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Joint Resource Allocation for Max-min Throughputin Multicell Networks
Zhuo Li, Member, IEEE, Song Guo, Senior Member, IEEE, Deze Zeng, Member, IEEE,Ahmed Barnawi, Member, IEEE, and Ivan Stojmenovic, Fellow, IEEE
Abstract—We investigate the resource allocation problem inmulticell networks targeting at max-min throughput of all cells.A joint optimization over power control, channel allocation anduser association is considered and the problem is then formulatedas a non-convex mixed integer non-linear problem (MINLP). Tosolve this problem, we proposed an alternating optimizationbased algorithm, which applies branch-and-bound and simu-lated annealing in solving subproblems at each optimizationstep. We also demonstrate the convergence and efficiency ofthe proposed algorithms by thorough numerical experiments.The experimental results show that joint optimization over allresources outperforms the restricted optimization over individualresources significantly.
Index Terms—Resource Management, Channel Assignment,Power Allocation, User Association, Load Balancing, MulticellNetworks
I. INTRODUCTION
IN order to eliminate dead-spots and to improve quality ofservice (QoS) in wireless networks, multicells are usually
deployed in practical systems like wireless local area networks(WLAN) and cellular networks. Recently, femtocell network,as a new form of multicell network, has been widely adoptedin the new data-minded cellular standards. [1]. Because densedeployment of small cells is beneficial for both users andoperators, special attentions for developing multicell solutionshave been received recently from both industry and academia[2].
The communication resources in a multicell network consistof access points (AP), power and frequency bands (codes).For convenience, we denote the frequency band or code as“channel” (CH for short) in this paper. To achieve desirednetwork performance in supporting a group of users, theallocation of these communication resources should be jointlyoptimized. On one hand, the link capacity between an AP anda receiver is an increasing function of the transmission power.On the other hand, high transmission power would generatestrong interference on the links nearby using the same channel,
Z. Li is with Beijing Key Lab of Internet Culture and Digital DisseminationResearch, Beijing Information Science & Technology University, Beijing,China, and State Key Lab of Novel Software Technology, Nanjing University,Nanjing, Jiangsu, China. E-mail: [email protected]
S. Guo is with the School of Computer Science and Engineering, TheUniversity of Aizu, Japan. E-mail: [email protected]
D. Zeng is with School of Computer Science, China University of Geo-sciences (Wuhan), China. E-mail: [email protected]
A. Barnawi is with King Abdulaziz University, Jeddah, Saudi Arabia. E-mail: [email protected]
I. Stojmenovic is with the SIT, Deakin University, Melbourne, Australia;King Abdulaziz University, Jeddah, Saudi Arabia; and SEECS, University ofOttawa, Canada. E-mail: [email protected]
and thus decrease their capacity. Power control handles suchtrade-off by choosing appropriate transmission power of APs.Because there are multiple channels that can be used, espe-cially under the widely-supported cognitive radio technologies[3], channel allocation tries to assign different channels toneighboring links to reduce interference. In densely deployedmulticells networks, a user may be covered by several APs.It should select one to connect with. The more users an APis connected with, the heavier load it will experience. Theusers should associate with the APs appropriately in order toguarantee the load balance.
Power control, channel allocation and user association arethree optimization issues for the resource allocation in wirelessmulticell networks. Most existing work focuses on one at atime. In this paper, we apply a joint optimization approachfor resource allocation in a multicell network, which can beinvoked at the network planning stage or when the resource s-tatus changes. We target at maximizing the minimal throughput(max-min throughput) of the multicells and formulate it as anon-convex mixed integer non-linear problem (MINLP). Thisgoal simultaneously reflects the capacity of whole networkand the fairness among different cells. Due to the hardnessof MINLP, we propose an alternating optimization basedalgorithm to achieve the goal. The basic idea is to use iterativesteps to optimize the objective jointly over all resources until astop condition (i.e., either the maximum number of alternatingoptimization rounds is reached or the quality of the solutionis acceptable) is satisfied. We alternate restricted optimizationover the individual resources. Each step of the method consistsof two subroutines. Under a fixed power assignment, theoptimal channel allocation and user association is achieved bya simulated annealing based subroutine. This result is the inputof the second subroutine that solves the power optimizationproblem using a branch-and-bound method. We have also eval-uated the proposed algorithms by comprehensive simulations.Our main contributions are summarized as follows:
1) For the sub-problem of channel allocation and user asso-ciation, we prove they are both NP-complete. Further-more, we prove that when the number of channels isequal to or larger than 3, there is no polynomial-timealgorithm with approximation ratio ρ ≥ 1 for the channelallocation problem unless NP = P . Similarly, we alsoprove that there is no polynomial-time algorithm withapproximation ratio 1 ≤ ρ < 2 for the user associationproblem unless NP = P . We then propose simulatedannealing based heuristic algorithm for this sub-problem.
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2) For the sub-problem of power control, we show it isessentially a D.C. (Difference of Convex Functions) Pro-gramming problem [4] and propose a branch-and-boundalgorithm. In particular, we use a linear relaxation methodto estimate the lower bound and prove the convergenceof the algorithm.
3) We propose an alternating optimization based algorithmfor the joint allocation of power, channel and AP for eachuser. The high performance and efficiency of our algorith-m makes it suitable for both initial network planning andsubsequent resource reallocation when necessary (e.g.,user join-in/leave and channel update).
4) In our simulation studies, we analyze the impact of dif-ferent network parameters on convergence and efficiencyof our proposed algorithms. We find that, in an unplannednetwork, user association and channel allocation playa more important role than power control in the opti-mization of max-min throughput. We have also observedthat jointly optimization over all resources significantlyoutperforms the traditional optimization over one kind ofresource.
The remainder of the paper is organized as follows. SectionII reviews the related work. Section III presents the networkmodel and problem formulation. The hardness of each sub-problem is analyzed and an alternating optimization algorithmis then proposed for the max-min throughput problem inSection IV. Various simulated performance results confirmingthe proposed algorithm are presented in Section V. Finally,concluding remarks are given in Section VI.
II. RELATED WORK
In multicell wireless networks, user association with APs,communication channel allocation and power control are thethree most influential issues to the network performance. Mucheffort has been contributed to orchestrating the communicationresources in multicell networks for various optimization pur-poses.
A. Single-Resource based Optimization
On user association, Bejerano et al. [5] propose an efficientsolution to determine the user-AP associations for max-minfair bandwidth allocation with a constant-factor approxima-tion. Later on, Xu et al. [6] propose a game theoretical modelfor the user association problem using the congestion metric“airtime cost” to alleviate that load imbalance among APs.
Ji-Hoon Yun et al. [7] propose a power control algorithmfor the femtocell users. The maximum power is boundedin order to guarantee the load at the macrocell is below athreshold. Nash equilibrium is achieved among the femtocellusers through distributed power control. Imran Ashraf et al.[8] propose a novel power saving procedure according to thepattern of user activities.
For channel allocation, Taeyoung Lee et al. [9] investigatea static channel allocation problem in femtocell networksunder the assumption that the channels of the macrocell usersare fixed. Benmesbah et al. [10] propose a filled functionbased decentralized spectral resource allocation on downlinks.
Recently, Xu et al. [11] model the channel assignment tobalance the traffic load of APs operating on different channelsas a non-cooperative game, whose Nash equilibrium is proved.
