erlang capacity of multiclass tdma systems with amc

8/13/2019 Erlang Capacity of Multiclass Tdma Systems With AMC

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Erlang Capacity of Multi-class TDMA Systemswith Adaptive Modulation and Coding

Hua Wang and Villy B. IversenDepartment of Communications, Optics & Materials

Technical University of Denmark, Lyngby, Denmark Email: {huw, vbi }@com.dtu.dk

Abstract Erlang capacity is traditionally dened as the max-imum value of offered trafc among different service classes thatthe system can support when the blocking probabilities at the calladmission control (CAC) level do not exceed certain thresholds.That is valid when a xed amount of bandwidth is allocated toeach user in each frame during the whole service time. However,with the introduction of adaptive modulation and coding (AMC)scheme employed at the physical layer, outage might occur dueto the fact that the allocation of bandwidth is dynamic basedon the time-varying wireless channel conditions. In this paper,we evaluate the Erlang capacity of a TDMA system with AMCsupporting voice and data trafcs, by taking both the blockingand the outage probabilities into account. The analytical modelsfor calculating the blocking and the outage probabilities aredeveloped separately, and a joint algorithm for determining theErlang capacity of the system is proposed with some numericalexamples.

I. INTRODUCTION

Future generation wireless communication systems areevolving to provide a wide range of services, including voice,data and multi-media applications, with different quality of service (QoS) requirements, such as delay and throughput.

The economical usefulness of a system is effectively mea-sured by the Erlang capacity, which is generally dened asthe maximum trafc load that the system can support with acertain blocking probability requirement. Many models havebeen proposed at separate layers, such as the Rayleigh, Ricianand Nakagami fading models at the physical layer [5][7], andqueuing models at the data link layer [4]. In traditional chan-nelized multiple access systems, e.g., TDMA and FDMA, eachuser is assigned a xed amount of bandwidth during the wholeservice time, and the Erlang capacity can be easily obtained byusing the well-known Erlang-B formula. However, traditionalqueuing models do not consider the time-varying nature of wireless channels due to multipath fading and Doppler shift.Unlike wired networks, even if large bandwidth is allocated toa certain wireless connection, the QoS requirements may notbe satised when the channel experiences deep fades.

In order to enhance the spectrum efciency while main-

taining a target packet error rate (PER) over wireless links,adaptive modulation and coding (AMC) scheme has beenwidely adopted to match the transmission rate to time-varyingchannel conditions. With AMC, the allocation of bandwidthto each user is no longer deterministic (i.e., a xed amountof bandwidth), but in a dynamic behavior. Therefore, thecalculation of the Erlang capacity should take both the call

admission control (CAC) module at the data link layer as wellas the AMC module at the physical layer into considerations.

An analytical model to investigate the performance of transmissions over wireless links is developed in [1], wherea nite-length queuing is coupled with AMC. However, theauthor only concentrates on a single-user case. Reference [2]calculates the Erlang capacity of WiMAX systems with xedmodulation scheme, where two trafc streams, streaming andelastic ows, are considered. In this paper, we investigatethe Erlang capacity of multi-user multi-class TDMA Systemsunder the joint effect of CAC and AMC, by coupling theanalysis of the blocking probability and the outage probability.We rst present analytical models to calculate the blocking andthe outage probabilities respectively. Based on that, a jointalgorithm to determine the Erlang capacity of the system isproposed. The Erlang capacity in this paper is dened as themaximum trafc load among different service classes that thesystem can support when both the blocking and the outageprobabilities do not exceed certain thresholds.

The rest of the paper is organized as follows. In Section II,we introduce the system model and the call admission controlpolicy. In Section III, analytical models for calculating theblocking and the outage probability with AMC are presentedtogether with the joint algorithm. Numerical results are shownin Section IV. Finally, a conclusion is drawn in Section V.

I I . S YSTEM M ODEL

We consider an infrastructure-based wireless access net-work, where connections are established between base station(BS) and mobile stations (MSs). Two types of services, voiceservice and data service, are supported by the system. Voiceservice is not tolerant to packet delay, thus requires a constantbit rate. Data service is more elastic in terms of being able tovary the transmission rate according to the channel conditions,but also requires a minimum throughput.

