Connection-based Cross-layer Design in Wireless

Cellular NetworksJie Chen, Minjian Zhao and Shiju Li

Department of Information Science and Electronic EngineeringZhejiang University

Hangzhou, 310027, ChinaChenjieO422 @yahoo.com.cn

Abstract- Making efficient use of network resources is ofgreat importance in wireless networks. In this paper, we proposea cross-layer algorithm to minimize the energy consumptionand provide QoS guarantee jointly for various connections withdifferent QoS requirements in wireless cellular networks. Weformulate an admission control problem and a multiple objectiveprogramming problem which allocates the rate and networkresource for each connection in the network simultaneously. Wederive efficient solutions to the two problems which attain anefficient usage of the transmission powers, and guarantee therate and delay requirements of every admitted connection inthe network. The simulation results show that our algorithmmakes use of the network resource more effectively than layeredapproaches and our algorithm can be implemented in variouswireless cellular networks with fast convergence.

I. INTRODUCTION

Cross-layer design methods can couple different layers inthe network protocol and optimize the performance of thenetwork. In wireless networks, the channel quality of everylink is time varying due to fading, noise, interference andshadowing, and the energy of the nodes is constrained. Thetime-varying link quality allows opportunistic usage of thechannel by cross-layer design, which means the transmissionparameters can be dynamically adjusted according to thevariations in the channel[l].

Researches in [2]-[7] give various proposals to allocate theresources in wireless network by cross-layer design method.Approach [4] formulates the resource allocation problem as aconvex optimization problem and solves it by Lagrangian du-ality [11]. Approaches [2] [3] [5] adopt similar method in theirrespective models. Approach [2] also provides a distributedalgorithm and analyses the throughput loss.

Meanwhile, supporting different kinds of services withdifferent quality of service (QoS) requirements such as real-time video streams, online games, file transfer services anddistributed computing services is an important design goal inwireless cellular networks. The traditional layered approachesin [8] [9] are mainly concerned about the queuing policysuch as WFQ and do not use the knowledge of the channelstatus. Although these algorithms can also guarantee the QoSrequirements in the network, they don't make full use thevaluable network resources.On the other hand, the approaches in [4][5] can also be

used as the resource allocation algorithm in wireless cellular

networks, but they are regardless of the requests on applicationlayer and try to calculate the optimal resource allocation inthe network just according to the channel status. Moreover,the models in these works only support one connection oneach link and do not support QoS guarantee. However, in acellular network such as the WiMAX system[13], there willbe different kinds of connections on each link to guaranteethe QoS requirements of different applications from the upperlayers. So these algorithms are not fit for such complexconditions. Approach [7] guarantees the transmission rate andbit error rate requests in its model, but does not guarantee thedelay requests.

In this paper, we propose a cross-layer approach to makeefficient use of the network resources and provide QoS guar-antee for all the connections in wireless cellular networks. Theconnections on each link can have different QoS requirements.We first construct the model of a wireless cellular network

with various connections on different links. Then we formulatean admission control problem and a multiple objective pro-gramming problem to allocates the rate and network resourcesfor each connection in the network simultaneously. And wedecompose the latter problem into two sub-problems: a rateallocation problem and a network resource allocation problem.We derive efficient solutions to the sub-problems. Finally, wepresent the performance of our algorithm and compare it withthe performance of other layered approaches by simulation.

In the next section, we present the model of the cellularnetwork and the formulated problems. In section III,we introduce our algorithm. In section IV, we show thesimulation results of the proposed algorithm. And section Vis the conclusions.

II. SYSTEM MODEL

Consider a cellular network with a base station (BS) and Nsubscribe stations (SS). Each SS can only communicate withthe BS and the BS can communicate with all the SS. The BScan establish an uplink and a downlink with each SS. So thereare N uplinks and N downlinks in all.The total bandwidth ina cellular network is W(Hz).

