On Cell Association and Scheduling Policies in Femtocell Networks
Hui Zhou, Donglin Hu, Saketh Anuma Reddy, Shiwen Mao, Prathima Agrawal
aa r X i v : . [ c s . N I] D ec On Cell Association and Scheduling Policies inFemtocell Networks
Hui Zhou, Donglin Hu, Saketh Anuma Reddy, Shiwen Mao, and Prathima Agrawal
Department of Electrical and Computer Engineering, Auburn University, Auburn, AL, USA
Abstract —Femtocells are recognized effective for improvingnetwork coverage and capacity, and reducing power consumptiondue to the reduced range of wireless transmissions. Althoughhighly appealing, a plethora of challenging problems need to beaddressed for fully harvesting its potential. In this paper, weinvestigate the problem of cell association and service schedulingin femtocell networks. In addition to the general goal of offloadingmacro base station (MBS) traffic, we also aim to minimize thelatency of service requested by users, while considering both openand closed access strategies. We show the cell association problemis NP-hard, and propose several near-optimal solution algorithmsfor assigning users to base stations (BS), including a sequentialfixing algorithm, a rounding approximation algorithm, a greedyapproximation algorithm, and a randomized algorithm. For theservice scheduling problem, we develop an optimal algorithm tominimize the average waiting time for the users associated withthe same BS. The proposed algorithms are analyzed with respectto performance bounds, approximation ratios, and optimality,and are evaluated with simulations.
I. I
NTRODUCTION
A femtocell, as shown in Fig. 1, is a relatively smallcellular network with a femtocell base station (FBS), usuallydeployed in places where signal reception from the macrobase station (MBS) is weak due to long distance or obstacles.An FBS is typically the size of a residential gateway orsmaller and connects to the service provider’s network viabroadband connections. FBS is designed to serve approvedusers within its coverage to offload wireless traffic from MBS.Due to shortened wireless transmission distance, femtocell isshown very effective in reducing transmit power and boostingsignal-to-interference-plus-noise ratio (SINR), which lead toprolonged battery life of mobile devices, improved networkcoverage, and enhanced network capacity [1].Femtocells have gained a lot of attention from bothacademia and industry in the recent past. The three largest cel-lular network operators in the United States (i.e., AT&T, Sprintand Verizon) have offered commercial femtocell products andservice recently. Although highly promising, a plethora ofproblems with both technical and economic natures have notbeen fully addressed yet. In [1], a comprehensive discussion isprovided of the challenging technical issues in femtocell net-works, ranging from synchronization, cell association, networkorganization, to quality of service (QoS) provisioning.Unlike the MBS, whose placement is planned and opti-mized by operators, FBS’s are usually randomly deployed byusers. When the chaotic femtocell placement meets randomlydistributed mobile users, cell association (or load balancing)becomes a critical problem for the performance of femtocellnetworks. For example, an FBS might be deployed at a placewith high user density. With an inappropriate cell association
InternetFemtocell Base Station Macrocell Base Station
Fig. 1. Illustration of a two-tier femtocell network. strategy, this FBS may have to serve all the users within itscoverage, leading to very high load at this FBS and highservice latency for its users. An effective cell associationscheme should be used in this case to evenly distribute the loadamong neighboring FBS’s and/or MBS. The cell associationproblem is particularly prominent in femtocell networks dueto the unreliability of FBS’s. The operation of an FBS may beinterrupted by its owner (e.g., turned off after office hours); itmay also experience power outage or any other faults. Then allthe users initially associated with this FBS should be quicklyassigned to other neighboring FBS’s or the MBS. It is a loadbalancing problem on how to effectively associate these userswith neighboring BS’s without introducing a load burst andperformance degradation at a particular BS.In this paper, we investigate the problem of cell associationand service scheduling in a two-tier femtocell network. Inaddition to the general goal of offloading wireless traffic fromthe MBS, we also aim to minimize the latency of servicerequested by users, while considering both open and closedaccess strategies. In particular, we consider one MBS andmultiple FBS’s serving randomly distributed mobile users.Users request to the BS’s for downlink transmission of datapackets. Without loss of generality, we assume that each useris allowed to connect to either the MBS or an FBS. The cellassociate problem is to assign the users to the BS’s such thatthe transmission of all the data packets can be completed assoon as possible. When multiple users are associated with oneBS, we also aim to develop a service scheduling scheme suchthat the average waiting time for the users will be minimized.We provide a general framework for the cell associationproblem for both open and closed access scenarios, whichcan be reduced to the classic load balancing problem andis NP-hard [2]. Therefore, we develop effective near-optimalalgorithms with guaranteed performance. In particular, we1rst provide a sequential fixing algorithm based on a linearprogramming (LP) relaxation, which can achieve the bestperformance among the proposed