Distributed medium access control for energy efficient transmission in cognitive radios

0.45 MB·

by  Oshri Naparstek

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Distributed medium access control for energy
efficient transmission in cognitive radios
Oshri Naparstek1, Amir Leshem1 (Senior Member, IEEE) and Eduard Jorswieck2 (Senior Member, IEEE)
Abstract—In the last few years the issue of energy-efficient the consumed power as the sum of the transmit power plus
network design has gained increasingly more importance [1], a constant term, independent of the transmit power, which
[2], both in academia and in industry. Information and com-
models the circuit power needed to operate the terminal.
munication technologies (ICT) represent about 2% of the entire
Following [12], in [13], [14], [15] the consumed power is
world’s energy consumption, and the situation is likely to reach
a point where ICT equipments in large cities will require more also defined as the sum of the transmitted power and the
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energy than is actually available [3]. In this paper we develop circuit power. Moreover, in these papers the throughput is
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0 and investigate methods for energy efficiency optimization for replaced by the achievable rate in the definition of the energy
wirelessnetworkswithcognitiveunits.Weshowthattheproblem
2 efficiency. In [14], [15] a single user channel is considered
can be solved without explicit message passing using a modified
and the optimization is carried out through transmit power
n distributedauctionalgorithm.Thenweintroduceafastconverg-
a ing algorithm that approximates the solution to the maximal control. By contrast, in [13], multiuser interference channels
J energy efficiency problem. We state conditions under which the are considered and, as well as in references [4], [5], [6], [7],

8 fast algorithm is asymptotically optimal in terms of the number [8], [9], [10], [11], a competitive scenario in which users
of users. Finally, we provide simulated examples of the methods selfishly aim at individual EE maximization is addressed.
] developed in this article.
I Each transmitter k is not only interested in maximizing
N
its own performance in terms of achieved SINR γ , but also
k
. in saving as much battery energy as possible. This trade-off
s I. INTRODUCTION
c is well modeled by defining the EE of a given terminal k,
[ While for decades communication networks have been
as the ratio between the so-called efficiency function which

designed to optimize performance measures such as bit-error-
1 measurestheSINR-basedperformanceofuserkandthepower
rate, latency, data-rate, etc., in the last few years the issue
v consumed to attain this performance level [13], [12], [14],
1 of energy-efficient network design has gained increasingly [15]; namely
7 more importance [1], [2], both in academia and in industry. f(γ (p ))
6 Information and communication technologies (ICT) represent EEk = p +k Pk . (1)
1 k c,k
about 2% of the entire world’s energy consumption, and the
.
1 situation is likely to reach a point where ICT equipment in In (1), Pc,k is the power that is required by the transmitter
0 large cities will require more energy than is actually available electronic circuitry to operate the device, and which is dissi-
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[3].Asfordatanetworks,contrarytointuitionmoreenergyis patedevenduringnon-transmissionperiods.Forfurtherdetails
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consumed in access networks than in core networks, because on the circuitry power term, we refer the reader to [16] and
:
v the number of devices in access networks (i.e. mobile termi- references therein, where several power consumption models
i
X nals, base stations, and data modems installed on customers’ for wireless networks are developed. As for f(γ), in principle
r premises), is orders of magnitude larger than the number of itcanbeagenericincreasingfunctionofthek-thuser’sSINR,
a communicationdevices(routers,multiplexers,etc.)inthecore with f(0)=0 and such that (1) tends to zero for growing pk.
network. For this reason, research in the field of wireless Two widely used efficiency functions are
networkshasfocusedontheproblemofoptimizingtheEnergy 1) f(γk) = R(1−e−γk), where R is the communication
Efficiency (EE) at the physical layer, by maximising the ratio rate and (1−e−γk) an approximation of the probability
of a SINR-based function over the consumed power. of correct symbol reception. A similar approximation
The first and most widely used definition of EE is the ratio was used in [5], [6]. Thus, f is the number of bits that
between the throughput and the transmit power [4], [5], [6], arecorrectlydemodulatedatreceiverk perunitoftime.
[7], [8], [9], [10], and references therein. Another proposed 2) f(γ )=Wlog(1+γ ),whereW isthecommunication
k k
metric uses the goodput in place of the throughput[11]. In all bandwidth. For strictly static channels f represents the
oftheaboveworks,asfarasthecomputationoftheconsumed k-thuser’sachievablerate.Forquasi-staticchannels,the
power is concerned, only the transmit power is considered, use of f for resource allocation purposes is still well-
whereas the power that is dissipated in the electronic circuitry motivated in view of the assumption that the channel
of each terminal in order to keep the terminal active is coefficientsremainconstantforlongerthantheresource
neglected. This assumption was relaxed in [12], by defining allocation phase.