B. Multiple-Resource based Optimization
The joint optimizations on power control and user associ-ation are addressed in [12], [13]. The former takes a methodbased on Bender’s Decomposition, and the latter uses Gibbsbased sampling. However, both of them are just based on asingle-channel downlink communication.
The additional consideration of channel allocation has justreceived attention. For example, Torregoza et al. [14] study theoptimization problem to maximize the aggregate throughputin femtocell networks during operation stage, while the loadbalancing among different cells is not considered.
For user association and channel allocation, Fooladivanda etal. [15] formulate a non-linear integer programming problemto maximize the minimum user rate by jointly consideringthese two factors under a fixed power setting. Fallgren et al.[16] relax this limitation for similar optimization problem thatis to maximize either the minimum user throughput or themulticell sum throughput. They find that a simple “Greedy”link allocation algorithm performs very well to maximizemulticell sum throughput. Its basic idea is to always assigna user the AP with the highest path gain until all the channelson the same AP are used up.
The joint optimization of all three essential resources inmulticell networks has been also investigated more recent-ly. Chen et al. [17] propose an algorithm based on GibbsSampling, which is more suitable for the problems withdiscrete, instead of continuous, power settings. Borst et al. [18]propose a distributed resource allocation algorithm frameworkfor nonconcave utility maximization based on an improvedGibbs Sampler, i.e., Constrained Gibbs Sampler (CGS). Theirresults highly rely on a strong assumption: the global objectivein a sum of all local objectives, that does not apply to ournetwork model. Similar to our approach, the proposal in [19]also performs an alternating optimization technique. However,it targets an optimization goal different with ours. Furthermore,the algorithms proposed for power control and user associationhave no performance guarantees.
III. NETWORK MODELS AND PROBLEM FORMULATION
We consider a fixed multicell network with n APs and musers. In this paper, we mainly focus on downlink commu-nication. Each user associates with only one AP. The overallassociation status is denoted by an n×m matrix A, in whicheach entry is defined as
ai,j =
{1 user j is associated with AP i0 otherwise
andn∑i=1
ai,j = 1 for any user j. In our network model, there
are c non-overlapping (i.e., orthogonal) channels, each with thesame bandwidth B. The channel allocation for all the userscan be denoted by a m× c matrix C, in which each entry isdefined as
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cj,k =
{1 user j is assigned channel k0 otherwise
andc∑
k=1
cj,k = 1. Each AP will be allocated a subset of
channels. These channels, once allocated, will be occupied allthe time by the AP to support its associated users. If differentchannels are assigned to the users, the AP will communicatewith them simultaneously. Otherwise, the users assigned thesame channel will be served in turn, i.e., in a TDMA fashion,by sharing the transmission opportunities evenly.
We denote the power assignment of the APs as P = {pi}and the power of each AP pi can be adjusted continuouslyin [Pmin, Pmax]. Let the 1 ×m matrix N = {nj} denote thethermal noise nj at user j and the n×m matrix H = {hi,j}denote the channel gain hi,j between the AP i and user j.Since the network is fixed, the values of N and H can beestimated in advance. We apply an AP-level power controlscheme [20] that is usually adopted for down-link transmissionin traditional mutlicell networks such as WLAN and cellularnetworks. Under such model, for any AP i, it communicateswith its associated users using the same power level pi. Forthe transmission from AP i to user j, the interference madeby AP l will be pl ·hl,j if it serves its associated users on thesame channel as j. Therefore, the signal-to-interference-plus-noise ratio (SINR) at the user j with respect to AP i can beexpressed as
SINRi,j(P,A,C) =pi · hi,jnj + Ii,j
,
where
Ii,j =
c∑k=1
n∑l=1,l 6=i
pl · hl,j · xl,k · cj,k,
xl,k =
{1,
∑mv=1 al,v · cv,k ≥ 1;
0, otherwise.
Note that the binary xl,k represents whether AP l allocateschannel k for its user(s) or not.
According to Shannon-Hartley theorem [21], if we considerinterference as noise in narrow-band systems, the link capacitybetween AP i and user j is:
Ci,j(P,A,C) = ai,j ·B · log2(1 + SINRi,j(P,A,C)).
Without loss of generality, in this paper we regard B as unitbandwidth and di,j as the transmission delay of a packeton the link between i and j. Furthermore, we consider thatthe downlink traffic is always saturated between any pair ofAP and user. Thus the throughput of multicell AP i can becalculated as
Ti =
m∑j=1
1
di,j=
m∑j=1
Ci,j(P,A,C)∑mv=1 ai,v ·
∑ck=1 cv,k · cj,k
. (1)
Note that the denominator∑mv=1 ai,v ·
∑ck=1 cv,k · cj,k in the
above formula represents the number of users associated withthe same AP i and allocated the same channel as user j. Theseusers will utilize the channel capacity Ci,j evenly.
When the user association is fixed, the throughput Ti canbe optimized through power control and channel allocation. Ahigher Ti always means better service for the users associatedwith i. On the other side, when the power assignment andchannel allocation is fixed, Ti could be improved if more usersassociate with it. However, this may cause load unbalance andmake some APs overloaded. From a network deployer’s per-spective, the load balancing among different cells is desired.We formulate it as a max-min cell throughput problem asfollows.
max min Ti (2)
s.t.n∑i=1
ai,j = 1, 1 ≤ j ≤ m
c∑k=1
cj,k = 1, 1 ≤ j ≤ m
ai,j , cj,k ∈ {0, 1}, 1 ≤ i ≤ n, 1 ≤ j ≤ m, 1 ≤ k ≤ cPmin ≤ pi ≤ Pmax, 1 ≤ i ≤ n
Such optimization should be made at both network planningstage and when a network update (e.g., user join/leave orchannel update events) happens.
IV. ALGORITHMS
A. Simulated Annealing Based Channel Allocation and UserAssociation
In this section we first analyze the hardness of sub-problemsof channel allocation and user association with the same max-min throughput objective as formulated in (2), and then weprovide a metaheuristic algorithm.
Definition 1 (D-Max-Min-CA): INSTANCE: A multicel-l network with n APs, m users and c non-overlapped channels,in which the power assignment for each AP and user associ-ation are fixed, and a positive constant δ.QUESTION: Is there a channel allocation scheme for themulticell network such that δ ≤ Ti for any 1 ≤ i ≤ n, wherethe throughput Ti of multicell i is defined in (1)?
Theorem 4.1: If the number of non-overlapped channels cis equal or larger than 3, the problem D-Max-Min-CA is NP-complete.
Proof: We note that the D-Max-Min-CA problem is in NPas the objective function in (2) can be evaluated in polynomialtime. So we only need to prove it NP-hard. It will be doneby reducing the well-known NP-hard k-colorability problem(k ≥ 3) [22] to D-Max-Min-CA. Given a graph G with nnodes, we construct a multicell network with n APs, n usersand c = k channels. Each node i (1 ≤ i ≤ n) in G correspondsto a multicell, which consists of an AP i and an associateduser i. Other input parameters P , A, N and H of the D-Max-Min-CA problem are set as follows accordingly. (1) For any1 ≤ i ≤ n, we set pi = p > 0. (2) For any 1 ≤ i ≤ n and1 ≤ j ≤ n, we set ai,j = 1 if i = j, or ai,j = 0 otherwise.(3) For any 1 ≤ i ≤ n, we set ni = 1. (4) For any 1 ≤ i ≤ nand 1 ≤ j ≤ n, we set hi,j = (2δ − 1)/p, i = j;
hi,j > 0, the link (i, j) is in G;hi,j = 0, otherwise.
