Ordered Packet Length Based Groupwise Transmission Scheme for Rate Scheduling in Burst Switching...

1426 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 4, JULY 2005

Ordered Packet Length Based GroupwiseTransmission Scheme for Rate Scheduling in Burst

Switching DS/CDMA SystemMeejoung Kim, Chung Gu Kang, Member, IEEE, In-Chan Choi, and Ramesh R. Rao, Member, IEEE

Abstract—This paper proposes a new packet rate schedulingscheme for a non-real time data services over the uplink of aburst switching based direct sequence code division multiple ac-cess (DS/CDMA) system to support integrated voice/data services.We consider the most general form of optimization problem for-mulation to determine the optimal number of transmission timegroups along with their data rates, which minimizes the averagepacket transmission delay. An ordered packet length based group-wise transmission (OLGT) scheme is proposed as a simple heuristicsolution to this problem, and we present some analytical results forperformance comparison with other possible schemes.

Index Terms—Direct sequence CDMA, optimal scheduling, or-der statistics, packet scheduling algorithm, radio resource man-agement.

I. INTRODUCTION

THE objective of supporting a reasonable level of qualityof service (QoS) requirements for integrating service in

a CDMA based wireless system in which the scarcity of radiospectrum is the primary concern introduces efficient and fairresource sharing issues. For efficient management of limitedwireless resources, a new concept of medium access control(MAC) protocol called burst switching technique has been in-troduced [1] in which a burst of packets is admitted on the basisof resource availability governed by the interference limitedcharacteristics of the CDMA system and, subsequently, a highspeed traffic channel is reserved for it on a demand basis. Sincea dedicated control channel is allocated to each user during aburst mode, an obvious advantage of burst switching is to reducethe complexity of signaling and the associated overhead for theindividual packet for reservation request as compared with con-ventional packet switching. When a high speed data transfer isrequired, a resource negotiation procedure is executed with theresource request and assignment messages over the dedicated

Manuscript received June 21, 2002; revised March 17, 2004. This researchwas supported by the MIC (Ministry of Information and Communication), Ko-rea, under the ITRC (Information Technology Research Center) support programsupervised by the IITA (Institute of Information Technology Assessment). Thereview of this paper was coordinated by Prof. J. Shea.

M. Kim is with the Research Institute for Information and CommunicationTechnology, Korea University, Seoul, Republic of Korea (e-mail: [email protected]).

C. Kang is with the Department of Radio Communication Engineering, KoreaUniversity, Seoul, Republic of Korea.

I.-C. Choi is with Department of Industrial, Systems and Information Engi-neering, Korea University, Seoul, Republic of Korea.

R. R. Rao is with the Department of Electrical and Computer Engineering,the Irwin and Joan Jacobs School of Engineering, University of California, SanDiego (UCSD).

Digital Object Identifier 10.1109/TVT.2005.851300

control channel. A typical MAC state transition diagram forburst switching is given in Fig. 1. Especially in cdma2000, asupplemental code channel request message contains informa-tion such as message length and pilot strength information forcells in its neighbor list while an assignment message containsthe scheduling parameters, including duration, start time, andrate of packets [2]. This motivates the development of a sched-uler that can enhance the performance of packet services bymeans of proper coordination of scheduling parameters.

Related research deals with rate scheduling schemes fordownlink [3]–[5] and uplink [6]–[9]. A single server modelis considered in [3], where only one user is served for each timeslot. In fact, Earliest Deadline Algorithm (EDA) is applied toassign all the power; i.e., to assign a maximum possible datarate, to the user for which the deadline of the packet at the headof the queue is closest. To maximize the resource utilization, onthe other hand, [4] proposes a Powered Earliest Deadline First(PEDF) scheme in which additional users can be served in theorder of earliest deadline as long as there is remaining power.A similar approach is considered in [5] by introducing a newmetric of dynamic priority, which is quantified as the weighteddeadline normalized by the packet length. The proposed metricindicates that the higher scheduling priority must be given tothe longer packet of the earliest deadline. The above schemesmay be suited to the high speed data service, which allows forthe peak rate transmission of a packet with the highest priority.As long as the interference limited constraint is not violated,additional packets can be served in the order of priority.

For the uplink under the burst switching scheme, multipleburst data requests can be admitted into the system and sched-uled to be transmitted at the same time. To enhance the averagethroughput for data users in a DS/CDMA system for circuitswitching, a rate based grouping transmission scheme is consid-ered in [6]. The transmission power and rate allocation strategythat maximizes non-real time throughput (channel efficiency)subjected to constraints on peak transmission power and totalinterference generated by non-real time sources is investigatedin [7]. In [8], the problem of minimizing the time span tosend all the packets with the effort of resource management isinvestigated. We proposed a new transmission rate schedulingalgorithm in [9], which is called Length based GroupwiseTransmission (LGT) scheme that minimizes the average packettransmission delay while using resource effectively.

Traffic scheduling algorithms are a critical component offuture integrated services packet networks that will provide abroad range of QoS guarantees. The design of a traffic schedule

0018-9545/$20.00 © 2005 IEEE

KIM et al.: ORDERED PACKET LENGTH BASED GROUPWISE TRANSMISSION SCHEME FOR RATE SCHEDULING 1427

Fig. 1. MAC state transition diagram for burst switching.

algorithm involves an inevitable tradeoff between delay, com-plexity of implementation, and fairness. Among the three, delayand implementation complexity are clearly the most importantcriteria for selection of an algorithm for use in a real system.Efficient support of multimedia traffic requires intelligent re-source allocation because the scarcity of radio spectrum is theprimary concern in wireless communication systems.

In this paper, we consider a multi server model for the uplinkburst scheme in which a subgroup of users can be scheduled totransmit over the same interval. Furthermore, their data rates areindividually determined while supporting integrated voice/dataservices. In general, an optimal solution to this problem is notimmediately found due to the interference limited constraintand, furthermore, lack of any simple expression for the aver-age transmission delay. As a heuristic approach, we propose anordered packet length based groupwise transmission (OLGT)scheme which schedules a start time of packet transmission andits data rate so as to minimize average transmission delay andtotal transmission span. While providing a mathematical frame-work to capture the characteristics of optimal packet schedulingproblem, the current work does not intend to conform to anyspecific standard system specification.

