Proceedings of the International Conference on Computer and Communication Engineering 2008 May 13-15, 2008 Kuala Lumpur, Malaysia

978-1-4244-1692-9/08/$25.00 ©2008 IEEE

Modeling and Analysis for Heterogeneous Wireless Networks by Using of Multi-Dimensional Markov Models

Hesham ElBadawy1

1Research Doctor, National Telecommunication Institute, Cairo, Egypt heshamelbadawy@gmail.com

Abstract

Nowadays, a lot of radio access technologies (RAT)

have been deployed. In order to achieve a seamless integration between these different RATs, Common Radio Resource Management (CRRM) algorithms are considered to be deployed in such heterogeneous wireless networks. This paper provides a complete analytical approach to determine the performance of different RAT with selection procedures in multi-RAT/multi-service environment. Particularly, multi-dimensional Markovian model will be used to perform a complete derivation for the system state probabilities. The allocation of voice and data services in both of GERAN/UTRAN systems, will be considered. In addition numerical results will be illustrated for different operational scenarios for such heterogeneous system.

I. INTRODUCTION Deploying different Radio Access Technologies

(RATs) may allow operators to jointly manage, in the most efficient way, the available radio resources provided by each of their individual RATs. This concept is usually referred as Common Radio Resource Management (CRRM) for heterogeneous wireless networks. A lot of researchers had discussed this problem e.g. [1] and [2]. The proposed methodology usually relies on system level analysis in order to extract some relevant performance metrics. A few analytical proposals have been developed up to date [3]-[5],[7],[11]. In [3], proposes an analytical approach to the problem of traffic overflowing between several RATs using an M-dimensional Markov model. However, in order to derive a closed form solution by means of applying independence between service types.

In this paper, the proposed analytical model is introducing a framework by considering that only the total offered traffic for each RAT is known. In addition the proposed model considers a 4D-state fully describing. The analytical approach will contain the

selection algorithm of the CRRM. Markov states in this model indicate the number of sessions of each service that are being carried in whole composite network, but not on which RAT each session is being carried out. The presented model, will give an in depth sight for each service of each RAT which will be detailed evaluated. This is different from the presented work in [5], because the presented work will give a complete derivation for the steady state probability for each state. Whereas in [5], they gave the state transitions only, and no final closed form had been introduced. In addition, the presented model will evaluate the heterogeneous network based on which RAT each session is being carried out. The paper is organized as follows. Section II presents the analytical model and the notation that will be used throughout the paper. In section III, a load balancing RAT selection policy is described by means of the proposed model. Section IV presents the performance metrics that will be used to evaluate the behavior of the CRRM allocation policy. Section V illustrates the obtained results and its analysis. Finally, section VI concludes for the presented work and its future directions.

II. MULTI-DIMENSIONAL MARKOV MODEL Multi-Dimensional Markov models are used in the

field of networking to model the behavior of communication networks under varying traffic load conditions. In this paper, the aim is to drive two RATs one for the WCDMA-based UMTS Terrestrial Radio Access Network (UTRAN) and the other is GSM/EDGE Radio Access Network (GERAN). In order to accommodate these services, voice and data, being served over the mentioned RATs, GERAN and UTRAN. A combination of two 2D Markov chains may be used to model the system’s behavior. Let S(i,j,k,l) represents the state in which the system is servicing: in GERAN, i voice users and j data users, (with nC is maximum number of data sessions, that are sharing the same time slot TSL); whereas in UTRAN, the system is servicing k voice users and l data users.