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BS-A
BS-B
BS-C
A
B
C
Fig. 1. An example to illustrate the reduction from the k-colorability problemto the D-Max-Min-CA problem.
Fig. 1 shows an example of such reduction, where onlychannels between an AP and a client with positive gainsare represented by dashed lines. By these settings, it isstraightforward to discover that the throughput Ti of multicelli achieves δ iff there is no interference at user i. Consideringa 1:1 mapping between node coloring over G and channelallocation over the corresponding multicell network, we notethat interference from AP i to user j happens iff (i, j) is in Gand nodes i and j have the same color. Therefore, a channelallocation scheme for the D-Max-Min-CA problem exists iffG is k-colorable. The final conclusion is achieved since theabove reduction can be done in polynomial time.
Definition 2 (O-Max-Min-CA): QUESTION: Find a chan-nel allocation scheme for the links in multicells such thatmin1≤i≤n Ti is maximized.
The approximation ratio of an algorithm for the O-Max-Min-CA problem is ρ iff the value of an optimal solution willnot be more than a factor ρ times the solution produced bythe algorithm for any instance.
Theorem 4.2: There is no polynomial-time algorithm withapproximation ratio ρ ≥ 1 for the O-Max-Min-CA problemwhen c ≥ 3 unless P = NP .
Proof: This proof is done by contradiction. Suppose thatfor some ρ ≥ 1, there is a polynomial-time algorithm Q withapproximation ratio ρ for the O-Max-Min-CA problem. Letδ∗ and δ be the optimal solution and the one obtained by thealgorithm Q, respectively. Then we have
δ∗ ≤ ρ · δ. (3)
We will show that Q can be used to solve the k-colorabilityproblem. Given a graph G with n nodes, we construct acorresponding multicell network in a similar way as in theprevious proof except that H is set as:
hi,j =
(2δ − 1)/p, i = j;( 2δ−1
2δ′−1− 1)/p, the link (i, j) is in G;
0, otherwise.
in which δ and δ′ satisfy
δ > ρ · δ′. (4)
By these settings, we can conclude that for any channelallocation, Ti (1 ≤ i ≤ n) satisfies{
Ti = δ, no interference at user i;Ti ≤ δ′, otherwise. (5)
Fig. 2. An example to show the improvement of minimal throughput byload-balancing.
Note that the equality will hold in the second case of (5)when the interference at user i comes from only one AP near-by. Considering the relation between G and the constructedmulticell network, we can conclude that G is k-colorable iffδ∗ = δ. In the following, we show that δ∗ = δ iff δ > δ′.In other words, the k-colorability problem can be solved bychecking δ > δ′, which can be answered by the Q algorithmin polynomial time.
We consider the if case first. Referring to (5), δ =min1≤i≤n Ti > δ′ means δ = δ. Since any solution of the O-Max-Min-CA problem under the constructed multicell networkcannot exceed δ, we thus obtain δ∗ = δ = δ. For the only ifcase, we combine (3) and (4) under the condition δ∗ = δ toachieve ρ · δ ≥ δ∗ = δ > ρ · δ′, i.e., δ > δ′.
Now we consider the user association subproblem. Thesolution of this problem can be used to improve the max-min throughput by load balancing under the fixed powerand channel assignments. Fig. 2 gives an example, wherethe capacity of links between APs and users is the same.We observe that the minimal throughput of the APs can beimproved after load-balancing. The hardness of the problemis analyzed in the following theorems.
Definition 3 (D-Max-Min-UA): INSTANCE: A multicel-l network with n APs, m users, and c channels, in whichthe power assignment for each AP and the channel allocationfor each user are fixed, and a positive constant δ.QUESTION: Is there a user association scheme for the mul-ticell network such that δ ≤ Ti for any 1 ≤ i ≤ n, where thethroughput Ti of multicell i is defined in (1)?
Theorem 4.3: The problem D-Max-Min-UA is NP-complete.
Proof: D-Max-Min-UA is obviously in NP and we showits NP-hardness by reducing the NP-hard partition problem toit. Given a set of integers S = {δ1, δ2, · · · , δm}, we constructa multicell network with two APs, m users, and m channels,each assigned to a different user. We simply set other inputparameters such that the capacity Ci,j is equal to δj , for anyi = 1, 2 and 1 ≤ j ≤ m, if user j is associated with AP i.A partition {S1, S2} of S corresponds to a user associationscheme, where Si = {δj | user j is associated with AP i} fori = 1, 2. If δ =
∑mj=1 δj
2 , S can be partitioned into two “halves”iff there is a user association scheme for the problem D-Max-Min-UA. The above reduction in polynomial time leads to theNP-hardness of the D-Max-Min-UA problem.
Definition 4 (O-Max-Min-UA): QUESTION: Find a userassociation scheme for the users in multicells such thatmin1≤i≤n Ti is maximized.
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Algorithm 1 Simulated Annealing based CA and UA1: Estimate the average performance decrease ∆η of the
move from initial solution by running a number of sim-ulations. Set T = − ∆η
ln 0.8 , iNum = 0, sucNum = 0,rejNum = 0 and finished = 0.
2: Choose an initial random solution S = {A0, C0}. Recordthe best solution that have been found S∗ = S.
3: while !finished do4: iNum = iNum+ 1.5: Construct a neighbor S′ of S as follows. Select a user
and randomly update its associated AP.6: Select a user and randomly update its channel.7: Calculate the values of (2) under the solutions S and
S′, which are denoted as η and η′ respectively.8: if η ≤ η′ then9: S = S′, S∗ = S, sucNum = sucNum +
1, rejNum = 010: else11: With probability p = e
η′−ηT , S = S′ and
sucNum = sucNum + 1. With probability 1 − p,rejNum = rejNum+ 1.
12: end if13: if iNum ≥ iNum∗ ‖ sucNum ≥ sucNum∗ then14: if T < stopTemp ‖ rejNum ≥ rejNum∗ then15: finished = 116: else17: γ · T → T, iNum = 1, sucNum = 118: end if19: end if20: end while21: Return S∗
Theorem 4.4: There is no polynomial-time algorithm withapproximation ratio 2 − ε (ε > 0) for the O-Max-Min-UAproblem unless P = NP .
Proof: It has been proven in [23] that there is nopolynomial-time algorithm with approximation ratio 2 − ε(ε > 0) for the Max-Min Allocation problem unless P = NP .The proof is made by transforming the Max-Min Allocationproblem to an equivalent O-Max-Min-UA problem. The Max-Min Allocation problem states given m indivisible items, nusers, and a utility ui,j associating the assignment of item ito user j, how to allocate the items to the users such that theminimum utility of the user is maximized. We show that it isequivalent to the O-Max-Min-UA problem under an instanceconstructed as follows. The corresponding multicell networkconsists of n APs, m users, and sufficient number of channelssuch that each user has a dedicated channel. We also set otherinput parameters such that the capacity Ci,j is equal to ui,j ,for any 1 ≤ i ≤ n and 1 ≤ j ≤ m, if user j is associated withAP i. Therefore, assigning item i to user j in the Max-MinAllocation problem corresponds to associating user j to APi in O-Max-Min-UA. It is straightforward to verify that theoptimal solutions of the two problems are equal. Finally, theconclusion is achieved because the above process is done inpolynomial time.
