 13 the following linear first order differential equation is solved for Gz: G z + Cz2 C + + z + µz 2 Gz = Π 0 1 z z 2 14 where = λ/µ and = µ /µ. The expression for the steady state probability for Subchain-1 is obtained by taking the inverse transform of Gz as Π i = Π 0 C i! j k=1 C i =0 C +i i j=0 + i k + 1 1 i+i j C i i j m + 1 15 m=1 where 1 i hc. The derivation details of the steady state probabilities will henceforth be omitted. Interested readers are requested to refer to [4]. Equation 12 can be used to derive the steady state probability Π hc+1 of S hc+1 in terms of Π hc whichin turn is given by equation 15. Let Π hc+1 = η 1 Π hc.the values of Π hc and Π hc+1 are the boundary cases for the recursive balance equation for Π i in case of Subchain-2 and given by Π i 1 C i + 1µ + Π i+1 λ = Π i λ + C iµ. 16 Proceeding in a similar manner as in the case of Subchain-1 we obtain Π i = Π 0 [A {η 1 1 C } { i k C + j k + k=1 j=1 }] i 1 k C + j k 1 17 k=1 j=1 where hc + 1 i X 1and { } A = 1 i 1 k C + j k 1 1 + k=1 j=1 1 1 i +j i e j! i i 1 i k! k j Z i Z {0} + 1 e i Z j Z j j! 1 2i + 1 2i + 1 k 2 2i +1 k. 18 Here Z is the set of positive integers and the variables i j i j will assume certain discrete values from Z such that the following two conditions are satisfied. 1 i and j will assume those integer values satisfying the equation k = i + i + j C for0 k i. 2 i and j will assume those integer values satisfying the equation k = C +2i j i+1 for 0 k 2i +1. Henceforth wherever A will be used it is assumed that A is given as in equation 18 satisfying the above conditions. Using equation 16 the boundary cases for the recursive solution for the steady state probabilities of Subchain-3 can be derived as in the case of Subchain-2. Let Π X = η 2 Π X 1. The recursive balance equation for Π i in Subchain-3 is given as Π i 1 C i + 1µ + Π i X µ + Π i+1 λ = Π i C iµ + λ. 19 Note that Π i X forx i hc + X are the steady state probabilities of Subchain-1. Hence the G-transform of Π i X is equal to the G-transform of Π i for 0 i hc which is derived earlier. Let A l denote the same expression as A with the parameter C replaced by the variable l. Substituting GΠ i 0 i hc for GΠ i X X i hc+x and proceeding similar to the cases of Subchain-1 and Subchain-2 we obtain { i k C + j k Π i = Π 0 [Aη 2 C + i 1 k k=1 j=1 k=1 j=1 C + j k 1 } + 1 l=0 η 3 A l l! ] 20 where X i hc + X and C C +i j + i k + 1 η 3 = i i =0 j=0 k=1 l i j m + 1 1 C+i j l. 21 m=1

S.K. Das et al. / Tele-traffic load balancing 337 Let the boundary cases for the recursive solution for the steady state probabilities of Subchain-4 be Π hc+x and Π hc+x+1 = η 4 Π hc+x derived from equations 20 and 19 respectively. Then the steady state probability Π i for Subchain-4 will have the same expression as that for Subchain-2 with η 4 replacing η 1. Hence Π i = Π 0 [A {η 4 1 C } { i k C + j k + i 1 k k=1 j=1 C + j k 1 }] k=1 j=1 22 for hc + X + 1 i C. Using equations 15 17 20 22 and the fact that C i=0 Π i = 1 we first derive the expression for Π 0. Expressions for the steady state probabilities Π i wherei>0 are then derived in terms of Π 0. Two important performance metrices for our load balancing algorithm are the probability of call blockade in a cell and the probability of a cell being hot. The steady state probability Π 0 gives the call blocking probability of a cell. The probability of a cell being hot is given by p h = hc Π i i=0 = Π 0 [1 + j k=1 hc i=1 1 C i! C C i =0 + i k + 1 i +i j=0 1 i+i j ] C i i j m + 1. m=1 6.2. Estimation of the probability µ Let us consider a cell in Ring j of a complete hot spot. By our channel borrowing strategy µ is the probability that sufficient number of channels are available in the system excluding the part of the hot spot from Ring 0 to Ring j to meet the channel demand D = {3jj + 1 + 1}X. Let H j denote the set of cells forming Rings 0 to j and H j is the complement set. A cell will only be able to lend channels if it is in the cold safe state. If N avail i bethe number of available channels in cell i of the system this implies that the actual number of channels available in the system for lending purpose is A = N avail i Hj hc + 1. i H j Thus µ = Prob[A D] = Prob [ total number of available channels D + ] Hj hc + 1. To compute µ let us consider the evolution of the entire system in the time period between two successive runs Figure 8. Markov model of the system between two runs of the load balancing algorithm. of the load balancing algorithm assuming that the system contains M cells. By assumption the call arrival and termination processes in each