At the physical layer, the data is transmitted frame by frame,where each frame is comprised of a xed number of time slots,

each of which contains a xed number of symbols. In thedownlink, the transmission to different users is scheduled on atime-division multiplexing (TDM) fashion, while time-divisionmultiple access (TDMA) is applied in the uplink. Adaptivemodulation and coding scheme is employed at the physicallayer, where multiple transmission modes are available, witheach mode representing a pair of specic modulation format

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

978-1-4244-2075-9/08/$25.00 2008 IEEE 11

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and a forward error correcting (FEC) code. The transmissionmode is determined by the instantaneous signal-to-noise ratio(SNR). We assume that the channel is frequency at, andremains constant within a frame, but may vary from frameto frame. Furthermore, we assume that the BS has perfectknowledge of channel state information (CSI) of each user.Based on these assumptions, the AMC scheme adjusts itstransmission mode on a frame-by-frame basis.

Voice and data users arrive at the cell in a random order.The CAC module decides whether an incoming call shouldbe admitted or not. To prioritize voice service, CAC reservesa xed amount of bandwidth dedicated for voice users, andallows both voice and data users to compete for the remainingbandwidths. Let K denote the total number of time slotsin a frame for data transmissions, of which K time slotsare reserved for voice users only. Let {nv , n d } denote thenumber of admitted voice users and data users in the cell,respectively. Each voice user is allocated cv time slots perframe on average to ensure a constant bit rate, and each datauser is allocated at least cd time slots per frame on averageto guarantee a minimum throughput. The remaining time slotscan be allocated to data users on demand to increase their

throughput. Under the above conditions, the state space of thesystem is given by:

S := {nv , n d } N N (1)

subject to:nv cv + nd cd K

n d cd K K

III . A NALYSIS M ODELS

In this section, we rst develop an analytical model basedon multi-dimensional loss systems [4] to calculate the statespace and the blocking probability of each trafc stream. Then,given the distribution of the number of voice and data users

admitted in the system, the outage probability caused by AMCcan be derived by analyzing the stochastic model of differenttransmission modes. Finally, a joint algorithm for determiningthe Erlang capacity of the system is proposed.

A. Analysis of Blocking Probability

We use the BPP (Binomial, Poisson & Pascal) trafc modelin our analysis [4]. This model is insensitive to the servicetime distributions, thus is very robust for applications. Eachtrafc stream i is characterized by the offered trafc Ai , thepeakedness Z i and the number of channels ci needed forestablishing one connection. The offered trafc A i is usuallydened as the average number of call attempts per meanholding time. Peakedness Z i is the variance/mean ratio of the

state probabilities when the system capacity is innite, andit characterizes the arrival process. For Z i = 1 , we have aPoisson arrival process, whereas for Z i < 1, we have a nitenumber of users and more smooth trafc (Engset case). ForZ i > 1, it corresponds to a more bursty Pascal arrival process.

In this paper, we assume that voice users arrive followingthe Engset case with a limited number of S sources. Each

source switches between the states of idle and busy, whichare both exponentially distributed with intensity v and v ,respectively. Thus the arrival process of voice users is a statedependent stochastic process with arrival intensity vi = ( S i) v , where i is the number of busy voice users at a givenpoint of time. The offered trafc of voice users is equal toAv = S v v + v and the peakedness is equal to Z v = 1 Av /S .Data users arrive according to a Poisson process with arrival

intensity d

, offered trafc A d , and peakedness Z d = 1 .The call-level characteristics of multi-class (voice/data)TDMA systems with the CAC policy described in Section IIcan be modeled by a two-dimensional Continuous TimeMarkov Chain (CTMC) shown in Figure 1, where state (i, j )represents the number of time slots occupied by voice usersand data users, respectively. As it is a reversible Markovprocess and has product form, the numerical evaluation canbe done by using the convolution algorithm [4].

+ + + + + + + +

+

+

+

+

+

(i, j )

00

i

j

n 1

n 2

n 1

2

i + j = n

blocking for stream 1blocking for stream 2

.............. ............... .............. .......................... ..........................

........

........

........

........

........

.............................

..........................

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

........................

.............

.............

.............

............

.............

..............

.............

.............

.............

.............

......

Fig. 1. Structure of the state transition diagram for two-dimensionaltrafc processes with class limitations. When calculating the equilibriumprobabilities, state (i, j ) can be expressed by state ( i, j 1) and recursivelyby state ( i, 0) , ( i 1, 0) , and nally by (0 , 0) . [4]

The probability of having a voice users and b data users inthe system can be represented as:

P (nv = a) = q (nv = a)iv q (nv = i)

a v (2)

P (nd = b) = q (n d = b)

id q (nd = i) b d (3)

where v and d are the sets of possible number of voiceand data users in each service class, x denotes the largestinteger not exceeding x .