Because of the inapproximability of the O-Max-Min-CAand O-Max-Min-UA problems, we make use of metaheuristicand propose a simulated annealing [24] based algorithm asdescribed in Algorithm 1. The algorithm starts searching forthe optimal solution from a random allocation scheme, inwhich the user chooses an AP and a channel to connect withrandomly.
There are two steps in choosing a neighbor S′ from acandidate solution S. Firstly, an arbitrary user is selected andits associated AP is randomly reset. Secondly, an arbitrary useris selected and its channel is randomly updated. We take thisrandom way to increase the probability that a global optimalsolution can be reached. Let the solution of problem (2) underS′ and S be η′ and η, respectively. Solution S′ will be acceptedas a candidate solution when η ≤ η′, or just with probabilityeη′−ηT otherwise. It is noted that Algorithm 1 can traverse from
a candidate solution to its neighbor with worse performance.In this way, a local optimal solution would be avoided.
Algorithm 1 applies a cooling scheme, in which parameterT is iteratively updated as T = γ ·T with 0 < γ < 1. Note thatthe setting of parameter T affects the performance. Accordingto [25], T should guarantee that the average acceptanceprobability is 0.8 when the selected neighbor of the initialsolution has a worse performance. Thus, we set T = − ∆η
ln 0.8as the initial value, where the average performance decrease∆η of the move from initial solution is estimated by runninga number of simulations. Finally, the algorithm terminateswhen T falls below a certain temperature stopTemp or itfails to find a candidate solution over rejNum∗ times. Thebest candidate solution will be returned as the final solution.Note that parameters iNum∗ and sucNum∗ are introducedto achieve a tradeoff between the computation complexity andthe optimization quality. The former denotes the maximumnumber of iterations, and the latter denotes the maximumnumber of neighboring solutions to be traversed. When theyare sufficiently large integers, Algorithm 1 converges almostsurely to a global optimal solution as proved in [26].
B. An Algorithm for Power Control
When ai,j and cj,k are fixed, the problem (2) is simplified tofind the optimal P . This is a non-convex optimization problem,which is hard to get an optimal solution. In the following, wepropose a branch-and-bound based algorithm.
By defining the constants Si,j and bil,j :
Si,j =ai,j∑m
v=1 ai,v ·∑ck=1 cv,k · cj,k
,
bil,j =
{0, l = ihl,j ·
∑ck=1 xl,k · cj,k, l 6= i,
the max-min problem formulated in (2) can be rewritten as a
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Fig. 3. Linear relaxation of constraints in problem (6).
min-max one in follows.
min δ (6)
s.t.m∑j=1
Si,j · log2(nj + Ii,j)−m∑j=1
Si,j · log2(nj + Ii,j
+ pi · hi,j) ≤ δ, 1 ≤ i ≤ n
Ii,j =
n∑l=1
pl · bil,j , 1 ≤ j ≤ m
Pmin ≤ pi ≤ Pmax, 1 ≤ i ≤ n
Note that δ in (6) refers to max1≤i≤n(−Ti) in problem(2) and thus problem maxP,A,C min1≤i≤n Ti and problemminP,A,C max1≤i≤n(−Ti) are equivalent. Actually, problem(6) is a D.C. Programming problem [4], [27]. For convenience,we denote
gi(P ) = −m∑j=1
Si,j · log2(nj + Ii,j + pi · hi,j),
hi(P ) =
m∑j=1
Si,j · log2(nj + Ii,j).
In the following, we shall derive a lower bound gli(·)and hli(·) for gi(·) and hi(·), respectively. According to thegeometric feature of convex function of gi(·), for P ≤ P ≤ P ,gi(P ) is lower bounded by
gli(P ) = gi(P + P
2) + (P − P + P
2) · ∇gTi (
P + P
2),
where ∇gi(P ) = (∂gi(P )∂p1
, . . . , ∂gi(P )∂pn
) is the gradient of gi atP . By applying mean value theorem over hi(·), we have
hi(P ) = hi(P ) + (P − P ) · ∇hTi (ζ),
where P ≤ ζ ≤ P ≤ P . Let Pi denote the i-th entry of a
Algorithm 2 Optimal Power Control Algorithm
1: Initialization: k = 1; Sk = [P k, Pk], P k =
(Pmin, . . . , Pmin), Pk
= (Pmax, . . . , Pmax); set the lowerbound as αk = F∗l (Sk) and the corresponding powerallocation as P k; set an upper bound as βk = F(P k);set the search space as Rk = {Sk}.
2: while αk < βk − ε do3: Partition Sk into Sk1 and Sk2 such that Sk1 = {P ∈
Sk | Pi ≤ Pki+Pki
2 } and Sk2 = Sk \ Sk1 , in which i =
arg max1≤j≤n{Pk
j − Pkj }.
4: Set Pi (i = 1, 2) as the optimal power allocation thatachieves F∗l (Ski | δl ≥ αk).
5: βk+1 = min{βk,F(P1),F(P2)}6: P k+1 = arg min{F(P k),F(P1),F(P2)}7: Rk+1 = Rk \ {Sk} ∪ {Sk1 , Sk2 }8: Rk+1 = Rk+1 \ {S ∈ Rk+1 | F∗l (S | δl ≥ αk) ≥
βk+1}9: if Rk+1 = φ then
10: Return P k+1
11: end if12: αk+1 = minS∈Rk+1 F∗l (S | δl ≥ αk)13: Sk+1 = arg minS∈Rk+1 F∗l (S | δl ≥ αk)14: k = k + 115: end while16: Return P k
given vector P . We can further derive
∂hi(ζ)
∂pl=
1
ln 2·m∑j=1
Si,j · bil,jnj +
∑nl=1 ζl · bil,j
≥ 1
ln 2·m∑j=1
Si,j · bil,jnj +
∑nl=1 P l · bil,j
=∂hi(P )
∂pl,
i.e., ∇hi(ζ) ≥ ∇hi(P ). Therefore a lower bound of hi(P ) isachieved as
hli(P ) = hi(P ) + (P − P ) · ∇hTi (P ).
Fig. 3 gives an illustration of the linear relaxation of theconstraints in Problem (6), in which f(P ) can be regarded asgi(P ) or −hi(P ).
Letting δ∗ denote the optimal solution of problem (6), wecan calculate a lower bound of δ∗ by solving the followinglinear programming problem over S ≡ [P , P ] ≡ {P |P ≤ P ≤ P}, in which P = (Pmin, . . . , Pmin) andP = (Pmax, . . . , Pmax) are m-dimensional vectors.
min δl (7)
s.t. gli(P ) + hli(P ) ≤ δl, 1 ≤ i ≤ nP ∈ S
A branch-and-bound subroutine using the results of problem(7) is then proposed in Algorithm 2, in which F(P ) denotesthe solution of problem (6) under a given feasible powerallocation P ∈ S, and F∗l (S | ?) denotes the optimal solutionof problem (7) over space S with additional constraint ? (itwill be omitted if no additional constraint). Essentially, these
7
two notations F(P ) and F∗l (S | ?) represent an upper boundand a lower bound of the optimal solution for the problem (6)over S, respectively. The additional constraint ? updated ateach iteration step in Algorithm 2 will generate tighter boundsuntil converging to the optimal solution.
