Modeling of Terminal Mobility to Evaluate the Number of Location Updates Min Young Chung, Tai Suk Kim, Ho Shin Cho, and Dan Keun Sung Department of Electrical Engineering Korea Advanced Institute of Science and Technology
[email protected] and
[email protected]
Abstract: Since mobile user mobility behaviors can vary according to time periods, e.g. office-going/closing and office hours, the location management traffic load for tracking user positions is also a function of time. The transient characteristics of location management traffic vary according to users' initial starting positions. In this paper, we propose a mobile user mobility model considering office-going/closing and office hours for evaluating the effect of user mobility on the performance of mobile networks. We analyze the transient characteristics of a mobile network in terms of the mean location update rate.
1 Introduction One of major problems of mobile communications is to characterize user mobility. Since users continuously change their positions, mobile networks should keep their location information and support handoff calls or location updates if needed. This user mobility affects the resource management as well as the location management for tracking users' positions. Terminal mobility problems have been studied in 1-, 2-, and 3-dimensional environments. A 1-dimensional mobility problem for highway microcells was studied by El-Dolil et al. [1] Hong and Rappaport [2] modeled the mobility with random direction motions in 2-dimensional environments. Cho et al.[3] studied the effect of terminal mobility in rectangular shaped urban microcellular systems. In 3-dimensional indoor building environments, the number of handoffs was estimated by Kim et al. [4] However, these studies were mainly concerned with handoff problems instead of location management problems. Several location management schemes to reduce the network load have been studied in [5]-[10]. Lin [5] and Jain et al. [6] introduced the local storage(i.e. cache) of user location information and analyzed the network signaling and database traffic loads in terms of the user's local callto-mobility ratio. Tabbane [7] proposed an alternative strategy for location trackings based on the observation that the mobility behavior of a majority of people can be predicted. Wang [8] and Kim et al. [9] proposed location registration strategies based on the hierarchical registration and evaluated the performance in terms of the number of database updates and signaling traffic. Pollini et al. [10] evaluated the signaling traffic volume for mobile and personal communications. They considered the simple terminal mobility model and evaluated the performance of mobile networks in terms of the mean number of location updates or handoff calls
at steady state. However, terminal mobility behaviors can vary according to time periods. During an office-going hour, most of mobile users move from their homes to offices. On the contrary, they return home from their offices during an office-closing hour. The mobile management traffic varies according to users' initial starting positions. Therefore, the characterization of mobile management traffic in the transient period may be sometimes more important than that in stationary state. In this paper, we propose a terminal mobility model considering time-varying terminal mobility behaviors. And, we evaluate the performance of the considered mobile system in terms of the mean location update rate. In section 2, we introduce a structure of location areas of mobile networks and describe terminal mobility. In section 3, we analyze the mean location update rate. In section 4, we evaluate the performance of a given mobile system in transient periods in terms of the mean location update rate. Finally, we have conclusions in section 5.
2 System Description We here assume squared location areas with length D for computational and geometrical simplicity. Fig. 1 shows a location area structure. A location register (LR) manages a location register coverage area (LRCA) with n n location areas (LAs), and N N LRs cover the whole location area considered in this figure. Mobile user mobility behaviors can vary according to time periods. During an office-going hour, most of mobile users generally move from their homes to offices, while during an office-closing hour, they return home from their offices. Especially, some mobile users including salesmen move from place to place in the daytime. We model a mobile user mobility considering these user behaviors. We have the following assumptions to model mobile user mobility:
A LA has the shape of a square with length D(m). Mobile users move with speed V (m/sec). Mobile users move straightly until the change of moving directions. Moving direction changes occur according to a Poisson with rate .
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Mobile users change their moving directions with probabilities px , qx , py , and qy (px + qx + py + qy = 1) for right, left, upper, and lower directions, respectively.
pressed as
All user portable stations are in power-on mode.