Proceedings of the 2007 15th IEEE Workshop on Local and Metropolitan Area Networks1-4244-11 00-9/07/$25.00 ©2007 IEEE 123

A. Connection modelEach link can maintain several connections with different

QoS requirements. We classify these connections into threeclasses, including constant bit rate connections (symbolizedby b), real time connections (symbolized by ,o) and non-realtime connections (symbolized by w).

The constant bit rate connections support real time datastreams consisting of fixed length data packets such as VOIP.The real time connections support real time data streamsconsisting of variable-sized data packets such as MPEG videostream. The non-real time connections support delay tolerantdata streams consisting of variable-sized data packets withminimum data rate requirements, such as FTP.

All the uplinks are labeled i = 1, . . ., N, and the downlinksare labeled i = N + 1, . . ., 2N. The connections are labeled1. The BS makes the scheduling periodically and the timeinterval is T(in second). The scheduling intervals are labeledby integer n. In every scheduling interval n, li(n) connectionsrequire to transmit data on each link i. Each connection 1has a data rate requirement a-i(in bps) and each real timeconnection has a delay requirement Mil T, 1 E ',, where milis a positive integer. Each connection has a weight eil(n). Inour model, we assume that ei,l¢ > eij,l, > ei,le,w. And for eachlink i, the price of unit data transmission is Ki(n). In thispaper, we assume that links with worse channel conditionshave higher Ki(n). So the links with better channel status willtransmit more data. This is just the meaning of opportunistictransmission [2].

B. Channel modelWe assume that the network is under additive white Gaus-

sian noise (AWGN) and shadowing propagation model. We useadaptive M-QAM modulation and assume all the BS and SSin the network have the ability of power control. The approachin [10] gives the relationship between the SNR and the BERwhen using M-QAM modulation. So we can get the capacityof every connection (in bps) by

cil(n) = wil(n) 10g2(1 + 215pi(n)Gi(n) ) (1)ln(5Fi)(I1(n) + wil(n) o-a(n))

And the throughput(in bit) of every connection in eachscheduling interval is

bil(n) = til(n) cil(n) (2)

where for each link i, pi(n)(in watt) is the transmission powerof the transmitter and is constrained by pmax G. is the channelgain, o-2(n)(in watt/Hz) is the terminal noise density, Ii(n)(inwatt) is the interference power from other links. Gi(n), oi2(n)and Ii(n) can be derived from the channel estimation modulein the physical layer. Here we assume that the channel statusremains unchanged in each scheduling interval n. Fi is the bitrate error (BER) demand. wil(n)(in Hz) and til(n)(in second)are the bandwidth and transmission time arranged for each

connection 1 on link i in every scheduling interval n. Noticethat several variables are suffixed by (n), which means thesevariables may vary in different scheduling intervals.

For all the connections in the network, til(n) and wil(n) arelimited by Ei El>til(n) wil(n) < TW. This can be regardedas the network resource constraint for each connectionand can be simplified according to the access mode of thenetwork. For example, in TDMA mode, the resource for eachconnection is only the til and wil = W, Vi, 1, so the resourceconstraint can be rewritten as Ei El til < T.

C. Problem Description

As we want to guarantee the QoS requirements of all theconnections in the network, there should be an admissioncontrol module on the BS to decide whether a new connectioncan be admitted to the network without harming the QoS re-quirements of all the admitted connections. Here we symbolizethe new connections by q and admitted connections by i. Andwe can formulate the admission control problem as:

Max: >E >E eil(n)ai77dil(n)i l1e

s.t.: ZZE di(n) iii< cth(n) - ai-, Vii l1e i Ie1

(3)

where dil(n) is the result of admission control, dil(n) = 1 meansthe new connection is admitted to the network, dil(n) = 0otherwise. cth(n)(in bps) is the threshold of admission control.And in this paper, we choose cth(n) as the total channelcapacity when all the links in the network transmit at thethreshold power level pmax 0,0 < 0 < 1. 0 is the powerreservation ratio. And when 0 = 1, we define congestionthreshold bth(n)(in bit) as the total throughput in the networkof interval n. So if the total throughput of all the links exceedbth(n) in scheduling interval n, the network will congest.