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So, the system traffic model may be described as follows:

(i) In GERAN RAT, it may be assumed that the arrival rate is (λv

G) for both of new voice calls and handoff voice calls. On the other hand, the arrival rate is (λd

G) for both of new data sessions and handoff data sessions. The proposed model will consider the combination of multi-services, which is controlled by means of the available radio resources (namely, available time slots per carrier C). By assuming the GERAN service rates for both of vice and data calls are μv, and μd respectively. So, the number of the users who are sharing the GERAN resources will be restricted according to the following relation:

CnjiC

≤+≤0 (1)

So, the process may be done according to the 2-dimentional Markov Process as in [3]. This process may be solved by means of a combination of two single dimensional Birth-death processes. So, the solution of GERAN Markov Process will be according to the product form solution of two single dimensional Birth-Death Processes [6],[12]. For the GERAN voice calls, the steady state probability may be expressed as follows:

CiPi

iP Gv

i

v

GvG

v ≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛= 0;)0(

!1)(

μλ (2)

Also, for GERAN data sessions, the steady state probability will be:

CnjP

jjP

C

Gd

j

d

GdG

d ≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛= 0;)0(

!1)(

μλ (3)

Hence, the product form solution after applying the Chapman-Kolmogorov Criteria [8], it may be found that the Markov process is reversible. So, the product form is applicable. Due to resource sharing in the radio resources, the steady state aggregated service rate is a dependant upon both of the voice, and data service rates. So the probability of having i voice users and j data users is as follows [3],[8]:

)0,0(!!)!(),( G

j

d

Gd

i

v

GvG P

jijijiP ⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+=

μλ

μλ (4)

Where PG(0,0) is the GERAN probability of having no usage of time slots. By considering the offered traffic of voice to be Tv

G and for data is TdG

respectively, and so :

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

d

GdG

dv

GvG

v TandTμλ

μλ

(5)

So, the steady state probabilities in the GERAN system may be expressed as follows:

( ) ( ) )0,0(!!)!(),( GjG

diG

vG PTT

jijijiP +

= (6)

(ii) In UTRAN RAT, it may be assumed that the

arrival rate is (λvU) and the service rate is μv for both of

new voice calls and handoff voice handoff calls. On the other hand, the arrival rate is (λd

U) and the service rate is μd for both of new data sessions and handoff data sessions. By considering the offered traffic of voice to be Tv

U and for data is TdU, and so :

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

d

UdU

dv

UvU

v TandTμλ

μλ

(7)

Due to the source sharing of the same UMTS NodeB site/sector having capacity UC available physical links (WCDMA codes). As UTRAN is WCDMA based system, so the system capacity may be determined as a function of the processing gain (W/Rb) and the operating energy per bit-to-interference ratio (Eb/Io) as follows:

( ) 1+=oIE

RWUb

bC

(8)

Where, W is the chipping (spreading) rate, and Rb is the payload baseband rate. So, the limitations on both of k, and l are as follows [9]:

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+≤≤⎟⎟

⎠

⎞⎜⎜⎝

⎛+≤≤ 1010

db

bd

vb

bv

IERWland

IERWk

oo

(9)

So, the number of the users that are sharing the UTRAN resources will be restricted according to the following equation:

( ) ( )

111

0 ≤

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

≤

db

bd

vb

bv

IERWl

IERWk

oo

(10

)

Again, the process may be done according to the 2-dimentional Markov Process as in [6],[12]. This process may be solved by means of a combination of two single dimensional Birth-death processes. So, the solution of UTRAN Markov Process will be according to the product form solution of two single dimensional Birth-Death Processes [3],[6],[12].

Hence, the product form solution after applying the Chapman-Kolmogorov Criteria [8], it may be found that the Markov process is reversible. So, the product form is applicable. Due to Resource sharing of the Radio resources so the probability of having k voice users and l data users is as follows:

( ) ( ) )0,0(!!)!(),( UlU

dkU

vU PTT

lklklkP +

= (11)

Where PU(0,0) is the UTRAN probability of having no usage of its physical channels.

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III. SERVICE ALLOCATION METHODOLOGY AND LOAD BALANCING

The load balancing policy [10], intends to allocate users to the suitable RAT that undergoes a lower load situation at a given time. In particular, transitions between a source state and possible destination states will depend on the measured load at each destination state. In GERAN, the time slots utilization factor LG(i,j), [5], may be used as an indication for the load in a given state S(i, j,k,l) as:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛=

C

G

njiC

CjiL ,min1),(

(12)

The load LU(k,l) in UTRAN is calculated by means of the uplink load factor as:

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

11),(

db

bd

vb

bv

U

IERWl

IERWklkL

oo

(13)

