An Adaptive Distance-Based Location Update Algorithm for PCS Networks

Transcription

1 An Adaptive Distane-Based Loation Update Algorithm for PCS Networks Abstrat - In this paper, we propose a stohasti model to ompute the optimal update boundary for the distane-based loation update algorithm. The proposed model aptures some of the real harateristis in the wireless ellular environment. The model an adapt to arbitrary ell topologies in whih the number of neighboring base stations at different loations may vary. The ell residene time an follow general distributions whih aptures the fat that the mobile user may spend more time at ertain loations than others. The model also inorporates the onept of a trip in whih the mobile user may follow a partiular path to a destination. For implementation, the deision of loation update an be made by a simple table lookup. Numerial results indiate that the proposed model provides a more aurate update boundary in real environment than that derived from a hexagonal ell onfiguration with random walk movement pattern. The proposed model allows the network to maintain a better balane between the proessing inurred due to loation update and the radio bandwidth utilized for paging between all arrivals. I. INTRODUCTION In reent years, there has been a signifiant inrease in the number of Personal Communiations Servie (PCS) subsribers around the world. One of the issues in mobility management is to trak the loation of the users [I]. Sine mobile users are free to move within the overage area, the network an only maintain the approximate loation of eah user. When a onnetion needs to be established to a partiular user, the network has to determine the user s exat loation within the ell granularity. The operation of the mobile terminal informing the network about its urrent loation is known as loation update, and the operation of the network determining the exat loation of the mobile user is alled terminal paging or searhing. Loation update algorithms an be divided into two main groups: stati and dynami. In a stati algorithm, loation update is triggered based on the topology of the network. Examples inlude the onventional loation area (LA) based sheme used in GSM systems. In a dynami algorithm, loation update is based on the user s all and mobility patterns. Examples inlude the distane-based, timer-based, and the movement-based shemes. In the distane-based sheme [2][3], a mobile terminal transmits an update signal whenever its distane (in terms of the number of ells) exeeds a ertain threshold. In the timer-based sheme 141, a mobile terminal updates its loation periodially. In the movement-based sheme [SI, loation update is performed when the number of boundary rossings between ells exeeds a ertain value. A number of novel loation update algorithms have been This work was supported in part by the Natural Sienes and Engineering Researh Counil of Canada under a Postgraduate Sholarship and Grant No. OGP44286; sholarships from the Communiations Researh Centre, Industry Canada, and the University of British Columbia. Vinent W.-S. Wong and Vitor C. M. Leung Department of Eletrial and Computer Engineering University of British Columbia 2356 Main Mall, Vanouver, BC, Canada V6T vinentw@ee.ub.a, vleung@ee.ub.a proposed reently. In the preditive distane-based update sheme [6], the mobile terminal reports both its loation and veloity during the update proess. The network used this information to predit the mobile s loation in future time. In [7], a state-based update sheme is analyzed where the system state inludes the urrent loation and the time elapsed sine the last update. The LeZi update algorithm proposed in [8] an be onsidered to be a path-based update sheme in whih the movernetit histoty rather than the urrent loation is sent in an update message. In [9], the onepts of boundary loation area and boundary loation register were proposed to failitate roaming between different wireless networks. The update distane is a funtion of the QoS fator, veloity ratio, and predefined threshold. Although various loation update algorithms have been proposed in the literature, the evaluations of these algorithms are often performed under ertain unrealisti assumptions: 1. Strutured ell onfigurations: Mesh or hexagonal ell onfigurations are used in two-dimensional models [3][5], and a linear model is used in the one-dimensional ase [2][6]. Although these ell topologies simplify the analyses, they do not give an aurate representation of a realisti ellular network topology where the number of neighboring ells varies from ell to ell. 2. Exponential ell residene time distribution: Most of the work assumed the ell residene time follows a geometri (or exponential) distribution [2][6][1], whih is assumed to be independent and identially distributed (i.i.d.) for all ells. The drawbak of this assumption is that it does not apture an aurate representation of individual user mobility patterns, where a user may stay at ertain loations (e.g., his home or offie) for a relatively long period of time. 3. Symmetri random walk movement