Multicast Beam Forming Data
Transcript of 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)
DPEQPIFFDASPMEBCASPGREPdGRASP2
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)
DPEQPIFFDASPMEBCASPGREPdGRASP2
0 500 10006
7
8
9
10
11
12
13
14
15
16802.11b
Switch delay (us)
Multicasttime(ms)
DPEQPIFFDASPMEBCASPGREPdGRASP2
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.
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