Given the above assumptions, Cho[11] studied the effect of user mobility and calculated the location update rate for the same probabilities of moving directions, i.e. px = qx = py = qy . The location update rate RLA due to crossing the location area boundary can be written as
V: RLA = D
i;j )
((v;w )
8p V for upper side; > > V >qx DV for left side; :
(2)
px D for right side:
The location update rates due to crossing the boundaries of location register coverage areas are simply given by
RLR = n V D :
(3)
The location update rates through the upper, lower, left, and right sides of location register coverage area are calculated as
8p V for upper side; > > V > :qx nVD for left side;
1
2 (1 2
1 3
(1
)
)
3
(1
1 2
(1)
The location update rates can be decomposed according to the moving direction of mobile users. The location update rates through the upper, lower, left, and right sides are simply obtained as
8V , ); , V > D (1 , n ) , nD (1 , > > for (v = 0 or v = N , 1) > > and (w = 0 or w = N , 1); > > V (1 , , ) , V (1 , , ); > > < Dfor (v = inand (wnD= 0 or w = N , 1)) => or (w = j and (v = 0 or v = N , 1)); > , ); , V V > D (1 , n ) , nD (1 , > > for (v 6= i and (w = 0 or w = N , 1)) > > or (w 6= j and (v = 0 or v = N , 1)); > > : DV , nVD ; otherwise:
(5)
)
2
Let 1 = 41 nVD , 2 = 1,3 nVD , 3 = 1,2 nVD , 1 = nVD , (i;j ) and 2 = 2 nVD . The leaving rate, (v;w)(k;l) , from the area of LR (v; w) to that of LR (k; l) is different from according to the positions of the current LR (v; w), the next LR (k; l), and home LR (i; j ). For example, if (i = v = k , j = w, and l = j 1), (i = v , w = l, and k = v 1), (v < i, w < j , k = v , 1, and l = w), (i = v = k, w > j , and l = w , 1), and (v < i, w < j , ) k = v + 1, and l = w), ((i;j v;w)(k;l) becomes 1 , 2 , 3 , 1 , and 2 , respectively.
3 Mathematical Analysis We assume LR identifier (i; j ) as the state of a user's current location. We also assume that the leaving events of LRs and LAs occur according to a Poisson process. If a user's home and current location are in state (i; j ) and (v; w), respectively, the leaving (i;j ) (i;j ) rates of LRCA and LA are (v;w)(k;l) and (v;w) , respectively. We introduce the following notations:
M (t): number of location updates in (0; t]. Sr (t): geographical row position of a user's current location at time t.
(4)
Sc (t): geographical column position of a user's current loca-
px nD for right side:
tion at time t.
When a user's current location is covered by the LR (v; w), and his home LR is denoted by (i; j ), the probabilities of selecting his moving directions are given in Table 1. The probabilities of selecting moving directions represent user mobility behaviors and vary according to time periods and user's current location. During an office-closing hour, a user generally returns home from his office. The probability of selecting moving directions to his home is greater than that to others. If his current location is on the right/left of his home, the probabilities of selecting moving directions to his home and others are and 1,3 (1=4 < 1), respectively. However, if his current location is on the upper/lower right/left of his home, the probabilities of selecting moving directions to his home and others are 2 and 1,2 (1=2 < 1), respectively. Location updates are classified into two types: a mobile user changes his LA either within a LRCA or to another LRCA. Location update rates within the coverage area of LR (v; w) are ex-
0
-
7
Table 1: Moving direction selection probabilities of a user with home location register (i; j ) Visited location register Moving direction selection probabilities
(i; j ) (i; w) for w 6= j and w > j (i; w) for w 6= j and w < j (v; j ) for v 6= i and v > i (v; j ) for v 6= i and v < i (v; w) for v > i and w > j (v; w) for v > i and w < j (v; w) for v < i and w > j (v; w) for v < i and w < j
px = qx = py = qy = 14 px = py = qy = 1,3 , qx = qx = py = qy = 1,3 , px = px = qx = qy = 1,3 , py = px = qx = py = 1,3 , qy = qx = py = 2 , px = qy = 1,2 px = py = 2 , qx = qy = 1,2 qx = qy = 2 , px = py = 1,2 px = qy = 2 , qx = py = 1,2
8
3
0
3
-
9
2
8
-
2
The process f(M (t); Sr (t); Sc (t)) : t 0g is then a Markovian arrival process, and Fig. 2 shows the state transition diagram of user location. The infinitesimal generator of written as follows
(M (t); Sr (t); Sc (t))
can be
0C D 0 : : 01 Q = B @ 0. C. D. .: : 0. CA ; ..
..
..
..
(6)
..
where C and D are N N matrices and each element of C and D is N N matrices. Each element of C , Ckl is written as
0,c 0 :: k B 0 , c k :: Ckl = B B .. .. .. @ . . . 0
1
0
0 0 .. .