Variationsofoption1)arealsoavailableintheliteratureinthe
1 FacultyofEngineering,Bar-Ilanuniversity,Ramat-Gan,52900,Israel.
2 SchoolofEngineeringSciences,TUDresden,01062Dresden,Germany. formoff(γk)=R(1−e−γk)M andf(γk)=R(1−e−γk/2)M,
ThisresearchwaspartiallysupportedbyISFgrant903/2013 and in this case the function f(·) is an approximation of
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the probability of error-free reception of a data packet of M on the local prices an algorithm which can be implemented
symbols. An EE that considers both the case of M > 1 and using multichannel opportunistic CSMA is presented and its
the circuit power P was considered in [12] for a single-hop optimality is proved.
c,k
system.There,itwasshownthatanequilibriumforthepower In this paper we present fully distributed MAC protocols
allocationalgorithmexisted,buttheconvergencecouldnotbe to maximize the energy efficiency of wireless networks. The
proved. The techniques developed in this paper could be used protocols does not require any explicit communication be-
to extend the results of [12] to the relay-assisted scenario, as tween the wireless devices and are asymptotically optimal.
well. However, in the following we choose to focus on the The paper is organized as follows: in Section II we define
equally well-motivated case of M =1. Thus, for any M, the the maximal energy efficiency problem. In Section III we
resulting EE (1) is a measure of the number of bits that are present the algorithm and show how it can be implemented
correctly decoded at the receiver, per unit of time and per in a fully distributed manner using opportunistic CSMA. In
Joule of energy drained from the battery of the transmitter. Section IV we present a relaxation to the maximal energy
Moreover, all the efficiency functions that we consider result efficiencyproblemandshowthatthealgorithmfortherelaxed
in an EE (1) which is measured in bits over Joules, thus problem terminates within O(Nlog(N)) iterations with high
representing a natural measure of the efficiency with which probability. We then show that the algorithm can be used to
each Joule of energy drained from the battery is being used. yieldanasymptoticallyoptimalsolutiontothemaximalenergy
Two pertinent social welfare performance metrics are the efficiencyproblem.InSectionVIwediscusssimulatedresults
average EE and the system global EE (GEE), respectively for the proposed algorithm.
defined as [6], [8], [7], [10]
K II. PROBLEMFORMULATION
EEav = 1 (cid:88) f(γk) (2) We consider the case of N cognitive transceiver pairs
K P +p
c,k k
k=1 sharing a time slotted unlicensed frequency band divided into
and K subbands.WeassumeK ≥N.Eachusernispreassigned
GEE= (cid:80)Kk=1f(γk) . (3) a target rate Rn. The transmission power P(n,k) is defined
(cid:80)K P +p as the minimal power needed by the n’th user to achieve
k=1 c,k k
his target rate in the k’th channel. We assume all the users
Customarily, the GEE is used to describe the overall system
have continuous sensing over all channels. We also assume
EE.Centralisedanddecentralisedresourceallocationinmulti-
that only one user can transmit on each channel in each time
hop networks for energy-efficiency maximization was studied
slot and consider out of cell interferences as noise. We seek
in [17], [18].
a fully distributed method to maximize the overall energy
Unlikecellularnetworkswheretheenergyefficiencycanbe
efficiency of the system. We assume that the coherence time
optimized in a centralized manner, in many cases centralized
of the channel is long enough for the algorithm to converge.
optimization is not possible. In this case distributed protocols
Underthisassumption,P(n,k)isthesolutiontothefollowing
are needed to maximize the energy efficiency of the network.
equation
Several approaches had recently been suggested for optimal
distributedallocation.These approachesutilizeanopportunis- (cid:18) |H(n,k)|2P(n,k)(cid:19)
R =log 1+ (4)
tic version of carrier sensing to determine an orthogonal n 2 σ2
n
allocation which is optimal or near optimal in the sense of
where H is the channel matrix and σ2 is the noise variance
exploiting channel diversity. The first example is the use of n
of the n’th receiver. The solution to the above equation is
the well known Gale-Shapley stable marriage theorem [19] to
allocatespectainamultichannelsetup[20],[21].Inthatwork, (cid:0)2Rn −1(cid:1)σ2
P(n,k)= n. (5)
Leshem et al. demonstrated how a stable channel allocation |H(n,k)|2
canbeobtainedwithoutanyexplicittransmissionusingcarrier
P(n,k) can also be chosen according to a goodput require-
sensing as a mechanism to prioritize channel. Analysis of
ment. Alternatively, instead of a rate requirement R per user
this technique for Rayleigh fading channels appeared in [27]. n
we can choose P(n,k) to fulfill a goodput requirement
Otherapproachestothedistributedchannelallocationproblem
include game theoretic bargaining solutions [22], [23], [24] −|H(n,k)|2P(n,k)
and distributed allocation using multichannel ALOHA [25], Tn =R(1−e σn2 ) (6)
[26]. While the stable allocation which turns out to be the for fixed R>0 and T . Solve for P(n,k) leads to
n
greedy assignment is almost optimal for Rayleigh fading
log(cid:0)1− Tn(cid:1)σ2
channels, it is desirable to obtain the optimal allocation. It P(n,k)= R n (7)
iswellknownthatthisallocationcanbecomputedbysolving |H(n,k)|2
a linear programming problem [28]. However, in order to using the formulation of equation 5. The utility for the n’th
compute the optimal distributed solution [29], [30] revised user in the k’th channel in the computation of the EEav
the auction technique of Bertsekas [31] which requires a criterion is given by:
shared memory or price exchange between the bidders and
R
the auctioneer. Instead of knowledge of the highest price U (n,k)= n (8)
this technique only requires knowledge of local prices. Based av (2Rn−1)σn2 +c(n)
|H(n,k)|2
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TABLEI
where c(n) is the minimal amount of energy the n’th device
DISTRIBUTEDAUCTIONALGORITHM
requires to operate at any given time. Under the assumption
of preassigned target rates the GEE simplifies into a power Select(cid:15)>0,setalltheusersasunassignedandset
C(n,k)=0,∀n,k
minimization problem under rate constraints. We further as-
Repeat
sumethattheinstantaneoustransmissionpowerislimited.We 1.Eachunassigneduserncalculateshisownmaximumprofit:
denote the maximum instantaneous transmission power per γn=maxk(U(n,k)−C(n,k))
2.Eachunassigneduserncalculateshissecondmaximumprofit:
user as P . For simplicity, we formulate the utility of the
max k˜n=argmaxk(U(n,k)−C(n,k))
GUGEEEE(ans,ka)mfaoxrimthiezaGtiEonEpcrroibtelreimon. Wase define the utility matrix 3.ωEnach=umnaasxsikg(cid:54)=nek˜dn(uUse(rnn,ku)p−daCte(snt,hke)p)riceofhisbestchannelk˜n
(cid:26) tobeC(n,k˜n)=C(n,k˜n)+γn−ωn+(cid:15)
U (n,k)= Pmax−P(n,k), P(n,k)≤Pmax (9) 4.Alltheusersbid.Theunassignedusersbidontheirnewbest
GEE 0, P(n,k)>P . channelwiththeupdatedbid.Theassignedusersbidonthelast
max
channeltheybidonandwiththesameprice.
