Analysis of Direct Communication Considering
Mobility and Localized Distribution of Terminals
Tatsuya KABASAWA 1 and Keisuke NAKANO 2121Nagaoka College of Technology, Japan
Dept. of Information Engineering, Niigata University, Japan

Introduction

In cellular systems, a mobile terminal is always connected
to a base station (BS) during communication even if it
communicates with another mobile terminal in the same cell. A
new type of a cellular system is proposed, in which a mobileterminal is directly connected to another terminal if theseterminals are close to each other only using one channel.
In [1], a theoretical technique for analyzing trafficcharacteristics of circuit switching calls in cellular networks
with direct connection is proposed in a circular cell, whereterminals are independently and uniformly distributed over thecell. In [2], we present theoretical analysis of directcommunication and performance improvement in a onedimensional cell with localized distribution. In [3], authors
present theoretical analysis of direct communication andperformance improvement in a two-dimensional cell withlocalized distribution. However, the mobility of terminals is notconsidered in the analysis. Mobility is an important factor. A
direct connection between terminals can potentially fail due tomobility because the distance between them may become
greater than the communication range for direct communication
before the end of communication. In such a case, improvementof the efficiency of channel use depends on how long directcommunication is possible before the direct link breaks. In [4],theoretical analysis with the effects of mobility is presented.
However, it is assumed that a source terminal does not movefrom the point after call origination and only destination
terminal moves in the cell. We extend our study to the analysisof the teletraffic characteristics in the model that two terminals
move in a cell. We consider the model that there exist the points
that terminals stay and tend to gather in a cell. The mean
holding time of direct communication is important factor forteletraffic analysis. Hence, we theoretically evaluate the mean of
the holding time for direct communication with the effect ofmobility of terminals.
2

Definitions and Assumptions

Let us assume circular cell with radius R, as shown in Fig. 1.
In this figure, BS means a base station. We assume thefollowing: the received power levels are always sufficient for
direct communication if the distance between the two terminals
is not longer d. If the distance between the two terminals is not
greater than d, these terminals can directly communicate witheach other using one channel. Otherwise, these terminals arerelayed by the BS, using two channels.
We consider a single cell system. We assume that twoterminals can judge by themselves whether directcommunication between them is possible or not based on thereceived power levels of control signals which are sent to eachother. In this paper, we assume that the received power levels
are always sufficient for direct communication if the distance
between the two terminals is not longer d. We also assume thatthe terminals notify a BS of the result of the judgment. Then, we
assume the following: Let Y be the distance between twoterminals. If Y is not greater than d, these terminals can directly
communicate with each other using one channel. Otherwise,
these terminals are relayed by the BS, using two channels.
Channel assignment is always done by the BS. The BS assigns achannel to a call which is carried by direct communication andassigns two channels to a call which is relayed by the BS basedon the result of the judgment which is sent from the terminals.
We assume that a channel is not spatially reused in a cell.

BS

Relayed byBase Station

RdDirectFig. 1. Model of cell and communication range.
33.1

Mean Holding Time for Direct Communication

Definitions and Assumptions
We consider a one-dimensional model. Assume that eachof the places is a point on the one-dimensional cell asrepresented in Fig. 2. Let xi be the point i, where i = - n, ..., n.
Distance between xi and xi+1 is equal to 1, for n  i  n  1 .
We use a walk model of terminals. In this model, aterminal exists at one of the (2n+1) points. It remains static for apause time tp at the point and then starts moving one step to theright or one step to the left. This movement is repeatedaccording to the same rule. We assume that a terminal leaves thecell when it moves to the point xn+1 or the point x-n-1.
Let us denote a source terminal with A and a destinationterminal with B, as shown in Fig. 2. Denote the range in which aterminal can directly communicate with the terminal A with d.
For simplicity, we assume that source A moves one step to theleft and destination B moves one step to the right. Furthermore,
we assume that terminal A and B stay at the same point x0 at thetime at which the terminal A aims to communicate with B.
We assume following: The pause time tp obey an
exponential distribution with a mean of ts. The lifetime of a callobeys an exponential distribution with a mean of h0.simulation results. From these results, we confirm the validity of
the analysis.

dAx -n･･･ x -Bx 01x 14x 2x 3･･･ xnFig. 2. One-dimensional model.

Conclusion

We theoretically evaluate the mean of the holding time fordirect communication with the effect of mobility of terminals
using a walk model of terminals. In the future, we plan to extend
the theoretical methods to teletraffic analysis of carried traffic,
two-dimensional scenarios, and other motion patterns.

3.2Analysis
We assume that source terminal A and destination terminal
B exist at the same point x0 after call origination. In thissituation, source A can directly communicate with destination B.
Let us consider the mean of the holding time for directcommunication, which is the time interval between the time atwhich direct communication begins and the time at which directcommunication ends.
First, we consider the mean holding time for directcommunication in the case that d is equal to 0. Let si, j denote thestate that terminal A exists at the point xi and terminal B exists
at the point xj ,where i = - n, ..., n and j = - n, ..., n. In this case,direct communication between terminals fails when terminal Amoves one step to the left or terminal B moves one step to theright. Let td1 be the mean of the time in this case. The
probability that transition from s0, 0 to s-1, 0 or s0, 1 occurs is equal
to 1/2. Therefore, td1 is represented as follows:

ttd1  s .
2

Mean Holding Time h1 (s)

40

11

h0 t d 1

3.3

1.
11
h0 t d 2

Simulation (h0=60s)
Simulation (h0=90s)

30405060708090100605040

Theory (h0=30s)

30Theory (h0=60s)
Theory (h0=90s)

20

Simulation (h0=30s)
Simulation (h0=60s)

10Simulation (h0=90s)

0(3)

20(4)

Results and Discussions
The numerical results of the mean holding time h1 for
h0 = 30, 60, and 90s are shown in Fig. 3. The horizontal axis is
the mean pause time ts. The results of computer simulations arealso plotted in the figure. h1 increases as h0 and ts increase. The
theoretical results almost agree with the simulation results. Fromthese results, we confirm the validity of the analysis.
The numerical results of the mean holding time h2 are also
shown in Fig. 4. The theoretical results almost agree with the30405060

70

80

90100

Mean Pause Time ts (s)Fig. 4. Mean holding time (d=1).

References
[1]

[2]

Let h2 be the mean holding time for direct communication.
h2 can be computed by the following equation :h2 Simulation (h0=30s)

10Fig. 3. Mean holding time (d=0).

(2)

In second case, we consider the mean holding time fordirect communication in the case that d is equal to 1. In this case,direct communication between terminals fails when the distance
between two terminals becomes 2. Let td2 be the mean of thetime in this case. The mean of the time interval from the time atwhich direct communication begins to the time at which
terminal A moves one step to the left or terminal B moves one
step to the right is equal to ts. The probability that transition
from s0, 0 to s0, 2 or s-1, 1 occurs is equal to 1/4. The probability
that transition from s0, 0 to s-2, 0 or s-1, 1 occurs is equal to 1/4.
Therefore, td2 is represented as follows:t5ttd 2  t s  s  s .
44

Theory (h0=90s)

Mean Pause Time ts (s)(1).

Theory (h0=60s)

20

Mean Holding Time h2 (s)1Theory (h0=30s)

200Let h1 be the mean holding time for direct communication.
We assume that the lifetime of a call and the pause time obey anexponential distribution. Therefore, h1 can be computed by thefollowing equation :h1 30[3][4]

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