multicast beam forming data

8/3/2019 Multicast Beam Forming Data

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Wireless Data Multicasting with Switched

Beamforming Antennas

Honghai Zhang*, Yuanxi Jiang*, Karthik Sundaresan*, Sampath Rangarajan*, Baohua Zhao

*Mobile Communications and Networking Research, NEC Laboratories AmericaDept. of Computer Science and Technology, University of Science and Technology of China

AbstractUsing beamforming antennas to improve

wireless multicast transmissions has received considerable

attention recently. The work in [20] proposes to partition

all single-lobe beams into groups and to form composite

multi-lobe beam patterns to transmit multicast traffic.

Depending on how the power is split among the individual

beams constituting a composite beam pattern, two power

models are considered: (i) equal power split (EQP), and(ii) asymmetric power split (ASP).

This work revisits the key challenge - beam partitioning

in the beamforming-multicast problem considered in [20]

and makes significant progress in both algorithmic and

analytic aspects of the problem. Under EQP, we propose

a low-complexity optimal algorithm based on dynamic

programming. Under ASP, we prove that it is NP-hard

to have (32

)-approximation algorithm for any > 0.For discrete rate functions under ASP, we develop an

APTAS, an asymptotic (32

+ )-approximation solution(where 0 depends on the wireless technology), andan asymptotic 2-approximation solution to the problem

by relating the problem to a generalized version of the

bin-packing problem. For continuous rate functions under

ASP, we develop sufficient conditions under which the

optimal number of composite beams is 1, K, and arbitrary,respectively, where K is the total number of single-lobe beams. Both experimental results and simulations

based on real-world channel measurements corroborate

our analytical results by showing significant improvement

compared to state of the art algorithms.

I. INTRODUCTION

Designing efficient link layer multicast solutions is

becoming increasingly important in data disseminationfor group communications (such as mobile TV, Sports

Telecast, Video Teleconference, etc.). While the shared,

broadcast nature of the wireless medium provides natural

support for wireless multicast services, the multicast

transmission rate is limited by the client with the worst

channel conditions in the group (e.g., [3]). Beamforming

antennas, by virtue of their ability to focus energy in a

specific direction, provide a natural solution to improve

the received signal strength at the weakest client, which

can potentially improve multicast performance.

However, it is challenging to apply beamforming

technologies to multicast transmissions because of the

inherent tradeoff between multicasting and beamform-

ing. While beamforming increases the signal energy in

a particular direction, it also reduces the energy in otherdirections, thereby restricting the wireless broadcast

advantage, which is a key component in multicasting.

Recently, Sen et al. [17] considered the problem of

integrating multicast with beamforming and proposed to

transmit with an omni-directional beam first, followed

by one or several sequential single-lobe transmissions.

Several other works [1], [12], [20] pointed out that

in a strong line-of-sight (LOS) environment, such as

indoor channels at 60 GHz or outdoor wireless systems,

the beamforming gain is significant, especially when

the number of antenna elements is large (e.g., Fidelity

Comtech [6] provides antennas with eight elements and

a typical 60 GHz system allows 32-64 antenna elements

[9], [23]). With a large number of antenna elements in

a LOS environment, the antenna can form very narrow

single-lobe beams that can be roughly viewed as non-

overlapping [20]. In other words, the received energy

at any location from one particular single-lobe beam

dominates that from all other single-lobe beams.

Under such a context, Sundaresan et al. [20] showed

that composite multi-lobe beam patterns are needed to

address the multicast-beamforming tradeoff efficiently.

[20] formulated the problem as minimizing the aggregatetransmission time in disseminating a common message.

This is achieved by partitioning single-lobe beams into

multiple groups and forming a composite beam for

each group with sequential transmission on each of the

composite beams. Depending on how the power is split

among the individual beams constituting a composite

beam pattern, two models are considered: (i) equal power

split (EQP), and (ii) asymmetric power split (ASP). The


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key challenge is how to partition the beams into groups.

Several algorithms are reported in [20] to address the

challenge.

In this work, we revisit the key challenge of beam

partitioning considered in [20] and make significant

progress on this problem. We accomplish this through

rigorous complexity analysis and designing new low-

complexity algorithms with performance guarantees for

both EQP and ASP models. Our main contributions can

be summarized as follows.

Under the EQP model, we provide a low-complexity, dynamic-programming-based optimal

solution for both continuous and discrete rate func-

tions. The complexity of our algorithm is O(K2),in contrast to the O(K7) complexity of the optimalsolution in [20], where K is the total number ofnon-overlapping single-lobe beams.

Under the harder ASP model, currently there exists

no hardness results or approximation solutions. Wepresent several key results in this context.

1. We prove that it is NP-hard to have (32 )-approximation solution for a general rate function

for any > 0.2. For discrete rate functions, we show that multicast-

beamforming problem can be converted to a gen-

eralized version of the bin-packing problem. This

allows us to leverage and apply generalized-bin-

packing algorithms to obtain an APTAS (Asymp-

totically Polynomial Time Approximation Scheme)1

as well as an asymptotic (3

2 + )-approximationsolution for the multicast-beamforming problem,where 0 depends on the discrete rate functionused by the wireless technology. We also develop

a novel asymptotic 2-approximation solution that

applies to all discrete rate functions. In retrospect,

this also yields an asymptotic 2-approximation

solution for the generalized bin-packing problem,

which is of independent interest.

3. For continuous rate functions, we derive generic

sufficient conditions for the rate functions under

which it is optimal to have (i) one, (ii) K, and (iii)

arbitrary number of composite beams, respectively.In particular, we show that if the rate is a non-

decreasing concave function of SNR, it is optimalto have one composite beam containing all single-

lobe beams, which coincides with the result in

[20] for the special case of Shannon-capacity rate

1For the definition of APTAS and approximation algorithms, please

refer to Appendix A and [15].

function.