schemes but with a relativelyhigh computational complexity. To reduce the complexity, wepropose a rounding approximation algorithm that ensures an ( ρ + 1) -approximation of the optimal solution, and a greedyapproximation algorithm that ensures a (2 ρ ) -approximation ofthe optimal solution. To further reduce the requirement onfrequently updated channel state information (CSI), we thendevelop a randomized algorithm that allows a user to randomlypick a BS to connect to from a reduced BS list. Once thereduced BS list is generated by the randomized algorithm,no information exchange is required among users. An upperbound for the maximum expected service time achieved bythe randomized algorithm is then derived.After the users are assigned to the BS’s, we next address theservice scheduling problem for determining the transmissionorder of the data packets requested by the users associatedwith the same BS. We develop a simple algorithm to minimizethe average waiting time for the users, and prove its optimal-ity. In addition rigorous analysis of the proposed algorithmswith respect to performance bounds, approximation ratios,and optimality, we also evaluate the proposed schemes withsimulations, where superior performance is observed.The remainder of this paper is organized as follows. Therelated work is discussed in Section II. We present the systemmodel in Section III. Cell Association problem formulationand solutions are presented in Section IV. The schedulingproblem is studied in V. The proposed algorithm are evaluatedin Section VI. Section VII concludes this paper.II. R ELATED W ORK
Femtocells have been acknowledged as an effective solutionto the capacity problem of wireless networks. Ref. [1] providedcomprehensive discussions of the technical issues, regulatoryconcerns, and economic incentives in femtocell networks.There are three different access control strategies in femtocellnetworks, open access, closed access and hybrid access. Thepros and cons of these strategies were studied in [3].Deploying femtocells also means introducing interference ifno appropriate mitigation strategy is incorporated. Consider-able research have been conducted on interference mitigationby assigning users to proper orthogonal channels [4].Apart from the studies on interference mitigation, there arean increasing number of papers on cell association or cellselection under various scenarios [5]–[10]. Dhahri and Ohtsukiin [5] proposed a learning-based cell selection method for anopen access femtocell network. The authors in [6] describednew paradigms of cell association in heterogeneous networkswith the help of third-party backhaul connections. Their simpleand lightweight methodologies and algorithms incur very lowsignaling overhead. In [7], a convex optimization problem wasformulated for cell association and a dynamic range extensionalgorithm was proposed to maximize the minimum rate ofusers on the downlink of heterogeneous networks. However,this paper did not directly optimize the load balancing inHeterogeneous Networks (HetNet), but rather focused on thesum rate and min rate. In [8], a cell association and accesscontrol scheme was presented to maximize network capacitywhile achieving fairness among users. In [9], the authors provided an analytical framework for evaluating outage prob-ability and spectral efficiency with flexible cell association inheterogeneous cellular networks. Mukherjee in [10] analyzedthe downlink SINR distribution in heterogeneous networkswith biased cell association.There are also some interesting prior work on load balancingin cellular networks. A theoretical framework was presentedin [11] for distributed user association and cell load balancingunder spatially heterogeneous traffic distribution. A distributed α -optimal algorithm was proposed and it supports differ-ent load-balancing objectives, which include rate-optimal,throughput-optimal, delay-optimal, and load-equalizing, as α is set to different values. In [12], the authors developedan off-line optimal algorithm for load balancing to achievenetwork-wide proportional fairness in multi-cell networks.They considered partial frequency reuse (PFR) jointly withload-balancing in a multi-cell network to achieve network-wide proportional fairness. An on-line practical algorithm wasalso proposed and the expected throughput was taken as thedecision making metric. On-line assignments when users ar-rive one at a time was studied extensively in computer scienceliterature. The competitive ratio analysis in [13] showed thatany deterministic on-line algorithm can achieve a competitiveratio of log n , where n is the number of servers.We find most of the related research was focused on offload-ing MBS traffic and improving network capacity with FBS’s.In the following sections, we propose several cell associationand transmission scheduling schemes with the objective ofminimizing service latency in femtocell networks.III. S YSTEM M ODEL
We consider a two-tier femtocell network with M basestations: one MBS (indexed by ) and M − FBS’s (indexedfrom to M ). The All the BS’s are connected to the Internetvia broadband wired connections. There are N mobile usersrandomly located within the coverage of the femtocell net-work. We assume the MBS and FBS’s are well synchronizedand they share the same spectrum. Assume each user requestsa fixed-length data packet from one of the M BS’s. Theproblem is to assign the users to the BS’s and schedule thetransmission of their requested data packets at each BS, suchthat the transmissions can be finished as earlier as possible.
