A New Random Walk Model for PCS Networks
Ian F. Akyildiz, Fellow, IEEE, Yi-Bing Lin, Senior Member, IEEE, Wei-Ru Lai, and Rong-Jaye Chen
Abstract—This paper proposes a new approach to simplify the
two-dimensional random walk models capturing the movement of mobile users in Personal Communications Services (PCS) networks. Analytical models are proposed for the new random walks. For a PCS network with hexagonal configuration, our approach reduces the states of the two-dimensional random walk from (3 2+ 3 5) to ( + 1) 2, where is the layers of a cluster. For a mesh configuration, our approach reduces the states from(2 2 2 + 1) to ( 2+ 2 + 4) 4 if is even and to ( 2 + 2 + 5) 4 if is odd. Simulation experiments are con-ducted to validate the analytical models. The results indicate that the errors between the analytical and simulation models are within 1%. Three applications (i.e., microcell/macrocell configuration, distance-based location update, and GPRS mobility management for data routing) are used to show how our new model can be used to investigate the performance of PCS networks.
Index Terms—Cell, mobility management, personal
communi-cations services, random walk.
I. INTRODUCTION
I
N Personal Communications Services (PCS) systems, the service areas are covered by radio base stations (BS’s) [12]. The radio coverage of a BS is called a cell. A mobile phone or mobile station (MS) moves from one cell to another. Most PCS performance studies assume that the cells are configured as a hexagonal network given in Fig. 1 or a mesh network given in Fig. 2. To investigate the MS movements is a challenging problem in PCS. For example, starting from a particular cell, the destination cell of an MS after movements is determined in [2]. Another example is given in [15], which studied how many steps an MS should make to leave a region.A two-dimensional random walk model with absorbing states [9] can be used to study the movements of an MS. In this model, a state represents a cell where the MS may reside. Fig. 1 shows a 6-subarea hexagonal cluster. The cell at the center of the cluster is called subarea-0 cell. The cells surrounding the subarea (
) cells are called subarea cells. There are cells in sub-area except that exactly one cell is in subarea 0. An -subarea cluster contains cells from subarea 0 to subarea ( ). The cells surrounding the subarea ( ) cells are referred to as boundary
neighbors, which are outside of the cluster. Fig. 2 shows a
5-sub-Manuscript received October 6, 1999; revised February 23, 2000. The work of I. F. Akyildiz was supported by the National Science Foundation (NSF) under Grant NCR-97-04393.
I. F. Akyildiz is with Broadband and Wireless Networking Lab., School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: Ian.Akyildiz@ee.gatech.edu).
Y.-B. Lin and R.-J. Chen are with the Department of Computer Science and Information Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C. (e-mail: liny@csie.nctu.edu.tw; rjchen@csie.nctu.edu.tw).
W.-R. Lai is with the Department of Information Management, Chin-Min College, Tou-Fen, Miao-Li, Taiwan, R.O.C. (e-mail: wrlai@mis.chinmin.edu.tw).
Publisher Item Identifier S 0733-8716(00)05415-9.
Fig. 1. Hexagonal PCS cell structure.
Fig. 2. Mesh PCS cell structure.
area mesh cluster. In this network, every cell has four neighbors and an MS can only move to one of the four neighboring cells.
To compute the number of movements, whenever an MS moves out of the region, i.e., moving to a boundary cell, the random walk enters an absorbing state. A potential problem of this model is that the number of states increases rapidly as the size of the region increases. To reduce the computational com-plexity, we modified the two-dimensional random walk model
TABLE I THEACRONYMLIST
given in [2]. However, the modified model has been simplified, which introduces some inaccuracy to the two-dimensional random walk models.
In this paper, we show how to reduce the two-dimensional random walks for hexagonal and mesh planes such that the sim-plified random walks behave exactly the same as the original random walks. The paper is organized as follows. Section II de-scribes the new random walk for hexagonal configuration. Sec-tion III shows how to group cells with the same routing pat-terns so that the states in the random walk can be reduced. tion IV validates our model with simulation experiments. Sec-tion V elaborates on how to use a similar technique to reduce the states for a mesh random walk model. Section VI illustrates three applications that benefit from our random walk model. The acronyms used in the paper are listed in Table I.