The maximum energy efficiency problem can be formulated 5.Assignchanneltothehighestbidder(channelswithnobids
stayunassigned)
as an integer programming problem
Untilallusersareassigned
N K
(cid:88)(cid:88)
max U(n,k)η(n,k)
η profitable channel ω
n=1k=1 n
s(cid:80).t η(n,k)=1, ∀n=1,2,..,N (10) k˜n =argmax(U(n,k)−C(n,k))
k k
(cid:80)nη(n,k)=1, ∀k =1,2,..,K γn =U(n,k˜n)−C(n,k˜n) (11)
η(n,k)∈{0,1}, ∀n,k ω = max(U(n,k)−C(n,k))
n
k(cid:54)=k˜n
The constraint matrix of this problem is totally unimodular.
Each unassigned user raises the price on his most profitable
This means that the solution to the relaxed problem where
channel by
we replace the integer constraint by 0 ≤ η(n,k) ≤ 1 is also
the solution to the original problem. The relaxed problem is a C(n,k˜ )=C(n,k˜ )+γ −ω +(cid:15) (12)
n n n n
linearprogramming(LP)problemandcanbesolvedefficiently
in a centralized manner by LP solutions methods such as where (cid:15) is a predetermined constant that can be seen as the
the Hungarian Method [28]. In the next section we review priceofparticipatingintheauction.Aftertheunassignedusers
distributed algorithms for the channel assignment problem. updater their prices, all the users bid on their most profitable
channels. If user n gets assigned to channel k he continues
to bid on that channel without raising his bid. If a user n
III. DISTRIBUTEDPROTOCOLFORMAXIMALENERGY is unassigned he bids on k˜ with the new bid C(n,k˜ ). In
n n
EFFICIENCY
the assignment stage each channel is assigned to the highest
In this section we propose the use of a fully distributed bidding user. A channel without bids stays unassigned and
channelassignmentalgorithmthatdoesnotrequireanyexplicit users who were not assigned to channels become unassigned.
message passing or a shared memory between the users. The bidding and assignment stages proceed in iterations until
The algorithm relies on the auction algorithm [31] and the alltheusersareassignedtochannels.Oncealloftheusersare
distributed algorithm suggested in [30] for sum-rate maxi- assigned to channels, no one raises his bid and as a result the
mization. The distributed protocol consists of a bidding stage assignmentbecomesstatic.Whenalltheusersareassignedwe
and an assignment stage. The description of the algorithm say that the algorithm has converged. The distributed auction
is as follows. The utility matrix U is an N ×K matrix of algorithm appears in Table I. It was proven in [30] that the
energyefficiencyindexesandCisanN×K costmatrix.The distributed auction algorithm converges in finite time to a
cost of a channel C(n,k) is a unitless number that merely solution within N(cid:15) from the optimal solution. The distributed
represents how much user n wants channel k in comparison auctionalgorithmcanbeappliedusinganopportunisticCSMA
totheotherusers.Wedefinetheprofitofusernfromchannel protocol.
k as the reward (i.e. energy efficiency index) minus the price One can implement the distributed protocol without the
of the channel U(n,k)−C(n,k). In the initialization stage use of explicit message passing using opportunistic CSMA.
each user sets the cost for all of the channels to be 0; Opportunistic CSMA [32] is a distributed transmission pro-
i.e., C(n,k) = 0,∀n,k, select (cid:15) > 0 and sets his state to tocol suggested for wireless sensor networks. Opportunistic
unassigned.k˜ isdefinedasthemostprofitablechannelofthe CSMA is composed of carrier sensing and a waiting strategy.
n
n’th user. The distributed protocol proceeds in iterations. In Continuous sensing of all channels by all users is assumed.