pattern: This assumption is ommonly used to haraterize individual movement behavior [2][3][5]. Although the random walk model simplifies the analysis, the diretion of the mobile user is not taken into aount. In general, a mobile user may travel with a destination in mind. Thus, the mobile s loation in the future is likely to be orrelated with its movement history. In this paper, we fous on determining the update boundary for the distane-based sheme. Our goal is to develop an analytial model whih an eliminate some of the unrealisti assumptions ommonly used. We formulate the loation traking problem as a semi-markov deision proess [ 1 I]. There is a ost funtion assoiated with loation update and another ost funtion assoiated with terminal paging. The objetive is to determine the optimal update boundary so as to minimize the expeted total ost between all arrivals. Distint features of our model inlude [ 121: I. Appliable to arbitrary ell topologies: This feature aptures the fat that the number of neighboring base stations at different loations may vary in real life. Some /1/$1. 21 IEEE 21

2 base stations may only have two neighboring base stations while others an have as many as six. Thus, our model is not restrited to strutured ell onfiguration suh as mesh or hexagonal. Cell residene time an follow general distributions: This aptures the fat that a mobile user may spend more time at ertain ell loations (e.g., home or offie) than other loations. In addition, various distributions an be used to model different ell sizes (e.g., maroell, miroell, or pioell). The average residene time in eah ell an be different. Inorporate the movement history: The probability that the mobile user moves to a partiular neighboring ell an depend on the loation of the urrent ell or a list of ells reently visited. This movement pattern an inorporate the onept of a trip in whih the user may hoose a partiular path to a destination. The struture of this paper is as follows: In Setion 11, we introdue the notations and desribe the model formulation. In Setion 111, we analyze the distane-based algorithm under a given ell residene time distribution and desribe its implementation in arbitrary ell topology. Numerial results are presented in Setion IV. Conlusions are given in Setion V. 11. MODEL FORMULATION In this setion, we desribe how to formulate the loation traking problem as a semi-markov deision proess. A Markov deision proess model onsists of five elements: deision epohs, states, ations, transition probabilities, and osts. In our model, the deision epohs orrespond to the time instants when the mobile terminal moves from one ell to another (i.e., rossing the ell boundary). When the mobile terminal rosses the ell boundary, it has to deide whether to update its loation or not. We let the ation set A = (, 11, where I represents the ation of performing loation update and represents the null ation of no intervention. The mobile terminal hooses the ation based on its urrent state information. For the model formulation, we let the state s = (i, j ), where i represents the identifier of the urrent ell that the mobile terminal is residing, and j represents the identifier of the ell in whih the mobile terminal performed its last update. Two ost funtions are introdued to aount for the network resoures used for loation update and terminal paging. The loation update ost is assumed to be fixed and is denoted by C. It reflets the update proessing inurred on the database. $\e paging ost funtion is given by h(i, j). The paging ost reflets the onsumption of radio bandwidth utilized in determining the exat loation of the mobile terminal. The paging ost is a funtion of the number of ells being paged and the number of searh iterations performed. In this paper, we assume that the paging strategy follows the shortestdistane-first order. That is, when a paging event ours, the searh is onduted first at the user s last reported ell. If it is not found there, then the searh is onduted in the next ring of ells in inreasing distane order from the last reported ell, until the user is loated. The maximum paging delay orresponds to the maximum number of searh iterations allowed. At eah searh iteration, a set of ells are being paged simultaneously. This paging strategy has also been used in onjuntion with other loation update algorithms (e.g., [2][3][6]). For a partiular mobile user, different ell residene time distributions an be assigned to different ell loations. The average residene time in eah ell an be different. This is ahieved by letting G(rli) denote the umulative distribution funtion of the ell residene time, given the identifier of the urrent ell is i. This aptures the fat that a mobile user may spend more time at ertain ell loations than the others. Different ell sizes (e.g., maroell, miroell, pioell) an have different ell residene time distributions. Thus, the usual i.i.d. exponential ell residene time assumption an be relaxed. In this paper, we use G(dr1i) to represent the timedifferential. That is, G(dt1i) = dg(t1i). For the movement pattern in an arbitrary