A. Link Capacity
Let P m be the transmit power of BS m and G m,n thechannel gain between the BS and user n . According to theShannon Theorem, the network capacity of user n connectedto BS m is given by C m,n = B log (cid:18) G m,n P m σ + I m,n (cid:19) , (1)where B is network bandwidth, σ is noise power density,and I m,n is the interference from all other BS’s. We have that I m,n = M X i =1 G i,n P i − G m,n P m = I n − G m,n P m , (2) It is well-known from queuing theory that a single server single bufferqueue has the lowest delay than splitting the service capacity to multipleservers or maintaining multiple queues. I n is the sum of interference from all BS’s to user n .It does not depend on which BS user n is connected to andis a constant for each user. Substituting (2) into (1), we have C m,n = B log (cid:18) G m,n P m σ + I n − G m,n P m (cid:19) = B log (cid:18) − η m,n (cid:19) , (3)where η m,n is signal to interference plus noise ratio (SINR),the same ratio of the received power in I n at user n . B. Service Time
We assume each user requests a fixed-length data packetfrom one of the BS’s. For simplicity of notation, we assumeall the packets have the same length, denoted as L . Then theprocessing/service time of BS m for user n is given by t m,n = L/C m,n . (4)The service time depends on the link capacity C m,n as givenin (3). Note that the service time defined here is actuallythe transmission delay, i.e., the time it takes to finish thetransmission of the data packet. The propagation delay isnegligible due to the short distance and is ignored. C. Femtocell Access Control
The type of access control for femtocells can be classifiedinto two categories: closed access and open access. Theopen-access strategy allows all mobile users of an operatorto connect to the FBS’s; in this case, femtocells are oftendeployed by an operator to enhance coverage in an area wherethere is a coverage hole. With the closed access strategy, onlya specific user group can get service from the FBS’s [14].Although closed access has been shown to decrease systemthroughput by 15%, surveys suggest that closed access isusers’ favorite option [15].In this paper, we consider both access strategies. Let A m denote the set of users that can connect to BS m and B n the set of BS’s that user n can connect to. Both open andclosed access strategies can be easily modeled by these twosets. Specifically, for open access, we have A m = { , · · · , N } and B n = { , · · · , M } .IV. C ELL A SSOCIATION P ROBLEM F ORMULATION AND P ROPOSED S CHEMES
To make the complex problem tractable, we divide theproblem into two steps. First, we assign each user to one ofthe M BS’s with the objective of minimizing the total servicetime on each BS. Second, we schedule the service order ateach BS to minimize the average waiting time of users.
A. Problem Statement
The cell association problem can be formulated as a loadbalancing problem. Given a set of N users and a set of M BS’s. Each user n has a service time t m,n if it is connectedto BS m . Let C m denote the set of users assigned to BS m .Then it takes a total amount of time T m = P n ∈C m t m,n forBS m to transmit all the packets. For optimal network-wide performance, we seek to minimize the maximum load amongall the BS’s, i.e., min T = max m { T m } = max m ( X n ∈C m t m,n ) . (5)We find the cell association problem is similar to a loadbalancing problem. However, our problem is more challengingthan the classic load balancing problem, where the service timeof a user is identical when connecting to any BS. In our cellassociate problem, the service time is a function of the linkcapacity as in (4). Its solution depends on not only user n ,but also BS m . This cell association problem is easily seen tobe NP-hard: when all the t m,n ’s are identical for any BS m ,the problem is reduced to the classic load balancing problem,which is NP-hard [2].In the remainder of this section, we develop effectivealgorithms to solve the cell association problem. In particular,we present a sequential fixing algorithm, an approximationalgorithm, as well as a randomized algorithm, and deriveseveral approximation ratios and performance bounds. B. Sequential Fixing Algorithm
