UMTS Network Planning - The Impact of User Mobility · UMTS Network Planning - The Impact of User ... One question to answer here is, ... UMTS Network Planning - The Impact of User

7

UMTS Network Planning - The Impact of User Mobility

S.A. van Gi ls, o.v. Iftime, H. van der Ploeg

Abstract

The impact o f user mobi lity on network planning is investigated. For a ystem of tw o base station the stationary distribution o f a Markov chain, including mobility, is computed.

Keywords

Markov chain, mobility, limiting distribution, Poisson dis tribution.

7.1 Introduction

For the }'o_ generation mobile communication system UMTS (Universal Mobile Telecommunication System) a network has to be planned . Therefore locations of lhe base stations have to be chosen. One question to answer here is, what should be the base station density. This is an opti mi zatjon problem with a trade o ff between on the one hand the investment costs, and on the other hand the quality of service, here expressed by 0) the probability o/blocking, i.e. the probabili ty that a new request for a call has to be blocked because there are no channels available and (ii) the handover blocking probability, i.e. the probability that fo r an existing call, moving from one base station to another, a so called hand over, there is no channel available at the new base station.

We have to deal with two effect . There arc fresh calls arriving in a cell. Thi s is typically a Poisson process . Secondly, we assume mobility in the system. While maki ng a call, the user might move from one cell to another. We reserve capacity in the base stations to serve hand over calls. From a quality of service point of vi ew it is much worse to drop an existing call when the user moves form one cell to another, than to block a new call.

The main problem to be addressed here is to quantify thc effects of user mobili ty on the density of the base stations.

79

Figure 7. 1: Hexagonal covering of the area. In this paper we focus on the two-base­station ystem indicated by the light-grey ce lls

7.2 A simple model

We assume, in the planning stage, that the area we are interested in is covered by hexagons. In each hexagon there is a base station that can accept a number of calls. We drastically simplify the analysis in this paper by concentrati ng on a two-base­station system. In addition we assume these cells to be eq ual. See Figure 7. 1. The state of the system at a particular time instant is characterized by the two-tuple (nt, n2 ) , where n ; is the number of call s at cell i . We assume that for both cells the capacity is N. The state space of the system i s depicted in Figure 7.2.

OUf approach is to calculate the stationary di stribution p (n [ , n2 ) ' In the station­ary di stribution tbe transition rates coming from a state balance wi th the transi tion rates entering into that state. There fore the distributi on doesn' t change in time. T he stationary distributi on tells us the average fract ion of lime the system is in a certain state.

Once the distribution is known the probability of blocking, Pbl ock , can be obtained by identifyi ng the states that don't allow an extra fresh eall and adding up their prob­abili ties. In Figure 7.2 these states can be found on the lower and right boundaries.

The probabi li ty of dropping a call Pdrup is the frac tion of the calls that, when moving from one cell to another, is dropped. We take into account that difrerent states have different weight by the number of calls in th at state. T his probability follows from the transi tion rales due to mobili ty and is given ex plicitly in Formula (7.4) .

If we make no distincti on in the treatment of fresh call s and hand over call , we obtain a Markov process for which we can easily solve the stationary distribution.

We assume that the arrivals of fre sh calls in the cells is a Poisson process. Let A be the arrival rate of fresh call s and let -j; be the mean of the exponentially di tributed ca\1-length . The transition rate due to terminated calls is proportional to the number of existing calls in the cell. Therefore it is expressed as /l-n ;. Without mobili ty we have the following transitions around the state with (nl' nz) = (i. j) : To obtain the

80

UMTS Ne twork Planning - The Impact of User M obility SWI 2000

Number of calls in cell 2 ~

Figure 7.2: Stale space fo r the two-base-station system. N denotes the max. imum number of calls accepted in a cell

I. ) .

stationary distribution w need to solve the balancc equations;

)..(p (i ­ 1, j) + p(i, j - 1» + FW + 1) p(i + I , j) + (f + 1) p(i, j + 1»

= (2).. + FU + j »p(i, j). (7. 1 )

A solution is given by

P ( I , ) . .) =

(! )i f- j

c-.JL,-.,-, (7.2)

where C is the nonnalizing constant. From this distribution we can calculate the probabi li ty of blocking by

N (A)N R = '"' ( N .) = -;;. (7 .3)block ~ P , ) - ,- -"N:--_-' -(!-).......,j.

) = 0 N. Lj: O j l jJ.

7.2.1 Including mobility

Next we allow a call 10 move from one cell to the other. We assume for this process an exponenti al distribution wi th mean 1, i.e. the average li me [hal a call resorts under tbe same antenna is ~ . Apart from th~ transitions in Figure 7.3 there art: a number of ex. tra possible transitions. In the interior of the state space there are two diagonal 'arrows' leaving eaeh slale, corresponding lO calls moving from one celllo another. Their lransitions rales are proporlionallo the num ber of calls in the cell thal the caller is leaving. Likewise there are two additional tran itions towards each slale. Note lhat these four Iran ilions balance each other if the distribution (7.2) i used.