The selection methodology must be applied in order to determine whether the incoming user demanding a given service should be allocated to GERAN or to UTRAN. This may be classified into two cases: the first one of having new voice call and load factor of the GERAN is higher or lower than that of UTRAN. The other case is considered for new data session. As a mathematical representation, this classification may be defined by introducing two variables (α,β) for both of voice and data cases respectively. This may be illustrated in mathematical form as follows:

⎩⎨⎧

+>++≤+

=

⎩⎨⎧

+>++≤+

=

)1,()1,(;0)1,()1,(;1

),,,(

:),1(),1(;0),1(),1(;1

),,,(

:

lkLjiLlkLjiL

lkji

SessionsDataForlkLjiLlkLjiL

lkji

CallsVoiceFor

UG

UG

UG

UG

β

α

(14)

Assume that the voice traffic to be serviced by factor η in the GERAN RAT, and similarly for data traffic to be serviced by a factor ζ in the UTRAN RAT.

So, the previously mentioned traffic loads may be linked by means of the functions based on (η,ζ) and (Tv,Td) as follows:

dUdd

Gd

vU

vvG

v

TTandTT

TTandTT

)1(

)1(

ζζ

ηη

−==

−== (15)

Where Tv, and Td are the total offered traffic for both of voice calls and data sessions respectively.

So, the joint steady state probabilities for the heterogeneous wireless GERAN and UTRAN systems may be done by means of the common radio Resource Management (CRRM) system. Hence, the user may be served by GERAN or UTRAN at the same time. On that basis, the joint probability density function of the whole system may be expressed as follows:

)1,(),()1(),()1,(),1(),()1(),(),1(),,,(

−−+−+−−+−=

lkPjiPlkPjiPlkPjiPlkPjiPlkjiP

ββαα

(16)

So, the complete derived P(i,j,k,l) may be written as follows:

),(),(

))(1()1(

))(1()1(

)()(),,,( lkPjiP

lkTl

lkTk

jiTj

jiTi

lkjiP

dv

dv

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

+−−

++−

−+

++

+=

ζβ

ηαζ

βη

α (17)

By applying the normalization process for the system at steady state, so:

( )( )

11

0

1

0 0 0

),,,()0,0,0,0(

−+

=

+

= = = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛= ∑ ∑ ∑ ∑

db

bd

vb

bv

CIERW

l

IERW

k

Cn

j

C

i

lkjiPPo o (18)

It is important to notice that, this definition of load balancing implies that a data user will not be forced to share a time slot in GERAN unless there will no resources are available in UTRAN [5].

IV. PERFORMANCE METRICS In order to compute the steady state system

performance, the equations from (12) up to (18) in consistence with the limiting conditions in equations (3), (4), (9), and (10) must be solved simultaneously. This may be carried out using numerical methods. Then, performance metrics may be directly derived from the steady state probabilities, P(i,j,k,l). The proposed performance metrics is in consistence with that introduced in [5],[9]. In this paper, we will concentrate on the blocking and loading performance factors, which may be evaluated as follows [2],[5], and [9]:

A. Carried Traffic (NvG, Nd

G, NvU, and Nd

U) The average carried traffic (average number of

users) may be computed from the steady state probabilities P(i,j,k,l). The fractional average number of users with specific service (voice or data) in both of GERAN or UTRAN RATs can be derived as follows:

∑∑

∑∑•=•=

•=•=

kji

Ud

lji

Uv

lki

Gd

lkj

Gv

lkjiPlNlkjiPkN

lkjiPjNlkjiPiN

,,,,

,,,,

),,,(;),,,(

),,,(;),,,( (19)

B. Blocking Probabilities (Pb,v , Pb,d) As the definition of having equal opportunities for

servicing both of new calls or handoff calls. It may be calculated by collecting all the possible cases that may become out of service due to traffic arrivals (but we consider here perfect power control situation, so the block/drop calls will be due to only traffic collisions). Making use of the blocking state sets defined by the unavailability of having physical links in both of the current service in each RAT. So, the blocking/call drop probabilities may be expressed as follows:

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( )