Each admitted connection manages its own queue withlength qil(n)(in bit). In every scheduling interval n, ail(n)bitsdata enter each queue with mean rate ail, while ril bits dataleaving each queue, transmitting to the other nodes. Noticethat ail(n) is constant for 0, and is stochastic for 'o and w.So in the latter case, ail is the mean value of ail(n) in severalintervals.

For 'o, we set a time-stamp for the incoming ail(n)bits data ineach interval with a initial value mil which is the connection'sdelay requirement. After each scheduling interval, the time-stamp minus one. So we should send the data out before itstime-stamp is less than zero to guarantee the delay require-ments. And we define rtil(n)(in bit) as the amount of datawith zero time-stamp in interval n. For 0, rtil(n) = ail. And forw, to guarantee their minimal data rate requirements, we shallalso give them a large time-stamp and guarantee the data ratein the long term.We exhibit all the variables in our model as well as their

units and definitions in table 1. The meaning of all thevariables can be found in this section.

124

TABLE I

THE DEFINITIONS AND UNITS OF ALL THIE VARIABLES IN OUR MODEL

We formulate the problem (4)-(9) to allocate the rate andnetwork resources for each connection in the cellular simul-taneously. And the results can guarantee different QoS re-

quirements of every admitted connection while optimizing theresource allocation in the network. We adopt utility functionlog(ril) here to ensure the fairness among all the admittedconnections.

Max: eil(n) log(ril(n)) (4)i lepUSOuw

- EKi(n) *E ril(n)ilepufcuw

s.t. : ril(n) < qil(n), Ii,cE 0U Uw (5)

ril(n) > rtil(n), Vi, 1 E U io U (6)

Ei E ril(n) < bth(n), Vi (7)i lcoUSouw

Ei E til(n) *wil(n) < TrW,Vi (8)i lcouSouc

pi(n) < pmax, Vi (9)The objective function (4) maximizes the difference

between the total utility of different kinds of connections andthe expense of transmission in the network. The economicsmeaning of (4) can be the profit of the transmission in thenetwork. It also ensures that the most important connectionscan transmit more data while not starving the non-real timeconnections by the utility function. The constraint (5) ensures

that the data to be sent does not exceed the queue size, so

no meaningless data will be transmitted in the network. (6)ensures the real time requirements of 0 and 'o as well as theminimal data requirements of (7) ensures the network willnot congest, which we have discussed above. (8) ensures thatthe network resource is enough as discussed in subsectionB. (9) ensures the transmission power of every link doesnot exceed its constraint. By solving the multiple objectiveprogramming problem (4)-(9), we want to get ril(n), pi(n),wil(n) and til(n) in each scheduling interval n.

III. PROPOSED ALGORITHM

In section II, we have formulated an admission controlproblem (3) and a multiple objective programming problem(4)-(9). The latter problem has multiple objectives includingril(n), til(n), wil(n) and pi(n). It is difficult to solve the prob-lem directly. So we decompose it into two sub-problems as

(11)(12). And we also rewrite the admission control problem(3) as sub-problem (10).

Max: eil(n)adidil(n)i l1e

st:E E di,(n)ai < Clh(n) - ail, Vile i lej

Max: (ei,(n) log(ril(n)) - Ki(n) * ril(n))i lefcuw

s.t. : ril(n) < qil(n), vi, 1 E io U

ril(n) > rtil(n), Vi, I E io U Xi

ril(n) < bth(n) - ril(n), Vii lefcuw i /eG

ril(n) = ail(n) , Vi, 1 E 0

(10)

(1 1)

(12)Min: ( E til(n)) Pi(n)i lcouSouw

s.t. : Yiril(n)= Li bil(n), ViI lleupu

Y til(n) wil(n) < TW, Vi, 1 0 UiO U (

i I

Pi <p max, Vi

A. Description of the sub-problems

With the knowledges of channel status Gi, -i Ii and prede-fined 0, we can decide cth. When we get cth and requirementsail eil from the application layer, we can decide which ofthe new connections can be admitted to the network by (10).