II. THETWO-DIMENSIONALHEXAGONALRANDOMWALK
Let us consider a hexagonal plane. We assume that an MS resides in a cell for a period, then moves to one of its neighbors with the same probability, i.e., with probability 1/6; see Fig. 3. We derive the number of moving steps (a step represents an MS movement from a cell to another) from the starting cell until the MS moves out of the cluster.
Based on the assumption that the routing probabilities are equal, we observe that the cells in a cluster can be classified into several types, where a type represents a state in the new random walk. The term “type” is defined as follows.
Definition 1: Two cells and are of the same type if the multiset of types for ’s neighbors is the same as that for ’s neighbors.
A multiset is a collection of objects that are not necessarily distinct. The multiplicity of elements has significance in the type definition.
The 6-subarea cluster is shown in Fig. 4 where lines 1–3 di-vide the cluster into 6 equal pieces. Exchange of any two pieces has no impact on the structure of the cluster. If two cells, for ex-ample, the cells marked with , are at the same relative position on different pieces, then they are grouped together and assigned to the same type. MS’s in the cells of the same type will leave the cells with the same routing pattern. It is intuitive that cells in different subareas should have different types. In the next sec-tion, we describe a type classification algorithm based on the 3-line symmetry concept, which satisfies Definition 1.
Fig. 3. The hexagonal routing pattern.
Fig. 4. The type classification in a 6-subarea cluster.
III. THETYPECLASSIFICATION FORHEXAGONAL
RANDOMWALK
Here we describe a type classification algorithm which sat-isfies Definition 1. This algorithm recursively assigns types for cells in an -subarea cluster. Basically, every cell is marked as type , where “ ” represents that the cell is in subarea- , and “ ” represents the st type in subarea- . The al-gorithm is described as follows.
The Type Classification Algorithm for an -Subarea Cluster:
Step 1) The subarea-0 cell is assigned to type . is the subarea of cells being labeled).
Step 2) . If , then Stop.
Step 3) Find unmarked subarea- cells that have one neighboring cell. Label them by type .
represents the st type in subarea- . Step 4) Let . If , then go to Step 2.
Step 5) Find unmarked subarea- cells that are the neighbors of cells in the clockwise direction. Mark them with type . Go to Step 4.
It is easy to verify that the classification algorithm has the following three properties:
1) For , the cell has six neighboring cells. For , it has six boundary neighbors.
2) For a cell (where , the multiset of types of its six neighboring cells is , ,
mod , , , . For
, the multiset of is , mod
, , boundary , boundary , boundary mod , where “boundary ” represents the type of the boundary neighbor out of the cluster.
3) For a cell (where , if ,
the multiset of types of its six neighboring cells is mod
mod . For , the
multiset of is mod
mod , boundary ,
boundary mod .
The above properties ensure that the classification algorithm satisfies Definition 1.
Fig. 4 illustrates the types of cells for a 6-subarea cluster clas-sified by the algorithm. Cells are assigned to types , where “ ” indicates that the cell is in subarea- , and represents the st type in subarea- . The cell in subarea-0 is of the type . The six subareas-1 cells are of the same type because they have the same neighboring types (i.e., one , two cells and three boundary neighbors). For a 2-subarea cluster, a subarea-2 cell may have 2 or 3 boundary neighbors and is assigned to type or , respectively. By following the same method, we mark all cells subarea by subarea. It is easy to verify that the resulting type assignment satisfies Definition 1. For example, consider cells and in Fig. 4. These cells are assigned to type . Both multisets of types for ’s and ’s
neighbors are , , , , , . Thus,
and are grouped together and are assigned to type . Based on the type classification and the concept of absorbing states, the state diagram of the random walk for an -subarea cluster (where is shown in Fig. 5. In this state diagram, state represents that the MS is in one of the cells of type
, where and . State
represents that the MS moves out of the cluster from state
, where . For and ,
states are transient and for , states are absorbing.