II. SYSTEM MODEL AND FORMULATION OF OPTIMAL

SCHEDULING SCHEME

A. System Model

We consider the uplink of a single cell DS/CDMA system witha fixed chip rate and a spreading bandwidth of W that supportsthe integrated services of voice and data traffic. The voice trafficservice is supported by a typical circuit switching scheme witha fixed voice rate of Rv (bps) subject to voice activity detection.The data traffic service is supported by a burst switching schemewith a data rate of Rd (bps), which may vary dynamically de-pending on the interference level caused by the number of activeusers in the system. When a new packet arrives, it waits for thenext scheduling instance at which a maximum allowable trans-mission data rate is determined subject to the multiple accessinterference (MAI)-limited resource constraint. Once the datarate is determined, the corresponding traffic channels are indi-vidually reserved and maintained for the active data users untiltransmission is completed or a new scheduling instance starts.

Since there is a tradeoff between the length of the schedul-ing period and resource waste, the shorter scheduling periodwill lead to a longer delay or the higher packet loss while thelonger scheduling period will incur the high resource waste, thelength of scheduling period would need to be either dynamically

Fig. 2. Burst admission control and scheduling process.

updated. However, the tradeoff between these two aspects is be-yond the scope of this paper, and so we assume that the lengthof the scheduling period is sufficient to guarantee transmittingthe packet over that period. In addition, when a burst arrives atthe system, burst admission control (BAC) is performed so thatthe queueing delay can be negligible. Burst admission controland the scheduling process are described in Fig. 2.

Let Nv and Nd denote the numbers of voice and active datausers in the system, respectively. Since the voice and data ser-vices are subject to different levels of BER requirements toensure their individual service qualities, the required energy perbit per noise power densities, denoted by ηv and ηd, respectively,are individually imposed. Ignoring additional background noiseother than thermal noise and assuming perfect power control, thenormalized power constraint for the user group is representedas follows [10]:

Nv

Wαv Rv ηv

+ 1+

Nd

Wαd Rd ηd

+ 1≤ 1 (1)

where αv is the voice activity factor for voice service subject tocircuit switching. Two terms in (1) represent the power portionsassigned to voice and data users, denoted by ∆v and ∆d, re-spectively. Note that the activity factor of αd = 1 is assumed fordata service, since data traffic channel is continuously reserveduntil the burst transmission is completed.

B. Typical Transmission Scheduling Schemes and Formulationof Optimal Transmission Rate Scheduling Problem Final Stage

In this paper, we consider the scheduling scheme that min-imizes overall transmission delay and total time span. Thereare two types of transmission schemes: one is to serve all theusers at the same time (all-at-once transmission) and the otheris to serve one user at a time (one-at-a-time transmission) asdescribed in Fig. 3.

1) Typical Transmission Scheduling Schemes: For all-at-once transmission, an individual data rate for each user mustbe configured in each transmission group. One special case iscalled a nongrouping transmission (NGT) scheme, which as-signs the same rate that is determined so as to minimize trans-mission delay while satisfying the power constraint (1) to allactive data users in each transmission group. In this case, the


Fig. 3. Types of transmission: All at-once transmission, one-at-a-time trans-mission, and hybrid transmission.

maximum completion time is determined by the longest packetwhich is generated before the scheduling algorithm is applied,and the optimal transmission rate RNGT (bps) is given by

RNGT =W

ηd

1Nd/(1−∆v)− 1

. (2)

On the other hand, it is well known that a shortest processingtime first (SPTF) rule, which serves a user in the order of theshorter processing time, each with a full capacity, is optimal forone-at-a-time transmission type. The optimal transmission rateR (bps) is given by

R =W

ηd

11/(1−∆v)− 1

. (3)

However, this is no longer true when each user is constrainedto a maximal allowable transmission rate RM (bps), which istypically imposed by a physical layer of terminal or time varyingchannel condition. In this case, the optimal transmission ratesin (2) and (3) will be determined by min{RM , RNGT} andmin{RM , R}, respectively. In the case of the SPTF rule, itis probable that extra interference can still be tolerated afterthe maximum rate is assigned, leaving some resources wasted.Meanwhile, the first type of transmission scheme can achieveoptimal resource utilization as long as there are a sufficientlylarge number of active users. Even if it is simple, the NGTscheme never achieves optimal resource utilization, as will beshown later.

2) Formulations of Optimal Scheduling: We conjecture thatthe optimal approach will lie somewhere between the two ex-treme types described above. In other words, there may exist anoptimal number of data users that can transmit at the same timeunder a given performance objective. Toward this end, we con-sider a hybrid type, which combines two types considered aboveand is described in Fig. 3; i.e., active data users are grouped (inorder) in term of a given priority metric and, furthermore, allusers in each group are served simultaneously with specifictransmission rates determined to minimize the transmission de-lay. For simplicity of presentation and analysis, packet length ischosen as a simple priority metric in the following discussion.

Let N be the number of active data users in a group, which isdetermined by transmission priority metric and transmit simul-taneously. The NGT and SPTF can be treated as special cases.The former is the case of N = Nd while the latter is of N = 1.In general, we note that 1 < N < Nd for the proposed hybridscheme.

Denoting the packet length of ith active data users by Li andthe corresponding data rate by R

(d)i , i = 1, . . . , N , two different

types of problem formulation can be stated as follows:

minR∈F0

1N

N∑i=1

(Wi +

Li

R(d)i

)(P0)

minR∈F0

max1≤i≤N

(Wi +

Li

R(d)i

)(P1)

where Wi is the waiting time from the instant of the packetarrival of user i with packet length Li to the instant of thescheduling period begins and the feasible set F0 is given by

F0 =

{R(d) =

(R

(d)1 , R

(d)2 , . . . , R

(d)N

): 0 < R

(d)i ,∆v

+N∑

i=1

Pi ≤ 1, Pi = 1/(

W/

R(d)i ηd + 1

), i = 1, . . . , N

}.

(P0) intends to determine the data rates so as to minimize theaverage transmission delay of all simultaneous packets, while(P1) attempts to minimize the maximum possible completiontime of transmitting all the packets. In this paper, for a closedform solution, we assume Wi = 0 for all i, and thus we havethe following propositions for (P0) and (P1):

Proposition 1: The optimal solution to problem (P0),(R(d)

1 , R(d)2 , . . . , R

(d)N ) satisfies the following conditions:

∆v +N∑

i=1

Pi = 1,L1

P 21

=L2

P 22

= · · · = LN

P 2N

. (4)

Proposition 2: The optimal solution to problem (P1),(R(d)

1 , R(d)2 , . . . , R

(d)N ) satisfies the following conditions:

∆v +N∑

i=1

Pi = 1,L1

R(d)1

=L2

R(d)2

= · · · = LN

R(d)N

.