To corroborate the theoretical analysis, we evaluate

the algorithms based on real traces from signal mea-

surements of an eight-element phased array antenna

in an outdoor testbed, as well as real-world outdoor

experiments. Comprehensive evaluations indicate that the

proposed algorithms significantly improve the state of

the art in literature. The multicast delay reduction for

802.11a and 802.11b is up to 20% and 25%, respectively,

compared to the algorithms in [20] under the ASP model.

The rest of the paper is organized as follows. In

Section II, we discuss the background and the optimiza-

tion framework. The proposed algorithms are presented

in Sections III and IV for EQP and ASP models,

respectively. Evaluation of the proposed solutions based

on real-world traces is presented in Section V and the

experimental results are presented in VI. We discuss the

related work in Section VII, followed by conclusion in

Section VIII.

II . BACKGROUND AND OPTIMIZATION FRAMEWORK

A. Background and motivation

A smart antenna system combines multiple antenna

elements in an array with signal processing capability

to optimize its transmission and/or reception pattern. In

a beamforming antenna system, each antenna element

can be pre-coded with a complex weight, forming a

beam pattern, such that the total energy of all beams

along a certain direction in the physical or signal space

is maximized. Beamforming can be either adaptive, or

switched. The former generates the pre-coding weights

dynamically based on the receiver channel conditions

and the latter provides a set of pre-computed beams to

be used at any time instant. Although adaptive beam-

forming provides better antenna gain, it also requires

sophisticated signal processing capability and complex-

valued channel feedback. On the other hand, switched

beamforming achieves better performance-complexity

tradeoff and is thus considered in this work.

In a switched beamforming system, the antenna

typically provides a set of K single-lobe beam patterns

of degree 360/K covering the entire azimuth of 360owhere K is the number of antenna elements in the array.In environments with strong LOS (line-of-sight) such as

in-door channels at 60 GHz or out-door wireless systems,

at any given client location, the power received from one

single-lobe beam (that is closest to the line-of-sight to the

client) typically dominates that from all other single-lobe

beams. This work considers such line-of-sight scenarios

and assumes that the received energy from all other

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single-lobe beams are negligible compared to that from

the beam with the strongest signal.

Under this assumption, Sundaresan et al. [20] pro-

posed to partition the single-lobe beams into groups, with

each group of beams forming a composite beam, and to

transmit on each composite beam. The key challenge

is then to determine the optimal partitioning of beams

into groups. Several algorithms were presented in [20] to

solve the problem. This work makes significant progress

over [20] on both the algorithmic and analytic aspects of

the beam partitioning problem. While this work focuses

on the non-overlapping beam patterns, we consider the

overlapping beam patterns for video delivery in a parallel

work [25].

B. Network model

We consider a single-cell environment where an AP

with a smart antenna system serves multiple clients using

link layer multicast. All clients measure the SNR valuesfrom each beam and feedback the best SNR and the

index of the best beam back to the AP. Assume that yiis the SNR value of client i when it is served by the APwith a single best beam pattern and bi is the best beamof client i. Let Ck denote all clients who are best servedby beam k, i.e., Ck = {i : bi = k}. Define the effectiveSNR of a beam k as k = min{yi : i Ck} representingthe minimum SNR among all clients served by beam k.When the AP transmits at a rate that corresponds to kusing a single beam pattern k, all clients associated withbeam k can decode the packet. Note that the usage of

all K beams (at appropriate rate) will cover all clients.

C. Optimization framework

We denote R() as a general non-decreasing ratefunction of an SNR value . Assume that there aretotally K switched beams and N users in the system,and the multicast data size is L bytes. The objective isto partition the beams into G groups and transmit L bytessequentially on each beam group, such that the aggregate

transmission delay to deliver L bytes to all clients isminimized. Assuming that there is a switching delay Wfor each transmission to a beam group, the objective can

be written as

min

Gg=1

(W +L

R(effg )) (1)

where effg is the effective SNR value of group g (tobe decided). When the AP transmits on a group of

beams simultaneously, the transmit power on each beam

decreases because the net power is split among multiple

beams. Depending on how the power is split among

multiple beams in a given group, two possible models

are considered (as in [20]): EQP (EQual Power) model

and ASP (ASymmetric Power) model.

III. EQP MODEL

A. Problem formulationUnder the EQP model, power is equally split among

multiple beams. While this method of power allocation

is not optimal, it is a simple, yet reasonable choice. Let

Bg denote the set of beams of group g. Due to equalpower splitting, the SNR value of each beam in group greduces by a factor of |Bg|. In order that all clients servedby a group of beams can decode the packet, the AP

transmits at a rate corresponding to the lowest SNR value

among all beams in the group. Therefore, the effective

SNR value for the group g under EQP model is effg =

minkBg{k}/|Bg|. Now the objective in Eq. (12) canbe written as

min

Gg=1

(W +L

R(minkBg{k}/|Bg|)). (2)

The number of partitions G and the partitions Bg(g =1, , G) are the variables to be optimized.

Sundaresan et al. [20] showed that this problem under

the specific Shannon capacity rate function (i.e., R() =log(1+ )) can be solved, albeit at a high complexity ofO(K7). In the following, we develop an optimal solution

to problem (2) via dynamic programming with a lowcomplexity of O(K2) and it is applicable to any non-decreasing rate function R().

B. Optimality

Denote T(P) as the total transmission time of partitionP. If P contains only one group Bg, its transmission time(including delay between switching beams) is

T(Bg) = W +L

R(minkBg{k}/|Bg|). (3)

Assume that the rate function R() is non-decreasingwith respect to the SNR value . Let be a list of allbeams sorted in the decreasing (or increasing) order of

their effective SNR (i.e., k). Now, we can establish thefollowing lemma.