: : ,ck(N ,1)
0
1 CC CA for k = l:
(7)
00 : : 01 Ckl = B @ ... . . . ... CA for k 6= l:
(8)
0 :: 0
The diagonal element of Ckl
(k = l) is given by
i;j) + (i;j) ) cku = ((k;u + ((i;j ) (k;u)(k+1;u) k;u)(k,1;u) ) (i;j ) +((i;j k;u)(k;u+1) + (k;u)(k;u,1) ; where (i1 ;j1 )(i2 ;j2 ) = 0 for (i1 ; i2 ; j1 ; or j2 j2 > N , 1). Each element of D, Dkl is expressed as:
0 i;j
k; B i;j B k; k; B B . B @ ..
( ) ( 0) ( ) ( 1)( 0)
Dkl =
0 for k = l:
0 i;j k; l; B B 0 B B . B @ .. ( (
Dkl =
) 0)( 0)
< 0) or (i1; i2 ; j1 ; or
) ((i;j 0 :: k;0)(k;1) )
((k;i;j1)) ((i;j k;1)(k;2) : :
.. .
.. .
0
0
(9)
..
.
0 0 .. .
i;j ) : : ((k;N ,1)
1 CC CC CA
(10)
0
:: ::
0 0
.. .
..
.. .
) ((i;j k;1)(l;1)
0 for k = l 1:
0
.
) : : ((i;j k;N ,1)(l;N ,1)
(11)
00 : : 01 Dkl = B @ ... . . . ... CA for k 6= l and k 6= l 1: 0 :: 0
1 CC CC CA
(12)
The infinitesimal generator of the underline Markov chain is given by
Q = C + D: The steady state probability vector of Q, ,1 N ,1 (i;j) using Q = 0 and Nv=0 w=0 (v;w)
P
P
((0(i;j;0)) : : : (0(i;j;N) ,1) (1(i;j;0)) : : : ((Ni;j,) 1;N ,1) ).
(13)
can be found by = 1, where =
The mean number of location updates in time (0; t][12] can be given by
(
Qt ,1 E [M (t)] = t + p(e , I )(Q + e ) t
for p 6= for p = (14)
where p and e denote the initial position vector of user's location and the unit column vector, respectively, and = De. Finally, we can obtain the mean location update rate at time t.
(t)] Qt ,1 R(t) = dE [M dt = + pQe (Q + e ) :
(15)
4 Numerical Examples To evaluate the mean location update rate, we consider a mobile network with 9 LRs (N = 3) each of which manages the user roaming data for 4 LAs (n = 2). We assume that users move with speed 40km=hour, and the length of all LAs is 4km. User mobility patterns can vary according to three types of time periods: office-going/closing and office hours. We can model user mobility according to time periods for selecting the appropriate values of and . During an office-going hour, and are selected by (1=4; 1] and (1=2; 1], respectively. And, during an office-closing hour, and are selected by [0; 1=4) and [0; 1=2), respectively. During office hours, most of mobile users stay in their offices but some of them, e.g., salesmen, continuously move from place to place. In this case, we may model user mobility as = 1=4 and = 1=2. To characterize the location update rate during an office-going hour, and are assumed to be 0.1 and 0.1, respectively. Fig. 3 shows the mean location update rate during an office-going hour. The mean location update rate depends on both the limiting probabilities of user locations and the location update rate in each state. The location update rate in each state is different due to boundary conditions. When = 0:1 and = 0:1, the limiting probability of users locating in corner states, i.e., (0; 0), (0; 2), (2; 0), and (2; 2), is larger than that in other states. However, the location update rate in corner states is smaller than that in other states. When a user's initial position is in either state (1; 0) or state (1; 1), the mean location update rate decreases as time passes because the location update rate in the user's current state is larger than the weighted sum of location update rates in neighboring states of current state. However, if the user's initial position is in state (0; 0), the mean location update rate increases as time passes. When the user's initial position is in state (1; 1), the mean location update rate at the initial time is about 1.67 times larger than that in stationary state. However, when the user's initial position is state
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(0; 0), the mean location update rate at the initial time becomes 92% of that in stationary state. During office hours, most of mobile users stays in their offices, but some people moves from place to place. If a user moves with the same probabilities of selecting moving directions, then we may model this user mobility as = 1=4 and = 1=2. Fig. 4 shows the mean location update rate for = 1=4 and = 1=2. When = 0:25 and = 0:5, the limiting probabilities of user positions are constant for all states but the location update rates in all states are different according to location positions. The LR boundary crossing rate in each corner state for states (0; 0), (0; 2), (2; 0), and (2; 2), becomes 2=3 and 1=2 in each boundary state for states (0; 1), (1; 0), (1; 2), and (2; 1) and the boundless state (1; 1), respectively. When the user's initial position is in either state (1; 0) or state (1; 1), the mean location update rate decreases as time passes because the location update rate at the initial time is larger than that in stationary state. On the contrary, if the user's initial position is in state (0; 0), the mean location update rate increases as time passes. To model user mobility during an office-closing hour, we assume = 0:9 and = 0:9. Fig. 5 shows the mean location update rate during an office-closing hour. Since a user returns his home(state (1; 1)) from his office, the limiting probability of the user location in his home state is larger than that of other states. When the user's initial position is in state (1; 1), the mean location update rate decreases as time passes because the location update rate in initial state is larger than the weighted sum of location update rates for all states in stationary state. On the contrary, if the user's initial position is in either state (0; 0) or state (1; 0), the mean location update rate increases as time passes.