The implication of Propositions 1 and 2 is that the resource(power) must be fully utilized without any waste. Moreover,


to minimize the average transmission delay, transmission rateshave to be assigned so as to satisfy (4), while the rates have tobe assigned so that the transmission of all the packets may endat the same time to minimize the maximum completion time. Itis intuitive, since otherwise power waste is obvious. Proofs ofPropositions 1 and 2 are presented in Appendices A and B. Sincethe conditions given in Proposition 2 involve solving N th ordernonlinear equations, (P1) does not have a closed form solution,while (P0) does have the following corollary.

Corollary 1: The transmission rate Ri, which is assigned tothe user with packet length Li, and average transmission delayE(D) for the optimal solution to the problem (P0) are as follows:

Ri =(1−∆v)W

√Li

ηd

∑Ni=1

√Li − (1−∆v)

√Li

,

E(D) =ηd

NW

(∑N

i=1

√Li

)2

1−∆v−

N∑i=1

Li

.

Verification of Corollary 1 can be found in Appendix A.Formulations (P0) and (P1) have an optimal solution when all

terminals in the system transmit simultaneously; i.e., N = Nd,which do not represent a general optimal scheduling schemeincluding an SPTF scheme. Therefore, we need the follow-ing generalized optimal scheduling formulation that relaxes theconstraint that transmission of all Nd packets starts at the sametime.

min(t,R)∈F1

max1≤i≤Nd

(Wi + ti +

Li

Ri

)(Q0)

In (Q0), we assume that the packets are indexed by the orderof transmission instances, t1 ≤ · · · ≤ tNd

, without loss of gen-erality; Wi is the same parameter described in (P0)–(P1), andthe feasible set F1 is given by

F1 ={(t,R) = ((t1, . . . , tNd

), (R1, . . . , RNd)) :

t1 ≤ · · · ≤ tNd, Pi = 1

/(W/

R(d)i ηd + 1

), i = 1, . . . , N

}.

In the above expression, Li’s in the formulation are in theorder which is determined by priority metric, nt is the numberof packets that are transmitting at time t, and Pi is the power thatis assigned for ith packet at that time, which is given by Pi =1/(W/Riηd + 1). We note that (Q0) is an NP-hard problem. Todevelop a heuristic solution to (Q0), we first consider a new formof problem formulation (Q1) which considers transmission timegroup:

min(t,R,n)∈F2

max1≤i≤N,1≤j≤n+1

(Wji + tji +

Lji

Rji

)(Q1)

where Wji and tji are the waiting time from the instant ofpacket arrival of the ith packet of length Lji in jth transmissiontime group to the instant the scheduling period begins, and thewaiting time from the instant the scheduling period begins tothe instant that actual packet transmission starts, respectively;

the feasible set F2 is defined by

F2 ={(t,R,(n, r))

=((

t11, . . . , t1N , . . . , tn1, . . . , tnN ,

t(n+1)1, . . . , t(n+1)r

),(

R11, . . . , R1N , . . . , Rn1, . . . , RnN ,

R(n+1)1, . . . , R(n+1)r

),(

N, . . . , N︸︷︷︸n

, r)): tji ≥ 0, tji ≤ t(j+1)i, tji ≤ tj(i+1),

0 < Rji ≤ RM ,Pe ≤ (1−∆v)e, nN + r = Nd,

j = 1, . . . n, , i = 1, . . . , N, r = 0, . . . , N − 1}

where e is N × 1 column vector with all of components isconsisting of 1, and the matrix P, which satisfies the powerconstraints:

∆v +nt∑

i=1

P•i ≤ 1, P•i = 1/(W/R•iηd + 1).

In the above, nt is the number of packets transmitted at timet, and P•i is given as follows:

(P)•i ={

P•i, i = 1, . . . , N0, otherwise

where • denotes an index of the group in which a packet trans-mitted at time t is included. Depending on r, the number oftransmission time groups is either n or (n+ 1). In (Q1), weform the transmission time group, which is defined as a sub-set of packets transmitted in the identically indexed group anddetermined by priority metric. The same number N of packetsis served in all the transmission time groups and the transmis-sion time group means the transmission order; that is to say thepackets in the prior transmission time group must start to trans-mit prior to the packets in next prior transmission time group.Note that formulation (Q0) and (Q1) are equivalent, which isintuitively obvious by establishing one-to-one correspondencebetween two vectors that minimize (Q0) and (Q1). The detail ofthe correspondence is presented in Appendix C. As implied byProposition 2, transmission of all packets in each transmissiontime group must end at the same time for optimal scheduling.Since optimal rates cannot be immediately determined fromProposition 2, however, we resort to the heuristic approach bysolving the following problem:

min(t,R,n)∈F2

1n+ 1

n+1∑j=1

max1≤i≤N

(Wji + tji +

Lji

Rji

)(OPT)

The above formulation is intended to separate the overallminimization problem into many successive minimization sub-problems on a transmission time group basis. In other words,it minimizes a total span of transmitting all the packets for agiven number of transmission time groups while using the op-timal data rate to minimize the average transmission delay ineach transmission time group. In fact, (OPT) is formulated by


carefully combining the aforementioned formulations (P0) and(Q1). While the equivalence of (Q1) and (OPT) is not clear at themoment, it is intuitively true that the successive minimizationof each group along with the minimum delay rate assignmentmay result in a good solution.

III. PROPOSED TRANSMISSION SCHEDULING SCHEMES

In this section, we describe the ordered length based group-wise transmission (OLGT) scheme, which is a heuristic ap-proach to solving the optimal formulation (OPT). In the currentdiscussion, we only consider the packet length as the prior-ity metric and, furthermore, we assume Wi = 0, for all i, formathematical analysis. Even though many other parameters canbe incorporated into the metric, e.g., waiting time, delay con-straint, packet length, and channel state, it will be challengingto determine the transmission delay in an analytic manner.

A simple rate assignment for a subgroup of users that aretransmitting at the same time is considered in [9]. As impliedby Proposition 2, optimal rate configuration is computation-ally involved. Instead, in [9], we considered the length basedgroupwise Transmission (LGT) scheme, which provides a sim-ple means of determining the transmission rates of the indi-vidual users in each transmission time group so as to mini-mize transmission delay. The LGT scheme will be incorporatedas a simple rate scheduling approach into the ordered lengthbased groupwise transmission scheme, which is introducedbelow.

