Dropping Rate Simulation for a Handover Scheme Using Importance Sampling

doi:10.4236/cn.2013.53B2078

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Communications and Network, 2013, 5, 426-429

http://dx.doi.org/10.4236/cn.2013.53B2078 Published Online September 2013 (http://www.scirp.org/journal/cn)

Dropping Rate Simulation for a Handover Scheme Using

Importance Sampling*

Dong Liang, Gan Ding, Wuling Qin, Mugen Peng

Wireless Signal Processing and Network Lab, Key Laboratory of Universal Wireless Communications

(Ministry of Education), Beijing University of Posts and Telecommunications, Beijing, China

Email: [email protected], [email protected], qi[email protected], pmg@bupt.edu.cn

Received June 2013

ABSTRACT

The process of changing the channel associated with the current connection while a call is in progress is under consid-

eration. The estimation of dropping rate in handover process of a one dimensional traffic system is discussed. To reduce

the sample size of simulation, dropping calls at base station is considered as rare event and simulated with importance

sampling - one of rare event simulation approaches. The simulation results suggest the sample size can be tremendously

reduced by using importance sam pling.

Keywords: Handover; Importance Sampling; Monte Carlo; Dropping Rate

1. Introduction

Handover is the process of changing the channel (fre-

quency, time slot, spreading code, or combination of them)

associated with the current connection while a call is in

progress [1]. Usually, continuous service is achieved by

supporting handover from one cell to another [2,3]. As

shown in Figure 1, it is often initiated eith er by crossing

a cell boundary or by deterioration in quality of the sig-

nal in the current channel [4].

The handover process starts when the power received

by the mobile station from a neighboring cell’s base sta-

tion (BS) exceeds the power received from the BS of the

current cell by a certain amount, called handover thre-

shold. This is the threshold in the received power, below

which acceptable communication with the BS of the cur-

rent cell is no longer possible [5]. If the power level from

the current BS falls below the receiver threshold prior to

the mobile being assigned to a channel by the target BS,

the call is terminated and the handover attempt fails.

Queuing priority schemes give possibility to reduce the

blocking probability of new calls, where the calls queuing

in handove r queues. Que uing pri ori ty channel assignment

strategy is described in [6]. Analysis of a mobile cellular

system with handover priority and hysteresis control is

given in [7].

Queuing of handover requests is possible, because the

mobile station spends some time in handover area, where

communications with the current BS decrease in depen-

dence of the speed of moving of the mobile station. Each

next request into the handover queue can be served ac-

cording to certain service discipline.

In nowadays broadband wireless networks probabilis-

tic parameters of Quality of Service (QoS) like probabil-

ity of dropped calls because all channels at the BS are

busy is very small, less than10−9. In such cases the Monte

Carlo simulation, which is implemented for probabilities

not less than 10−5 [8] is useless and for estimation of

handover QoS parameters as blocking probabilities is

suggested implementation of rare event simulation.

Rare event simulation helps to speed up the simulation

process, as studied probabilistic parameters of quality of

service have very small probability between 10−8 and 10−12,

and they can’t be reached with standard Monte Carlo.

2. Modeling of Handover Scheme

A simplified handover mechanism is shown in Figure 2,

Figure 1. Handover in cell edge.

*This work was supported in part by the Fundamental

Research Funds

for the Central Universities (Research on efficient algorithm and con-

fidence probability in system level simulatio ns in wireless communica-

tions) (2012RC0113), the Stat e Major Science and Technology Special

Projects (Grant No. 2011ZX03003-002-

01, 2012ZX03001028), the

Beijing Natural Science Foundation (Grant No. 4131003).

D. LIANG ET AL.

427

………

Handover Queue

………

Figure 2. Handover Scheme.

where A is the channel array for BS and B is the han-

dover queue. Here, the number of the BS channels is N,

and the hand over traffic coming rate for B is v.

For arbitrary time instant k, denote

()

1, the -th channel is available

() 0, the -th channel is ocuppied

ak i



=



(1)

and denote

()

to be the occupied time for the i-th

channel till time instant k,

()1, 2,

bk=

. The numerical

relations between

( 1)

ak+

and

()

can be written

[ ]

(1)( )

akhb k+=

(2)

Usually, when

()

increase,

[ ]

Pr(1) 1

ak+=

also

increase, but unfortunately the closed form solution of h

is always hard to achieve.