Lemma 1: If Pnc is a non-contiguous partition of resulting in a total transmission time T(Pnc ), thereexists a contiguous partition Pc of such that T(P

c)

T(Pnc ).

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This lemma generalizes the Lemma 1 in [20] where

the Shannon capacity rate function is assumed. The proof

is similar to that in [20] and is omitted.

This lemma reduces problem (2) to a contiguous

partition problem, where we first sort the beams in

the decreasing or increasing order of their effective

SNR (i.e., k), and then divide the ordered set intogroups consisting of contiguous elements so as to

minimize the total transmission time. We will now

show that the optimal solution to our problem can be

constructed recursively from the optimal solutions to its

sub-problems, thereby enabling a dynamic-programming

based solution.

Theorem 1: Let be a list of all the beams sortedin the decreasing (or increasing) order of their effective

SNR. If P is an optimal contiguous partition of (i.e., with the minimum transmission time) and P =(P\BG , BG), where BG is the last (most recent) group

of beams determined, then P\BG is an optimal partitionfor the set of beams in \BG (which denotes the set ofbeams in but not in BG).

Proof: We will prove this by contradiction. If P\BGis not an optimal partition for the set of beams in \BG,we can construct an optimal partition P\BG for it.

Denote the new partition of as P = (P\BG

, BG).

Now the total transmission time of P is

T(P) = T(P\BG

) + T(BG)< T(P\BG) + T(BG) = T(P).

This leads to a contradiction, given that P is an optimalpartition.

C. Optimal Dynamic Programming Solution: DP-EQP

Based on the optimality principle in Theorem 1,

we design the following dynamic-programming-based

approach to compute the optimal solution. Assume that

= (1, 2, , K) is the complete list of beamssorted in the decreasing order of their effective SNRs.

Denote Sk as the total transmission time of the optimalpartition of the first k beams in . From Theorem 1, Skcan be recursively computed as

Sk = min1jk

(Sj1 + T({j , , k})) (4)

where T({j+1, , k}) is the transmission delay ofthe last group and is calculated using Eq. (3). Since the

beams are sorted in the decreasing order of their SNRs,

the computation can be simplified as

T({j+1, , k}) = W +L

R(k/(k j)).

The initial condition is

S1 = T({1}) (5)

The complete algorithm (DP-EQP) based on dynamic-

programming is illustrated in Algorithm 1.

Algorithm 1 DP-EQP:Dynamic-programming-based al-

gorithm for EQP

1: Sort the beams in the decreasing order of

their SNR. Denote the resulting permutation as

(1, 2, , K).2: Compute S1 using Eq. (5)3: for k = 2 to K do4: Compute Sk using the recursive equation (4).5: end for

6: Return the optimal multicast transmission time SKand the optimal partition.

Complexity of DP-EQP: Step 1 of the algorithm in-

volves sorting and can be computed with O(Klog K)complexity, while steps 3-5 require O(K2) complexity.Therefore, the total complexity of the algorithm DP-EQP

is O(K2).Remarks: 1) In order to return the optimal beam parti-

tioning, DP-EQP needs to keep track of the intermediate

optimal state at each step 4, or use back-tracking after

obtaining the optimal cost. Both approaches are standard

methods in dynamic-programming and are omitted. 2)

DP-EQP can be applied to arbitrary rate functions R,

including both continuous and discrete functions, as longas they are non-decreasing with respect to the SNR

values.

IV. ASP MODEL

A. Problem formulation

Under the ASP model, power can be optimally

allocated among multiple beams in a group such that

the minimum SNR value across all beams in the group is

maximized. In [20], the authors showed that the optimal

power allocation to beam k within a group is given byk

= k

, where

=1kBg

1k

(6)

is the effective SNR (i.e., effg ) of group g. Under thismodel, the objective in (12) can be written as

minGg=1

(W +L

R(1/kBg

1k

)). (7)

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Again, the optimization variables are the number of

groups G and the set of beams (i.e., Bg) in each groupg.

B. Hardness of the Problem

We first prove that problem (7) is NP-hard for a

general rate function.Theorem 2: There is no approximation algorithm

with a guarantee of 3/2 - for Problem (7) for > 0unless P = NP.

Proof: It was shown in [21] that it is NP-hard to

have a (32 )-approximation algorithm for the bin-packing problem. We reduce the bin-packing problem

to a special case of problem (7).

Bin packing problem: Given n items with sizes s1, s2, sn (0, 1], find a packing in unit-sized bins thatminimizes the number of bins used.

Consider the following special case of problem (7).

There are n beams with the effective SNR of beam ibeing i = 1/si. Further, let L = 1 and the switchingdelay W = 0. Let the rate function be

R() =

1, 10, otherwise

(8)

First note that the optimal solution to this problem takes

a finite value because if we let each beam occupy one

group, the resulting partition has a finite cost. Therefore,

for each group g in the optimal partition, its effectiveSNR is at least 1 (so that the resulting cost is finite).

We next establish a one-to-one relationship between

a solution for the bin-packing problem and that for our

beam partitioning problem. For each non-empty bin Bjin the bin-packing solution, we can construct a group

of beams in our problem, where each beam i in thegroup corresponds to the item i in the bin. Since thetotal size of all items in a bin cannot exceed 1, i.e.,iBj

si 1, this indicates that the effective SNRof the corresponding beam group (which is equal to

1/(iBj

1/i) = 1/(iBj

si)) is greater than one.As a result, the cost of the corresponding beam group

is 1 from Eq. (8). Therefore, the objective value of anyfinite solution to our problem is equal to the number

of non-empty bins in the bin-packing solution. Clearly,

the mapping between the bin-packing solution and the

solution to the beam partition problem is one-to-one

and the transformation can be done in polynomial time,

which completes the proof.