We assume that the packets lengths {Li}Ndi=1 of active data

users are independent and identically distributed with an ex-ponential distribution of parameter µ. Assuming that there areno packets with the same length, we consider the following or-der statistics {L(i)} by arranging the packet lengths {Li} inan ascending order: L(1) < L(2) < · · · < L(k) < L(k+1) < · · ·< L(Nd ). Then, it is well known that the random variablesL(1), L(2) − L(1), . . . , L(Nd ) − L(Nd−1) are independent witha probability density function P (L(k+1) − L(k) = t) = (Nd −k)µe−(Nd−k)µt [11].

Note that with these random variables Li’s the average trans-mission delays ENGT(D) and ESPTF(D) for NGT and SPTFschemes, respectively, are given by

ENGT(D) =1

µRNGT,

ESPTF(D) =1

µR

Nd∑j=1

j−1∑i=0

Nd − j + 1Nd − i

=Nd + 34µR

.

A. Proposed Packet Transmission Scheme: Ordered LengthBased Groupwise Transmission (OLGT) Scheme

The LGT scheme is assigning two different transmissionrates R

(LGT)min and R

(LGT)max simultaneously to all active data

users with packets in Gmin = {L(1), . . . , L(k)} and Gmax ={L(k+1), . . . , L(Nd )}, respectively (N = Nd in (P0); i.e., 1transmission time group). Depending on a specific priority met-ric (packet length in this paper), users are divided into two

different subgroups called rate groups, each of which will beassigned the different data rate so as to minimize transmissiondelay while satisfying interference limited resource constraintsuch as (1).

Ordered length based groupwise transmission (OLGT)scheme is of packet transmission scheduling with several dif-ferent transmission time groups and the order of transmissionof packets therein. In other words, the OLGT scheme deter-mines which transmission time group and which packet in thegroup are transmitted first, while the LGT scheme allows trans-mission of all the packets simultaneously. Depending on thepriority metric, packets are divided into several transmissiontime groups, and users with packets in each transmission timegroup are again divided into two rate groups. Each rate group isassigned the same transmission rate, which is determined by theLGT scheme. Therefore, the objective of the OLGT scheme isto determine the transmission time group and the correspondingrate of each rate group therein, such that overall transmissiondelay and total span time are minimized. There may exist manytransmission time groups transmitted in a specific order, whichincurs a queueing delay for each group. In the sequel, the OLGTscheme is detailed.

Without loss of generality, consider the case of two trans-mission time groups without any constraint on the maximumpossible data rate (Ri < ∞). In other words, data users aredivided into two transmission time groups, and those in eachgroup are again divided into two rate groups following the LGTscheme. For notational simplicity in this case, we define ki,min

and ki,max as follows:

k1,min = �b�Nd/2 , k1,max = �Nd/2 k2,min = k1,max + �b(Nd − �Nd/2 ) , k2,max = Nd

where b = k∗/Nd ∼ 0.54, which is obtained by the LGTscheme. Two different transmission time groups are de-noted by the set of packets with different lengths, G1 ={L(1), . . . , L(k1,max)} and G2 = {L(k1,max+1), . . . L(k2,max)}where the packet lengths are in ascending order. For each Gi, therate groups, Gi,min and Gi,max, i = 1, 2, are given as follows:

G1,min : L(1) < · · · < L(k1,min),

G1,max : L(k1,min+1) < · · · < L(k1,max),

G2,min : L(k1,max+1) < · · · < L(k2,min),

G2,max : L(k2,min+1) < · · · < L(k2,max).

For each data user with a packet in Gi,max and Gi,min, we

assign the transmission rate R(OLGT)max and R

(OLGT)min , respec-

tively. Let∆v,∆min, and∆max be the power portions assignedto Nv voice users, k1,min data users with packets in G1,min, and(k1,max − k1,min) data users with packets in G1,max, respec-tively, satisfying

∆v +∆min +∆max = 1 (5)


Fig. 4. Illustration of transmission time and rate groups.

where ∆min and ∆max are, respectively, given by

∆min =k1,min

W

R(OLGT)min ηd

+ 1and ∆max =

k1,max − k1,min

W

R(OLGT)max ηd

+ 1.

Then the transmission rate R(OLGT)max , assigned to the data

users with packets in G1,max, is obtained by (5). Once the data

transmission rate R(OLGT)min (R(OLGT)

max ) for each user with packetlength in G1,min (G1,max) is determined, the packets in G1

are simultaneously transmitted, and this rate is also applied tosubsequent transmission time groups.

We now consider a scheduling scheme to determine whichpacket is transmitted first in each transmission time group. Tominimize total transmission completion time, the basic idea isto combine the shortest processing time first (SPTF) scheme ineach rate group, and optimality of the equal completion timegiven by Proposition 2. Once all the packet lengths are arrangedin an ascending order, the first transmission time group con-sists of the k1,max shortest packets, which will be divided intotwo rate groups, one with shorter packets and other with longerpackets. Within each rate group of the first transmission timegroup, packets are transmitted in an ascending order of packetlength. In the subsequent transmission time group of next short-est (k2,max − k1,max) packets, however, packet transmission ineach rate group follows the descending order of packet lengths,which makes the maximum transmission completion time ofeach transmission time group as small as possible. This simpleidea forms a basis of our heuristics solution to (OPT) along withthe LGT scheme.

The above rule can be simply extended to the general caseof n > 2. In other words, packets in the odd numbered trans-mission time groups are transmitted in the order of the shorterpacket while, for packets in the even numbered transmissiontime groups are transmitted in the order of the longer packet ineach rate group while applying the same data rate determined bythe the LGT scheme to each rate group. Fig. 4 illustrates an ex-ample with Nd = 6n packets and each transmission time groupwith 6 packets. Fig. 5 illustrates how packets of the differentlengths are arranged with n = 3.