cell i are Poisson and denoted as λ i and µ i respectively. Applying results from queueing theory the call arrival and termination processes for the entire system are also Poisson with the rates λ = M i=1 λ i and µ = M i=1 µ i respectively. Our system can then be modeled as an M/M/k/k queue with state S i = i where 0 i MC is the total number of available channels in the system see figure 8. From known queueing theoretic results the steady state probability of S i for such a birth and death Markov chain is given by where Π 0 = Π i = λ /µ i Π 0 23 i! [ MC l=0 ] 1 λ /µ l. l! Thus the expression for the transition probability µ can be derived as µ = Prob [ total number of available channels D + Hj hc + 1 ] D+ H j hc+1 1 = 1 Π i i=0 where Π i is given by equation 23. 6.3. Markov chain model for a complete hot spot This is a simple birth and death Markov chain as shown in figure 9. Let the stochastic process describing this Markov chain take up the discrete set of values {n: 0 n N} where n denotes the number of Rings in the hot spot and N M denotes the maximum number of rings which a hot spot can contain in an area of M cells. The stochastic process is said to be in state S i if n = i. Let us compute the forward transition probability P i 1i from state S i 1 to S i where i 1. In a complete hot spot this gives the probability that all the cells in Ring i 1 are hot. Since there are 6i 1 cells in Ring i 1 for i>1 and only one cell in Ring 0 we have { ph i = 1 P i 1i = p 6i 1 24 h 1 <i N.

338 S.K. Das et al. / Tele-traffic load balancing Figure 9. Markov model for a complete hot spot. Similarly the reverse transition probability P ii 1 from state S i to S i 1 denotes the probability that all cells in Ring i 1 are cold. Hence { 1 ph i = 1 P ii 1 = 1 p h 6i 1 25 1 <i N. The steady state probability Π i of the state S i is obtained by solving the recurrence Π i = thus yielding { ph 1 p h Π 0 i = 1 ph 1 p h 6i 1Πi 1 1 <i N 26 Figure 10. Blocking probability versus size of the hot spot. ph 3ii 1+1 Π i = Π 0. 27 1 p h The probability Π i gives the steady state probability that the hot spot consists of i Rings. 7. Simulation experiments Simulation experiments are carried out emulating a real time cellular mobile environment in an urban area. For example the downtown area of the city is chosen as the hot spot while the suburbs comprise the outer rings of cells. A considerable reduction in the blocking probability of the system with load balancing is observed as compared to the system without load balancing. Also the performance of our scheme is compared with the CBWL channel borrowing without load balancing scheme proposed in the literature. In our simulation model call arrivals and terminations are modeled as Poisson processes with rates λ and µ respectively and time is equivalent to the number of iterations. A fixed channel assignment scheme with an initial allocation of C channels per cell is assumed. The hot spot density and the number of rings comprising the hot spot are variable. 7.1. Performance results 7.1.1. Impact of size and density of hot spot The goal of this experiment is to measure the stability of our load balancing scheme under most severe tele-traffic demands. For this we define a parameter called spot density sd varying which we can control the number of hot cells in the hot spot. For example sd = 0.1 gives only 3 hot cells in a 4 Ring hot spot while sd = 0.5 gives16 and sd = 0.9 gives 32 hot cells in the same hot spot. The size of the hot spot is varied by the number of rings that Figure 11. Blocking probability for various call arrival rates. comprise hot cells. For this particular run the spot density was fixed at sd = 0.5 and the size was varied. The total size of the system was 100 rings. It is observed from figure 10 that the percentage reduction of blocked calls attains a maximum of 62.50% with a very small hot spot number of rings = 10 and a minimum of 14.38% with almost the entire system being a hot spot number of rings = 90. Hence under very high tele-traffic demand there is still about 15% improvement in system performance with the introduction of load balancing. 7.1.2. Impact of call arrival rate This experiment evaluates the performance of the load balancing strategy under various call generation rates. With a low call generation rate of λ = 0.1 the average percentage reduction in the number of blocked calls is 85.9%. With a very high call generation rate of λ = 0.9 the performance of the algorithm suffers but still an improvement of about 7% is observed as compared to the system without load balancing.