We next consider two cases depending on whether the

rate function R() is discrete or continuous.

C. Discrete Rate Function

In this subsection, we consider problem (7) withdiscrete rate functions. As the general beam multicastingproblem is NP-hard, we turn to approximation solutionsto solve the problem. A general discrete rate functionR() can be represented using a step function as,

R() =Rm if m < m+1, for 1 m < M

RM if M. (9)

where m typically represents a Modulation CodingScheme (MCS), and m, Rm are the SNR threshold andthe transmission rate with MCS mode m. We assumethat the effective SNR (k) of each beam k is at least 1(otherwise, some users under beam k cannot be servedwith any MCS by any beam, and have to be dropped

from consideration).

For such discrete rate functions, we can convert the

beam partition problem into the following generalized-

cost variable-sized bin packing (GCVS-BP) problem [7].

Mapping to Bin Packing: GCVS-BP problem: We are

given L types of bins, each with infinite supply. A bin oftype l has size 0 < bl 1 and cost cl > 0. Without lossof generality, we assume that b1 b2 bL. Itemsof sizes in (0, b1] are to be partitioned into J subsets.Each subset j of items are then packed into a bin oftype lj such that the size of the bin type lj is at least aslarge as the total size of all items in the subset j. Thetotal cost of the packing is

Jj=1 clj . The goal is to find

a feasible packing such that the total cost is minimized.

We next convert beam partitioning problem (7) under

ASP to the GCVS-BP problem. For each beam patterni with effective SNR i, we construct an item with size1/i. For each MCS mode m, we create a bin typewith size bm = 1/m and cost cm = W +

LRm

. There

are totally M types of bins. Each group Bg of beamscorresponds to a bin. If the effective SNR (defined inEq. (6)) of the group of beams satisfies m(g), itcan choose transmission rate Rm(g), which correspondsto packing the items derived from Bg to a bin of typem with a cost of W + LRm(g) (as m(g) implies

iBg1i

1/m(g) = bm(g), corresponding to the bin-

size constraint). The resulting total cost of all bins isGg=1

W +

L

Rm(g)

,

which is exactly the aggregate transmission delay. Table

I shows the mapping from the beam multicast problem

to the GCVS-BP problem.

With the above mapping, we can apply algorithms

developed for the GCVS-BP problem to solve the beam

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TABLE I

MAPPING BETWEEN A BEAM MULTICAST PROBLEM TO A

GCVS-BP PROBLEM

Variables in beam multicast Variables in GCVS-BP

beam SNR i object size si = 1/iSNR threshold m of MCS m bin size bm = 1/m

Rate Rm of MCS m bin cost cm = W+ L/RmTotal delay

PGg=1(W+

LRm(g)

) Total costPGg=1(W+

LRm(g)

)

Algorithm 2 General algorithm for beam multicasting

1: Convert the beam multicast problem (7) to a GCVS-

BP problem with item sizes 1/i, i = 1, , N,bin sizes 1/m, m = 1, , M and bin costs W +LRm

, m = 1, , M.2: Apply a generalized bin-packing algorithm to solve

the converted GCVS-BP problem.

3: Map the items in each bin in the bin-packing solution

to a group of beams for simultaneous transmission

to obtain a beam multicast solution.

partitioning problem for multicast. Algorithm 2 shows a

framework for beam multicasting based on a generalized

bin-packing algorithm.

While the bin-packing problem and even the variable-

sized bin-packing problem are well studied, the study

of GCVS-BP is very limited. In the following, we first

discuss an APTAS algorithm in [7] for the GCVS-

BP problem, which is of very high complexity. We

then show that the IFFD algorithm in [11], which was

designed for a special class of GCVS-BP problems, canbe applied under relaxed conditions to obtain a weaker

asymptotic approximation factor of (1.5+). Finally, wedevelop a new algorithm that solves the general GCVS-

BP problem and achieves asymptotic 2-approximation

guarantee.

Epstein and Levin [7] provided an APTAS (Asymp-

totic Polynomial Time Approximation Scheme) for

a generalized bin-packing problem, which achieves

asymptotic performance ratio (1 + ) in polynomial timefor any > 0. Therefore, if we apply this APTAS scheme

in the framework Algorithm 2, we obtain an APTASsolution for the beam-multicast problem, as captured by

the following corollary.

Corollary 1: There exists an APTAS scheme for the

general ASP problem (7).

However, the solution in [7] requires high complexity

(which is exponential in 1/, representing the tradeoffbetween sub-optimality and complexity). Besides the

high complexity, the procedure in [7] is very involved

and not amenable to implementation. Hence, more

efficient solutions are desired.

An Asymptotic (32 + )-approximation Algorithm: Kangand Park [11] studied the GCVS-BP problem with

variable cost functions satisfying

ci

bi

cj

bjfor any bi < bj , (10)

and proposed an algorithm with asymptotic performance

ratio of 32 . They show that their proposed algorithm

obtains a cost which is less than 32C(B) + c1, where

C(B) is the optimal cost and c1 is the cost of thelargest bin if the problem satisfies the requirement in Eq.

(10). For the sake of completeness, we list the algorithm

(IFFD ) in [11] in Algorithm 3.

Algorithm 3 IFFD

1: Assume that the bins are sorted in the decreasing

order of their sizes.2: Allocate all the items into bins of type 1 using

the first-fit decreasing manner. Denote the resulting

sets of bins as B1 = {B11 , B12 , , B

1k1

} where B1irepresents the set of items packed in the ith bin.Denote C1 as the total cost of B1. Let l = 1.

3: while l < m and max{wj : j Blkl

} bl+1 do4: Allocate all the items in Blkl to bins of type l + 1

using the first-fit decreasing manner.