Now consider the case of Ri ≤ RM . As long as thereare a sufficiently large number of data users in the system,R

(OLGT)max = RM will be assigned to �bNOLGT data users while

R(OLGT)min = aRM to (NOLGT − �bNOLGT ) data users, which

will give the minimum transmission delay. In this case, the num-

Fig. 5. Illustration of packet scheduling (ordering).

ber NOLGT of packets in a transmission time group is obtainedby

NOLGT =

(1−∆v)(

WaRM ηd

+ 1)(

WRM ηd

+ 1)

{ab+(1−b)}WaRM ηd

+ 1

(6)

where a ≈ 0.44 and b ≈ 0.54 are derived from the LGT schememinimizing the average transmission delay. Therefore, the op-timal number of transmission time groups is given by n =�Nd/NOLGT .

In the proposed OLGT scheme, the arrival packets have tobe ordered and grouped in advance according to their length,which will cause some overhead. However, since the schedulingproblem is already involved with prohibitively high complexity,the proposed OLGT scheme is still simple enough if it involvessuch overhead.

IV. PERFORMANCE ANALYSIS

A. Analysis of Average Transmission Delay

Case 1: Not considering the maximal allowable transmissionrate: Ri < ∞.

For the case of two transmission time groups (i.e., n = 2),the transmission delay Ti for the group Gi, i = 1, 2, is given by

T1 =k1,min∑j=1

L(j)

R(OLGT)min

+k1,max−k1,min∑

j=1

L(k1,min+j)

R(OLGT)max

,

T2 =k1,min∑j=1

(L(j)

R(OLGT)min

+L(k2,min−j+1)

R(OLGT)min

)

+k1,max−k1,min∑

j=1

(L(k1,min+j)

R(OLGT)max

+L(k2,max−j+1)

R(OLGT)max

).

Since each data user with packet in G2 has to wait to transmitfor the duration of the transmission of each packet in G1 group,T2 contains queueing delay. Therefore the average transmissiondelay EOLGT(D) is given by

EOLGT(D) =1

Ndµ

1

R(OLGT)min

×k1,min∑j=1

2

j−1∑i=0

1Nd − i

+k2,min−j∑

i=0

1Nd − i


+1

R(OLGT)max

k1,max−k1,min∑j=1

2

k1,min+j−1∑i=0

1Nd − i

+k2,max−j∑

i=0

1Nd − i

.

For the case of n different transmission time groups, n =�Nd/k1max , let Tj,min, Tj,max be the transmission delays forrate groups Gj,min, Gj,max of jth transmission time group

Gjwith assignment of rates R(OLGT)min , R

(OLGT)max , respectively,

then they satisfy the following recursive equations

T1,min

=k1,min∑k=1

L(k)

R(OLGT)min

,

Tj,min

=kj,min∑

k=kj−1,max+1

(Tkj,min−k+kj−2,max+1,min +

L(k)

R(OLGT)min

),

T1,max

=k1,max∑

k=k1,min+1

L(k)

R(OLGT)max

,

Tj,max

=kj,max∑

k=kj,min+1

(Tkj+1,min−k+kj−2,max+1,max +

L(k)

R(OLGT)max

),

j ≥ 2.

Therefore, the average transmission delay is given by

EOLGT(D)

=1

Ndµ

1

R(OLGT)min

k1,min∑j=1

{n

j−1∑i=0

1Nd − i

+ (n − 1)k2,min−j∑

i=0

1Nd − i

+ (n − 2)k2,max+j−1∑

i=0

1Nd − i

+ · · ·+kn ,min−j∑

i=0

1Nd − i

+1

R(OLGT)max


n

k1,min+j−1∑i=0

1Nd − i

+ (n − 1)k2,max−j∑

i=0

1Nd − i

+ (n − 2)k3,min+j−1∑

i=0

1Nd − i

+ · · ·

+kn ,max−j∑

i=0

1Nd − i

.

Case 2: Considering the maximal allowable transmissionrate: Ri ≤ RM < ∞.

As in Case 1, the average transmission delay is given as

EOLGT(D)

=1

Ndµ

1

aRM

k1,min∑j=1

{n

j−1∑i=0

1Nd − i

+ (n − 1)k2,min−j∑

i=0

1Nd − i

+ (n − 2)k3,min+j−1∑

i=0

1Nd − i

+ · · ·+kn ,max−j∑

i=0

1Nd − i

+1

RM


n

k1,min+j−1∑i=0

1Nd − i

+ (n − 1)k2,max−j∑

i=0

1Nd − i

+ (n − 2)k3,min+j−1∑

i=0

1Nd − i

+ · · ·+kn ,max−j∑

i=0

1Nd − i

where n = �Nd/NOLGT with NOLGT in (6).

B. Performance Comparison of Different Scheduling Schemes

We compare the performances of different schedulingschemes when Ri < ∞. The following observations are ob-tained by an LGT scheme and an OLGT scheme.

Observation 1: 2R(LGT)min ≤ R

(LGT)max .

Observation 2: nR(LGT)min ≤ R

(OLGT)min and nR

(LGT)max ≤

R(OLGT)max , when OLGT scheme is considered with n different

transmission time groups.Observation 2 is trivial since the interference among users of

the OLGT scheme is reduced by a factor of 1/n as compared tothat of the LGT scheme.

Proposition 3: ESPTF(D) ≤ EOLGT(D) ≤ ELGT(D).Proof of Proposition 3 can be found in Appendix D.Observation 3: For the case of Ri ≤ RM , Proposition 2 does

not hold in general.When there are few data users in the system so that the power

may be sufficient to serve all data users with RM , it is not nec-essary to use the OLGT scheme. For example, when there is onevoice user and 10 data users in the system with RM = 64 kbps,we obtain ELGT(D) = 0.4102 and EOLGT(D) = 0.4598. Asimilar result is shown in Fig. 8.


Fig. 6. Performance of the different scheduling schemes (Ri < ∞).

Fig. 7. Performance of the different scheduling schemes (Ri ≤ RM ).

V. NUMERICAL RESULTS

We first present the numerical results for the proposedrate scheduling scheme with the following system parameters:W = 5 MHz, αv = 3/8, Rv = 9.6 kbps, ηv = 7 dB, ηd = 10dB, and 1/µ = 25000 bits. In Fig. 6, we compare the aver-age packet transmission delay performance of the different ratescheduling schemes with two transmission time groups for theOLGT scheme without any rate constraint (i.e., Ri < ∞). Asexpected, the SPTF scheme shows the best delay performance,since a single server model achieves the least MAI, and thus itprovides the best possible throughout without any limit on thedata rate. In other words, it may play a role in the referenceperformance, which can be considered the best possible perfor-mance. The LGT and OLGT schemes are always better thanthe NGT scheme; moreover, the OLGT scheme is significantlybetter than the LGT scheme. The average delay performance ofthe LGT and OLGT schemes outperform the NGT scheme by asmuch as approximately 11%–14% and 60%–70%,respectively.As expected, the packet transmission delay increases as the num-ber of voice and data users increases. Furthermore, performanceimprovement of the OLGT scheme becomes more significant asthe number of data users increases.