v := {0, 1, . . . ,K cv

}

d := {0, 1, . . . ,K K

cd}

(4)

q (nv ) and q (nd ) are the relative state probabilities, calculatedas follows:

q (n v = a) = pv (acv ) K K j =0 pd ( j ) if a

K c v

pv (acv ) K a cvj =0 pd ( j ) if a > K c v

q (nd = b) = pd (bcd ) K bc d

j =0

pv ( j )

(5)


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P vblock =

Kc v

i = Kc v

vi pv (i cv ) pd ( K i c vc d cd ) Kc v

i =0 vi pv (i cv )

K K j =0 pd ( j ) +

Kc v

i = Kc v +1

vi pv (i cv ) K i cvj =0 pd ( j )

(8)

P dblock =

K Kc d

1i =0

d pd (i cd ) pv ( K i cdc v cv ) + d pd ( K K cd cd )

K K Kc d c dj =0 pv ( j )

K Kc d

i =0 d pd (i cd ) K i c dj =0 pv ( j )

(9)

where pv () and pd () are the state probabilities of each trafcstream as if it is alone in the system:

pv (i cv ) = (Si ) i (1 )S i

j v (Sj ) j (1 ) S j

if i v

pv () = 0 else(6)

where = 1 Z v .

pd (i cd ) =A id

i !

j d

A jdj !

if i d

pd () = 0 else(7)

Based on the state probabilities, the closed-form of the callblocking probability of each trafc stream can be obtained byusing the convolution algorithm, shown in Eqn. (8) & (9).

B. Analysis of Outage Probability

In AMC scheme, the modulation mode and coding rate ischosen depending on the time-varying channel conditions. Asa consequence, the number of time slots allocated to eachuser is varying on a frame by frame basis. Outage is denedto occur when the total number of time slots required by theadmitted users exceeds the total available time slots.

1) AMC: For at fading channels, we adopt the generalNakagami- m model to describe the received signal-to-noiseratio (SNR) statistically, which is a random variable withGamma probability density function [1]:

p ( ) = mm m 1

m (m) exp

m

(10)

where := E [ ] is the average received SNR, (m) :=

0 tm 1 exp( t)dt is the Gamma function and m is the

Nakagami fading parameter ( m 1/ 2).The design objective of AMC is to maximize the data rate

by adjusting the transmission parameters according to channelconditions, while maintaining a prescribed packet error rate(PER) P 0 . Let N denote the total number of transmissionmodes available ( N = 5 in this paper). Assuming constantpower transmission, we partition the entire SNR range intoN + 1 non-overlapping consecutive intervals with boundaries

denoted as { n }N +1n =1 . Specically, mode n is chosen when [ n , n +1 ). Therefore, mode n will be chosen with probability:

P r (n) = n +1

n p ( )d

= m, m n m,

m n +1

(m)

(11)

where (m, x ) :=

x tm 1 exp( t)dt is the complementary

incomplete Gamma function.The closed-form of the average PER corresponding to mode

n is obtained as [1]:

PER n = 1P r (n)

n +1

nan exp( gn ) p ( )d (12)

where an , gn are the mode dependent parameters shown inTable I.

The algorithm for determining the threshold { n }N +1n =1 with

the prescribed PER P 0 is described in details in [1].

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

Modulation BPSK QPSK QPSK 16QAM 64QAM

Coding rate 1/ 2 1/ 2 3/ 4 3/ 4 3/ 4

R n (bits/sym) 0.5 1.0 1.5 3.0 4.5

a n 274.7229 90 .2512 67 .6181 53 .3987 35 .3508

gn 7.9932 3 .4998 1 .6883 0 .3756 0 .0900

pn (dB) 1.5331 1 .0942 3 .9722 10 .2488 15 .9784

TABLE I

T RANSMISSION MODES WITH CONVOLUTIONALLY CODED

MODULATION [1 ]

2) Outage Probability: Assume that there are Gv voiceusers and Gd data users admitted in the system, respectively.Each user has an ON/OFF activity model, represented by arandom variable , with probability P ( v = 1) = v for voiceusers, and P ( d = 1) = d for data users. We further assumethat voice users require a constant bit rate of Rv bits per frame,and data users require a minimum throughput of R d bits perframe. Hence, the total amount of required bandwidth in termsof time slots per frame is given by:

Z =G v

i =1

v,i N v,i +G d

j =1

d,j N d,j (13)

where v,i and d,j are the activity factors for the ith voiceuser and the j th data user, respectively. Gv and Gd are randomvariables representing the number of voice and data usersadmitted in the system. N v,i and N d,j denote the number of time slots required by the i th voice user and the j th data user


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respectively, depending on the current channel conditions.