5: l = l +1; Let Cl =l1i=1 C(B

i{Biki})+C(Bl)

6: end while

7: let i = arg min1il

Ci.8: Repack each bin in i1i=1 (B

i Biki) Bi with the

cheapest bin that can contain all items in the original

bin. Let B denote the resulting solution.

We next examine whether the cost functions for

802.11a and 802.11b satisfy the required conditions in

Eq. (10) to help us leverage the asymptotic guarantee

of 32 . Table II shows different transmission rates, their

corresponding SNR threshold, and the converted bin

sizes and costs for 802.11a and 802.11b2 where the

rate table is taken from [20], and the switching delay

is assumed to be 0.

From Table II, we can see that 802.11a almost satisfies

the required conditions in [11] (except the transition from

24Mbps to 36Mbps), and 802.11b does not satisfy the

conditions in general. This prevents us from directly

leveraging the performance guarantee in [11] for 802.11a

2Note that the SNR in the first column is in log scale but the SNR

used in the third column is in linear scale.

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ship, may be of independent interest in the area of bin-

packing problems and their applications.

Theorem 4: Let C(B) be the optimal cost of binpacking and C(BMEBC) be the cost using the MEBC al-gorithm. We have

C(BMEBC) < 2C(B) + c1

The proof is shown in Appendix C.

D. Continuous Rate Function

We now consider the multicasting problem (7) under

ASP for a continuous rate function R(), where weassume that the rate function R() is continuous andnon-decreasing. We develop sufficient conditions under

which (i) one group, (ii) K groups (where K is the

number of beams), or (iii) any number of groups, forms

the optimal partitioning scheme.

Theorem 5: Assume that R() 0 is non-decreasingover [0, ). Define g() = (W + L

R()

). If g() is anincreasing function of SNR value , the optimal partitionhas one group including all single-lobe beams. If g()is a decreasing function of SNR value , the optimalpartition has K groups and each group contains onesingle-lobe beam. If g() is a constant, any partition isoptimal.

Before we prove the theorem, we show some special

examples of the rate functions. The following corollary

shows that for a wide class of continuous rate functions,

the optimal partition is to have only one beam group.

Corollary 3: If the rate function R() 0 is non-

decreasing and concave over [0, ), it is optimal tohave only one group. Special examples that fall in this

category include the Shannon channel capacity, R() =B log2(1 + ), and the modified Shannon capacity,R() = B log2(1 + /) where > 1 represents thegap between the channel capacity and the actual coding.

If W = 0 and R() = C (which can be viewed as theapproximate Shannon channel capacity at the low SNR

regime), any partition is optimal.

Proof: Assume R() 0 is concave over [0, ).In order to show that the optimal partition is to have one

beam group, it is sufficient to show that g() 0. Since

g() = W + L R() R()

R2(),

it is sufficient to show that R() R() 0.By the convexity theory (e.g., Proposition B.3 in [2]),

0 R(0) R() + (0 )R(). (11)

Therefore, the sufficient condition for having one beam

group is satisfied.

If W = 0 and R() = C, we obtain that g() =L/C is a constant, so any partition is optimal.

Proof of Theorem 5: For continuous rate functions,

Problem (7) can be viewed as a continuous version of

GCVS-BP problem, in which there are infinitely many

types of bins and the bin sizes can take any continuous

positive real value. Now for any SNR and rate functionR(), it corresponds to a bin size s = 1/ and costW + L/R().

As R() is a non-decreasing function of , indicatingthe bin cost is non-decreasing as the bin size increases,

in the optimal solution, the bin size should be equal to

the total size of all items in the bin. As there is no extra

space left in a bin, it is optimal to choose bins that have

the lowest cost-to-size ratio (i.e., (W + L/R(1/s))/s).Therefore, if the cost-to-size ratio of bins is a non-

increasing function of bin size s, the best choice is tohave only one bin containing all items, with a size that is

equal to the total size of all items. If the cost-to-size ratioof bins is an non-decreasing function of s, the optimalpacking is to choose as small bin sizes as possible. Thus,

the optimal solution in this case is to put one item in each

bin, with a size equal to the size of the item. If the cost-

to-size ratio of bins is a constant c, it does not matterwhich bins are chosen. Thus the total cost is always given

by the total size of all items to be packed multiplied by

c, regardless of how they are packed.

Finally, notice that the cost per unit size of bin, (W+L/R(1/s))/s, is increasing (decreasing, or constant)

with respect to s if and only if (W + L/R()) isdecreasing (increasing, or constant) with respect to .This completes the proof.

V. TRACE-DRIVEN PERFORMANCE EVALUATION

In this section, we evaluate the performance of the

proposed algorithms through trace-driven simulations

where the signal SNR trace is obtained from an experi-

mental beamforming system. The experimental testbed

consists of an eight-element Phocus Array [6] from

Fidelity Comtech as the access point (AP) and multiple

laptops with omni-directional antennas as mobile clients.The Phocus Array contains 8 antenna elements and is

capable of providing eight 45-degree single-lobe beam

patterns that are approximately non-overlapping. Figure

1 shows the testbed in an outdoor parking lot as well

as the locations of both the AP and clients used in the

experiments.

We compare our algorithms with the GREPd and

GRASP2 algorithms in [20] for the EQP model and ASP

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Fig. 1. Outdoor testbed and AP/client locations.

model, respectively. We do not include the other well-

known beamforming multicasting algorithm beamcast

[17] as it was shown in [20] that GREPd and GRASP2

outperform beamcast. We carry out simulations based

on the SNR measurements at all receivers locations in

the testbed but randomly pick a subset of users for each

simulation run. We simulate both the continuous and thediscrete rate cases, but only report the results for the

discrete case as almost all practical wireless systems use

discrete rate tables. In the resulting figures, all delays are

calculated based on transmitting a 1500-byte multicast

message. All results are averaged over ten simulation

runs.