eachiterationtwostagesaresequentiallyperformed,abidding Each user in the network calculates a fitness measure ψ
n
stage where users raise the price on their most profitable and maps it into a waiting time τ based on a predetermined
n
channel and an assignment stage where channels are assigned commondecreasingfunctionf(ψ ).Anexampleofamapping
n
to the users who proposed the highest prices. In the bidding function from fitness to waiting times is shown in Figure 1.
stage, each unassigned user n find his most profitable channel Each user waits until his waiting time ends and if no one
k˜ and the profits from that channel γ and his second most transmitted on his most wanted channel then he is allowed
n n
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TABLEII
DISTRIBUTEDAUCTIONALGORITHMUSINGOPPORTUNISTICCARRIER
SENSING
Select(cid:15)>0,eachusersetshimselfasunassigned
andsetC(n,k)=0,∀k
Repeat
1.Eachunassigneduserncalculatesitsownmaximumprofit:
γn=maxk(U(n,k)−C(n,k))
2.Eachunassigneduserncalculatesitsownsecondmaximumprofit:
k˜n=argmaxk(U(n,k)−C(n,k))
ωn=maxk(cid:54)=k˜n(U(n,k)−C(n,k))
3.Eachunassignedusernupdatesthepriceofhisbestchannelk˜n
tobeC(n,k˜n)=C(n,k˜n)+γn−ωn+(cid:15)
4.Eachunassignedusernmapsthenewbestprofitinto
backofftimeτn=f(C(n,k˜n))
5.Eachuserwaitsforτn milliseconds.Iftheuserdetectsthat
k˜n isfreewhenitsbackofftimehasexpired,he
transmitshispacketonk˜n andsetshisstatetoassigned.
Untilallusersareassigned
Fig.1. Anexampleofafunctionthatmapsbetterfitnesstoshortertime
U ∈ {0,1}N×K the problem is reduced to finding matchings
on bipartite graphs. To formulate the maximum energy effi-
to transmit on it. The outcome is that the user with the
ciencyassignmentproblemasamatchingproblemonbipartite
highest ψ transmits in the channel. This can be seen as a
n
graphs we use the following definitions [33]:
distributed winner determination algorithm where the winner
Definition 1: Let G=(V,E) be a graph with a vertex set
gets the channel. In the distributed auction algorithm the
V and an edge set E. The neighbor set of vertex v ∈ V is
only coordination needed between users is an auctioneer to
given by
decide which user made the highest bid. The opportunistic
n ={u∈V :(u,v)∈E}. (15)
CSMAcanbeusedasanauctioneer.TheopportunisticCSMA v
protocol does not require explicit message passing between Definition 2: Let G=(Vˆ,E) be a graph with a vertex set
users and hence it is suitable for the implementation of the Vˆ and an edge set E. If Vˆ can be divided into two subsets
fully distributed solution on the maximum energy efficiency U,V such that
problem. As seen above, the resulting assignment is within
n ∩U =∅, ∀u∈U
N(cid:15) from the optimal solution as in the auction algorithm. We u
n ∩V =∅, ∀v ∈V.
can define the reward that each user n gets from channel k v
to be the energy efficiency of that channel U(n,k). Using the we say that G is a bipartite graph and we denote it by
opportunistic CSMA scheme, each user n tries to access his G(U,V,E).
best profit channel defined by Definition 3: Let G = (U,V,E) be a bipartite graph with
k˜ =argmax(U(n,k)−C(n,k)) (13) a vertex sets |U|=|V|=N and an edge set E. Let M ⊆E
n k and let G˜ = (U˜,V˜,M) be a bipartite subgraph of G with
with a backoff time of vertexsets|U˜|,|V˜|=N andanedgesetM.M isamatching
on G if
τn =f(ρn,k˜n) (14) max |nv|=1. (16)
v∈U∪V
where f(x) is a positive monotonically decreasing function.
Definition 4: Let G = (U,V,E) be a bipartite graph with
The price C(n,k) is determined and updated if necessary as
vertex sets |U|=|V|=N and an edge set E.M is a perfect
describedintableI.Thepricesandtheircorrespondingwaiting
matching if M is a matching and |M|=N.
times must converge in a finite number of iterations as in
Definition 5: Let G = (U,V,E) be a bipartite graph with
the distributed auction algorithm. The distributed auction al-
vertexsets|U|=|V|=N andanedgesetE.LetM ⊆E and
gorithmforcognitiveradiosystemsusingopportunisticCSMA
let G˜ = (U,V,M) be a bipartite subgraph of G with vertex
is described in Table II. It was shown in [30] that the number
sets |U|,|V|=N and an edge set M. A vertex v ∈U∪V is
of iterations needed until the distributed auction algorithm is
converged is bounded by O(cid:0)N3(cid:1) with respect to N. In the free if |nv|=0 otherwise we say it is not free.
Using the above definitions we define the maximum car-
next sections we suggest a relaxation to the maximal energy
dinality matching problem as follows: Let G = (U,V,E) be
efficiency problem. We show that the suggested relaxation
bipartite graph with vertex sets |U| = |V| = N and an edge
is asymptotically optimal and can be solved with Nlog(N)
set E. Find a matching M such that |M| is maximal.
expected number of iterations.