S.K. Das et al. / Tele-traffic load balancing 339 borrower cell and the co-channel cells of the lender. Detailed analytical modeling of the system with our load balancing scheme captured certain useful performance metrices like call blocking probability the probability of a cell being hot and the evolution of the hot spot size. Exhaustive simulation experiments are carried out to validate that our load balancing algorithm is robust under severe load conditions. Comparison of our scheme with the CBWL strategy demonstrates that under moderate and even very high load conditions a performance improvement of as high as 12% in terms of call blockade is achievable with our load balancing scheme. Figure 12. Comparison of our scheme with CBWL and no load balancing. 7.2. Comparison with CBWL The proposed load balancing scheme is compared with the CBWL scheme [11]. In [3] we observed that CBWL outperforms every other existing load balancing scheme under moderate tele-traffic load while under heavy load only LBSB Load Balancing with Selective Borrowing performs better. Hence CBWL is chosen as the most suitable candidate for comparison with our new load balancing scheme in this paper. Figure 12 compares the performance of our scheme with CBWL with respect to the call blocking probability for various call arrival rates. The results show that our scheme performs better than CBWL both for moderate and high tele-traffic loads in the system. This is expected because in case of a dense hot spot the interior cells tend to starve in the CBWL channel assignment scheme. Our scheme adopts a layered structured approach of channel migration from the exterior cold cells to the interior cells of the hot spot region. Performance improvements over a scheme without load balancing are 12% for our scheme and 7% for CBWL for a high arrival rate of 0.9 whereas for an arrival rate of 0.3 the improvements are 95.3% for our scheme and 98.7% for CBWL. 8. Conclusions In this paper we proposed a load balancing strategy for the tele-traffic hot spot problem in cellular networks. A hot spot is viewed as a stack of hexagonal rings of cells and is termed complete if all the cells within it are hot. We first propose a load balancing scheme for a complete hot spot and then extend it to the general case of incomplete hot spots. Load balancing is achieved by a structured borrowing mechanism whereby a hot cell can borrow a fixed number of channels depending on its relative position within the hot spot only from adjacent cells in the next outer ring. In this way unused channels are migrated into the hot spot from its peripheral rings. The structured borrowing mechanism also reduces the amount of interference between the References [1] L.J. Cimini Jr. and G.J. Foschini Distributed algorithms for dynamic channel allocation in microcellular systems in: IEEE Vehicular Technology Conf. 1992 pp. 641 644. [2] D.C. Cox and D.O. Reudink Increasing channel occupancy in large scale mobile radio systems: dynamic channel reassignment IEEE Transactions on Vehicular Technology 22 November 1973 218 222. [3] S.K. Das S.K. Sen and R. Jayaram A dynamic load balancing strategy for channel assignment using selective borrowing in cellular mobile environment in: Proceedings of ACM/IEEE Conference on Mobile Computing and Networking Mobicom 96 New York November 1996 pp. 73 84; Wireless Networks 35 1997 333 347. [4] S.K. Das S.K. Sen and R. Jayaram A structured channel borrowing scheme for dynamic load balancing in cellular networks in: Proceedings of 17th IEEE International Conference on Distributed Computing Systems Baltimore MD May 1997 pp. 116 123. [5] S.K. Das S.K. Sen R. Jayaram and P. Agrawal An efficient distributed channel management algorithm for cellular mobile networks in: Proceedings of IEEE International Conference on Universal Personal Communications ICUPC San Diego CA October 1997. [6] M. Duque-Anton D. Kunz and B. Ruber Channel assignment for cellular radio using simulated annealing IEEE Transactions on Vehicular Technology 42 February 1993. [7] B. Eklundh Channel utilization and blocking probability in a cellular mobile telephone system with directed retry IEEE Transactions on Communications 344 April 1986. [8] S.M. Elnoubi R. Singh and S.C. Gupta A new frequency channel assignment in high capacity mobile communication systems IEEE Transactions on Vehicular Technology 313 August 1982. [9] K.K. Goswami M. Devarakonda and R.K. Iyer Prediction-based dynamic load-sharing heuristics IEEE Transactions on Parallel and Distributed Systems 46 June 1993. [10] D. Hong and S.S. Rappaport Traffic model and performance analysis for cellular mobile radio telephone systems with prioritized and non-prioritized hand-off procedures IEEE Transactions on Vehicular Technology 35 August 1986. [11] H. Jiang and S.S. Rappaport CBWL: A new channel assignment and sharing method for cellular communication systems IEEE Transactions on Vehicular Technology 432 May 1994. [12] T.J. Kahwa and N.D. Georganas A hybrid channel assignment scheme in large scale cellular structured mobile communication systems IEEE Transactions on Communications 264 April 1978. [13] J. Karlsson and B. Eklundh A cellular mobile telephone system with load sharing an enhancement of directed retry IEEE Transactions on Communications 375 May 1989. [14] W.C.Y. Lee Elements of cellular mobile radio systems IEEE Transactions on Vehicular Technology 35 May 1986. [15] V.H. Macdonald Advanced mobile phone service: The cellular concept Bell System Technical Journal 58 January 1979 15 41.