Fig. 8. Performance of the different scheduling schemes (Ri < ∞).

Fig. 7 compares the average transmission delays of four differ-ent rate scheduling schemes with two transmission time groupsfor the OLGT scheme with data users only, under the givenconstraint on the maximum possible data rate (i.e., Ri ≤ RM ).For the cases of RM = 64 kbps and RM = 144 kbps, the LGTand OLGT schemes are better than the SPTF scheme. This isattributed to the fact that the maximum rate constraint resultsin the remaining power in the SPTF scheme. In the case ofRM = 384 kbps, however, which is sufficiently large to providethe optimal data rate, the SPTF scheme is always better than theother two schemes while the OLGT scheme is better than theLGT scheme. In other words, performance of the SPTF schemeis strictly governed by the maximum data rate constraint.

Fig. 8 shows the performance of LGT and OLGT, with twotransmission time groups for the OLGT scheme, compared withthe lower bound that is obtained by Corollary 1. Obviously thelower bound is the best, which is approved by Proposition 1,when all the active users are transmitting simultaneously. Asexpected, the OLGT scheme shows performance improvementcompare to the LGT and NGT schemes. Furthermore, the packettransmission delay increases as the number of voice and datausers increases, and performance improvement of the OLGTscheme becomes more significant as the number of data usersincreases.

Fig. 9 compares the SPTF and OLGT schemes with the opti-mal number of transmission time groups for the different trafficmixes with RM = 384 kbps. In these cases, regardless of thenumber of concurrent voice and data users, the OLGT schemeis always better than the SPTF scheme. Table I summarizes theresults of optimal transmission time grouping for the varyingnumber of voice and data users with the given maximum trans-mission rate constraint RM and furthermore, provides someperformance comparison of OLGT vs. SPTF for the differenttraffic mixes. It is obvious from these results that there existssome region in favor of the OLGT scheme, which justifies ourgeneralized formulation of the rate scheduling problem.

Fig. 10 shows the average transmission delays and vari-ance of the transmission completion time for the case ofRM = 384 kbps. In particular, variance of the transmission


Fig. 9. Performance with the different traffic mixes (RM = 384 kbps).

TABLE IOPTIMAL TRANSMISSION TIME GROUPING AND PERFORMANCE COMPARISON

Fig. 10. Average transmission delay and variance of transmission completiontimes (RM = 384 kbps, 1/µ = 25 000).

completion time indicates robustness against the optimality ofthe equal completion as implied by Proposition 2. As the num-ber of voice and data users increases, variance of SPTF steadilyincreases while that of OLGT does not. It implies that the opti-mality of the equal completion time can be rather consistentlyreached by the proposed scheme as designed.

VI. CONCLUSION

It has been shown that the OLGT scheme is more effectivethan any other scheduling scheme for a sufficiently large num-ber of data users, regardless of the maximum allowable datatransmission rate. Compared to the optimal scheduling schemethat cannot be readily solved, the proposed scheme still guaran-tees reasonable performance, while providing a computationallytractable framework of optimal resource management for burstswitching CDMA networks. Due to the oversimplified systemmodels and assumptions, however, this work cannot be imme-diately applied to the currently available standardized system.Moreover, as the physical layer constrained data rate is one of themost important factors that governs overall performance, it willbe essential to evaluate the current scheme under a more realisticsituation, especially under a time varying channel condition.

APPENDIX A

PROOF OF PROPOSITION 1

To find the optimum value of (P0) while assigning transmis-sion rate to N data users simultaneously, we find Pi’s that satisfythe following equations:

∆v + P1 + · · ·+ PN = 1, R(d)i = PiW/{ηd(1− Pi)}.

(A1)Using R

(d)1 , we rewrite transmission delay by

D =1N

N∑i=1

Li

Ri=1N

N∑i=1

ηdLi(1− Pi)PiW

.

By using the Lagrange multiplier method, for Lagrange mul-tiplier λ, we have

∂D

∂Pi= − ηd

NW

Li

P 2i

= λ, i = 1, . . . , N.

Therefore, we obtained the following result:

L1

P 21

= · · · = LN

P 2N

. (A2)

Proof of Corollary 1: By (A1) and (A2) the transmissionpower and rate assigned to the user with packet length Li aregiven by

Pi = (1−∆v)√

Li

/N∑

j=1

√Lj ,

Ri =(1−∆v)W

√Li

ηd

(∑Ni=1

√Li − (1−∆v)

√Li

) , i = 1, . . . , N.

(A3)

Therefore, we have the average transmission delay as follows:

E(D) =ηd

NW

(∑N

i=1

√Li

)2

1−∆v−

N∑i=1

Li

.


APPENDIX B


To prove Proposition 2, we need the following two lemmas.Lemma 1: For the optimal solution to (P1), the equality must

holds in (1).Proof: Let R(1) = (R(1)

1 , . . . , R(1)N ) be the transmission vec-

tor for (P1). Assume that the minimum value is found with thepacket with length Lk such thatminR∈F0 max1≤i≤N (Li/R

(1)i )

= Lk/R(1)k . If the remained power is not fully used, Lk/(R(1)

k +δ) < Lk/R

(1)k holds for some δ > 0. Then R(2) = (R(1)

1 , . . . ,

R(1)k + δ, . . . , R

(1)N ) is the vector that solves (P1). This contra-

dicts to our assumption.Lemma 2: Let R(1),R(2) ∈ F0 be the transmission rate vec-

tors that are fully using the power while satisfying the followingrelation:

max1≤i≤N

(Li

/R

(1)i

)≤ max

1≤i≤N

(Li

/R

(2)i

).