A. Performance without Switching Delay

In this section, we compare the performance in the

ideal situation where there is no delay when switching

beam patterns.

Uniform node distribution: We first consider the case

where the users are uniformly distributed across all 8

beam sectors. Figure 2 shows the aggregate delay for

802.11a and 802.11b systems. It can be seen that our

DP-EQP algorithm consistently outperforms the greedy

algorithm GREPd, with a delay reduction of about 10%

for both 802.11a and 802.11b. For the ASP algorithms, it

is interesting to observe that both IFFD-ASP and MEBC-

ASP obtain significant improvement over GRASP2. The

average improvement of both IFFD-ASP and MEBC-

ASP, compared to GRASP2, is over 15% and 20% for802.11a and 802.11b systems, respectively.

From Fig. 2, we can also see that IFFD-ASP and

MEBC-ASP perform similarly in most scenarios and

MEBC-ASP slightly outperforms IFFD-ASP when the

number of clients increases. Moreover, we can see that

when the number of clients increases, the improvement

of IFFD-ASP and MEBC-ASP over other schemes

increases. This is not surprising as when the number

0 10 20 30 401

1.5

2

2.5

3

3.5

4

4.5

5802.11a

Number of users

Multicasttime(ms)

DPEQPIFFDASP

MEBCASPGREPdGRASP2

0 10 20 30 402

4

6

8

10

12

14802.11b

Number of users

Multicasttime(ms)

DPEQPIFFDASP

MEBCASPGREPdGRASP2

Fig. 2. Uniform user distribution in all beam sectors.

2 4 6 81

1.5

2

2.5

3

3.5

4802.11a

Number of beam sectors (K)

Multicasttim

e(ms)

DPEQPIFFDASPMEBCASPGREPdGRASP2

2 4 6 82

3

4

5

6

7

8

9802.11b

Number of beam sectors (K)

Multicasttim

e(ms)


Fig. 3. Clustered user distributions where users are clustered in Kbeam sectors.

of clients increases, the number of possible partitions

increases and so does the potential for improvement.

Clustering node distribution: We also investigate the case

where mobile clients locations are clustered. To model

user clustering, we randomly draw 30 users from a

randomly selected K beam sectors where K 8. Figure3 compares the aggregate delay of transmitting a 1500-

byte message. Similar patterns are observed as in the

case of uniform node distributions. It can also be seen

that when the system is less clustered (i.e., larger number

K of beam sectors), the improvement of IFFD-ASPand MEBC-ASP (compared to other schemes) increases.

Therefore, we conjecture that the proposed algorithmsare more beneficial to systems with larger number of

beams such as 60GHz systems [9], [23].

B. Performance with Switching Delay

We next show how the switching delay affects the

performance of all algorithms. Figure 4 plots the multi-

cast transmission time vs. switching delay for different

algorithms. It can be seen that the multicast transmission

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0 500 10002

3

4

5

6

7

8

9802.11a

Switch delay (us)

Multicasttime(ms)


0 500 10006

7

8

9

10

11

12

13

14

15

16802.11b

Switch delay (us)

Multicasttime(ms)


Fig. 4. Comparison of multicast time with switching delay.

time increases almost linearly as the switching delay

increases for all algorithms. As the switching delay

increases, IFFD-ASP shows 20% and 25% improvement

over GRASP2 for 802.11a and 802.11b consistently.While IFFD-ASP and MEBC-ASP achieve nearly the

same delay in 802.11a systems, IFFD-ASP obtains much

smaller delay than MEBC-ASP in 802.11b systems with

a large switching delay. A little investigation shows that

the performance gain is due to Steps 3-6 in IFFD-ASP,

which attempt to break the last bin into multiple smaller

bins in anticipation that it may not be sufficiently filled.

However, we note that these steps can also be applied to

MEBC-ASP.

C. Uncertainty of Channel Conditions

In real (especially mobile) environments, wireless

channel conditions vary and cannot be estimated pre-

cisely. In this set of simulations, we generate the wireless

channel SNR (in dB) according to a Gaussian distribu-

tion based on the measured mean and variance values

at each client, but the algorithms only use the mean

value of the SNR. To emulate the packet transmission

errors, we record a packet loss if the randomly generated

SNR of a client is smaller than the SNR threshold

of the MCS computed by each partitioning algorithm.Figure 5 shows the packet loss rate of our algorithms

for the uniformly distributed user distributions. It can be

observed that the IFFD-ASP is most robust among the

three algorithms evaluated. This is because, IFFD-ASP

always tries to use the largest bin to pack items in the

first step, which corresponds to using the lowest MCS

to transmit packets. Thus, it is the most reliable under

varying (fading) wireless channel conditions.

0 10 20 30 400

1

2

3

4

5

6

7

802.11a

Number of users

Packetloss(%)

IFFDASPMEBCASP

DPEQP

0 10 20 30 400.5

1

1.5

2

2.5

3

3.5

4

4.5

5

802.11b

Number of users

Packetloss(%)

IFFDASPMEBCASP

DPEQP

Fig. 5. Impact of channel state uncertainty.

VI. EXPERIMENTAL EVALUATION

In this section, we conduct small-scale experiments to

evaluate the developed algorithms.

A. Experimental setupTestbed: We perform experiments using a testbed con-

sisting of an 802.11b/g access point with a beamforming

antenna from Fidelity-Comtech [6] and three clients with

D-Link DWL-AG660 802.11a/b/g cards in the outdoor

environment shown in Figure 1. Both the AP and clients

run Ubuntu 8.04 and Madwifi WLAN drivers. We use

iperf as the traffic generator and the athstats Madwifi

utility to obtain packet statistics (e.g., packet loss rates).