TheMCMproblemcanalsobeformulatedasamax-energy
efficiency problem (10) where the reward matrix is a binary
IV. FASTMATCHINGALGORITHM matrixwith0,1values.Wenowpresentanalgorithmthatfinds
InSectionIIweformulatedthemaximumenergyefficiency amaximumcardinalitymatchingonbipartitegraphswhichcan
channel assignment problem for arbitrary values. However, if be implemented in a fully distributed manner.
5
TABLEIII
fastmatchingalgorithmcanbeimplementedtofindasymptot-
ALGORITHMFORMAXIMALMATCHING
ically optimal solutions to the energy efficiency maximization
1) Initializehv =0,∀v∈V andsetM =∅ problem with a small number of iterations.
2) While|M|<N do
a) Chooseu∈Ufree
b) j=argminv∈nuhv V. FASTENERGYEFFICIENCYMAXIMIZATIONALGORITHM
c) M =M∪(u,j)
d) uold={u∈U :(u,j)∈M} InthissectionweshowhowthealgorithmdescribedinSec-
e) M =M\(uold,j) tion IV can be used to give asymptotically optimal solutions
f) Ufree=Ufree∪(uold,j)
g) hj =hj+1 to obtain maximally energy efficient channel assignment. The
3) Return main idea is to transform U into a binary 0,1 matrix U˜ and
apply the matching algorithm on U˜. The transformation from
U to U˜ is done by applying a threshold athresh ≥ 0 for each
n
Weproposeaniterativealgorithmthatassignsanunassigned row.
user to a channel according to the following scheme: Each U˜(n,k)=(cid:26)1, U(n,k)≥atnhresh (19)
channel k ∈ K is assigned a value h that represents how 0, U(n,k)<athresh.
k n
many times the channel was reassigned to different users.
To ensure asymptotically optimal solutions to the max-energy
Let h(i) = [h ,h ,...,h ] be the vector of the values of the
1 2 K efficiency problem, athresh must satisfy the following require-
channelsonthei’thiteration.Atthebeginningofthealgorithm N
ments:
all the values of the channels are initialized to 0; i.e.,
1) Only the best channels of each user should be above
h(0) =0, ∀k =1,2,...,K. athresh.
k N
2) U˜ should contain a perfect matching with high proba-
Let U be the set of all free users. On each iteration, an
free bility.
unassigned user u ∈ U is chosen. u is then assigned to
free The first condition is to ensure that the solution to U˜ will
the channel with a minimal value he can access and raises its
provide a good solution to the max-sum problem and the
value by 1. The algorithm is depicted in Table III.
second condition is to ensure that with high probability all
of the users will get assigned by the algorithm. Assume that
A. Expected number of iterations of the matching algorithm eachrownofUconsistsofi.i.drandomvariablesandassume
Inthissectionweanalyzetheexpectednumberofiterations that the cumulative distribution function (CDF) of each entry
until the algorithm converges for random bipartite graphs. A of U(n,k) is given by Fn. A proper choice of atnhresh would
random bipartite graph is defined as follows: be
(cid:18) (cid:19)
αlog(N)
Definition 6: G = (U,V,E) is called a bipartite random athresh =F−1 1− , (20)
n n N
graph if G is a bipartite graph and the edges in E are
independently chosen with probability p; i.e., where α > 1. The choice of the threshold satisfies the first
condition since only αlog(N) best channels of each user
Pr((u,v)∈E)=p, ∀u∈U, ∀v ∈V.
are above the threshold. Theorem 1 ensures that with high
Denote the set of all random bipartite graphs with vertex sets probability U˜ contains a perfect matching if atnhresh ≥0 for all
|U| = |V| = N and probability p for an edge by B(N,p). n=1,2,...,N. Hence, the target rates should be chosen such
The following known result on perfect matching in random that with high probability atnhresh ≥0 for all n=1,2,...,N.
bipartite graphs was proven by Erdo˝s and Re´nyi in [34]:
Theorem 1: Let p= (1+(cid:15))Nlog(N) and G∈B(N,p) then A. Target rates for Rayleigh channels
lim Pr(G contains a perfect matching)−e−2N−(cid:15) =0. In this section we analyze the fast matching algorithm
N→∞ for Rayleigh fading. We model the channels of each user
(17)
as Rayleigh fading channels; i.e., the channel attenuation
The next theorem proven in [36] shows that for random
bipartitegraphswithp≥ (1+(cid:15))log(N) thenumberofiterations |H(n,k)|2 is an exponential random variable given by:
N
until the convergence of the algorithm is less than cNlog(N) 1
log(Np) |H(n,k)|2 =G ·F · (21)
with high probability and c>0 is some constant. n n rα
n
Theorem 2: LetG=(U,V,E)bearandombipartitegraph
where G is a global normalizing factor, F is an exponen-
with |U| = |V| = N and p ≥ (1+(cid:15))log(N). Let T be the n n
N tially distributed gain (due to the Rayleigh fading channel
number of iterations until the algorithm converges then:
with a multipath effect),r is the distance between the n’th
n
(cid:18) cNlog(N)(cid:19) transmitter from its receiver and α is the path loss exponent;
lim Pr T ≤ =1. (18)
N→∞ log(Np) therefore:
. f|H(n,k)|2(x)=λne−λnx (22)
Theorem 2 ensures that the fast matching finds a per-
where
fect matching with a probability that approaches 1 in rα
λ = n (23)
O(Nlog(N))iterations.Inthenextsectionweshowhowthe n G
n
6
The rate of user n in channel k is given by [35]: B. Asymptotical optimality
(cid:18) |H(n,k)|2P(n,k)(cid:19) Inthissectionweshowthatthefastassignmentisasymptot-
R (k)=log 1+ (24)
n 2 σ2 icallyoptimalforRayleighfadingwithproperlychosentarget
n
We now derive sufficient conditions on R such that the rates as shown in V-A. For the analysis we use some known
n
fast algorithm will converge within O(Nlog(N)) expected results from order statistics [37].