The superscript k for a notation represents its intermediatevalue at the k-th iteration step. Given a possible powerallocation Sk, it will be split into new subspaces Sk1 and Sk2as defined in line 3 of Algorithm 2. Firstly, we find AP isuch that its corresponding power range in Sk is widest, i.e.,i = arg max1≤j≤n(P
k
j − Pkj ). Then,
Sk = [(P k1 , · · · , Pki , · · · , P
kn), (P
k
1 , · · · , Pk
i , · · · , Pk
n)]
is partitioned into{Sk1 = [(P k1 , · · · , P
ki , · · · , P
kn), (P
k
1 , · · · ,Pki+P
ki
2 , · · · , P kn)],
Sk2 = [(P k1 , · · · ,Pki+P
ki
2 , · · · , P kn), (Pk
1 , · · · , Pk
i , · · · , Pk
n)].
After the optimal solutions over the subspaces are obtained,the tightest upper bound βk+1 will be updated as given inline 5, and any subspace leading to a lower bound largerthan the current upper bound βk+1 will be discarded as givenline 8. The subspace that achieves the minimal lower boundis selected as Sk+1. This recursive procedure stops when asolution within ε to the optimum is found or no more subspacesplitting is possible.
Theorem 4.5: If ε = 0, Algorithm 2 converges to theoptimal solution of the problem (6).
Proof: Let δ∗ be the optimal solution of problem (6).Note that Algorithm 2 satisfies αk ≤ δ∗ ≤ βk at each stepk. If it is terminated at line 10 or line 16, i.e., αk ≥ βk, wecan immediately conclude that Algorithm 2 finds the optimalsolution. Otherwise, we consider the case after significantnumber of iteration steps. Due to the partition over Sk asshown in line 3, we notice that it will shrink to an arbitrarysmall one denoted as P ∗ ∈ Sk, i.e.,
limk→∞
Sk = {P ∗}.
Considering gli(P ) and hli(P ) approach to gli(P∗) and
hli(P∗), respectively, when P approaches to P ∗, we have the
following derivation based on the definition of problem (7).
limk→∞
F∗l (Sk+1) = limk→∞
minP∈Sk+1
max1≤i≤n
(gli(P ) + hli(P ))
= max1≤i≤n
(gli(P∗) + hli(P
∗))
= max1≤i≤n
(gi(P∗) + hi(P
∗))
= F(P ∗) ≥ δ∗
On the other hand, according to lines 13 and 14 in Algo-rithm 2, we have
F∗l (Sk+1 | δl ≥ αk) = αk+1 ≤ δ∗.
Because F∗l (Sk+1) ≤ F∗l (Sk+1 | δl ≥ αk), we combine theabove results to finally achieve
δ∗ ≤ limk→∞
F∗l (Sk+1) ≤ limk→∞
F∗l (Sk+1 | δl ≥ αk)
= limk→∞
αk+1 ≤ δ∗.
In other words, the lower bound limk→∞ αk+1 calculated byAlgorithm 2 converges to the optimal solution.
Algorithm 3 Alternating Optimization based Joint ResourceAllocation
1: l = 0, h = 02: Choose an initial setting for the power allocation,
channel assignment and user association. Denote it as〈P0, A0, C0〉. τ = minTi(〈P0, C0, A0〉). 〈P,C,A〉 =〈P0, C0, A0〉.
3: repeat4: Fix the power settings. Optimize the channel assign-
ment and user association by Algorithm 1. Get thenew channel assignment C ′ and user association A′.τ ′ = minTi(〈Pl, C ′, A′〉).
5: if τ ′ > τ then6: C = C ′, A = A′, Cl+1 = C ′, Al+1 = A′
7: if τ ′ − τ < µ then8: h = h+ 19: else
10: h = 111: end if12: τ = τ ′
13: else14: Cl+1 = Cl, Al+1 = Al15: end if16: Fix the channel assignment and user association. Opti-
mize the power control by Algorithm 2. Get the newpower allocation P ′ and τ ′ = minTi(〈P ′, Cl, Al〉).
17: if τ ′ > τ then18: P = P ′ Pl+1 = P ′
19: if τ ′ − τ < µ then20: h = h+ 121: else22: h = 123: end if24: τ = τ ′
25: else26: Pl+1 = Pl27: end if28: l = l + 129: until h > hmax ∨ l > lmax
30: Return 〈P,C,A〉
C. Alternating Optimization Based Joint Resource AllocationUsing the results of subproblems on restricted optimization
over individual resources, we propose an alternating optimiza-tion [28] based algorithm as summarized in Algorithm 3. Thebasic idea of the algorithm is to solve the restricted optimiza-tion problems over the individual resources alternatively. Eachstep of the method consists of two subroutines. At first, theoptimal channel allocation and user association is obtainedby Algorithm 1 under a fixed power assignment. This resultis then taken as the input of Algorithm 2 that achieves theoptimal power control.
Parameter µ is a given minimum performance improvement.If such minimum improvement cannot be achieved in hmax
8
5 10 15 200
5
10
15
20
25
The Number of APs (n)
Ave
rage
Min
imum
Thr
ough
put (
T)
SA
Greedy+
GibbsGreedy
(a) The minimum throughput v.s. the number of APs when the numberof channels c = 3.
20 30 40 50 60 70 800
2
4
6
8
The Number of Users (n)
Ave
rage
Min
imum
Thr
ough
put (
T)
SA
Greedy+
GibbsGreedy
(b) The minimum throughput v.s. the number of users when the numberof channels c = 3.
5 10 15 200
5
10
15
20
25
30
The Number of APs (n)
Ave
rage
Min
imum
Thr
ough
put (
T)
SA
Greedy+
GibbsGreedy
(c) The minimum throughput v.s. the number of APs when the numberof channels c = 5.
20 30 40 50 60 70 800
2
4
6
8
10
The Number of Users (n)
Ave
rage
Min
imum
Thr
ough
put (
T)
SA
Greedy+
GibbsGreedy
(d) The minimum throughput v.s. the number of users when the numberof channels c = 5.
Fig. 4. The minimum cell throughput as a function of the number of APs (n) and the number of users (m) under various number of channels (c = 3 or 5)and fixed transmission power (Pmax = 0.25).
number of consecutive iterations, Algorithm 3 is consideredas trapped at a local optimum, which will be returned as thefinal solution. Furthermore, Algorithm 3 can make alternatingoptimization by running Algorithms 1 and 2 at most lmax
rounds, resulting in a complexity factor of O(lmax).Note that Algorithms 1 - 3 are all centralized since they are
invoked in the network planning stage. They usually performbetter than the distributed algorithms [29], [30] and do notrequire to run in real time for the following considerations.The movement of indoor users is usually in small range andthus the channel gain matrix H can be regarded as changinglittle [18], [19]. Our research model focuses on the closedaccess mode for user association. In other words, the userassociation status will not change after the network planningis done. Our proposed algorithms need re-run only when theresource changes, e.g., when the available channel is updatedby the underlying cognitive radio detection techniques.