To solve the above problem, we first define an indicatorvariable x m,n as x m,n = (cid:26) , if user n is connected to BS m , otherwise . (6)Then we reformulate the problem as follows: min T (7)s.t. X m x m,n = 1 , for all users X n t m,n x m,n ≤ T, for all BS’s x m,n ∈ { , } , for all n ∈ A m x m,n = 0 , for all n / ∈ A m . In the formulated problem (7), all the indicator variable x m,n ’sare binary, while T is a real variable. Thus it is a mixed integerlinear programming problem [2], denoted by MILP, which isusually NP-hard.The original MILP is next relaxed to a linear programming(LP) problem, denoted as RLP. Specifically, we allow binaryvariable x m,n ’s to take real values in [0 , . Then, the MILPproblem can be converted into RLP as follows: min T (8)s.t. X m x m,n = 1 , for all users X n t m,n x m,n ≤ T, for all BS’s x m,n ≥ , for all n ∈ A m x m,n = 0 , for all n / ∈ A m . Since the sum of x m,n ’s is already upper bounded by inthe first constraint, we remove the upper bounds of x m,n ’sin the third constraint of MILP. Obviously, the solution tothe RLP problem is a lower bound of the original MILP3 lgorithm 1: Sequential Fixing for Cell Association Initialize N = { , · · · , N } ; Relax x m,n to real numbers ; while N is not empty do Solve the RLP problem ; Find x m ′ ,n ′ that is the closest to integer ; x m ′ ,n ′ = min n ∈A m ∩N { x m,n , − x m,n } ; Set x m ′ ,n ′ to the closest integer ; if x m ′ ,n ′ is set to then Set x m,n ′ = 0 for all m = m ′ ; Remove n ′ from N ; else Remove n ′ from A m ′ ; end end problem because it is obtained by expanding the solutionspace. Unfortunately, it is usually an infeasible solution to theoriginal MILP problem. Therefore, we develop a sequentialfixing (SF) algorithm [16] to find a feasible solution to theMILP problem, which is presented in Algorithm 1.Algorithm 1, we solve the RLP problem iteratively. Duringeach iteration, we find the x m ′ ,n ′ that has the minimum valuefor ( x m,n − ) or ( − x m,n ) among all fractional x m.n ’s, andround it up or down to the nearest integer. Setting x m ′ ,n ′ to means user n ′ is connected to BS m ′ . Therefore, user n ′ cannot be connected to any other BS’s and the rest of x m,n ′ ’sare set to , for all m . This procedure repeats until all the x m,n ’s are fixed.The complexity of SF depends on the specific LP algorithm.With Karmarkar’s algorithm, the worst-case polynomial boundfor solving LP problems is O ( n v . L b ) , where n v is thenumber of variables and L b is the number of bits of inputto the algorithm. We have the following proposition. Proposition 1.
The computational complexity of the sequentialfixing algorithm is O (( M N ) . L b ) .Proof: The number of binary variables in MILP is at most
M N , so the number of loops in sequential fixing problem isat most
M N . In each iteration, the complexities of Steps , and the rest of the steps are O (( M N ) . L b ) , O ( M N ) and O (1) , respectively. Besides, in each iteration, the number ofvariables is reduced by . Therefore, the complexity of SF isgiven by P MNi =1 O (( M N − i + 1) . L b ) = P MNi =1 O ( i . L b ) = O (( M N ) . L b ) . Therefore, the complexity of SF is upperbounded by O (( M N ) . L b ) . C. Approximation Algorithm
Although the sequential fixing algorithm can solve theMILP problem within polynomial time, its complexity maybe high even for small femtocell networks. In this section, wepropose an approximation algorithm with low complexity tosolve the MILP problem. Before we introduce the approxima-tion algorithm, we first give the lemma below.
Lemma 1.
The optimal solution, denoted by T ∗ , to the MILPproblem is lower bounded by T ∗ ≥ M P Nn =1 t n where t n =min m ∈B n t m,n . Proof: Given the optimal allocation C ∗ m for BS m , wehave T ∗ = max m P n ∈C ∗ m t m,n . Then we have T ∗ ≥ max m X n ∈C ∗ m t n ≥ M M X m =1 X n ∈C ∗ m t n = 1 M N X n =1 t n . The first inequality is due to the definition of t n . The secondinequality is due to the fact that the maximum value is alwaysgreater than the mean value. The last equality is because allusers have to be connected to one of the BS’s and ∪ Mm =1 C ∗ m is the set of all users.Intuitively, the maximum total service time is at least theservice time of any one user. We have the following lemma. Lemma 2.
The optimal solution, denoted by T ∗ , to theMILP problem is lower bounded by T ∗ ≥ max t n , where t n = min m ∈B n t m,n . These lemmas will be used in analyzing the approximationratio of the proposed approximation algorithms, which arepresented in following subsections.