ell topology, we let P(kli) denote the probability that the mobile user will move to neighboring ell k when it leaves ell i. This aptures the orrelations of the user movement between two neighboring ells. Note that orrelations between the diretions of suessive moves of the mobile user an also be inorporated if the state inludes a history of the ells visited. Thus, the model an inorporate the onept of a trip in whih the mobile user may follow a partiular path to a destination. In addition, the above formulation does not assume any partiular ell onfigurations. Therefore, the proposed model an adapt to arbitrary ell topologies DISTANCE-BASED UPDATE ALGORITHM A Optimality Equations Sine the network must trak the user s loation perfetly during a all, the user s loation is known to the network when a all terminates. Thus, the time interval requiring mobility traking is between the termination of the last all and the arrival of the next one. The total ost for loation update and paging is alulated within this time interval. Let v(i, j ) denote the minimum expeted total ost between all arrivals given state (i, j ). If the time between all arrivals is exponentially distributed with rate A, then the optimality equations are given by: where 1 2 i, j 5 N and N denotes the number of base stations within the overage area. Due to spae limitations, please refer to [ 121 for the proof of the optimality equations. The first term in (1) denotes the expeted total ost if no update is performed at state (i, j ), while the seond term denotes the expeted total ost if loation update is performed at state (i, j ). B Value Iteration Algorithm There are a number of algorithms available to solve equation (I). Examples inlude the value iteration or poliy iteration algorithms [I I].The following value iteration algorithm finds a stationary deterministi optimal poliy and the orresponding expeted total ost. The solutions of the optimality equations orresponds to the qinimum expeted total ost v(i, j) aqd the optimal poliy 6 (i, j). Note that the optimal poliy 6 (i, j ) indiates the deision whether to update its loation or not at state (i, j ). O /1/$1. 21 IEEE 22

3 Algorithm 1. Set \p(i, j) = for eah state (i, j) where 1 2 i, j 2 N. Speify E > and set n =. 2. For eah state (i, j ) where 1 I i, j 2 N, ompute v"(i, j ) by vn+'(i,j) = min{(l -e -11 )/i(i,j)g(dtli) a mg h C Cell (a) The (b) Graph model showing the interonnetions of the ells. Figure 1 Cellular network topology represented by a graph model. 3. If l/~"+ ' - v ~ I / < E, go to step 4. Otherwise inrement n by 1 and return to Step For eah state (i, j ) where 1 I i, j I N, the stationary optimal poliy S (i, j) = if Otherwise, Sx(i,j) = stop. There are a number of definitions for the funtion norm In this paper, the funtion norm is defined as IIvII = max v(i, j) for 1 5 i, j 2 N. Convergene of the value iteration algorithm is ensured sine the operation in Step 2 orresponds to a ontration mapping. Thus, the funtion v"(i, j ) onverges in norm to v(i, j). C Iinplerneiitation Considerations Having identified the different parameters involved in the model, we are now in a position to explain the steps that need to be taken in order to implement the model. First of all, the network ontroller has to assign ost funtions for loation update and terminal paging. It also has to maintain the mobility profile of eah mobile user (i.e., its movement history and all history). Based on these information, the average residene time in eah ell as well as the average all arrival rate an be estimated [ 131. Given the input parameters (i.e., ost funtions and various distributions), the value iteration algorithm an be used to alulate the optimal poliy 6 (i, j). Based on the optimal poliy, the optimal update boundary an be determined. The optimal update boundary is then stored in a database (e.g., loation register). After eah loation update or a all termination, the mobile terminal needs to download the list of the update boundary ell identifiers that orresponds to its urrent loation. Whenever the mobile terminal moves to another ell, it ompares the new ell identifier with the list of the update boundary ell identifiers. Loation update is performed if the new ell is one of those update boundary ells. The optimal update boundary stored in the loation register needs to be revised whenever there is a hange of the movement history, all history, or network topology (e.g., t!"ntal hllanee Figure 2 The nodes represent the loation of the base stations. An edge between two nodes represents those two base stations are neighbors to eah other. installation of new base stations). The alulation of the optimal update boundaries an be performed off-line, e.g., whenever spare proessing apaity is available at the network ontroller. A IV. NUMERICAL RESULTS AND DISCUSSIONS Simulation Assumptions and Speifiations In our simulation environment, a graph model is used to represent the topology of a ellular network. In general, the interonnetion of the ells an be modeled as a onneted graph G = (N, E), where the node set N represents the