Then VarR(1)(Li/R(1)i ) ≤ VarR(2)(Li/R

(2)i ) holds and the

reverse is also true.Proof: For the optimal solution to (P1), it is obvious that

the higher transmission rate has to be assigned to the packetwith longer length. Let Lk be the length of the longest packet.Let R

(1)k and R

(2)k be the components of the transmission

vectors R(1) and R(2), respectively, which are assigned thelongest packet. Then by the hypothesismax1≤i≤N (Li/R

(1)i ) ≤

max1≤i≤N (Li/R(2)i ), we have R

(1)k ≥ R

(2)k . Since R(1) and

R(2) are using the resource fully, there exists j such that R(1)j ≤

R(2)j . Therefore, we obtain Lj/R

(2)j ≤ Lj/R

(1)j ≤ Lk/R

(1)k ≤

Lk/R(2)k and thus, VarR(1)(Li/R

(1)i ) ≤ VarR(2)(Li/R

(2)i ).

Conversely, if we assume VarR(1)(Li/R(1)i ) ≤ VarR(2)

(Li/R(2)i ), i.e., the degree of disperse of the transmission com-

pletion time by using R(1) is smaller than that of by using R(2),then it implies the higher power has to be assigned to the longestpacket in the components of R(1) as compared to that of R(2).Therefore, the longest transmission time obtained by R(2) islarger than that obtained by R(1).

Lemma 2 implies that the transmission rate vector that solves(P1) is the one that minimize the variance of the completiontimes of packet transmission.

By Lemma 2, it is sufficient to induce a relation that minimizesthe variance of the completion times of packet transmission:

minR∈F0

Var(Li

/R

(d)i

)= min

R∈F0

E{

Li

/R

(d)i − E

(Li

/R

(d)i

)}2

= 0.

This equation holds when Li/R(d)i = E(Li/R

(d)i ) for all i.

Therefore, (P1) has an optimal solution for the transmissionvector R = (R(d)

1 , . . . , R(d)N ) that satisfies

L1

/R

(d)1 = L2

/R

(d)2 = · · · = LN

/R

(d)N = E

(Li

/R

(d)i

).

APPENDIX C

PROOF OF THE EQUIVALENCE OF (Q0) AND (Q1)

Since we are assuming the number of voice users is invariantduring the scheduling period, we may assume that the resourcethat can be used by data users is also invariant during the period.

Case 1. Not considering the maximal allowable transmissionrate: Ri < ∞

In this case, the SPTF rule will be the best scheme andt = (t1 . . . , tNd

) in (Q0) satisfies t1 < · · · < tNd. For t =

(t11, . . . , tNd 1) in (Q1), let t11 = t1, . . . , tNd 1 = tNd, Ri =

Rj1, all of the resource that can be used by data users, and(n, r) be (1, . . . , 1︸︷︷︸

Nd

).

Case 2. Not considering the maximal allowable transmissionrate: Ri ≤ RM < ∞

Since RM is imposed in the system, first of all, determine thenumber N of packets that can be transmitted simultaneously.

a) Let (t,R) be the vector minimizing (Q0) which isrepresented by

t1 = · · · = tN = 0, tN+1 = min1≤i≤N

{Li/RM}

tkN+j = min1≤i≤N

{Li/RM : 0 �= Li/RM �= tm,

m = 1, . . . , kN + j − 1}

and Ri = RM , for all i = 1, . . . , Nd. Then the cor-responding vector (t,R, (n, r)) in (Q1) consists ofthe following factors:

t =((t11 = t1, . . . , t1N = tN ), . . . ,(tn1 = t(n−1)N+1, . . . , tnN = t(n−1)N+N

),(

t(n+1)1 = tnN+1, . . . , t(n+1)r = tnN+r

)),

Ri = RM , (n, r) = (N, . . . , N︸︷︷︸n

, r).

It is obvious that this vector solves (Q1).b) Conversely, to find the vector that solves (Q1), we

determine the number of packets that are transmit-ting simultaneously at the beginning of schedulingperiod, just as the same in a), and N be the num-ber again. Let (t,R,(n, r)) be the vector that solves(Q1) and given by

(t,R,(n, r)) =((

t11, . . . , t1N , . . . , tn1, . . . ,

tnN , t(n+1)1, . . . , t(n+1)r

),

Rji = RM ∀j, i, (N, . . . , N, r)).

Then it is noticed that the packets in a transmission time groupdo not start transmission at the same instant since the completiontime of each packet in a group is different. By considering thereverse process of a), we can find a vector (t,R) for (Q0) asfollows:

t(j−1)N+i = tji, R(j−1)N+i = Rji = RM ∀j, i.


Obviously it is an element of F1 and furthermore it solves(Q0).

APPENDIX D


Since it is the most efficient to assign all the power to only oneuser when there is no power limit to the user, the first inequalityis trivial. To show the second inequality, we assume two differenttransmission time groups for the OLGT scheme for simplicity.Then the average transmission delay is as follows:

EOLGT(D)

=1

Ndµ

2

R(OLGT)min

k1,min∑j=1

j−1∑i=0

1Nd − i

+1

R(OLGT)min

k2,min∑j=k1,max+1

j−1∑i=0

1Nd − i

+2

R(OLGT)max

×k1,max∑

j=k1,min+1

j−1∑i=0

1Nd − i

+1

R(OLGT)max

×k2,max∑

j=k2,min+1

j−1∑i=0

1Nd − i

≤ 1Ndµ

2

2R(LGT)min

k1,min∑j=1

j−1∑i=0

1Nd − i

+2

2R(LGT)min

×k1,max∑

j=k1,min+1

j−1∑i=0

1Nd − i

+2

2R(LGT)max

×k1,max∑

j=k1,min+1

j−1∑i=0

1Nd − i

+1

2R(LGT)max

k2,max∑j=k2,min+1

j−1∑i=0

1Nd − i

≤ 1Ndµ

1

R(LGT)min

k1,max∑j=1

j−1∑i=0

1Nd − i

+1

R(LGT)max

k2,max∑j=k1,max+1

j−1∑i=0

1Nd − i

= ELGT(D).

The inequalities come from Observations 1 and 2 in SectionIV, and �Nd/2 < �bNd , which is the result of LGT.

REFERENCES

[1] M. O. Sunay, S. Tekinay, and S. Z. Ozer, “Efficient allocation of radioresources for CDMA based wireless packet data systems,” in Proc. IEEEGlobecom., vol. 1, Dec. 1999, pp. 638–643.

[2] C.2005-A, Mar. 2000.

[3] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, and P. Whiting,“Data rate scheduling algorithms and capacity estimates for the CDMAforward link,” Bell Labs. Tech. J., Sep. 13, 1999.

[4] A. C. Varsou, H. C. Huang, and L. Mailaender, “Rate scheduling forCDMA downlink mixed traffic networks,” in Proc. WCNC, vol. 1, 2000,pp. 370–375.