number of iterations. Definition 7: Let A be a random variable with CDF FA(r)
Theorem 3: The fast matching algorithm will converge and let A1:N < A2:N < ... < AN:N be random variables
within O(Nlog(N)) expected number of iterations in obtained by taking N samples from A and ordering the
Rayleigh fading if the following requirement is satisfied for samples in an increasing order. Ak:N is called the k’th order
all n : statistics of A with N samples.
 (cid:16) (cid:17) Definition 8: Letk beafunctionofN suchthatk →∞
P log N N N
Rn ≤log21+ max λnσαn2log(N) . (25) AaskNN:N→are∞calalenddinlitmerNm→ed∞iatkNeNor=der0sttahtiesnticAs.N−kN+1:N and
Definition 9: LetF (x)beadifferentiable,absolutelycon-
Proof:Asshowninequation(5)theminimalpowerneeded A
tinuous distribution function. If
to achieve rate R is given by:
n
P(n,k)= (cid:0)2|RHn(n−,k1)(cid:1)|2σn2. x→lFim−1(1)ddx(cid:18)1−f(Fx()x)(cid:19)=0 (33)
For Rayleigh channels |H(n,k)|2 is exponentially distributed then the third Von Mises condition is satisfied.
with CDF An important result on intermediate order statistics was
F|H(n,k)|2(x)=1−e−λnx. (26) found by Falk [38].
Theorem 4: (Falk1989).LetF beanabsolutelycontinuous
As a result, the CDF of P(n,k) is given by:
CDF satisfying one of the Von Mises conditions. Suppose
FP(n,k)(x)=e−λnσn2(2xRn−1). (27) kN → ∞ as N → ∞ and limN→∞ kNN = 0. Then there
exist norming constants α and β >0 such that
N N
The expected number of iterations will be O(Nlog(N))
only if there exists a perfect matching in the graph with a AN−kN+1−αN −−d→N(0,1). (34)
probabilityofatleast1− 2 .FromTheorem1,thereexists β
Nα−1 N
iafptehrefeecxtpmecattecdhinngumwbitehraofpreodbgaebsilictoynonfecatteldeatsot1ea−chNαv2−er1teoxnliys where αN =F−1(cid:0)1− kNN(cid:1) and βN = N√f(kαNN).
at least αlog(N) for α > 1. Hence, in order to fulfill this We now state the main result of this section:
requirement the expected number of channels in which each Theorem 5: Let AGEE be the optimal solution to the max-
opt
user is able to transmit without violating his power constraint energy efficiency problem for the GEE and let AGEE be the
fast
is at least αlog(N). A user can transmit on a channel only if solutionobtainedbythefastassignmentalgorithm.Weassume
the power needed on the channel to achieve the target rate is thattheratesareproperlychosenandaperfectmatchingexists.
less than P . Then for Rayleigh channels
max
F (P )≥ αlog(N) (28) E(cid:0)AGEE(cid:1)
P(n,k) max N lim fast =1 (35)
(cid:0) (cid:1)
Hence, the target rates for each user must satisfy: N→∞E AGopEtE
(cid:18) (cid:19)
F−1 αlog(N) ≤P , (29) Proof: We start by formulating the probability distribution
P(n,k) N max andquantilefunctionofU(n,k)giventhatthepowerislower
The inverse CDF of FP(n,k)(x) is given by than Pmax.