Let

()xk

to be the total number of available channels

in time instant k, obviously

() ()

xka k

∑

(3)

Let

()sk

to be the number of the incoming handover

traffic at time instant k, the average coming rate for han-

dover traffic is defined as

()vsfsds= ⋅

∫

(4)

Here

()fs

is the probability density function (PDF)

of s.

() ()sk xk<

, all the incoming handover traffic can

be allocated with no latency, otherwise, some need to be

stored in Queue B temporarily and the number is

()() ()rksk xk= −

(5)

For Queue B, denote

()

to be the waiting time for

the j-th element till time instant k. If

()

≥

(6)

thej-th handover traffic in Queue B is dropped at time

instant k, where

is the dropping threshold. The final

question here is to find the average dropping rate under

given conditions (included but not limited to N, v, α,

()h⋅

, and

()f⋅

), but unfortunately the closed form solu-

tion is hard to achieve, we always use Monte Carlo si-

mulation to find the numeric results.

3. Monte Carlo Simulation

By Monte Carlo method, the average dropping rate can

be estimated as

(7)

Here L is the total number of handover traffic (also

called as sample size) and K is t he dropped n umber.

For the l-th sample, defi ne 2 -value variable

1, the -th sample droped

0, else

Z

=



(8)

The average dropping rate p also can be expressed as

∑

(9)

The variance of p can be calculated as

ˆ2

pl l

D ZDZ

= =

 

= =

 

 

∑∑

(10)

If all the samples are independent,

can be simpli-

fied as

( )

ˆ2

(1 )

LDZp p

=⋅⋅= ⋅⋅−≈

(11)

The accuracy of

is always defined as

ˆp

(12)

then

= =

(13)

In order to assure the accuracy, for very small p, we

need to run large number of L to find enough K, which

means large amount of simulation time.

4. Improved Simulation Using Importance

Sampling

The most famous approach for rare event simulation is

Importance Sampling. Importance Sampling is connected

with change the probability density distribution for in-

creasing the frequency of appearance of more “signifi-

cant” for simulation events.

The basic purpose of this simulation technique is to

reduce dispersion or other estimation function, received

as a result of computer simulation. During the simulation,

process is expected to receive samples proportional of

their importance to expected results.

D. LIANG ET AL.

428

The Importance Sampling estimators can receive in

advance given accuracy and in this way the simulation

time can be shorten. For generation of significant sample

is used limited number of independent variables with

normal distribution [9]. Then the conditional probability

of appearance of rare event is changed with conditional

probability of appearance less rare event with similar

distribution [10].

In this research, the average dropping rate p also can

be ex pressed as

() ()pq sfsds=∫

(14)

Where

()qs

is the probability for certain s.

By using importance sampling, the above equation can

be re-expressed as

()

() ()()

pqsfsds

′

=′

∫

(15)

Here

()fs

′

is the importance sampling PDF for s,

while

()

() ()

ws fs

=′

(16)

is the weighting f unction. Here, s follows negative expo-

nential distribution, and

()fs

′

is chosen to increase the

probability of dropping through change the exponent.

5. Simulation Results

The simulation parameter is shown in Table 1. The si-

mulation results between dropping rate p and average

handover traffic coming rate v is shown in Figure 3. It

can be seen p decreases with increasing of v.

The numerical relations between Monte Carlo simulation

sample size LMC and average handover traffic coming

rate v is shown in Figure 4. Here LMC is the least sample

size to assure the simulation accurac y of p is smaller than

10%.

Define sample size reduction efficiency as

(17)

the numerical relations between

and average han-

dover traffic coming rate v is shown in Figure 5. It can

be seen

decreases with increasing of v. Typically,

when

0.1v=

approaches to 104, which is a tre-

mendous reduction.

6. Conclusions

The processes of changing the radio channel associated

with the current connection, while a call in progress is

under consideration. A queue handover scheme for broad-

band mobile communication is suggested.

Table 1. Simulation parameter.

N 64

()f⋅

negative exponential distribution

α 3

()h⋅

negative exponential distribution

Figure 3. p ~ v.

Figure 4. LMC ~ v.

Figure 5. β ~ v.

A simulation approach using importance sampling for

estimation of probabilistic parameters of handover drop-

ping rate at broadband wireless networks with rare event

estimation is suggested.

D. LIANG ET AL.

429

The simulation results show that the dropping rate de-

decreases with increasing of average handover traffic

coming rate v. By using importance sampling, the sample

size reduction efficiency

decreases with increasing

of v. typically, when

is very small and

appro-

aches a tremendous reduction.

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