125

|Name(Unit) Definition

W(Hz) total bandwidth of the network

T(S) the scheduling interval lengththe index of the links in the network

the index of the connections on each link

Ki the price of unit data transmission

cil(bps) the capacity on each connection

bil(bit) the throughput on each connection

pi(watt) the transmission power on each link

p'ax(watt) the transmission power threshold

Gi the channel gain on each link

o:r(watt/Hz) the terminal noise density on each link

Ii(watt) the interference power on each link

til(s) the transmission time allocated for each connection

wil(Hz) the transmission bandwidth allocated for each connection

Cth(bps) the threshold of admission control0 the power reservation ratio

bth(bit) the threshold of congestionn the index of scheduling interval

di, the admission control decision

qil(bit) the queue length of each connection

ail(bit) the amount of data entering each queue of the connec-tions

ril(bit) the amount of data leaving each queue of the connections

rtil(bit) the amount of data should be transmitted in each schedul-ing interval

aji(bps) the data rate requirement of each connection

Mil the delay requirement of each connection

eil the weight of each connection

Among the new connections i, we admit the connections withhigher weights eil(n) first. And the admitted connections havehigher priority to reserve their connections.

All the connections in (4)-(9) have been admitted to thenetwork based on the results of (10). We shall allocate thetransmission rate for them first. Notice that the transmissionrate is allocated per interval, so the unit of it is bit. Forthe admitted constant bit rate connections, we can allocatethem with ril(n) = ail T directly. For real time connections'o and non-real time connections Co, we use the objectivefunction in (11) to allocate the rate. Each connection mustbe allocated more than rtil(n) bits per interval to guarantee thedelay requirements of 'o and minimal rate requirements of Co.Otherwise, a QoS violation occurs. And the fairness among allthe connections with different weights are guaranteed by theutility function log(.). So we can see that the objective functionand the last constraint of sub-problem (11) correspond to theobjective function (4), and the first three constrains of (11)correspond to the constrains (5)-(7).

For a cross layer approach, it has the flexibility to access

with the physical layer, and achieve performance gain over

the whole network. So with ril(n) from (11), we can go on

to calculate the transmission time til(n) or bandwidth wil(n)for each connection, and the transmission power for each linkpi(n) by (12). For the same amount of data, different networkresource allocations incur different energy consumption. Soin this paper, we minimize the energy consumption in thenetwork. Notice that the results of (12) are related to the access

mode as discussed in section II. So the solution to (12) isdifferent according to the access mode.

B. Solution to Sub-Problems

Sub-problem (10) can be solved by greedy algorithm. Wecan sort eil(n), and admit the connections with higher weightuntil the total transmission rate requests of the admittedconnections approaches the admission threshold cth(n). And as

mentioned in section II, we reserve some transmission power

to avoid QoS violation. This method is more efficient thanreserving the time slots or bandwidth, which will be shownby the simulation results in the next section.

(11) is a convex optimization problem. We can solve it byLagrangian duality[ll ]. So we introduce Lagrangian multipli-ers uil, vil, z, and get ril(n) by solving the following Lagrangiandual function:

Min: fi (u, v, z) (eil log(ril(uil, vil, z)) -

i I

(Ki + uil - vil + z) ril(uil, vil, z) + (uilqil - vilrtil)