[5] A. C. Varsou and H. V. Poor, “HOLPRO: A new rate scheduling algorithmfor the downlink of CDMA networks,” in Proc. Veh. Technol. Conf., vol. 2,Fall 2000, pp. 948–954.

[6] H. Kang and K. Kim, “Throughput enhancement scheme for integratedvoice/data DS-CDMA system: Rated-based grouping transmission,” Elec-tron. Lett., vol. 35, no. 17, pp. 1437–1438, Aug. 1999.

[7] S.-J. Oh and K. M. Wasserman, “Optimality of greedy power control andvariable spreading gain in multi-class CDMA mobile networks,” in Proc.ACM/IEEE MobiCom., 1999, pp. 102–112.

[8] R. Jantti and S.-L. Kim, “Transmission rate scheduling for the non-real-time data in a cellular CDMA system,” IEEE Commun. Lett., vol. 5, no. 5,May 2001.

[9] M. Kim, C. G. Kang, and R. R. Rao, “Group-wise transmission ratescheduling scheme for integrated voice/data service in burst-switchingDS/CDMA system,” in IEICE Trans. Commun., vol. E85-B, no. 8, Aug.2002.

[10] T. Ito et al., “A wireless packet transmission with adaptive process-ing gain and transmitter power control scheme for circuit-switched andpacket-switched modes integrated DS/CDMA system,” in Proc. IEEE Veh.Technol. Conf., 1999, pp. 2039–2043.

[11] W. Feller, An Introduction to Probability Theory and Its Applications,2nd ed., New York: Wiley, 1971, vol. II.

Meejoung Kim received the B.S. degree in mathe-matic from the Korea University, Seoul, Republic ofKorea, in 1986, and the M.S. and Ph.D. degrees inmathematics from the Korea University, Universityof Minnesota, Twin Cities, and the Korea Universityin 1988, 1993, and 1996, respectively.

In 1996, she was a Research Fellow and Lecturerin the Department of Mathematics, and from 2000to August 2004 she worked as a Research Fellowand Assistant Professor in Brain Korea 21 Informa-tion Technology. Since September 2004, she has been

with the Research Institute for Information and Communication Technology atthe Korea University, Seoul, Republic of Korea, as an Assistant Professor. Sheteaches mathematics, and her research interests include wireless multimediacommunication systems, especially resource management and medium accesscontrol, wireless security, and white noise analysis.

Chung Gu Kang (M’96) received the B.S. degreein electrical engineering from the University of Cal-ifornia, San Diego in 1987, and the M.S. and Ph.D.degrees in electrical and computer engineering fromthe University of California, Irvine, in 1989 and 1993,respectively. While working on his Ph.D. dissertationfrom June 1991 to May 1992, he was also with theAerospace Corporation in El Segundo, California, asa part time Member of Technical Staff.

In 1993, he joined Rockwell International Inc. inAnaheim, CA. Since March 1994, he has been with

the Department of Radio Communication Engineering at the Korea University,Seoul, Republic of Korea, where he is currently a Full Professor. In the aca-demic year 2000–2001, he was a Visiting Associate Professor at the Universityof California, San Diego, where he was also affiliated with the Center for Wire-less Communications. His recent research is focused on MIMO/multiple accessschemes for mobile broadband wireless access systems and MAC/routing pro-tocols for mobile ad hoc networks. He has over 100 refereed publications ininternational journals and conferences in the areas of communications network,CDMA cellular systems, OFDM systems, and wireless local area/personal area


networks. He has been a consultant to wireless industries, including cellular ser-vice and content providers. He serves as an Editor of JCN (Journal of Communi-cation and Network) and as a technical program committee member of variousinternational conferences. He has also been involved with various standardiza-tion activities, including ITU-R JRG 8A-9B. He is currently a Vice-Chairmanof the 2.3 GHz Portable Internet Project Group (PG305), and a Chairman ofthe PG305-WG2 service/network working group in the TelecommunicationsTechnology Association of Korea.

Dr. Kang is a member of IEEE COMSOC, IT, and VT, and is a member ofKICS and KITE, currently serving as a Chairman of the KICS Mobile Commu-nication Technical Activity Group.

In-Chan Choi received the B.S. degree from KoreaUniversity, in 1982 and Iowa State University, Ames,in 1984 and the M.S. and Ph.D. degrees in operationsresearch from Columbia University, New York, NYin 1986 and 1990, respectively.

He is a Full Professor in the Department of Indus-trial, Systems and Information Engineering at KoreaUniversity. Prior to joining Korea University in 1996,he held a tenure track faculty position in the Depart-ment of Industrial Engineering at Wichita State Uni-versity, KS, and was employed as a Member of the

Technical Research Staff at Bell Communications Research (Bellcore). His pri-mary research interests are in optimization theory, including the developmentof algorithms and the study of polyhedral structures of optimization models,mainly for linear and discrete optimization models. He is also interested in themodeling and application of optimization theory to industries, specifically in

transportation, telecommunications, finance, and scheduling problems. His re-search has appeared in a wide range of professional journals.

Prof. Choi is a Member of honor societies Omega Rho (operations research),Pi Mu Epsilon (mathematical honor society), and Alpha Pi Mu (industrial en-gineering).

Ramesh R. Rao (M’85–SM’90–M’03) was born inSindri, India. He received the B.E. (Honors) degreein electronics and communications at the RegionalEngineering College of the University of Madras,Tiruchirapalli, India in 1980. He received the M.S.and Ph.D. degrees at the University of Maryland,College Park.

He is currently a Professor at the University ofCalifornia, San Diego (UCSD) in the Department ofElectrical and Computer Engineering, Irwin and JoanJacobs School of Engineering, where he has been a

member of the faculty since 1984. He is the former Director of the UCSDCenter for Wireless Communications (CWC), and is currently the Director ofthe San Diego Division of the California Institute of Telecommunications andInformation Technology [Cal-(IT)2], where he leads several interdisciplinaryand collaborative projects. His research interests include architectures, proto-cols and performance analysis of computer and communication networks, andhe has published extensively on these topics. Since 1984, he has authored nu-merous technical papers, contributed book chapters, conducted short courses,and delivered invited talks and plenary lectures. He is currently supervising bothmasters and doctoral students.

Prof. Rao was recently appointed the first holder of the Qualcomm EndowedChair in Telecommunications and Information Technologies.

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