Let be the one-step transition probability from state to state ; i.e., the probability that the MS moves from a cell to a cell in one step. Since all neighbors of the cell are cells, the process moves from state (0, 0) to state (1, 0) with probability . A cell has one neighbor, and the transition from state (1, 0) to state (0, 0) has probability . The transition back to a state itself occurs when the MS moves to a
Fig. 5. State diagram for a 6-subarea cluster.
cell of the same type. Since each cell has two neigh-bors, state (1,0) has a transition back to itself with probability
. For , is the
prob-ability that the MS moves from a cell to a neighbor out of the cluster in one step. The absorbing state loops back
to itself with probability , for .
Let be the total number of states for an -subarea cluster random walk. Then and if
The transition matrix of this random walk is an
matrix where
..
. ... ... ... ... . .. ... ... ...
For , , and in this matrix, the elements in each column and row are listed in the following order: (0, 0), (1, 0), (2, 0), (2, 1), (3, 0), (3, 1), , (6, 3), and (6, 4). We use the Chapman–Kolmogorov equation [18] to compute the probability for the number of steps that an MS moves from a cell type to another. For , let
if
An element in is the probability that the random walk moves from state to state with exact
steps. For , define as
for for
(2) Then is the probability that an MS initially re-sides at a cell, moves into a cell at the st step, and then moves out of the cluster at the th step. Our new hexagonal random walk model reduces the states from
to . In the next section, we validate the new random walk by simulation experiments.
IV. PERFORMANCECOMPARISONS
Suppose that an MS initially resides at a cell. Then the expected number of steps that the MS stays in a 6-subarea cluster is computed as
(3)
Equation (3) is validated by simulation experiments fol-lowing the procedure described in [19]. In the th simulation experiment, we simulate the movement of an MS in a 6-cluster to compute the number of steps that the MS moves from a cell to a boundary cell (outside the cluster). Let be the value obtained from simulation experiments, where
(4)
We calculate with simulation
experi-ments and use 200 truncated terms in to approximate the infinite summations. Table II shows the results from (3) and (4). The discrepancy between (3) and (4) is within 0.3% for all test cases.
The expected number of steps that the MS leaves the cluster through a cell is computed as
(5)
Equation (5)is also validated by simulation experiments. From the simulation experiments, we compute using an equation similar to (4). Table III shows the results derived from analytic computations and simulation experiments. The discrepancy between them is within 1% for all test cases.
V. TWO-DIMENSIONALMESHRANDOMWALK
The two-dimensional mesh random walk model can be sim-plified following the same concept described in the previous
TABLE II
COMPARISON OFK (ANALYSIS)ANDK~ (SIMULATION)FOR
6-CLUSTERHEXAGONALCONFIGURATION
TABLE III
COMAPRISON OFL (ANALYSIS)ANDL~ (SIMULATION)FOR
6-CLUSTERHEXAGONALCONFIGURATION
Fig. 6. The mesh routing pattern.
sections. The routing pattern for a mesh cell is shown in Fig. 6. We assume that an MS resides in a cell for a period, then moves to one of its neighbors with the same probability, i.e., with prob-ability 1/4.
Fig. 7 illustrates the types of cells. Following a type assign-ment procedure similar to the one described in the previous sec-tion, cells are assigned to types , where indicates that the cell is in subarea- , and 0 represents the st type in subarea- .
Based on the type classification and the concept of absorbing states, the state diagram of a 4-subarea mesh random walk is shown in Fig. 8. In this state diagram, state represents that the MS is in one of the cells of type . State (4, 0) is an absorbing state and represents that the MS moves out of the cluster.
Let be the one-step transition probability from state to state ; i.e., the probability that the MS moves from a cell to a cell in one step. Then the
Fig. 7. Type classification for a 4-subarea mesh cluster.