F−1 (x)= λnσn2(cid:0)2Rn −1(cid:1). (30) FUGEE(n,k)(x)=1−ePamnaxe−Pmaanx−x (36)
P(n,k) log(1)
Hence, the target rates must satisfy x where an =λnσn2(cid:0)2Rn −1(cid:1) and
FP−(n1,k)(cid:18)αloNg(N)(cid:19)= λlnoσgn2(cid:16)(cid:0)2RNn −(cid:17)1(cid:1) ≤Pmax (31) FU−G1EE(n,k)(ρ)=Pmax+ log(1−aρn)− Pamnax. (37)
αlog(N)
We now observe that this probability distribution satisfies the
which implies that if all R satisfy:
n
third Von Mises condition and as a result
 (cid:16) (cid:17)
P log N
max αlog(N) (cid:16) (cid:17) αlog(N)
Rn ≤log21+ λnσn2 . (32) Nl→im∞E UGNE−Eαlog(N)+1:Na =F−1(1− N )=
=P + n
then the fast algorithm converges with an expected max log(log(N))−log(N)+ an
O(Nlog(N)) number of iterations. Pmax (38)
7
WecannowobtainsimpleboundsonE(cid:0)AGEE(cid:1)andE(A ) TABLEIV
fast opt MATCHINGALGORITHMUSINGOPPORTUNISTICCARRIERSENSINGFOR
. THEn’THUSER
N
E(cid:0)AGEE(cid:1)≤ (cid:88)E(A )= Initializehk=0,∀k∈K,setassigned=false
opt N:N setbn tobetheindicesofαlog(N)best
n=1 (39) channelsofthen’thuser,setNiter=0
=NP −(cid:88)N an R1ep.eIfatassigned=falsethen
max n=1log(N)− Pamnax 11..12NFiitnerd=theNcithera+nne1lwithminimalvaluej=argminm∈bnhm
and 1.3Chooserandombackofftimeτn
1.4Ifchannelmwasaccessedbeforeτn then
N
E(cid:0)AGfaEstE(cid:1)≤ (cid:88)E(AN:N)= 1.t4i.m1eNsolottransmissionattemptbythen’thuserinthecurrent
n=1 (40) 1.4.2hm=hm+1
N 1.5Else
=NP −(cid:88) an 1.5.1Accessthej’thchannel.
max n=1log(N)−log(αlog(N))− Pamnax 11..55..23Shejt=asshigjn+ed1=true
it is now easy to see 1.6Endif
2.Else
NPmax−(cid:80)Nn=1 log(N)−log(αalnog(N))−Pamnax ≤ 2.s1etIfhcmha=nnehlmm+w1asaccessedbeforeτmax then
NPmax−(cid:80)Nn=1 log(Na)−n an (41) 22..23NIfimter==Njiter+1
E(cid:0)AGEE(cid:1) Pmax 2.3.1Notransmissionattemptbythen’thuserinthecurrent
≤ fast ≤1 timeslot
E(Aopt) 2.3.2Setassigned=false
2.4Elseaccessthej’thchannel.
However,
3.EndIf
Nl→im∞NPmaxN−Pm(cid:80)axNn=−1(cid:80)logNn(=N1)−lologg(N(αaa)l−nnog(aNn))−Pamnax =1. EUIfnnNdtriuilItnefarlt=lheu(sdNeirsst−raibr1eu)ta2esdsiaguncetdioonraNlgioterrit=hm(Nin−Ta1b)le2I.
Pmax (42)
and as a result
E(cid:0)AGEE(cid:1) 2 shows the ratio between the expected transmitted power
lim fast =1. (43)
N→∞E(cid:0)AGEE(cid:1) achievedbythefastmatchingprotocolandtheexpectedtrans-
opt
mitted power achieved by the distributed auction algorithm. It
Theorem 6: Let Aav be the optimal solution to the max-
opt
energy efficiency problem for the EEav and let Aav be the Ratio of the expected power by the fast algorith
fast and the distributed auction
solutionobtainedbythefastassignmentalgorithm.Weassume
6
thattheratesareproperlychosenandaperfectmatchingexists.
5.5
Then for Rayleigh fading
5
E(Aav )
lim fast =1 (44) 4.5
(cid:0) (cid:1)
N→∞E Aaovpt atio 4
Proof: The proof is identical to the proof of Theorem 5. er r3.5
w
The only difference is that the CDF FUav(n,k)(x) is given by: Po 3
2.5
FUav(n,k)(x)=1−e−λn(R2nR−nc−(n1))xσn2x (45) 2
1.5
It is easy to verify that all arguments applied to the CDF in 1
Theorem 5 also apply to this CDF.
0 50 100 150 200 250 300 350 400 450 500
Thefullydistributedmatchingalgorithmforcognitiveradio Number of users
systems using prioritized CSMA is depicted in Table IV.
Fig. 2. Ratio of the average optimal EE and the sum rate obtained by the
matchingalgorithm.
VI. SIMULATIONS
In this section we present simulated results for the methods is easy to see that the ratio converges to 1 with the number
described in the paper. The simulated results are consistent ofusers.ThiscoincideswithTheorem5whichstatesthatthis
withthetheory.Inthefirstsetofsimulationswecomparedthe ratioshouldconvergeto1asthenumberofusersandchannels
distributed auction algorithm and the fast matching algorithm. goes to infinity. In the next set of simulations we investigated
We generated N ×N random energy efficiency matrices for theexpectednumberofiterationsachievedbybothalgorithms.
independentRayleighchannelswithN varyingfrom22 to29. First, we verified Theorem 18 which states that the number of
ThetargetrateswerechosenaccordingtoSectionV-Aandthe iterations in the fast matching algorithm exceeds cNlog(N)
thresholdwaschosentobeathresh =F−1(1−2log(N)).Figure with a probability of less than 1. In Figure 3 we plotted the
n N N
8
empiricalprobabilitythatthefastalgorithmexceedsNlog(N) We then presented a fast algorithm to solve the problem.
iterationsagainstthetheoreticalboundof 1 forN =10...103. The algorithm was shown to converge within O(Nlog(N))
N
Again, the simulations support the theoretical results. iterations with high probability. We showed that under mild
assumptions, the fast algorithm can produce asymptotically
Probability of more than Nlog(N) iterations optimal solutions to the max energy efficiency problem. We
100
then showed how to implement the matching algorithm in a
Simulated results
Upper bound fully distributed manner using prioritized CSMA.