N v,i N v , N v := RvsR n

, n = 0 , , N

N d,j N d , N d := RdsR n

, n = 0 , , N (14)

where s is the xed number of symbols per time slot, Rndenotes the number of bits carried per symbol in transmissionmode n shown in Table I, and x denotes the smallestinteger larger than x. N v,i and N d,j are random variables withprobability mass function:

P (N v,i = RvsR n

) = P r (n) n = 0, , N

P (N d,j = RdsR m

) = P r (m) m = 0 , , N (15)

Since Z is a sum of random variables, we can approximateZ to be a Gaussian random variable by applying the centrallimit theorem. Therefore, the outage probability of the systemcan be approximated by:

P outage = P r {Z > K } QK E [Z ]

V ar(Z ) (16)

It is assumed that N v,i , N d,j , v,i and d,j are independentand identically distributed random variables. Furthermore, v,i , d,j , N v,i and N d,j are independent of each other. Gv and Gdfollow probability mass function shown in Eqn. (2) & (3),respectively. The mean and variance of Z are given by:

E [Z ] = E E [Z | (Gv , Gd )]

= E [Gv ]E [ v ]E [N v ] + E [Gd ]E [ d ]E [N d ](17)

V ar[Z ] = E V [Z | (Gv , Gd )] + V E [Z | (Gv , Gd )]

= E [Gv ] E [ 2v ]E [N 2v ] E 2[ v ]E 2[N v ]

+ E [Gd ] E [ 2d ]E [N 2d ] E 2[ d ]E 2[N d ]+ V [Gv ]E 2[ v ]E 2[N v ] + V [Gd ]E 2[ d ]E 2[N d ]

(18)

C. Calculation of Erlang Capacity

The Erlang capacity in this paper is dened as the maximumvalue of offered trafc in each trafc stream {Av , Ad } thatthe system can support when the blocking probability and theoutage probability do not exceed certain thresholds.

From the analytical models developed in the previous twosubsections, we note that the blocking probability is deter-mined by the trafc loads {Av , A d } and the average numberof time slots occupied per connection {cv , cd }, which dependon the outage probability. On the other hand, the outageprobability is determined by the distributions of the numberof voice and data users {Gv , Gd }, which depend on theblocking probability. Therefore, the analytical models of theblocking and the outage probabilities are tightly coupled. Inthis subsection, we propose a joint algorithm to calculate theErlang capacity of multi-class TDMA systems with AMC.

),,,( d vd

block v

block G G P P Calculate

outage P Calculate

P outage meetsrequirement?

P outage > P req

),( d block v

block P P Record

),( d v ccIncrease ),( d v ccDecrease

Change theoffered traffic?

),( d v A AChange

Y

N

Y

N

N

Y

),( d v A AExtract Erlang

Capacity Region

),,,( d vd v A AccInitialize

Fig. 2. Flow chart of the proposed joint algorithm

The joint calculation algorithm consists of iterations of two steps. In the rst step, given the trafc loads of voiceusers and data users {Av , Ad }, the blocking probabilities areobtained such that the outage probability requirement P outageis satised. Specically, the proposed algorithm rst sets theinitial values of cv = E [N v ] and cd = E [N d ], which arethe average number of time slots needed for establishingone connection of voice and data services, respectively, andcalculates the user distributions, the blocking probabilities, andthe outage probability by using Exp. (2), (3), (8), (9) & (16).Then it checks whether P outage meets its requirement or not

at the current values of cv and cd . The values of cv and cdare increased if P outage is larger than the threshold, and aredecreased otherwise. Unless P outage meets its requirement, cvand cd are further increased or decreased. Once P outage issatised, the blocking probabilities are recorded under giventrafc loads. Then the values of {Av , Ad } are modied andthe above procedure is repeated. After we got the blockingprobabilities under various trafc loads, the algorithm proceedsto the second step, where the Erlang capacity region isobtained by extracting the maximum values of {Av , Ad } thatmeet both requirements of blocking and outage probabilities.The ow chart of the joint algorithm is shown in Fig. 2.

IV. N UMERICAL R ESULTS

In this section, we present some numerical results based onthe analytical models developed in Section III. The systemparameters are listed in Table II.