Fig. 8. State diagram for a 4-subarea mesh cluster.
transition matrix of the 3-subarea random walk is a 7 7
ma-trix where
In this matrix, the elements in each column and row are listed in the following order: (0, 0), (1, 0), (2, 0), (2, 1), (3, 0), (3, 1), and (4, 0). It is clear that our new mesh random walk model
reduces the states from to if
is even and to if is odd. Following (1) and (2), we compute . The state diagram in Fig. 8 is validated by simulation experiments by comparing the and values defined in (3) and (4). Table IV shows that the discrepancy between and is within 0.2% for all test cases.
VI. APPLICATIONS FOR THENEWRANDOMWALKMODEL
This section describes three applications that can utilize our random walk model: microcell/macrocell PCS network
mod-TABLE IV
COMPARISON OFK (ANALYSIS)ANDK~ (SIMULATION)FOR
4-CLUSTERMESHCONFIGURATION
Fig. 9. Type assignment of microcells in three neighboring macrocells.
eling, distance-based location update modeling, and General
Packet Radio Service (GPRS) mobility management modeling.
In a microcell/macrocell PCS network [4], [11], [20], the ser-vice area is covered by both microcell BS’s and the macrocell BS’s. A macrocell overlays several microcells to increase the circuit capacity. An example of the microcell/macrocell config-uration is the dual-band GSM network deployed by Far EasTone in Taiwan [15]. In this network, there are two types of base
transceiver stations (BTS’s). The DCS 1800 BTS’s serve for
microcells, which operate at 1.8 GHz. The GSM 900 BTS’s serve for macrocells, which operate at 900 MHz. The typical coverage area of a microcell is between 0.5 and 3 km, and the area of a macrocell is between 3 and 10 km. In modeling mi-crocell/macrocell configuration, it is required to derive the MS residence time distribution at a macrocell based on the MS resi-dence time distribution at the microcells. Our new random walk model can be used for the derivation of the macrocell residence time distribution. The first step is to classify the types of micro-cells within a macrocell. Fig. 9 plots three neighboring macro-cells and type assignment of micromacro-cells in these macromacro-cells. For a specific type of microcell, we use (2) to compute the number of microcells that are visited before the MS moves out of the macrocell. The residence times of these microcells are
accumu-lated to derive the time before the MS leaves the macrocell. The modeling details can be found in [15].
Another application of the new random walk is the modeling of distance-based location update scheme. In existing PCS net-works, the service area is partitioned into several location areas (LA’s). Each LA consists of a group of cells and each MS per-forms a location update whenever it enters an LA [6], [14]. The location information is stored in the location databases such as home location register and visitor location register. When an in-coming call arrives, the network retrieves the location databases to identify the LA where the MS resides. All cells in the LA are paged to find the MS for call delivery. In [3], three loca-tion update schemes were proposed. In the time-based scheme, after a location update, a timer is set for the MS. When the timer expires, the MS performs the next location update. In the
movement-based scheme, after a location update, a counter is
set for the MS. When the MS moves across a cell boundary, the counter is incremented by one. When the counter value reaches a threshold, the MS performs the next location update. In the
dis-tance-based scheme, when the distance between the cell where
the previous location update was made and the current cell is longer than a threshold, the next location update is performed. Results demonstrated that distance-based scheme has the best performance [10], [1]. In [2], analytical models were proposed to study the costs of paging and location updates for the dis-tance-based scheme. A simple random walk was used to inves-tigate the MS movement, which introduces inaccuracy. With the new random walk proposed in this paper, accurate user moving behavior can be modeled.