+z(bth - >E ail)li I

uil(Ij + 1) = max([ui(II) - 61 (qil -

Ki + uil(Ij) - vil(I1) + z(II)' (15)

vil(Il + 1) = max([vil(II) - 1

edi(II) tl ,0 I6

Ki + uil(Ij) - vi1(I1) + z(II) rti1)], 0) (16)

z(Il + 1) = max([z(II) - 61 (bth - E ail -i I

(17)Ki + uil(Ih) - vii(I1) + Z(Il)

The solution to (12) is related to the access mode, and we

only give the solution in TDMA mode here. In TDMA mode,(12) can be rewritten as:

Min: Z( til(n)) Pi(n)i /epUSoUw

s.t.: ril(n) =( til(n)) W log2(1 +

I lcouSouwpi(n) * hi(n)), Vi,

til(n) < T,

i I

Pi < pi(n)na, Vi,hi(n)=

~ - 1 .5Gi(n)log(5Fi)(Ii(n) + W 2(n))

We can solve the following formula:

E-' /EI ril(n)

iY W ln(I + A hi(n))T

(18)

(19)

and get til(n) and pi(n) by:

pi(n) = min([(l + hi(n). A)'n(2) - 1]/hi(n), piax)

til (n)ril(n)

(=W . 1g2(l + pi(n) * hi(n))

(20)

(21)

Formula (19) can also be solved by Newton's method as

follows:

(13)

(14)s.t. : ril (uil, vil, z) = eilKi + Uil -Vil + Z

(13) can be solved by Newton's method[II] as follows:

126

[nput: qjj,rt,j,Kj, bth and ail at scheduling interval n.Step 1: Initialize uil, vil and z and step length 61; set iteration numbei

II = 0 in equations (15)-(17).Step 2: Update uil, vil, z by equation (15)-(17), and increase II by 1.Step 3: Calculate fi(Ii) in (13). If fi( fif('i 1) > A1, goto step 2.Wherefi (II)

A1 is the convergence threshold.Step 4: Get uil, vil, z and iteration number II; Calculate ril(n) by (14).

Input: hj,rjj,T and W at scheduling interval n.

Step 1: Initialize A and step length 62; set iteration number I2 = 0 inequations (22)(23).

Step 2: Update A by equation (23), and increase I2 by 1.Step 3: Calculate f2(I2) in (22). If f2(I 2) f2(I2 1) > A2, goto step 2.Wheref2(I2)

A2 is the convergence threshold.Step 4: Get A and iteration number I2.

50

45

40 ---:-- +1

, 30E

z

Z 25

a 20

15 -

10

0.1 0.2 0.3 0.4 0.5 0.6 0.7Power Reservation Ratio

0.8 0.9

Fig. 1. QoS violation probability under different power reservation ratio

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Power Reservation Ratio

0.9

Fig. 2. Iteration number under different power reservation ratio

25r

(22)f2(A(12)) =(4 W* ln(I+ r(I2) hi)

A(I2 + 1) = A(I2) - f2(1(2))i [ln(l+A(I2) hi)]2(1+A(I2) hi)

IV. SIMULATION RESULTS

To demonstrate the benefits of the proposed algorithm, we

consider a wireless cellular network with one BS and ten SS.The access mode of the network is TDMA. The SS are placedin a circle area with radius lkm and the BS is in the center.The bandwidth W is 4MHz. The channel gain Gi = K* Si(d°where K = 1, do = 500m, a = 4, di is the length of link i andSi follows log-normal distribution with mean 0 and variance6dB. oQ2 is 10-l'w/Hz. pm' is 0.2w and Fi is 10-6.The scheduling interval T = lOms. In every interval, each

SS and BS can generate different kinds of connections. For 0,ail(n) is constant, and for 'o and w, ail(n) follows exponentialdistribution with mean ail(n). aii(n) distributed randomly fromlOOkbps to 1Mbps.We simulated 20 times. The power reservation ratio in

admission control process increased 0.05 each time, from0.05 to 1. And each simulation lasted 20000 intervals. Ineach simulation, the traffic in the network was heavy andthe sum of data rate requests always exceeded the admissionthreshold cth(n). So our algorithm admitted the most importantconnections and arranged data rate, power and transmissiontime for them according to the network status.