1) Rounding Approximation Algorithm:
To ensure requiredSINR for each user, B n should not include all the FBS’s ina real femtocell network. For example, some faraway FBSshould not be considered by a user. Thus, we can use athreshold ρ to obtain the subsets A m and B n ( A m will beupdated when B n is determined). B ′ n = B n ∩ ( { m | t m,n /t n ≤ ρ } ) , A ′ m = { n | m ∈ B ′ n } . (9)Usually only a limited number of FBS’s will be taken intoconsideration for a user. After we adopt this threshold, notonly users’ SINR requirements will be satisfied, but also thecomputational complexity will be greatly reduced.Once A ′ m and B ′ n are determined, the following relaxed LPproblem can be solved by any LP solver. min T (10)s.t. X m x m,n = 1 , for all users X n t m,n x m,n ≤ T, for all BS’s x m,n ≥ , for all n ∈ A ′ m x m,n = 0 , for all n / ∈ A ′ m . We denote the solution obtained by solving this RLP programby T . Since x -variables are allowed to take fractional values,we have T ≤ T ∗ .Without sequentially fixing these fractional values, we adopta rounding method from [17] to obtain a feasible solutionfor the MILP problem. In this rounding method, a bipartitegraph is constructed according to the RLP solution, whichis constructed as a undirected bipartite graph G ( A ∪ B , E ) .In the disjoint set A , each node represents a user n , whilethe other disjoint set B consists of BS nodes. We create k m = ⌈ P n x m,n ⌉ nodes in B for BS m and these nodeare denoted by { b m, , b m, , · · · , b m,k , · · · , b m,k m } . The edgesare determined in the following way. For BS m , we sort theusers in the order of non-increasing service time t m,n and theusers are renamed { u , u , · · · } . Let X m,u j = P ji =1 x m,u i .4or each BS, we divide the users associated to it into k m groups, as G , G , · · · , G K m . User u j will be included ingroup k ( ≤ k ≤ k m ) if k − < X m,u j ≤ k or k − ≤ X m,u j − < k . If a user u j is included in two groups,the association x -variables need to be adjusted, such that x ′ b m,k ,u j = X m,u j − k + 1 and x ′ b m,k − ,u j = x m,u j − x ′ b m,k ,u j .Then we insert edges between BS node b m,k and all the usernodes in group k . Now the bipartite graph is created and wenext find a maximum matching M from each user to nodes inthe other disjoint set. This maximum matching M indicates afeasible solution for MILP problem: for each edge ( n, b m,k ) in M , we associate user n to BS m .Let T ( b m,k ) denote the total service time at node b m,k beforethe matching operation and T ′ ( b m,k ) the total service time atnode b m,k obtained by the above rounding method. We havethe following lemma. Lemma 3.
For each node b m,k , where k m ≥ k > , we have T ( b m ,k − ≥ T ′ ( b m ,k ) .Proof: First, observe that the minimum service time ingroup ( k − will be always no less than the maximum servicetime in group k , because we sort the users according to theirservice time in the non-increasing order.According the above bipartite graph construction, for any k < k m , we have P i ∈ G k x ′ b m,k ,u i = 1 ; for k = k m , we have P i ∈ G k x ′ b m,k ,u i ≤ . T ′ ( b m ,k ) will be no greater than the maximum ser-vice time in group k and will thus be no greater thanthe minimum service time in group ( k − , which isless than P i ∈ G k − x ′ b m,k − ,u i t m,u i . Since T ( b m ,k − = P i ∈ G k − x ′ b m,k − ,u i t m,u i , consequently, we have the conclu-sion that T ( b m ,k − ≥ T ′ ( b m ,k ) .Now we show that the solution produced by this roundingapproximation algorithm is at most ( ρ + 1) times greater thanthe optimal solution. Theorem 1.
The approximation algorithm based on linearprogramming and the rounding method ensures a ( ρ + 1) -approximation of the optimal solution.Proof: For each BS m , we create k m nodes for it andthere are k m corresponding groups of user nodes adjacent tothem. Thus the total service time is P k m k =1 T ′ ( b m ,k ) .According to Lemma 3, we have T ( b m ,k − ≥ T ′ ( b m ,k ) for k m ≥ k > . It follows that k m X k =2 T ′ ( b m ,k ) ≤ k m − X k =1 T ( b m ,k ) ≤ k m X k =1 T ( b m ,k ) ≤ T. In the first group, the maximum load will be the maxi-mum service time of users associated with m . According toLemma 2 and the definition of ρ in (9), we have T ′ ( b m , ≤ max t m,n ≤ ρ max t n ≤ ρT ∗ . Then, the total service timeon any BS computed by our association algorithm will be P k m k =1 T ′ ( b m ,k ) ≤ ρT ∗ + T ≤ ( ρ + 1) T ∗ . The last inequalitywas due to T ≤ T ∗ , since T is the solution of the relaxedproblem (10). Our proof is complete.The complexity to compute a maximum matching is O ( V E ) , where V and E are the number of nodes and Algorithm 2:
Greedy Approximation Algorithm for CellAssociation Initialize T m = 0 and C m = φ for all BS’s ; Set the user set N = { , · · · , N } ; while N is not empty do Find the BS m ′ that has the minimum T m : m ′ = arg min m ∈ ( ∪ n ∈N B n ) T m ; Find the user n ′ that has the minimum t m ′ ,n : n ′ = arg min n ∈{A m ′ ∩N} t m ′ ,n ; Set C m ′ = C m ′ ∪ { n ′ } ; Set T m ′ = T m ′ + t m ′ ,n ′ ; Set ρ m ′ ,n ′ = t m ′ ,n ′ t n ′ ; Remove n ′ from N ; end edges, respectively. Since we only need to run the matchingalgorithm once to obtain the association relationship, the totalcomputational complexity of this algorithm is O (( M N ) . L b ) ,which is better than that of the sequential fixing algorithm. Proposition 2.