A third application of the new random walk is the modeling of General Packet Radio Service (GPRS) mobility management. GPRS [7] provides data services for digital TDMA systems such as GSM [13], [16], [17] or Digital AMPS (IS-136) [12], [5]. To support GPRS, a new data protocol called GPRS Tunneling
Protocol (GTP) [8] is developed to route the GPRS packet data
to the external data networks. To accurately route data using GTP, traditional GSM mobility management [6] is modified. Besides LA’s, GPRS tracks the routing area (RA) of a GPRS MS. An RA is a group of cells, which is a subset of an LA. When a GPRS MS moves to a new RA, an RA update is performed. When the MS moves to a new LA, separate RA and LA updates or a combined RA/LA update is required. It is clear that LA/RA updates are more expensive than pure RA updates. Thus, it is desirable to configure an appropriate LA/RA layout based on the MS data/call traffic. Our random walk model can be used to determine the number of RA’s visited before the MS moves to the new LA. This piece of information is then used in GPRS LA/RA modeling to determine LA/RA layout.
VII. CONCLUSION
This paper proposed a new approach to simplify the two-di-mensional random walk models capturing the movement of mo-bile users in PCS networks. Analytical models were proposed for the new random walks with both hexagonal and mesh con-figurations. Our method significantly reduces the states in the random walks and thus, the execution times to derive the output measures. Specifically, for the hexagonal configuration, we
re-duce the number of states from to .
For the mesh configuration, the number is reduced from
to if is even and to
if is odd. Simulation experiments were conducted to validate the analytical models. The results indicated that the errors be-tween the analytical and simulation models are within 1%.
REFERENCES
[1] I. F. Akyildiz and J. S. M. Ho, “Dynamic mobile user location update for wireless PCS networks,” ACM-Baltzer J. Wireless Networks, vol. 1, no. 1, pp. 187–196, 1995.
[2] I. F. Akyildiz, J. S. M. Ho, and Y.-B. Lin, “Movement-based location update and selective paging for PCS network,” IEEE/ACM Trans. Net-working, vol. 4, no. 4, pp. 629–638, Aug. 1996.
[3] A. Bar-Noy, I. Kessler, and M. Sidi, “Mobile users: To update or not to update?,” ACM/Baltzer Wireless Networks, vol. 1, no. 2, pp. 187–196, 1994.
[4] R. Beraldi, S. Marano, and C. Mastroianni, “Performance of a reversible hierarchical cellular system,” Int. J. Wireless Inform. Networks, vol. 4, no. 1, pp. 43–54, 1997.
[5] EIA/TIA, “800 MHz TDMA cellular-radio interface-mobile sta-tion-base station compatability-digital control channel,” EIA/TIA, Tech. Rep. IS-136, 1994.
[6] ETSI/TC, “Mobile application part (MAP) specification, version 4.8.0,” ETSI, Tech. Rep. Recommendation GSM 09.02, 1994.
[7] , “GPRS service description stage 2,” ETSI, Tech. Rep. Recom-mendation GSM GSM 03.60 Version 7.0.0 (Phase 2+), 1998. [8] , “GPRS tunnelling protocol (GPT) across the Gn and Gp
inter-face,” ETSI, Tech. Rep. Recommendation GSM GSM 09.60 version 7.0.0 (Phase 2+), 1998.
[9] W. Feller, An Introduction to Probability Theory and Its Applications, Volume I. New York: Wiley, 1966.
[10] J. S. M. Ho and I. F. Akyildiz, “Mobile user location update and paging under delay constraints,” ACM-Baltzer J. Wireless Networks, vol. 1, no. 4, pp. 413–426, 1995.
[11] C.-L. I, L. J. Greenstein, and R. D. Gitlin, “A microcell/macrocell cel-lular architecutre for low- and high-mobility wireless users,” IEEE J. Select. Areas Commun., vol. 11, pp. 885–891, 1993.
[12] W. C. Y. Lee, Mobile Cellular Telecommunications Systems. New York: McGraw-Hill, 1995.
[13] Y.-B. Lin, “No wires attached: Reaching out with GSM,” IEEE Poten-tials, Oct./Nov. 1995.
[14] , “Mobility management for cellular telephony networks,” IEEE Parallel Distributed Technol., vol. 4, pp. 65–73, Nov. 1996.
[15] Y.-B. Lin, W. R. Lai, and R. J. Chen, “Performance analysis for dual band PCS networks,” IEEE Trans. Computers, 2000.