V. PERFORMANCE EVALUATION
The performance of our proposed algorithms has beenevaluated on Matlab 7.10.0. We consider a network, wheren APs and m users are uniformly deployed in a 200-meterby 200-meter square region. The number of non-overlappingchannels c = 3 or 5 is considered. The minimal transmission
power of each AP is set as 0.01W while the maximal valueis 0.1W or 0.25W. We apply the same path loss model as in[7], in which the channel gain hi,j is characterized as
hi,j = 10−3−S+Li10 · d−3.7
i,j .
In the above formula, di,j is the distance between AP i anduser j, S is the log-normal shadowing factor with a standarddeviation of 8 dB, and Li is the internal wall loss in cell iwhich is usually set to 5 dB. The thermal noise at each useris set to 4.0049 ∗ 10−15W .
Besides all the algorithms proposed in this paper, we alsoimplement the state-of-the-art Gibbs-sampling based algorithm[18] and the greedy algorithm proposed in [16]. To make a faircomparison with our proposal, the greedy algorithm is alsoadapted in a straightforward manner to optimize the same ob-jective that is to maximize the minimum cell throughput. Thethree competitor algorithms are named as “Gibbs”, “Greedy”and “Greedy+”, respectively. In the following, we first evaluatethe performance of Algorithm 1 (i.e., SA), Algorithm 2 (i.e.,PO) and Algorithm 3 (i.e., AO) under various network settings,respectively. For each network setting, the results are averagedover 200 instances. Then, the converge speeds of Algorithm 1and Algorithm 2 are investigated.
9
0 20 40 60 80 10015
20
25
30
35
40
iNum*
Min
imum
Thr
ough
put (
T)
Pmax
=0.25 c=5
Pmax
=0.1 c=5
Pmax
=0.25 c=3
Pmax
=0.1 c=3
(a) The effect of iNum∗.
0 5 10 15 200
10
20
30
40
50
sucNum*
Min
imum
Thr
ough
put (
T)
Pmax
=0.25 c=5
Pmax
=0.1 c=5
Pmax
=0.25 c=3
Pmax
=0.1 c=3
(b) The effect of sucNum∗.
−9 −8 −7 −6 −5 −4 −3 −2 −110
15
20
25
30
35
40
stopTemp (10x)
Min
imum
Thr
ough
put (
T)
Pmax
=0.25 c=5
Pmax
=0.1 c=5
Pmax
=0.25 c=3
Pmax
=0.1 c=3
(c) The effect of stopTemp.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.920
25
30
35
40
γ
Min
imum
Thr
ough
put (
T)
Pmax
=0.25 c=5
Pmax
=0.1 c=5
Pmax
=0.25 c=3
Pmax
=0.1 c=3
(d) The effect of γ.
Fig. 5. Impact of different values of control parameters on the performance of Algorithm 1 in networks with 80 users and 15 APs under different fixedtransmission power (Pmax) and number of channels (c).
0 2 4 6 80
0.2
0.4
0.6
0.8
1
Average Minimum Throughput ( T)
CD
F
Empirical CDF
Greedy (w/o PO)Greedy (w. PO)SA (w/o PO)SA (w. PO)
(a) CDF of the minimal throughput when the number of channels c = 3.
0 2 4 6 80
0.2
0.4
0.6
0.8
1
Average Minimum Throughput ( T)
CD
F
Empirical CDF
Greedy (w/o PO)Greedy (w. PO)SA (w/o PO)SA (w. PO)
(b) CDF of the minimal throughput when the number of channels c = 5.
Fig. 6. The CDFs of the minimal throughput with/without power optimization made by Algorithm 2 in 200 random topologies with m = 80 users andn = 15 APs under power range (0.01, 0.25).
A. Performance of Algorithm 1
We first evaluate our SA-based channel allocation and userassociation algorithm (i.e., Algorithm 1, denoted as SA) bycomparing it to “Gibbs”, “Greedy” and “Greedy+” algorithms.Parameters in Algorithm 1 are set as stopTemp = 0.1,iNum∗ = 400, sucNum∗ = 10, rejNum∗ = 500, Pmax =0.25 and γ = 0.5.
Fig. 4 shows the average minimum cell throughput as a
function of the number of APs (n) and the number of users (m)under different number of channels (i.e.,c = 3 or 5). When theeffect of the number of APs is studied, it varies in 5 ≤ n ≤ 20under fixed number of users m = 40. On the other hand, wefix n = 10 and study the effect of the number of users bysetting 20 ≤ m ≤ 80.
We find out that the average minimum cell throughputshows as a decreasing function of the number of APs and
10
5 10 15 200
10
20
30
40
50
60
70
The Number of APs (n)
Min
imum
Thr
ough
put (
T)
AO (c=5)SA−OP (c=5)AO (c=3)SA−OP (c=3)
(a) The average minimum throughput v.s. the number of APs when m =40 and Pmax = 0.1.
20 30 40 50 60 70 800
10
20
30
40
50
The Number of Users (m)
Min
imum
Thr
ough
put (
T)
AO (c=5)SA−OP (c=5)AO (c=3)SA−OP (c=3)
(b) The average minimum throughput v.s. the number of users when n =10 and Pmax = 0.1.
5 10 15 200
10
20
30
40
50
60
70
The Number of APs (n)
Min
imum
Thr
ough
put (
T)
AO (c=5)SA−OP (c=5)AO (c=3)SA−OP (c=3)
(c) The average minimum throughput v.s. the number of APs when m =40 and Pmax = 0.25.
20 30 40 50 60 70 800
10
20
30
40
50
60
The Number of Users (m)
Min
imum
Thr
ough
put (
T)
AO (c=5)SA−OP (c=5)AO (c=3)SA−OP (c=3)
(d) The average minimum throughput v.s. the number of users when n =10 and Pmax = 0.25.
Fig. 7. The minimum cell throughput as a function of the number of APs (n) and the number of users (m) under various number of channels (c) andmaximum transmission power (Pmax).
an increasing function of the number of users. This is becausewhen the ratio between m and n is large, more users perAP lead to a higher cell throughput. Furthermore, when thenumber of APs is less than 10, Algorithm 1 always achievesthe best performance due to its effective user association andchannel allocation scheme. For example, as shown in Fig. 4(a)with n = 6, m = 40, and c = 3, the average minimum cell-throughput obtained by SA is as high as 15.08, over two timesthe results achieved by either Greedy+ or Gibbs.
Next, we investigate how parameters iNum∗, sucNum∗,stopTemp and γ in Algorithm 1 affect the performance. Wefirst study the impact of iNum∗ to the performance by fixingstopTemp = 10−10, sucNum∗ = 10, γ = 0.5 and varyingiNum∗ from 10 to 300. The results are shown in Fig. 5(a).We notice that the performance can be significantly improvedby more iterations at the beginning, while the improvementbecomes limited after iNum∗ > 40 iterations. For the effectof sucNum∗, we then fix iNum as 100 and vary sucNum∗
from 5 to 50. Fig. 5(b) gives the evaluation results. Similarphenomena are observed that sucNum∗ = 10 is big enoughto exploit most performance improvement.
We then conduct simulations by setting stopTemp as10x, x ∈ [−9,−8, · · · ,−1] and fixing iNum∗, sucNum∗ andγ as 100, 10 and 0.5, respectively. The simulation results are
shown in Fig. 5(c). We notice that the minimum throughputdecreases by setting higher stopTemp. For example, whenstopTemp is set as 10−9, the average minimum throughput is4.4, while it drops to 2.0 when stopTemp increases to 10−1.This is because small stopTemp shrinks the searching space,making the results deviate from the optimal solution.