set of ell or base station identifiers and the edge set E represents the onnetivity between two neighboring ells. For example, referring to Figure 1, the node set N = {a, 6,, d, e, f, g} and the edge set E = {(a, b), (a, d), (b, ),..., (f, g)}. In partiular, we use a random graph model to represent the topology of a ellular network. The rationale behind using a random graph model is that: (1) the number of neighboring base stations for eah base station an be different; and (2) only the nodes that are lose together are onneted. This models the onnetivities of the neighboring base stations. The proedures of generating random graphs an be found in [ 141. In our model, we onsider a overage area that onsists of 1 base stations with an average node degree of 6. An example of a random graph model is shown in Figure 2. Sine the mobile user usually has a destination in mind, we model this behavior by hoosing one partiular node (or ell) in the random graph as the destination. Whenever the mobile /1/$1. 21 IEEE 23

4 . = m 2 $18 $ i lo- Call Arrival Rate (per minute) U) r w 12 X El oo 1 Loation Update Cost Figure 3 (a) Cost ratio versus all arrival rate h under different ell rossing rate I.( ;(b) Cost ratio versus loation update ost C,, different ell rossing rate p. under user leaves the urrent ell, it moves to a neighboring ell whih is losest to the destination. This aptures the behavior of moving towards the destination. If the mobile user is staying within the destination ell, after a ertain period of time it will move to one of the neighboring ells. This ontinues until the next all arrives. We now desribe the proedures of omparing the distane thresholds determined from our model to those derived from a hexagonal ell onfiguration. The goal is to show that the proposed model gives a more aurate update boundary in real wireless ellular environments ompared with that derived from a hexagonal ell onfiguration with random walk movement pattern. Given the ost and mobility parameters, we first use a hexagonal ell onfiguration with symmetri random walk movement pattern to obtain the optimal distane threshold. This optimal distane threshold is then applied to the random graph model with Markovian movement pattern. The expeted total ost of loation update and paging between all arrivals is then determined. This ost is denoted as Cost (hexagonal). The term hexagonal is used to remind us that the optimal distane threshold is derived from the hexagonal ell onfiguration. We also use the random graph model with the above movement pattern to determine the minimum expeted total ost by solving the optimality equations. This ost is denoted as Cost (optimal). The term optimal is used to remind us that the update boundary orresponds to the optimal poliy. The performane gain is the ost ratio whih is defined as Cost (hexagonal) / Cost (optimal). Unless stated otherwise, the parameters that we use in the following subsetions are: all arrival rate h =.1 per minute, the loation update ost C,, = 1, and the paging ost per ell C, = 1. We assume that the ell residene time follows an i.i.d. Gamma distribution with average tire l/p. For the value iteration algorithm, we hoose E = 1-. B Results Figures 4 (a) and (b) show the ost ratio versus the all arrival rate h and the loation update ost C,, under different ell rossing rate p (per minute). Note that the inverse (or reiproal) of the average ell residene time is the ell rossing rate. From these figures, we observe that the ost ratio inreases when p inreases or h dereases. And it approahes unity when h or C,, are large. In Figure 3 (a), when the average time between all arrivals is large (i.e., h is small), the optimal update boundary obtained from our model gives a lower ost than the distane threshold derived from the hexagonal model. However, when the average time between all arrivals is small (i.e., h is large), the mobile user does not travel muh before a all arrives. Thus, the distane thresholds derive from both methods give the same performane. In Figure 3 (b), the variation of the ost ratio with respet to C,, is due to the hanges of the optimal update boundary or distane threshold for different update ost values. When C,, is large, there is no inentive to perform loation update. The expeted total ost only onsists of the paging ost. Thus, the ost ratio approahes unity. These results imply that in real wireless ellular networks environments where the ell topology is not strutured and the user movement pattern is not random, our model an provide a more aurate update boundary than that derived from a hexagonal ell onfiguration with random walk movement pattern. And by using a more aurate update boundary, the network an maintain a better balane between the proessing inurred due to loation update and the radio bandwidth utilized for paging between all arrivals. C Sensitivity Analysis In order to determine the minimum expeted total ost and the update boundary, the optimal poliy needs to be alulated. The optimal poliy is a funtion of h, p, and other ost parameters. Although the ost parameters an be determined by the network, the values of h and p may not always be estimated orretly. If that is the ase, the poliy may not indeed be optimal. We are interested in determining the perentage hange of the expeted total ost to the variation of the average time between all arrivals and the average ell residene time. The proedures for the sensitivity analysis onsist of the following steps: Given the atual all arrival rate h and other ost and mobility parameters, we first determine the minimum expeted total ost, denoted as Cost (optimal). Let h denote the estimated all arrival rate and Ah denote the perentage hange of the average time between all arrivals. These parameters are related by the following equation: /1/$1. 21 IEEE 24