( )∑

∑=

=

availablearesourcesdataUTRANorGERANnodb

availablelaresourcesvoiceUTRANorGERANnovb

lkjiPP

lkjiPP

Re,

Re,

),,,(

),,,(

(20)

C. Load Recalling the load definitions as were introduced in

(12) and (13), the average load for each RAT may be calculated. For the GERAN, the average TSL utilization factor will be as follows:

∑ •=lkji

GG lkjiPjiLL,,,

),,,(),( (21)

In addition, For the UTRAN, the average uplink load factor may be computed as will be as follows:

∑ •=lkji

UU lkjiPlkLL,,,

),,,(),( (22)

V. RESULTS AND ANALYSIS The system performance is studied under different

offered voice and data traffic loads, Tv and Td. The system parameters are of practical background and in consistence with the published work in [5]. The total number of channels available for GERAN is C = 8, allowing up to nC =3 number of simultaneous users sharing the same time slot. Parameters for UTRAN consider a chipping rate of W = 3.84Mcps and target signal-to-interference ratios for voice and data services of 6dB and 5dB respectively. Achievable bit rates for both RATs are assumed to be equal with, Rbv = 12.2 kbps and, Rbd=44.8 kbps for voice and data services respectively. Under these assumptions, UTRAN exhibits a higher capacity in terms of maximum number of admissible voice and data users as compared to GERAN. So, in GERAN, the voice traffic factor η is 8 times that of the data traffic factor ζ in the UTRAN RAT. Tv and Td will be varied from rural area case studies and up to 3G hot spots very dense urban traffic case studies. In order that, Tv and Td will vary from few Erlang, as a rural area case study, up to 84 Erlang as the case of highly dense urban. These extracted results are in consistence with the results that had been published before in [2],[5], and [10]. But, here it may be noticed that the proposed technique is suitable to add both of another service (as video call services), or add some priority schemes such as the case of having contention between handoff call and new call request. Also, the presented model is a complete closed form to calculate the steady state probabilities and the different performance metrics.

In figure (1), the load balancing policy attempts to keep load levels equal in both systems for different offered loads. The load results are monotonically the same as the presented results in [5]. This is in spite of

the complete analytical derivation and more simplicity to find out the system key performance indices.

Figure 1. Average Load in each RAT under varying traffic

Figure (2), presents the blocking probabilities for voice services in each RAT. It may be noticed that as the offered traffic for the service under study increases, all the blocking probabilities will decayed. But the voice blocking probability for GERAN is decreased slowly with respect to that of UTRAN voice. This may be explained by means of there are more radio resources in the UTRAN than that of the GERAN.

Figures (3) & (4) illustrate the blocking probability for data services in both of GERAN and UTRAN respectively. The blocking probability is decreased as the data offered traffic is increased. This is due to the as the data traffic increased the opportunity to content with voice is increased and so the probability of blocking will decreases.

Figure 2. Blocking probability for voice

Figure 5 shows the number of served users of each service in UTRAN and GERAN. Since, GERAN offers a much lower capacity than UTRAN, so, many users are serviced in UTRAN. The number of data

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users in each RAT will decrease while offered voice traffic lies between 1.2 and 36 Erlangs. For higher Tv, data users are forced to share TSLs in GERAN, explaining the decrease of data users in both of GERAN and UTRAN respectively.

Figure 3. Blocking probability for GERAN data

Figure 4. Blocking probability for UTRAN data

Figure 5. Average number of served users in each RAT

VI. CONCLUSIONS AND FUTURE WORK In this paper, a complete closed form for Multi-

Dimensional Markov model has been derived for the performance evaluation of heterogeneous wireless network based on two different RATs. Multi-access using network comprising GERAN and UTRAN technologies had been modelled. The proposed analytical model enables detailed analysis for each RAT service performance parameter. Also, assessments of different performance metrics have been illustrated.

As a future work, more analysis extensions may be done based on the presented methodology for different directions such as: multi services, multi rates, and priority schemes.

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[9] G. Piao, "Radio Resource Management for Integrated Service in Multi-radio Access Networks", Doktor .Eng. Thesis, University of Kassel, Germany, Dec.2006.

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