The computational complexity of the roundingapproximation algorithm is O (( M N ) . L b ) .2) Greedy Approximation Algorithm: We next present a lowcomplexity approximation algorithm, where the BS with thelowest load is greedily chosen and the user whose completiontime at this BS is the smallest is assigned to this BS.By abuse of notation, we define ρ m,n = t m,n /t n and ρ = max { m,n } ρ m,n , which will be used in the optimalityanalysis. The greedy approximation algorithm is presented inAlgorithm 2. In Step , we find the candidate BS for usersthat has the minimum T m . Then we pick the user who hasthe minimum T m,n at the chosen BS in Step . Obviously, thecomputational complexity of the approximation algorithm is O ( M N ) , which is much lower than that of sequential fixing. Proposition 3.
The computational complexity of the greedyapproximation algorithm is O ( M N ) . We have the following lemma for the performance of thegreedy approximation algorithm.
Lemma 4.
The greedy approximation algorithm solution,denoted by T , is upper bounded by ρM P Nn =1 t n + ρT ∗ .Proof: We first consider the open access strategy whereeach user can connect to any of the BS’s. In the l -th iterationin Algorithm 2, we choose the BS with the minimum T m inStep . Thus we have T ( l − m ′ ≤ M M X m =1 T ( l − m = 1 M M X m =1 X n ∈C ( l − m t m,n = 1 M M X m =1 X n ∈C ( l − m ρ m,n t n ≤ ρ ( l − M M X m =1 X n ∈C ( l − m t n , where ρ ( l − = max { m,n ∈C ( l − m } ρ m,n . Note that C ( l − m is setof users that have been assigned to BS m in the ( l − )-thiteration.5n Step , we pick user n ′ and let user n ′ connect to BS m ′ .Since ρ ( l ) will always be greater than ρ ( l − and according toLemma 2, we have T ( l − m ′ + t m ′ ,n ′ ≤ ρ ( l ) M M X m =1 X n ∈C ( l ) m t n + ρ ( l ) t ′ n . The algorithm stops after N iterations. Since T ( l +1) =max { T ( l ) , T ( l ) m ′ + t m ′ ,n ′ } and T (0) = 0 , we conclude that T = T ( N +1) = max n T ( N ) , T ( N ) m ′ + t m ′ ,n ′ o ≤ ρM M X m =1 X n ∈C m t n + ρT ∗ = ρM N X n =1 t n + ρT ∗ . With the closed access stragegy, we set t m,n = ∞ , for BS m that user n cannot connect to, for all m , n . The proof followsthe same procedure and we have the same conclusion.Combining Lemmas 1 and 4, we have the following theoremregarding the performance of Algorithm 2. Theorem 2.