Under each power reservation ratio, we got the QoS viola-tion probability, average iteration number, average power andaverage throughput during 20000 intervals.

Figure 1 shows the QoS violation probability under differentpower reservation ratio. A QoS violation means that ril(n) isless than rtil(n) in any scheduling interval. This is possible

Cross layer design methodLayer design method with time slot reservationLayer design method with power reservation

20 _

(23);> 15 _

0:5

0.02 0.04 0.06 0.08Average Transmission Power(w)

0.1 0.12

Fig. 3. Performance comparison between cross layer and layered designmethod

because aij,ieuj(n) follows exponential distribution and isvariable. So when

rtil(n) bthi leGUepuw

happens, (11) will have no answer and the QoS requirementsof some connections have to be violated. We can see that whenpower reservation ratio is more than 0.6, the QoS violationprobability increases significantly. This is because when thepower reservation ratio is 0.6, there is no more than 20% ofthe total throughput in scheduling interval n reserved in corre-

spondence to ease the effects of the variance of ajji,Euj(n). Andwhen the power reservation ratio exceeds 0.6, the percentagebecomes more lower. Under such condition, not all the QoSrequirements of the connections can be guaranteed. So we

can choose a proper power reservation ratio in the regionbetween 0.3 and 0.6 to guarantee the QoS requirements ofall connections in the network according to figure 1.

Figure 2 shows that the iteration number I, in (15)-17)

127

0.2

0.18 _

0.16

0.14

, 0.12

012

o 0.1 _

.LD

>n 0.08 _

a0.06

0.04 _

0.02 _

OL_

0

increases basically with the power reservation ratio, and theiteration number I2 in equations (22)(23) decreases with thepower reservation ratio. This is because with the increment ofthe power reservation ratio, the the gap between rtil(n) andqil(n) increases, so I, increases. Meanwhile, the total rate oneach link becomes more balanced, so I2 decreases.

It is easy to see that the total iteration number I, + I2 isquite small when the power reservation ratio is between 0.3and 0.6, while this region can provide strong QoS guaranteeas shown in figure 1. So our algorithm can be implementedin realistic systems with fast convergence.

Figure 3 shows the performance comparison between crosslayer and layered design methods.The layered design methods we simulated here was in the

same simulation environment. The access mode of the networkis also TDMA. The main difference between the cross layerdesign and the layered design methods is that the MAC layercan cooperate with the physical layer for our cross layer designmethod. Here we introduced two admission control strategies:the power reservation and the time slot reservation. For theformer, it was the same as what we had used in our model,and the nodes could control their transmission powers either.For the latter, the nodes could only transmit with constantpower and they had to reserve some time slots in admissioncontrol. We had tried different time slot reservation ratios, andwhen the ratio was less than 0.8, QoS requirements could beguaranteed. So we chose 0.8 as the time slot reservation ratioand changed the transmission power of all the nodes in thenetwork in contrast with the change of power reservation ratio.

After each simulation, we sum the total throughput(in bit)and energy consumption(in joule) in the network during the20000 time intervals, and got the average throughput(in bps)and average energy consumption(in watt) for each test case.

In figure 3, the average throughput under different averagetransmission power is shown. We can see that in unit time, withthe same total transmission energy in the network, our crosslayer design method improves the throughput by 5% and 15%compared with the layered methods with power reservationand time slot reservation separately.

This is because for the cross layer method, the MAC layerdecide the rate allocated for each connection according tothe status of the physical layer, so connections with betterchannel conditions will transmit more data. And for thelayered methods with power reservation, it can only allocatethe rate for each connection according to the requirements ofthe application layer. So it is possible that the connections withworse channel conditions have to transfer more data as theirweights are higher. This means the cross layer design methodscan exploit the diversity gain of different links in the network.We shall also notice that the diversity gain is related to thevariety of channel status on each link.