10−1
APPENDIX
e
ur
ail Here we present an outline with the lemmas and theorems
of f used in the proof without the proof of the lemmas. The full
y 10−2
bilit proof is given in [36]. We start by noting the following trivial
a
b property of the fast matching algorithm:
o
Pr Lemma 7: Let T be the number of iterations until the
10−3
algorithm terminates and let h be the value of vertex v at
v
termination, then
N
(cid:88)
10−140 1 102 103 T = hv (46)
Number of users v=1
Proof: The proof is trivial since on every iteration the value
Fig.3. ProbabilitythatthealgorithmterminatesaftermorethanNlog(N) of exactly one vertex is increased by 1.
iterations.
Now, the theorem would be proven if we can prove that
After illustrating the correctness of Theorem 18 we com- with high probability the maximal value of hv is less than
pared the expected number of iterations of the fast matching αlog(N). This is done by proving the following claims:
algorithmwiththeexpectednumberofiterationsuntilconver- Lemma 8: Let G = (U,V,E) be a bipartite graph with
gence of the distributed auction algorithm. The comparison is vertex sets |U|=|V|=N and an edge set E. Let M(i)⊆E
shown in Figure 4. As predicted, the expected number of iter- be a non maximal matching obtained by the algorithm in the
ations needed by the fast auction algorithm was smaller than i’th iteration. Let hv(i) be the value of vertex v in the i’th
the expected number of iterations needed by the distributed iteration of the algorithm. Let Dl(i) be a subset of V defined
auction algorithm. by:
D (i)={v ∈V :h (i)≥l}. (47)
l v
Comparison between the distributed auction
and the fast matching algorithm If v0 ∈Dl(i) and (u,v0)∈M(i) then
1000
n ⊆D (i) (48)
900 u l−1
800 An illustration of lemma 8 is given in Figure 5.
ns 700 Lemma 9: LetG=(U,V,E)bebipartitegraphwithvertex
atio 600 sets |U| = |V| = N and an edge set E. Let M(i) be a
er of iter 500 nitoenratmioanx.imLeatl mu0at∈chiUngboebtaaifnreede bvyerttehxe saulgcohritthhamt iinn tthhee ii’’tthh
umb 400 iteration of the algorithm nu0 ⊆ Dl(i). Let u1 ∈ U be the
N 300 end point of an alternating path P with |P|=2 starting from
200 u0 then
Fast matching
100 Distributed auction v (cid:54)∈D (i)\D (i) ∀v ∈n (49)
l−2 l−1 u1
0
0 100 200Numbe3r0 o0f users400 500 600 Lemma 10: Let G = (U,V,E) be bipartite graph with
vertex sets |U| = |V| = N and an edge set E. Let M(i)
be a non maximal matching obtained by the algorithm in the
Fig. 4. Comparison between the expected number of iterations using the
distributedauctionalgorithmandthefastmatchingalgorithm i’th iteration and let u ∈ U be a free vertex such that in
0
the i’th iteration of the algorithm n ⊆ D (i); then every
u0 l
augmenting path of G on M(i) starting from u is at least of
0
VII. CONCLUSION length 2l+1.
In this paper we suggested fully distributed MAC protocols A well-known theorem by Berge [39] states that if a
for energy efficiency optimization for autonomous cognitive bipartite graph G contains a perfect matching there exists
units. We considered two measures for the energy efficiency, an augmenting path in G with respect to any non maximal
the GEE and the EEav criteria. To solve the optimization matching M.
problem we first suggested the use of the distributed Auction Theorem 11: [39] Let G = (U,V,E) be a bipartite graph
algorithm to maximize the energy efficiency of the network. with vertex sets |U| = |V| = N and an edge set E. If G
9
Definition 11: G ∈ B˜(N,p) if G ∈ B(N,p) and for any
non-maximalmatchingM thereexistsanaugmentingpathfor
M of length at most 2L+1 where L= c˜log(N) and c˜>0 is
log(Np)
some constant.
Lemma 15: Let G ∈ B˜(N,p) and let and let T be the
number of iterations until the algorithm terminates then
T ≤N(L+1). (52)
where L= c˜log(N).
log(Np)
The following theorem was proven in [40]:
Theorem 16: LetG∈B(N,p)wherep≥ (1+(cid:15))log(N) then
N
for every γ >0 there exists N such that for every N ≥N
γ γ
Pr(G∈B˜(N,p))≥1−N−γ. (53)
The following theorem was proven in [41]:
Theorem 17: Let G = (U,V,E) ∈ B(N,p) and p =
clog(N),c>2 then
N
lim Pr(G contains a perfect matching)=e−2N1−c. (54)
N→∞
Using the above theorems and lemmas Theorem 18 is
proven
Theorem 18: LetG∈B(N,p)bearandombipartitegraph
with N > N vertices on each side and p = clog(N),c > 2.
γ N
Then the expected number of iterations until the algorithm
terminates is:
(cid:18) (cid:19)
Nlog(N)
Fig.5. Anillustrationoflemma8 E(T)≤O (55)
log(Np)
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