The greedy approximation algorithm in Algo-rithm 2 ensures a (2 ρ ) -approximation of optimal solution.Proof: The proof is straightforward. We have T ∗ ≤ T ≤ ρM N X n =1 t n + ρT ∗ ≤ ρT ∗ , where T ∗ is the optimal solution and T is the greedy approxi-mation algorithm solution. Note that unlike in Section IV-C1,we have T ∗ ≤ T since there is no relaxation here.From Theorem 2, ρ is an important parameter to the perfor-mance of the greedy approximation algorithm. The smaller the ρ , the smaller the optimality gap. In order to make the greedyapproximation algorithm solution more competitive, we onlyallow users to choose from a subset B n of the original BS set.Then we have the new subsets B ′ n and A ′ m as B ′ n = B n ∩ (cid:18)(cid:26) m | t m,n t n ≤ Γ (cid:27) ∪ { } (cid:19) , A ′ m = { n | m ∈ B ′ n } , (11)where Γ is a predefined threshold and { } is the index of theMBS. Γ can also be used to indicate the SINR requirement ofusers. The set A m is replaced by A ′ m accordingly. This way,the greedy approximation algorithm solution will be T ∗ ≤ T ≤ T ∗ . (12) D. Randomized Algorithm
Both the rounding and greedy approximation algorithms arecentralized algorithms that require frequent CSI updates. Inthis section, we introduce a randomized algorithm for the cellassociation problem. With the randomized algorithm, each user n randomly chooses a subset of B n to connect to. Once thesubsets are determined, no information exchange is requiredamong the users. We assume user n connects to BS m withprobability p m,n and the expected service time for user n oneach BS is identical (i.e., by tuning the p m,n ’s), i.e., p m,n t m,n = H n , for all m ∈ B n . Since a BS with a smaller t m,n should have higher preference,we set p m,n proportional to /t m,n . Since each user has tochoose a BS to connect to, we have P m ∈B n p m,n = 1 for all n . It follows that H n = 1 P m ∈B n /t m,n , for all n. (13)The expected load on BS m , denoted by T m , is T m = E [ T m ] = X n ∈A m t m,n p m,n = X n ∈A m H n , for all m. (14)Since users are randomly connected to the BS’s, our objectiveis to minimize the maximum value of the expected load T max . min T max = min { max m T m } . (15)It can be seen from (14) that minimizing T m is equivalent toreducing the number of users in A m .The randomized algorithm consists of two phases. In PhaseI, we use a threshold Λ to obtain the subsets A m and B n . B ′ n = B n ∩ ( { m | t m,n ≤ Λ } ∪ { } ) , A ′ m = { n | m ∈ B ′ n } . (16)Note that the subsets A ′ m and B ′ n are different from thosedefined in (11): Λ is the upper bound of service time t m,n ,while Γ is the upper bound on the service time ratios. Thuswe have all t m,n ≤ Λ for all n and n ∈ A ′ m . Then we derivethe upper bounds for H n , T m and T max as H n = P m ∈B′ n /t m,n ≤ P m ∈B′ n / Λ = Λ |B ′ n | T m = P n ∈A ′ m H n ≤ |A ′ m | min n |B ′ n | Λ T max = max m T m ≤ max m |A ′ m | min n |B ′ n | Λ . (17)where |A ′ m | and |B ′ n | are the cardinalities of subsets A ′ m and B ′ n , respectively.In Phase II, we aim to further reduce the sizes of A ′ m and B ′ n . From (13), we find that H n ′ gets increased when BS m ′ is removed from set B ′ n ′ and user n ′ is removed from set A ′ m ′ simultaneously. The increase, denoted by ∆ m ′ ,n ′ , is given by ∆ m ′ ,n ′ = 1 P m ∈B ′ n /t m,n − /t m ′ ,n ′ − P m ∈B ′ n /t m,n = 1 /t m ′ ,n ′ ( P m ∈B ′ n /t m,n − /t m ′ ,n ′ )( P m ∈B ′ n /t m,n ) . (18)For those BS’s in the set { m | m ∈ B ′ n ′ , m = m ′ } , their T m ’sbecome larger when BS m ′ is removed from set B ′ n ′ and user n ′ is removed from set A ′ m ′ . On the other hand, T m ′ is reducedby H m ′ ,n ′ according to (14).The randomized algorithm is presented in Algorithm 3. InStep , we find the users that each has more than one BS ontheir BS list B ′ n . Then from Step to Step , we find the BS m ′ with the largest T m ′ and compute the possible maximumload T maxm ′ ,n on BS’s for all users that might be connected toBS m ′ , assuming user n is removed from A ′′ m ′ . In Step , wepick user n ′ with the minimum T maxm ′ ,n value. If the value is lessthan the original T m ′ , we remove the BS-user pair { m ′ , n ′ } from sets A ′′ m ′ and B ′′ n ′ . Otherwise, the algorithm is terminated.6 lgorithm 3: Randomized Algorithm for Cell Association Initialize A ′′ m = A ′ m , B ′′ n = B ′ n ; Set the user set N = { n ||B ′′ n | > } ; Compute T m according to (14) ; while N is not empty do Find the BS m ′ with m ′ = arg max m T m ; for user n in ( A ′′ m ′ ∩ N ) do Compute ∆ m ′ ,n according to (18) ; for m = 1 to M do if m = m ′ then Set T ′ m ′ = T m ′ − H n ; else if m in { m | m ∈ B ′′ n } then Set T ′ m = T m + ∆ m ′ ,n ; else Set T ′ m = T m ; end end Set T maxm ′ ,n = max m T ′ m ; end Find user n ′ with n ′ = arg min n T maxm ′ ,n ; if T m ′ ≥ T maxm ′ ,n ′ then Remove m ′ from B ′′ n ′ and n ′ from A ′′ m ′ ; Update all T m ’s ; if |B ′′ n ′ | = 1 then Remove n ′ from N end else The algorithm is terminated ; end end When the algorithm is executed, sets A ′′ m ′ and B ′′ n ′ are subsetsof A ′ m ′ and B ′ n ′ , respectively. Since the complexity from Step to Step is O ( M N ) in the worst case, the complexity ofthe entire randomized algorithm is O ( M × N ) . Proposition 4.