Parameter γ is another important factor to the performanceof Algorithm 1. We investigate its effect by fixing iNum∗,sucNum∗ and stopTemp to 100, 10 and 10−10, respectively.The performance under 0.2 ≤ γ ≤ 0.9 illustrated in Fig.5(d) shows that a larger γ (0 < γ < 1) usually improves theperformance. According to line 6 and line 9 in Algorithm 1, itcould be inferred that the larger γ is, the slower T reachesstopTemp. This also implies a larger number of possiblesearching actions towards the optimal solution. On the otherhand, we observe that the improvement becomes negligibleafter γ ≥ 0.5.
B. Performance of Algorithm 2
In this section, we study how power optimization (i.e., Al-gorithm 2, denoted as PO) improves the minimum throughputby comparing the results achieved by the Greedy algorithmand Algorithm 1 followed by Algorithm 2 (denoted as w. PO)or not (denoted as w/o PO). Parameter ε in Algorithm 2 is set
11
TABLE ICOMPARISON BETWEEN EXHAUSTIVE SEARCHING AND ALGORITHM 3
(a) ThroughputE1 E2 E3 E4 E5
Exhaustive 31.6659 32.3402 25.0992 31.7606 15.5494AO 31.4937 32.3402 25.0992 31.7606 15.3655
(b) Convergence Speed (s)E1 E2 E3 E4 E5
Exhaustive 21314 16333.96 4481.083 19166.34 3735.848AO 8.683363 8.12496 11.02827 11.07451 8.76277
20 50 100 15010
20
30
40
50
60
Min
imum
Thr
ough
put
lmax
P
max=0.25 c=5
Pmax
=0.1 c=5
Pmax
=0.25 c=3
Pmax
=0.1 c=3
(a) hmax = 5
10 20 50 100 15020
30
40
50
60
Min
imum
Thr
ough
put
hmax
P
max=0.25 c=5
Pmax
=0.1 c=5
Pmax
=0.25 c=3
Pmax
=0.1 c=3
(b) lmax = 150
Fig. 8. The expected performance of Algorithm 3 under various values of lmax and hmax when m = 80, n = 15 and γ = 0.5.
as 0.01. We run 200 simulation instances in a network with15 APs and 80 users under the power range (0.01, 0.25). Thenumbers of channels is set as individually. The CDFs of themax-min throughput obtained under c = 3 and c = 5 aredemonstrated in Fig. 6(a) and Fig. 6(b), respectively.
We observe that, even after a careful network planning bychannel allocation and user association, the performance canstill be clearly improved by just applying our power controlalgorithm. As shown in Fig. 6(a) with c = 3, about 8%instances achieve a minimum throughput at least 3.0 if onlySA is applied. This number goes to 13% when PO is appliedafterwards. The average minimum throughput increases from1.4808 to 1.6788 by applying Algorithm 2 after an initialchannel allocation and user association. When more channelsare available, the performance can be further improved asshown in Fig. 6(b) with c = 5. For example, the abovepercentage reaches 23%. The average performance also risesto 1.8718 with an additional 11.50% improvement.
C. Performance of Algorithm 3
In this section, we investigate the performance of ouralternative optimization approach (i.e., Algorithm 3), whichtakes both user/channel allocation and power optimization intoconsideration. We compare the results obtained by runningAlgorithm 1 and then Algorithm 2 iteratively by multipletimes or just once. They are denoted as AO and SA-OP,respectively. In both algorithms, we have the same parametersettings: ε = 0.01, stopTemp = 1e−8, iNum∗ = 400,sucNum∗ = 20, rejNum∗ = 1000 and µ = 0.01. InAlgorithm 3, the iteration controlling parameters lmax andhmax are set as 20 and 5, respectively. Fig. 7 shows the averageminimum throughput as a function of the number of APs andthe number of users in network with Pmax = 0.1 or 0.25 and
c = 3 or 5. Compared to the performance of Algorithm SA-OP, AO exhibits much improved performance. For example,when m = 80, n = 10 and c = 5, Fig. 7(d) shows that theaverage minimum throughput by SA-OP is 39.7 while thisperformance grows to 59.3 by AO. The improvement is ashigh as 26.7%.
We also evaluate the performance gap between Algorithm 3and the optimal results obtained by exhaustive searching. It ex-hausts all possible combinations of channel allocation and userassociation, and solves the resulting problem using Algorithm2 under each setting. In a small-scale network with 5 clientsand 3 APs deployed in a 20-meter by 20-meter square region,we randomly run five simulation instances. The correspondingcomparison results, denoted as Ei, i = 1, 2, · · · , 5, are shownin Table 1. It is observed that Algorithm 3 runs much fasterand performs almost as good as the optimal algorithm. Forexample, experimental instances 2-4 just achieve the optimum.
The number of iterations in Algorithm 3 to find the bestsolution is controlled by parameters lmax and hmax. To under-stand their effects, we run 50 simulation instances to obtainthe average minimum throughput under the setting m = 80,n = 15 and γ = 0.5. The simulation results when hmax =5, 20 ≤ lmax ≤ 150 and lmax = 150, 10 ≤ hmax ≤ 150 arepresented in Fig. 8(a) and Fig. 8(b), respectively. We find outthat the performance increases roughly with lmax but is notinfluenced much by hmax.
D. Convergence Speed Analysis
We have witnessed the high performance of the algorithmsproposed in this paper. Next, our evaluation focuses on theirconvergence speed. We fix the number of APs n = 10 andvary the number of users as 20 ≤ m ≤ 80.
12
20 30 40 50 60 70 800
500
1000
1500
2000C
onve
rgen
ce S
peed
The Number of Users (m)
Pmax
=0.25 c=5
Pmax
=0.1 c=5
Pmax
=0.25 c=3
Pmax
=0.1 c=3
(a) Average convergence speed when 20 ≤ m ≤ 80.
0 500 1000 1500 2000 2500 3000 35000
0.2
0.4
0.6
0.8
1
Convergence Speed of Alg. 1
CD
F
Pmax
=0.25 c=5
Pmax
=0.1 c=5
Pmax
=0.25 c=3
Pmax
=0.1 c=3
(b) CDF of convergence speed when m = 80.
Fig. 9. Convergence speed of Algorithm 1 with n = 10, stopTemp = 10−8, iNum∗ = 400, sucNum∗ = 20, rejNum∗ = 500 and γ = 0.5.
20 30 40 50 60 70 800
100
200
300
400
500
The Number of Users (m)
Min
imum
Thr
ough
put (
T)
Pmax
=0.25 c=5
Pmax
=0.1 c=5
Pmax
=0.25 c=3
Pmax
=0.1 c=3
(a) Average convergence speed when 20 ≤ m ≤ 80.
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Convergence Speed of Alg. 2
CD
F
Pmax
:0.25 c:5
Pmax
:0.1 c:5
Pmax
:0.25 c:3
Pmax
:0.1 c:3
(b) CDF of convergence speed when m = 80.
Fig. 10. Convergence speed of Algorithm 2 with n = 10 and ε = 0.01.