5 E _ ? = 1.2._ E n R f) - h =.1 + h =.2 e- h=.3? 115 = E $ 11 n Iṉ 8 15 I Perentage hange in average time between all arrivals (%) Perentage hange in average ell residene time ( A) Figure 4 (a) Cost ratio versus Ah under different all arrival rate h ; (b) Cost ratio versus Ap under different ell rossing rate p. Based on the estimated all arrival rate and other parameters, the sub-optimal poliy is determined. From this suboptimal poliy and other ost and mobility parameters (i.e., h, p, et), the sub-optimal expeted total ost, denoted as Cost (sub-optimal), is omputed. The hange of the expeted total ost with respet to the variation of the average time between all arrivals is haraterized by the ost ratio, whih is defined as: Cost (sub-optimal) / Cost (optimal). We also use the similar proedures desribe above to investigate the hange of the expeted ost to the variation of the average ell residene time. Figure 4 (a) shows the ost ratio versus Ah under different all arrival rate h and Figure 4 (b) shows the ost ratio versus AM under different ell rossing rate p. From these figures, we observe that the ost ratio is more sensitive to the under-estimation of both h and p. If the target ost ratio has to be less than 1.5 (i.e., 5% differene between the optimal and sub-optimal ost), then Ah has to be greater than -5% and A has to be greater than -6%. These results imply that if tiere is unertainty in estimating h or F, it may be better to over-estimate the values in order to redue the ost ratio differene. IV. CONCLUSIONS In this paper, we proposed a stohasti model to analyze the distane-based loation update algorithm. Numerial results indiate that the proposed model gives a more aurate update boundary in real wireless ellular environment than that derived from a hexagonal ell onfiguration with random walk movement pattern. And by using a more aurate loation update boundary, the network an maintain a better balane between the proessing inurred due to loation update and the radio bandwidth utilized for paging between all arrivals. Results from the sensitivity analysis show that if there is unertainty in estimating the all arrival rate or ell residene time, it may be better to over-estimate the values in order to redue the ost ratio differene. REFERENCES [l] V. Wong and V. Leung, Loation management for next generation personal ommuniation networks, IEEE Network, vol. 14, no. 5, Sept./Ot. 2. [2] U. Madhow, M. Honig, and K. Steiglitz, Optimization of wireless resoures for personal ommuniations mobility traking, IEEE/ACM Trans. Networking, vol. 3, no. 4, pp , De J. Ho and I. E Akyildiz, Mobile user loation update and paging under delay onstraints, ACM/Baltzer J. Wireless Networks, vol. 1, no. 4, pp , De C. Rose, Minimizing the average ost of paging and registration: A timer-based method, ACM/Baltzer J. Wireless Networks, vol. 2, no. 2, pp , June I. E Akyildiz, J. Ho, and Y.-B. Lin, Movement-based loation update and seletive paging for PCS networks, IEEE/ACM Tram. Networking, vol. 4, no. 4, pp , Aug B. Liang and Z. Haas, Preditive distane-based mobility management for PCS networks, in fro. IEEE INFOCOM 99, New York, NY, Marh C. Rose, State-based paging/registration: A greedy tehnique, IEEE Trans. Vehiular Tehnolog?; vol. 48, no. 1, Jan A. Bhattaharya and S. K. Das, LeZi-Update: An information-theoreti approah to trak mobile users in PCS networks, in Pro. ACMAEEE MobiCom 99, Seattle, WA, Aug W. Wang and I. E Akyildiz, Intersystem loation update and paging shemes for multitier wireless networks, in Pro. ACM MobiConz. Boston. August 2. [IO] S. K. Sen, A. Bhattaharya, and S.-K. Das, A seletive loation update strategy for PCS users, ACM/Baltzer J. Wireless Networks, vol. 5, no. 5, pp , Sept [ 1 I] M. L. Puterman, Markov Deision Proesses: Disrete Stohasti Dynami Programming, John Wiley & Sons In., [ 121 W.-S. V. Wong, Stohasti ontrol of inter-swith handoff and loation update in wireless ellular networks, fh. D. s Thesis, The University of British Columbia, Canada, April 2. [I31 M. Zonoozi and P. Dassanayake, User mobility modeling and haraterization of mobility patterns, IEEE J. Selet. Areas Commun., vol. 15, no. 7, pp , September [ 141 E. W. Zegura, K. L. Calvert, and M. J. Donahoo, A quantitative omparison of graph-based models for Internet topology, IEEE/ACM Trans. Networking, vol. 5, no. 6, De /1/$1. 21 IEEE 25