Phocus Array Antenna: An eight-element Phocus Array

Antenna provided by Fidelity Comtech [6] is employed

as the AP, as shown in Figure 6. The magnitude and

phase of signal on each element can be set separately

to form a special beam pattern. The antennas firmware

provides multiple pre-setting switched beamforming pat-

terns with various direction, power and width. Figure 7

shows one of these patterns. Table III is the configuration

of this pattern, where Mag represents the percentage of

maximum transmit power used on each element. Using

the SDK tools provided with the antenna, we can add

customized patterns to the antenna through a command

line interface.

Measurement: We use RSSI(Receive Signal StrengthIndication) to represent the SNR values in our experi-

ments. Latest Madwifi driver provides this measurement

by subtracting the measured noise level from the signal

strength (both in dBm). Packet loss rate is computed

as (1 trecvtsent ), where tsent is the number of packetssent by the antenna and trecv is the number of packetssuccessfully received at a client. Both numbers are taken

from the athstats Madwifi utility to obtain an accurate

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4 6 8 100

1

2

3

4Multicast Delay

Transmit power(dBm)

Multic

astDelay(ms)

4 6 8 10

94

96

98

100

Delivery Ratio

Transmit power(dBm)

Packets

DeliveryRatio(%)

4 6 8 10

0

0.5

1

1.5

2

2.5

3

3.5Number of Partitions

Transmit power(dBm)

Numb

erofPartitions

Omni

OptimalASP

IFFDASP

MEBCASP

DPEQP

Omni

OptimalASP

IFFDASP

MEBCASP

DPEQP

OptimalASP

IFFDASP

MEBCASP

DPEQP

(a) (b) (c)

Fig. 8. Experiment results

T packets. Let Ti be the number of packets

successfully received by client i. The ADR is then

ADR =

3i=1 Ti3T

, (13)

3. Number of partitions may have considerable

impact when switching delay of beam patterns is

taken into account. In that case, the more partitions

in the output of an algorithm, the longer time the

multicast procedure takes.

D. Evaluation Results

As it is difficult to eliminate all processing delay, con-

tention delay, and MAC-layer re-transmission backoff

delay, the delay values reported in Fig. 8(a) are those

generated by all algorithms. Nevertheless, the packet

delivery ratios in Figure 8(b) are obtained from the

measurements at all clients.

We evaluate the performance of five multicast algo-

rithms: simple omni-broadcast, IFFD-ASP, MEBC-ASP,

DP-EQP, and optimal-ASP. We conduct experiments

with transmit power ranging from 3 dBm to 11 dBm. For

each experiment, a file containing about 1000 packets

each with 1500 bytes is multicasted to three clients.

Figures 8 (a), (b), (c) show the multicast delay, averagedelivery ratio, and number of partitions, respectively.

It can be seen from Fig. 8 that, i) all algorithms

achieve the target delivery ratio 90% which is used

for constructing the rate-table, ii) both IFFD-ASP and

MEBC-ASP algorithms achieve similar delay as the

optimal algorithm (except for the 3dBm transmit power),

and iii) the delay of all ASP algorithms is smaller than

that of DP-EQP which is in turn smaller than that of

the omni-directional multicast scheme. It is interesting

to observe that while the optimal solution achieves lessdelay at 3dBm transmit power than the other ASP

algorithms, it also has lower delivery ratio because more

aggressive MCSs are used in the optimal solution. The

results for the omni-pattern algorithm at 3dBm transmit

power are not shown because not all clients are covered

by the omni-pattern at this power level. Finally, the

number of partitions of omni-broadcast is always 1 and

not plotted in the figures.

VII. RELATED WOR K

Wireless Multicasting in Communication Theory: In

the communication area, many researchers have studied

the problem of multicast/broadcast with adaptive/beam-

forming antennas [22], [18], [19], [24], [14]. Most of

the works assumed adaptive beamforming or MIMO

techniques, which require higher complexity to compute

the optimal beam or antenna weights and need detailed

complex-value channel feedback, as opposed to switched

beamforming, which is the focus of this work.

Link layer algorithms for wireless multicasting:

Many works have focused on link-layer algorithms for

enhancing wireless multicasting [16], [5], [10], [4], [13],

[3]. Parket al. proposed a new rate-adaptation algorithmfor multicasting multimedia content. The works [5], [10]

developed efficient feedback mechanism to improve the

multicasting reliability. Chaporkar et al. [4] designed a

scheduling policy for multicasting in ad hoc wireless

networks. Li et al [13] presented efficient resource

allocation algorithms for multicasting scalable video

streams. Chandra et al. [3] built a WiFi prototype

implementation of wireless multicasting and also solved

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several practical problems including the AP association

problem. Although all these works have their own merits,

they consider only omni-directional antennas.

Beamforming multicasting algorithms: Only a few

recent works looked at the integrated problem of beam-

forming and multicasting [8], [17], [20]. While Hou et

al. [8] developed new multicasting routing algorithms

exploiting beamforming antennas, the works [17], [20]

aimed to design link-layer multicast scheduling algo-

rithms with switched beamforming antennas. Sen et al.

[17] presented a first-cut solution, by performing a omni-

directional transmission followed by one or a few single-

lobe sequential directional transmissions to cover the

clients left behind from the initial omni-transmission.

Sundaresan et al. [20] provided a rigorous formulation

of the switched beamforming multicasting problem

and proposed several algorithms to solve the problem,

under different models. In this work, we adopt the

problem formulation in [20] but make several significantcontributions, including a much more efficient optimal

algorithm under the EQP model and several asymptotic

approximation solutions under the ASP model.