1) Convergence Speed of Algorithm 1: We first ana-lyze the convergence speed of Algorithm 1 under settings:stopTemp = 10−8, iNum∗ = 400, sucNum∗ = 20,rejNum∗ = 500 and γ = 0.5. The results about the averageconverge speed in terms of iteration number are shown in Fig.9(a). To detail the converge speed distribution, we also depictits CDF for the case n = 10 and m = 80 in Fig. 9(b).We notice that the number of iterations required exhibits apositive correlation with the number of channels c and thenumber of users m, and is not influenced much by the powerrange Pmax. For example, when m = 80, the expected numberof iterations increases by about by 45.2% if the number ofchannels changes from 3 to 5.
2) Convergence Speed of Algorithm 2: Next, we investigatehow fast Algorithm 2 converges to the optimal solution, interms of number of iterations, under various numbers of usersm, power ranges Pmax and numbers of channels c. For eachnetwork setting, we run 200 simulation instances, amongwhich the cases with idle APs (i.e., not associated by anyuser) are excluded. The average results as a function of mand the CDF when m = 80 are illustrated in Fig. 10(a) andFig. 10(b), respectively. It is observed that under the fixedPmax and c, the expected converge time increases with theuser size, e.g., about 343 and 386 iterations for m = 50 and80, respectively, when c = 5 and Pmax = 0.25. On the otherhand, we also find out that, when the user size is fixed, the
converge time is roughly a decreasing and increasing functionof c and Pmax, respectively. Furthermore, we discover that inmany cases, the algorithm stops after only one iteration. Asshown in Fig. 10(b), Algorithm 2 achieves the optimal solutionin only one iteration with probability around 20%.
VI. CONCLUSION
In this paper, we propose an alternating optimization basedresource allocation algorithm for multicell networks. Ouroptimization goal is to maximize the minimal throughput ofthe cells. In our method, we propose two algorithms. One isa simulated annealing based algorithm that solves the userassociation and channel allocation problem. The other is abranch-and-bound algorithm for power control. Furthermore,our solution can be extended in a straightforward manner bytaking the user data rate requirement into consideration. Thisis because the nature of the resulting formulation remains. Forexample, the power optimization problem under given channelassignment and user association is still a D.C. programmingproblem. Comprehensive evaluations indicate that our pro-posed method improves the max-min throughput in multicellnetworks significantly.
ACKNOWLEDGEMENT
The authors would like to thank the editor and the anony-mous reviewers for their helpful comments and suggestion-
13
s which have greatly improved the quality of this paper.This work was partly supported by the National NaturalScience Foundation of China (Nos.61202417, 61370065),the general program of science and technology develop-ment project of Beijing Municipal Education Commission(No.KM201411232013), the Fundamental Research Funds forthe Central Universities, China University of Geosciences(Wuhan), and the NSERC Discovery grant.
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Zhuo Li received the BSc, MSc and PhD degrees incomputer science from Nanjing University, China.He joined the school of computer at Beijing In-formation Science & Technology University as anassistant professor in 2012. His research interestsare mainly in wireless networks, mobile networksand distributed computing. He has published morethan 30 papers. He is a member of ACM, IEEE andCCF.
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Song Guo (M’02-SM’11) received the PhD degreein computer science from the University of Ottawa,Canada in 2005. He is currently a Full Professorat School of Computer Science and Engineering,the University of Aizu, Japan. His research interestsare mainly in the areas of protocol design andperformance analysis for reliable, energy-efficient,and cost effective communications in wireless net-works. He received the Best Paper Awards at ACMConference on Ubiquitous Information Managemen-t and Communication 2014, IEEE Conference on
Computational Science and Engineering 2011, and IEEE Conference on HighPerformance Computing and Communications 2008. Dr. Guo currently servesas Associate Editor of the IEEE Transactions on Parallel and DistributedSystems. He is in the editorial boards of ACM/Springer Wireless Networks,Wireless Communications and Mobile Computing, International Journal ofDistributed Sensor Networks, International Journal of Communication Net-works and Information Security, and others. He has also been in organizingcommittee of many international conferences, including serving as a GeneralChair of MobiQuitous 2013. He is a senior member of the IEEE and theACM.
Deze Zeng received his Ph.D. and M.S. degrees incomputer science from University of Aizu, Aizu-Wakamatsu, Japan, in 2013 and 2009, respectively.He received his B.S. degree from School of Comput-er Science and Technology, Huazhong University ofScience and Technology, China in 2007. He is cur-rently an associate professor in School of ComputerScience, China University of Geosciences (Wuhan),China. His current research interests include: cloudcomputing, software-defined sensor networks, datacenter networking, networking protocol design and
analysis. He is a member of IEEE.
Ahmed Barnawi is an associate professor at theFaculty of Computing and IT, King Abdulaziz U-niversity, Jeddah, Saudi Arabia, where he workssince 2007. He received PhD at the University ofBradford, UK in 2006. He was visiting professor atthe University of Calgary in 2009. His research areasare cellular and mobile communications, mobile adhoc and sensor networks, cognitive radio networksand security. He received three strategic researchgrants and registered two patents in the US. He isIEEE member.
Ivan Stojmenovic received his Ph.D. degree inmathematics. He is Full Professor at the Universityof Ottawa, Canada. He held or holds regular andvisiting positions in Saudi Arabia (DistinguishedAdjunct Professor at the King Abdulaziz University,Jeddah), China (Tsinghua University, DUT, Bei-hang), Serbia, Japan, USA, Canada, France, Mexico,Spain, UK (as Chair in Applied Computing at theUniversity of Birmingham), Hong Kong, Brazil, Tai-wan, and Australia. He published over 300 differentpapers, and edited seven books on wireless, ad hoc,
sensor and actuator networks and applied algorithms with Wiley. He is editor-in-chief of IEEE Transactions on Parallel and Distributed Systems (2010-3),and founder and editor-in-chief of three journals. He is Associate Editor-in-Chief of Tsinghua Journal of Science and Technology, steering committeemember of IEEE Transactions on Emergent Topics in Computing, and editorof IEEE Network, IEEE Transactions on Cloud Computing, IEEE Transactionson Computers, ACM Wireless Networks and some other journals. Stojmenovicis on Thomson Reuters list of Highly Cited Researchers (from 2013; ¡300computer scientist), has h-index 60, top h-index in Canada for mathematicsand statistics, and ¿14000 citations. He received five best paper awardsand the Fast Breaking Paper for October 2003, by Thomson ISI ESI. Hereceived the Royal Society Research Merit Award, UK (2006), and HumboldtResearch Award, Germany (2012). He is Tsinghua 1000 Plan DistinguishedProfessor (2012-5). He is Fellow of the IEEE (Communications Society, class2008), and Canadian Academy of Engineering (since 2012), and Memberof the Academia Europaea (The Academy of Europe), from 2012 (section:Informatics). He was IEEE CS Distinguished Visitor 2010-11 and received2012 Distinguished Service award from IEEE ComSoc CommunicationsSoftware TC. He received Excellence in Research Award of the Universityof Ottawa 2009. Stojmenovic chaired and/or organized ¿60 workshops andconferences, and served in ¿200 program committees. He was program co-chair at IEEE PIMRC 2008, IEEE AINA-07, IEEE MASS-04&07, foundedseveral workshop series, and is/was Workshop Chair at IEEE ICDCS 2013,IEEE INFOCOM 2011, IEEE MASS-09, ACM Mobihoc-07&08.