We set the target outage probability to be 2%, which isusually considered to be an acceptable QoS requirement.Fig. 3 shows the outage probability versus different trafcloads of voice and data services in Erlangs. We can see


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Parameter Value

Bandwidth 10 MHz

Frame duration 1 ms

Number of time slots in the downlink, K 800

Number of symbols per time slot, s 4

Reserved time slots for voice users, K 100

Nakagami fading parameter, m 1Average SNR, 15 dB

Target PER, P 0 10 4

Constant bit rate for voice users, R v 64 bits/frame

Minimum bit rate for data users, R d 128 bits/frame

Voice activity factor, v 0.4Data activity factor, d 0.6

Peakedness of voice trafc, Z v 0.5Peakedness of data trafc, Z d 1

TABLE II

SYSTEM PARAMETERS USED FOR THE NUMERICAL EVALUATION

from the gure that our proposed algorithm can well keepthe outage probability around the predened threshold undervarious trafc loads. This is achieved by properly adjustingthe values of {cv , cd }, which in tern changes the distributionof the number of voice and data users in the system.

A two dimensional Erlang capacity region at 2% blockingprobability is shown in Fig. 4 over different outage probabili-ties and over different bit rates of data users. We can see in allscenarios that the increase of data trafc will severely reducethe capacity for voice users, and this phenomenon becomesmore obvious when the bit rate of data users increases from128 bits/frame to 256 bits/frame. This is generally becausedata trafc usually has higher bit rate and activity factor,thus results in the loss of voice Erlang capacity more sharply.

Furthermore, we also observe that by changing the thresholdof the outage probability from 2% to 6%, the Erlang capacityregion of the system can only be marginally increased.

V. C ONCLUSION

Adaptive modulation and coding (AMC) has been widelyused to match transmission parameters to time-varying channelconditions. In this paper, we have investigated the perfor-mance of a TDMA system that carries both voice and datatrafcs with AMC in terms of the Erlang capacity. We havedeveloped analytical models to calculate the blocking and theoutage probabilities. As the outage probability depends on thedistribution of the number of users in the system, which istightly coupled with the blocking probability, we proposed

a joint algorithm to determine the Erlang capacity that thesystem can support when both the blocking and the outageprobabilities are within certain thresholds. Numerical resultshave shown that the voice trafc capacity is severely affectedby the data trafc. Moreover, we have also observed that theErlang capacity region of the system can only be marginallyincreased by loosing the target outage probability.

10

15

20

25

35

40

45

500

0.005

0.01

0.015

0.02

0.025

0.03

data traffic (Erlangs)voice traffic (Erlangs)

o u

t a g e p r o

b

Fig. 3. Outage probability versus different trafc loads, with a target outageprobability P outage = 2%

offered traffic of data users (Erlangs)

o f f e r e

d t r a

f f i c o

f v o

i c e u s e r s

( E r l a n g s

)

0 5 10 15 20 25 30 350

10

20

30

40

50

60

70

80outage=0.02, data=128 bits/foutage=0.06, data=128 bits/foutage=0.02, data=192 bits/foutage=0.06, data=192 bits/foutage=0.02, data=256 bits/f

outage=0.06, data=256 bits/f

data=128 bits/frame

data=192 bits/frame

data=256 bits/frame

Fig. 4. Two dimensional Erlang capcity region of the system, with a targetblocking probability P blocking = 2%

REFERENCES

[1] Qingwen L., Shengli Z., and Georgious B.: Queuing With Adaptive Modulation and Coding Over Wireless Links: Cross-Layer Analysis and Design , IEEE Transactions on Wireless Communications, Vol.4 Issue.3,pp. 11421153, 2005.

[2] Tarhini, C., and Chahed, T.: System capacity in OFDMA-based WiMAX ,International Conference on Systems and Networks Communication,ICSNC 06, Vol.4 Issue.3, pp. 7074, 2006.

[3] Ding Ling, and Lehnert James S.: Erlang Capacity of a Voice/Data Cellular CDMA Uplink System Using Prioritized Admission Control and Adaptive Power Control , International Journal of Wireless InformationNetworks, Vol.8 Issue.1, pp. 114, 2001.

[4] Villy B. Iversen: Teletrafc Engineering Handbook , COM department,Technical University of Denmark. 2005. 336 pp.

[5] Sarkar, T.K., Zhong J., Kyungjung K., Medouri, A., and Salazar-Palma,

M.: A survey of various propagation models for mobile communication ,IEEE Antennas and Propagation Magazine, Vol.45 Issue.3, pp. 5182,2003.

[6] Viterbi, A.M., and Viterbi, A.J.: Erlang capacity of a power controlled CDMA system , IEEE Journal on Selected Areas in Communications,Vol.11 Issue.6, pp. 892900, 1993.

[7] Hong Shen W., and Moayeri N.: Finite-state Markov channel - a useful model for radio communication channels , IEEE Transactions on Vehic-ular Technology, Vol.44 Issue.1, pp. 163171, 1995.


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erlang capacity of multiclass tdma systems with amc

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