[16] M. Mouly and M.-B. Pautet, The GSM System for Mobile Communica-tions Palaseau, France, 1992.
[17] S. Redl and M. Weber, An Introduction to GSM. Norwood, MA: Artech House, 1995.
[18] S. M. Ross, Stochastic Processes. New York: Wiley, 1983.
[19] , Introduction to Probability Models. New York: Academic, 1985. [20] L.-C. Wang, G. L. Stüber, and C.-T. Lea, “Architecture design, fre-quency planning, and performance analysis for a microcell/macrocell overlaying system,” IEEE Trans. Veh. Technol., vol. 46, pp. 836–848, 1997.
Ian F. Akyildiz (F’95) is a Professor with the School
of Electrical and Computer Engineering, Georgia Institute of Technology and Director of Broadband and Wireless Networking Laboratory. His current research interests are in Wireless Networks, Satellite Communication, ATM Networks, Next Generation Internet.
Dr. Akyildiz is an ACM Fellow. He received the Don Federico Santa Maria Medal for his services to the Universidad of Federico Santa Maria in Chile. He served as a National Lecturer for ACM from 1989 until 1998 and received the ACM Outstanding Distinguished Lecturer Award for 1994. He is the Editor-in-chief of Computer Networks (Elsevier). He received the 1997 IEEE Leonard G. Abraham Prize award for his paper entitled “Multi-media Group Synchronization Protocols for Integrated Services Architectures” published in the IEEE JOURNAL OFSELECTED AREAS INCOMMUNICATIONS
(JSAC) in January 1996. He was the program chair of the 9th IEEE Computer Communications workshop, and served as the program chair for ACM/IEEE MOBICOM’96 (Mobile Computing and Networking) conference as well as for IEEE INFOCOM’98 conference.
Yi-Bing Lin (S’80–M’96–SM’96) received the
B.S.E.E. degree from National Cheng Kung Uni-versity in 1983, and the Ph.D. degree in computer science from the University of Washington in 1990.
From 1990 to 1995, he was with the Applied Research Area at Bell Communications Research (Bellcore), Morristown, NJ. In 1995, he was appointed as a Professor of Department of Computer Science and Information Engineering (CSIE), National Chiao Tung University (NCTU). In 1996, he was appointed as Deputy Director of Microelec-tronics and Information Systems Research Center, NCTU. During 1997–1999, he was elected as Chairman of CSIE, NCTU. His current research interests include design and analysis of personal communications services network, mobile computing, distributed simulation, and performance modeling.
Dr. Lin is an Associate Editor of IEEE NETWORK, an Editor of IEEE J-SAC: Wireless Series, an Editor of IEEE PERSONALCOMMUNICATIONSMAGAZINE, a Guest Editor for IEEE TRANSACTIONS ONCOMPUTERSspecial issue on Mobile Computing, and a Guest Editor for IEEE COMMUNICATIONSMAGAZINEspecial issue on Active, Programmable, and Mobile Code Networking. He received the 1997 Outstanding Research Award from National Science Council, ROC, and Outstanding Youth Electrical Engineer Award from CIEE, ROC.
Wei-Ru Lai received the B.S.E.E and Ph.D. degrees
from the Department of Computer Science and Infor-mation Engineering, National Chiao Tung University in 1991 and 1999, respectively.
In 1999, she was appointed as an Assistant Pro-fessor and was elected as Chairman of Department of Information Management, Chin-Min College. Her current research interests include design and analysis of personal communications services network.
Rong-Jaye Chen was born in Taiwan in 1952. He
received the B.S. degree in mathematics from Na-tional Tsing Hua University in 1977, and the Ph.D. degree in computer science from the University of Wisconsin-Madison in 1987.
He is currently a Professor and Chairman of Com-puter Science and Information Engineering Depart-ment in National Chiao Tung University. His research interests include cryptography and security, personal communication service, algorithm design, and theory of computation.