References [1] S. A. El-Dolil, W. Wong, and R. Steele, “Teletraffic Performance of Highway Microcells with Overlay Macrocell,” IEEE J. Select. Areas Commun., Vol. 7, No. 1, pp.71-78, Jan. 1989. [2] D. Hong and S. S. Rappaport, “Traffic Model and Performance Analysis for Cellular Mobile Radio Telephone Systems with Prioritized and Nonprioritized Handoff Procedures,” IEEE Trans. on Veh. Technol., Vol. VT-35, No. 3, pp.77-92, Aug. 1986. [3] H. S. Cho, Y. W. Kim, D. K. Sung, and Y. K. Jhee, “The Effect of User Mobility in Rectangular Shaped Urban MicroCellular Systems,” Proc. of APCC' 95, pp.568-572, 1995. [4] T. S. Kim, M. Y. Chung, D. K. Sung, and M. Sengoku, “A Mobility Analysis in 3-Dimensional PCS Environments,” Proc. of VTC' 96, pp.237-241, 1996. [5] Y. Lin, “Determining the User Locations for Personal Communications Services Networks,” IEEE Trans. on Veh. Technol., Vol. 43, No. 3, pp.466-473, Aug. 1994. [6] R. Jain, Y. Lin, C. Lo, and S. Mohan, “A Caching Strategy to Reduce Network Impacts of PCS,” IEEE J. Select. Areas Commun., Vol. 12, No. 8, pp.1434-1444, Oct. 1994. [7] S. Tabbane, “An Alternative Strategy for Location Tracking,” IEEE J. Select. Areas Commun., Vol. 13, No. 5, pp.880892, June 1995. [8] J. Z. Wang, “A Fully Distributed Location Registration Strategy for Universal Personal Communication Systems,” IEEE J. Select. Areas Commun., Vol. 11, No. 6, pp.850-860, Aug. 1993.
5 Conclusions Location update rates during a transient period, i.e. officegoing hour and office-closing hour, may be sometimes more important than that in stationary state. The transient characteristics of mean location update rates depend on users' initial positions as well as user mobility activity, i.e. and . When ( ; ) is assumed to vary in the range of (0.1, 0.1), (1/4, 1/2), and (0.9,0.9), the mean location update rates in stationary state are given by 5.99, 8.33, and 9.90(1/hour), respectively. However, the mean location update rate at the initial time is quite different from that in stationary state. The proposed user mobility model can be applied to evaluating the transient characteristics of any location management schemes in terms of the mean location update rate. These results can be also utilized in the performance evaluation of mobile networks with different location management schemes.
[9] B. C. Kim, J. S. Choi, and C. K. Un, “A New Distributed Location Management Algorithm for Broadband Personal Communication Networks,” IEEE Trans. on Veh. Technol., Vol. 44, No. 3, pp.516-524, Aug. 1995. [10] G. P. Pollini, K. S. Meier-Hellstern, and D. J. Goodman, “Signaling Traffic Volume Generated by Mobile and Personal Communications,” IEEE Commun. Mag., Vol. 33, No. 6, pp.60-65, June 1995. [11] H. S. Cho, Analysis of Signaling Traffic Related to Location Registration/Updatings in Personal Communication Networks, MS thesis KAIST, Korea 1994. [12] D. M. Lucantoni, “New Results on the Single Server Queue with a Batch Markovian Arrival Process,” Commun. Statist. - Stochastic Models, Vol. 7, No. 1, pp.1-46, 1991.
Acknowledgement This work is supported in part by the Samsung Advanced Institute of Technology.