VIII. CONCLUSION

We have studied the problem of multicasting with

beamforming antennas in wireless networks. We con-

sider both the EQP model and ASP model. Under the

EQP model, we obtain optimal algorithms based on

dynamic programming for arbitrary rate functions. Under

the ASP model, we prove that the general problem isNP-hard and obtain approximation solutions for discrete

rate functions. For the continuous rate function under

ASP model, we also develop a set of sufficient conditions

under which the optimal solution has (i) 1 group, (ii) Kgroup (where K is the number of beams), and (iii) arbi-trary number of groups. In particular, we show that if the

rate function is continuous, non-decreasing and concave,

it is optimal to have only one group. The effectiveness of

our algorithms is evaluated through both experiments and

trace-driven simulations, and significant improvement is

observed over recently proposed algorithms.

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APPENDIX A

APPROXIMATION ALGORITHMS

We define some terminologies for approximation

algorithms used in this paper, largely following the

definitions in [15]. We consider the problem of mini-

mization of an non-negative objective function. For a

given problem instance I, we use A(I) and OPT(I)to denote the objective values of the solution of analgorithm A and the optimal solution, respectively. Iffor r > 1 and any instance I,

A(I) r OPT(I),

we call the algorithm A is an r-approximation algorithm,or A has performance ratio r. Let

r = inf{s > 1 : N0 such that for all I withOPT(I) N0, and A(I) s OPT(I)}.

We call the algorithm A is an asymptotic r-

approximation algorithm, or A has asymptotic perfor-mance ratio r. As a special case, if

A(I) r OPT(I) + C,

where C is a constant, the algorithm A has asymptoticperformance ratio r.

An asymptotic approximation scheme is an algorithm

A that takes as input both the instance I and an errorbound > 0, and has asymptotic performance ratio (1 +).

An APTAS (Asymptotic Polynomial Time Approxima-

tion Scheme) is an asymptotic approximation scheme{A} where each algorithm A has asymptotic perfor-mance ratio 1 + and runs in time polynomial in thelength of the input instance I.

APPENDIX B

PROOF OF THEOREM 3

Without loss of generality, we assume that b1 = c1 = 1(Otherwise, we can always normalize all bin sizes with

respect to b1 and all cost with respect to c1). Now weonly need to prove that C(BIFFD) (1+)

32C(B

)+1.As the steps in iterations 3-6 do not increase the cost, it is

sufficient to prove the theorem without them and simplylet i = 1 in step 7. As we only consider the bins inB1, we omit the superscript in the following discussionto ease the notations. For each allocated bin Bj , we usec(Bj), b(Bj), t(Bj) to represent the cost, size, and thetotal sizes of objects in the bin, respectively. First we

show a lemma.

Lemma 2: Assume that the conditions in Theorem 3

are satisfied. If the total size of the objects in a bin is

t(Bj), the minimum cost of holding these objects in the

optimal solution ist(Bj)1+ .

Proof. The minimum cost per unit bin size for all types

of bins is

mincibi

1

1 +

c1b1

=1

1 + .

Therefore, the minimum cost for a total object size oft(Bj) is

t(Bj)1+ .

We prove the theorem by conditioning on different

cases. Denote the optimal cost is C(B). If for everybin j except the last bin k1, t(Bj)

23c(Bj), then

k1j=1

c(Bj) 1 +3

2

k11j=1

t(Bj) 1 +3

2(1 + )C(B)

where the last inequality follows from Lemma 2. The

conclusion holds.

Therefore, we only need to consider the case that there

exists some k < k1 such that t(Bk) < 23c(Bk). Withoutloss of generality, we assume k is the largest index suchthat the condition satisfies (but still k < k1). We claimthat t(Bk) > 1/2. Because otherwise, any item in the binBk+1 should have been put into Bk. We also show thatBk only contains one item. Otherwise, Bk must containan item with size less than 1/3, since the total sizes ofthe objects in Bk is t(Bk) k have sizes largerthan 1t(Bk) (Otherwise, they should have been packedto the bin k). Therefore, none of these items can be putin the bins holding the previous big items from bins

j k Thus, the minimum cost of these items in bin

j > k isPk1

j=k+1 t(Bk)

1+ . As a result, the optimal cost is

C(B) 3

4k +

k1j=k+1 t(Bk)

1 + . (14)

The resulting cost from the algorithm IFFD is

C(BIFFD) k +k11j=k+1

c(Bj) + 1

k +3

2

k11j=k+1

t(Bj) + 1

< (1 + )(3

2(

3

4k +

k1j=k+1 t(Bj)

1 + )) + 1

(1 + )3

2C(B) + 1 (15)

where the second inequality is because for all k < j bj1 (otherwise, allitems in xj , zj can be put into the last bin of type j 1with MEBC). Since all items in zj1+xj+zj are largerthan bj+1, their cost-to-size ratios are lower bounded bycj/bj . Therefore, C

(zj1+xj+zj) bj1 cj/bj > cj .Summing up for all j = 1, , k, noting that C(x) 12C

(2x), and applying Lemma 3, we have

C(

kj=1

(zj + xj)) 1

2C(

kj=2

(zj1 + xj + zj)) >1

2

kj=2

cj (16

Let nj be the number of bins of type j used byMEBC. The number of bins containing items yj is thennj 1. Since type j is the most efficient bin with thesmallest size for all items in yj and all bins containing

yj are less than half-empty, we get

C(yj) yj cjbj

(nj 1) bj2

cjbj

(nj 1) cj2

.

Applying Lemma 3 and combining with Eq. (16), we

have C(kj=1 yj)

kj=1 (nj 1)cj/2 and

C(B) = C(kj=1

(xj + yj + zj))

kj=2

cj2

+kj=1

(nj 1)cj2

= (n1 1) c12

+kj=2

njcj2

.

Finally, note that the cost of the MEBC is just

C(BMEBC) =

kj=1

nj cj 2C(B) + c1.

This completes the proof.

15

multicast beam forming data

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