For the layered methods with time slot reservation, itsperformance is the worst because the energy is linear withthe transmission power and time, while the throughput is linearwith the transmission time and is convex with the transmissionpower. So with the ability of power control, our cross layer

design method can provide more throughput with the sameenergy consumption.And we can conclude that our algorithm allocate the

resource in the wireless network more efficiently.

V. CONCLUSION

We have presented a cross-layer approach to the QoSguarantee and network resource allocation problem inwireless cellular network. Our cross-layer algorithm couplesthe physic layer, the media access control (MAC) layer,the application layer and does opportunistic transmissionscheduling for different connections in the network. It isremarkable that the total iteration number of our algorithmis small while the algorithm provides global optimality ofresource allocation among the whole cellular. So the algorithmcan be implemented in various wireless cellular systems suchas the WiMAX system[13]. The simulation results showthat under proper power reservation ratio, our algorithm canprovide strong QoS guarantee. And our algorithm makes useof the network resource more efficiently than layered designmethods.

REFERENCES

[1] Srivastava.V. and Motani.M., "Cross-layer design: a survey and the roadahead", Communications Magazine, IEEE, vol.43, Issue 12, pp.112-119,Dec.2005

[2] Lin.X.,Shroff.N.B. and Srikant.R., "A Tutorial on Cross-Layer Optimiza-tion in Wireless Networks", Selected Areas in Communications,IEEEJournal on, vol.24,Issue 8, pp.1452-1463, Aug.2006

[3] S.Toumpis and A.J.Goldsmith, "Performance, optimization, and cross-layer design of media access protocols for wireless Ad hoc networks,"IEEE ICC, vol.3, pp.2234-2240, May 2003

[4] Lin Xiao,Johansson.M. and Boyd.S.P., "Simultaneous routing and re-source allocation via dual decomposition", Communications, IEEE Trans-actions on, vol.52,Issue 7, pp.1136-1144, July 2004

[5] Johansson.M. and Xiao.L., "Cross-layer optimization of wireless net-works using nonlinear column generation," Wireless Commun, IEEETrans, vol.5,Issue 2, pp.435-445, Feb.2006

[6] Neely.M.J., Modiano.E. and Chih-Ping Li, "Fairness and optimal stochas-tic control for heterogeneous networks", INFOCOM 2005. ProceedingsIEEE, vol.3, pp.1723-1734, March 2005

[7] Kozat U.C., Koutsopoulos I. and Tassiulas L., "A framework for cross-layer design of energy-efficient communication with QoS provisioningin multi-hop wireless networks", INFOCOM 2004, vol.2, pp.1446-1456,March 2004

[8] Hawa M., Petr D.W., "Quality of service scheduling in cable andbroadband wireless access systems", Quality ofService, 2002. Tenth IEEEInternational Workshop on, pp.247-255, May 2002

[9] Kitti Wongthavarawat and Aura Ganz, "Packet scheduling for QoS sup-port in IEEE 802.16 broadband wireless access systems", InternationalJournal Of Communication Systems, vol.16, pp.81-96, Feb.2003

[10] A.J.Goldsmith and Soon-Ghee Chua, "Variable-rate variable-powerMQAM for fading channels", Communications, IEEE Transactions on,vol.45,Issue 10, pp.1218-1230, Oct.1997

[11] S.Boyd and L.Vandenberghe, (2002) Convex Optimization [Online],Availale: http://www.stanford.edu/ boyd/cvxbook/

[12] Tingfang Ji, Chao.H.J., Nan Feng, "Wireless coexistence:Pareto opti-mality", GLOBECOM '04. IEEE, vol.6, pp.3803-3807, Dec.2004

[13] IEEE 802.16-2004, "IEEE Standard for Local and Metropolitan AreaNetworks Part 16: Air Interface for Fixed Broadband Wireless AccessSystem", 2004

128