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LRCA (0,j)
LRCA (i,0)
LA
LRCA (i,N-1)
LRCA (i,j)
LA
LA
LA
LA
LA
LA
9.5
LA
LA
LA
LA
LA
10
LA
LA
LA
LA
LA
LA
LRCA (0,N-1)
Mean location update rates(1/hour)
LRCA (0,0)
initial position: (1,1) (1,0) (0,0)
9 8.5 8 7.5 7 6.5 6 5.5 5
0
20
40
60
80
100
120
140
160
180
t(min.)
LA
LA
LA
LA
LA
LA
Figure 3: Mean location update rate during an office-going hour LRCA (N-1,j)
LRCA (N-1,0)
LA
LA
LA
LA
LRCA (N-1,N-1)
LA
LA 10
initial position: (1,1) (1,0) (0.0)
D
LA
LA
LA
LA
LA
LA
D n*D
Location Register Coverage Area(LRCA) Location Area(LA)
Mean location update rates(1/hour)
n*D
9.5
9
8.5
8
Figure 1: Location areas of mobile networks 7.5
0
20
40
60
80
100
120
140
160
180
t(min.)
γ (i,j)
γ (i,j)
γ (i,j)
(0,0)
(0,N-1)
(0,j)
(i,j)
λ (0,j-1)(0,j)
(0,0)
(0,j)
(0,N-1)
(i,j)
(i,j)
(i,j) (1,N-1)(0,N-1)
(i,j)
(i,j)
λ (0,N-1)(1,N-1)
λ (0,N-1)(0,N-2)
λ
(i,j)
λ (0,j+1)(0,j))
(i,j)
λ (0,j)(0,j-1))
λ (1,j)(0,j))
λ (0,0)(1,0)
(i,j)
λ (1,0)(0,0)
λ (1,0)(0,0)
(i,j)
λ (0,j)(1,j))
(i,j)
Figure 4: Mean location update rate during office hours
(i,j)
λ (0,0)(0,1)
(i,j)
λ (0,j)(0,j+1))
(i,j)
λ (0,0)(0,1)
10
λ
(i,j) (i,j)(i,j+1)
λ
(i,j) (i,N-2)(i,N-1)
(i,j)
(i,0)
(i,j)
(i,j)
λ (N-2,N-1)(N-1,N-1)
(i,j)
λ (N-1,j)(N-1,j-1)
(i,j)
λ (N-1,j)(N-1,j+1)
λ (i,N-1)(i+1,N-1)
λ (N-1,N-1)(N-2,N-1)
(N-1,j)
γ (i,j)
(i,j)
λ (N-1,N-2)(N-1,N-1) (N-1,N-1)
(N-1,j) (i,j)
λ (N-1,1)(N-1,0)
(i,j)
λ (i+1,N-1)(i,N-1)
γ (i,j)
(i,j)
(i,j)
λ (N-1,j-1)(N-1,j)
(N-1,0)
λ (i,N-1)(i,N-2)
(i,j)
(i,j)
λ (N-2,j)(N-1,j)
λ (N-1,j)(N-2,j)
(i,j)
λ (N-2,0)(N-1,0)
(i,j)
λ (N-1,0)(N-2,0)
(N-1,0) (i,j) λ (N-1,0)(N-1,1)
(i,j)
(i,j)
(i,j)
λ (i,0)(i+1,0)
(i,j)
λ (i+1,0)(i,0)
γ (i,j)
(i,N-1)
(i,N-1) (i,j)
(i,j)
λ (i,j+1)(i,j)
λ (i,j)(i+1,j)
(i,j)
λ (i,j)(i,j-1)
λ (i+1,j)(i,j)
(i,j)
λ (i,1)(i,0)
γ (i,j)
(i,j)
λ (N-1,j+1)(N-1,j)
(N-1,N-1)
Mean location update rates(1/hour)
(i,j)
(i,j)
γ (i,j)
λ (i-1,N-1)(i,N-1)
(i,j)
λ
(i,j) (i,j-1)(i,j)
λ (i,N-1)(i-1,N-1)
λ
(i,j) (i,0)(i,1)
(i,j)
(i,0)
λ (i-1,j)(i,j)
(i,j)
λ (i,j)(i-1,j)
(i,j)
λ (i-1,0)(i,0)
(i,j)
λ (i,0)(i-1,0))
γ (i,j)
9.95 9.9 9.85 9.8 9.75 9.7 9.65 initial position: (1,1) (1,0) (0.0)
9.6 9.55 9.5
0
20
40
60
80
100
120
140
160
180
(i,j)
λ (N-1,N-1)(N-1,N-2)
Figure 2: State transition diagram of user location
t(min.)
Figure 5: Mean location update rate during an office-closing hour
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