81

Figure 7.3: Transitions wi thout mobility around the state (i, j )

On the boundary of the state space things are more complicated, as a hand over call is not accepted by the new base station. Suppose a call moves from cell 1 into cell 2, which has got no capaci ty available. This corresponds to the arrow in Fig­ure 7.4 goi ng to the upper-righl. However since the ca]l is not accepted in cell 2, the nu mber of calt s in this ce ll is not raised. T he call di sappears fro m cell I , so instead of poi nting to the state (i - 1, j + 1), the arrow points to the state (i - 1, j ) . Similarly, there is anoUler transition fro m the state (i + I , j ) Lo the sLate (i, j). This leads Lo the transitions as depicted in Figure 7.4.

(b)

i-I,j )

Figure 7.4: Transitions and transition raLes with mobi li ty, (a) for a state in the interior of the slate space and (b) for a state on the boundary of the state space

82

The balance equations for states on the boundary arc differen t from those for states in the interior of the state space. Therefore the stationary distribution is altered. Qual itati vely the number of calls present in the cells is slightly lowered, but is not possib le to solve the new balance equations analytically. As a first approllimation we use distribution (7 .2) . This can be justi fi ed if the probabilities for the boundary states arc small, i.e. if b. « N. We obtai n for the probabil ity of dropping

11

N

Pdrop = ~ Lip(i, N) , (7.4) /I .- 0

where ii is the average number of calls in a cell

IN N

n= 2 LL(i + j) p(i, j ). (7.5) .- 0 J~O

To check whether thi s gives the correct numbers we have written a computer program to compute numericall y the stationary di trihulion. In the next section the results of these computations are compared to the resu lts obtai ned for tbe second scenario.

7.3 Distinguishing between fresh and hand over calls

In tbe previous section we have not distinguished between new incoming and hand over call s. As remarked before it i much worse to drop an existing call than to reject a new call. At each base station we want to reserve some space for hand over calls. The simplest, and defi nitely not the best, strategy is to lix a certain capaci ty for hand over calls only. Therefore we have

(7.6)

where the capacity of the base station, N, i divided into capacity for fresh calls, Nr• and capacity for hand over calls. Nh • At a certai n time instant the state of our

two-base-station system is characterized by the four tuple (1l{, 1l~ , n{. n1), where n{ denotes the number of fresh calls in station nu mber i and 1l~ denotes Lhe number of hand over calls in station number i. i = I , 2.

Like before we can wri te down the balance equations that yield the stationary distribution. From each SLate in the interior of the state space there are no less than ten possible transitions . T herefore we re frai n from listing the balance equations. The solution to the e balance equalions do not take a nice form like (7 .2).

However it is possible to obtain the probabil ity of blocking expliciUy. Thereto we look at the two dimensional subsystem (n{ ,1l{) consisting on ly of the parts of each cell reserved for fresh calls. To this subsystem a hand over call is exacUy the same as a temli naled cal l. After all the subsystem doesn' t sec where the call is going.

83

it j ust sees it disappear. Therefore the subsy tern has a state space as in Figure 7.2, wi th I.L replaced by I.L + LJ and N replaced by Nf . The stationary distribution in the subsystem is given by

-( ..) (;;1:-. t P' , ] = C ., . ,

j ' (7.7)

I. ] .

where jj (i, j ) denotes the stationary distribution in the subsystem, irrespective of the distribution over the capaci ty for hand over calls.

jj(i, j) = L p(i, k , j, m). (7.8) k .m

To compute the probabi lity of dropping we have written a MathcmaLica program', which computes numericall y the stationary distribution. This allows to play with the parameters to obtain reasonable tigures for bl cki ng and dropping rates .

As an example, the fo llowing table shows the results o f a number of computations on a small cell. Each computation was performed using the same total capacity, i.e. N = 10 and using the same parameter values, i.e. I.L = 0.1, A = 0 .4, LJ = 0.04. In Figure 7.5 we show the probabilities in onc cell for one of these computati ons.

Strategy Puro (%) According to (7 .2) Correction to (7.2) making no di tincti on Using di stinction, Nf = 10 U ing distinction, NJ = 8 Using di sti nction, Nf = 5 Using disti nction, Nf = 2

0.53 1 0.51 9 0.057 0.634 9.801

5 1.41 4

0.531 0.493

100. 20.64 1

0.263 0.000

In thi s example, we see that our second strategy can reduce the probability of dropping, but the probabili ty of blocking is increased enormously. Therefore we eoncl ude that for the model of two cell s this strategy reduces the probabi lity of drop­pi ng but in the same time increases the blocking probability. Th is is due to the fi xed number of available channels in an antenna. In practice we have interaction between one cell and the neighbouring cells. Si milar balance equations can be written for the system of seven cells or more. In order to keep the blocking probabili ty less then a desi red value using the proposed strategy, we have to increa e the den ity of antennas; thi s is the impact of the user mobil ity on the density of base stations .

7.4 Conclusions

• We have concentrated on a two cell system. To extend this to a seven hexagon system would still be doable, but the extension to a larger system requires large scale computations.

1This program is available on request.

84

Figure 7.S: Probabili ties in Lte first cell when = S, Nh = S, A = 0 .4, jJ., =Nf 0. 1, Ii = 0.04

• Setting up a small model with maple gives quickly some insight in the qualita­tive behavior.

• Additional modeli ng is requi red. We would li ke to investigate whether it is possible Lo treat a seven hexagon system as a si ngle antenna wilh adjusted parameters. This would scale done the complexity o f the compulalions.

Bibliography

[1] L. Klcinrock and R. Gail, Queuing systems: problems and solutions, New York, Wiley, 1996.