340 S.K. Das et al. / Tele-traffic load balancing [16] E. Del Re R. Fantacci and G. Giambene Handover and Dynamic channel allocation techniques in mobile cellular networks IEEE Transactions on Vehicular Technology 442 May 1995. [17] K.N. Sivarajan R.J. McEliece and Ketchum Dynamic channel assignment in cellular radio in: IEEE Vehicular Technology Conf. November 1990 pp. 631 637. [18] J. Tajima and K. Imamura A strategy for flexible channel assignment in mobile communication systems IEEE Transactions on Vehicular Technology 37 May 1988. [19] S. Tekinay and B. Jabbari Handover and channel assignment in mobile cellular network IEEE Communication Magazine November 1991. [20] M.H. Willebeek-LeMair and A.P. Reeves Strategies for dynamic load balancing on highly parallel computers IEEE Transactions on Parallel and Distributed Systems 49 September 1993. [21] Z. Xu and P.B. Mirchandani Virtually fixed channel assignment for cellular radio-telephone systems: a model and evaluation in: IEEE International Conference on Communications Chicago 1992. [22] K.L. Yeung and T.S.P. Yum The optimization of nominal channel allocation in cellular mobile systems in: IEEE International Conference on Communications 1993. [23] M. Zhang and T.S. Yum Comparisons of channel assignment strategies in cellular mobile telephone systems IEEE Transactions on Vehicular Technology 38 November 1989. Sajal K. Das received the B.Tech. degree in 1983 from Calcutta University the M.S. degree in 1984 from the Indian Institute of Science Bangalore and the Ph.D. degree in 1988 from the University of Central Florida Orlando all in computer science. Currently he is an Associate Professor of Computer Sciences at the University of North Texas Denton the Director of the Center for Research in Parallel and Distributed Computing and also the founder of the Center for Research in Wireless Computing CReW at UNT. He is a recipient of the Cambridge England Nehru Scholarship in 1986 Honor Professor Awards from UNT in 1991 and 1997 for best teaching and scholarly research and Developing Scholars Award in 1996 from UNT for outstanding research. Dr. Das has been a Visiting Scientist at the Slovak Academy of Sciences in Bratislava Slovakia and the Council of National Research in Pisa Italy. He has also been a Visiting Professor at the Indian Statistical Institute Calcutta. His current research interests include cellular communication and mobile computing parallel algorithms and data structures and distributed simulation. Dr. Das has published over 100 research papers in these areas in journals and refereed conference proceedings. He serves on the Editorial Boards of the Journal of Parallel and Distributed Computing Academic Press Parallel Processing Letters World Scientific Pub. and the Journal of Parallel Algorithms and Applications Gordon and Breach. He has been a Program Committee member of numerous international conferences and is the General Co-Chair of the International Symposium on Modeling Analysis and Simulation of Computer and Telecommunication Systems MASCOTS 98. Dr. Das is also a member of the IEEE and the ACM. E-mail: das@cs.unt.edu URL: http://www.cs.unt.edu/home/joy/crew.html Sanjoy K. Sen received his bachelor s degree in electronics and telecommunications engineering from Jadavpur University India in 1991 M.S. and Ph.D. degrees in computer science from the University of North Texas Denton TX in 1995 and 1997. He is a recipient of the Chancellor s Graduate Fellowship from UNT in 1995 1996. Between 1991 and 1993 he was involved with Siemens Telematik India Ltd. as a senior projects engineer in the switching hardware division. His current research interests include mobile computing and communications location management channel assignment call admission and control Quality-of- Service mobile data multiprocessor computer architecture distributed systems parallel discrete event simulation. Dr. Sen is a student member of the IEEE Communications Society Upsilon Pi Epsilon and was the vice-president of the Computer Sciences Graduate Student Association at UNT. E-mail: joy@cs.unt.edu sanjoy@nortel.ca Rajeev Jayaram received the bachelor s degree in computer science and engineering from the Bangalore University India in 1992. Until 1994 he worked on formal models for distributed systems at the Indian Institute of Science Bangalore. Since spring 1995 he has been a graduate student at the University of North Texas Denton working on wireless networks and mobile computing. Currently he is working as a software engineer in the wireless communications division of Ericsson Inc. in Richardson TX. His current research interests include channel assignment call admission and control Quality-of-Service provisioning in mobile computing and also distributed systems and networking. Mr. Jayaram is a student member of the IEEE Communications Society. E-mail: jayaram@cs.unt.edu