The computational complexity of the random-ized algorithm is O ( M × N ) . Finally, we have the following theorem on the performanceof the randomized algorithm.
Theorem 3.
The maximum expected service time achieved bythe randomized algorithm is upper bounded by T max ≤ max m |A ′′ m | min n |B ′′ n | × max n max m ∈B ′′ n t m,n . (19) Proof:
The proof is similar to the derivation of (17), butthe new upper bound of service time, max n max m ∈B ′′ n t m,n ,is used, instead of the service time bound Λ .V. S ERVICE S CHEDULING
Once the cell associate problem is solved as in Section IV,we then study how to schedule the transmissions of multipleusers connecting to the same BS. Since we assume thebandwidth B is fully utilized for transmitting a user’s datapacket (see (3)), the packets are transmitted consecutively.We need to determine the service order of the users that areassociated with the same BS. TABLE IS
IMULATION P ARAMETERS
Paramter Value
Number of BS’s Total network bandwidth MHzTransmit power of the MBS dBmTransmit power of the FBS . dBmPath loss model for MBS
28 + 35 log ( d ) Path loss model for FBS . ( d ) Shadowing effect dBPacket length KBytesThreshold ρ Consider a tagged BS to which K users are connected. Theuser service times are { t , t , · · · , t K } . If the service orderfollows the user index, the average waiting time is given by T wait = 1 K K X n =1 n X i =1 t i . (20)We have the following theorem to minimize the averagewaiting time T wait . Theorem 4.
Given K users with service times { t , t , · · · , t K } , the average waiting time is minimizedwhen the users are served in the increasing order of theirservice times.Proof: First we sort the users according to their servicetimes in the increasing order. The ordered service times aredenoted by { t ′ , · · · , t ′ K } . Consider two ordered users i and j , where ≤ i < j ≤ K . We have t ′ i ≤ t ′ j . If the positionsof i and j are swapped, it is obvious that the waiting timesof users from to i − and the users from j to K are notaffected and remain the same values. However, the awaitingtime for each user from i to j − is increased by t ′ j − t ′ i .Therefore, we conclude that the average waiting time is theleast when the users are served in the increasing order of theirservice times.VI. P ERFORMANCE E VALUATION
In this section, we evaluate the performance of the proposedcell association and service scheduling algorithms using MAT-LAB simulations. The channel models from [18] are adoptedin our simulations. The channel gain (in dB ) from the BS’sto users can be expressed as
10 log( G m,n ) = − P L m ( d m,n ) − u m , where d m,n is the distance from BS m to user n , and u m is the shadowing effect, which is normally distributed witha zero mean and variance δ m . The simulation parameters arepresented in Table VI. In the figures, each point in the averageof simulation runs; we included confidence intervalsas error bars to make the simulation results credible.We present simulation results for the following two scenar-ios: (i) open access femtocells; (ii) closed access femtocells.For comparison purpose, we also developed and simulated aselfish scheme and compared it with the proposed schemes.With the selfish scheme, every user simply chooses the BSwith the best channel condition to connect to. A. Open Access Strategy
In the first scenario, there are M = 6 BS’s, i.e., one MBSand five FBS’s. The number of users ranges from to M a x i m u m T o t a l S e r v i c e T i m e ( m s ) Greedy ApproximationSequential FixingRandomized AlgorithmSelfish UserRounding ApproximationLow Bound (a) Total service time vs. number of users
30 40 50 60 70 80100200300400500600700800 Number of Users A v e r age W a i t i ng t i m e ( u s ) Greedy ApproximationSequential FixingRandomized AlgorithmSelfish UserRounding Approximation (b) Average waiting time vs. number of users
30 40 50 60 70 800.160.170.180.190.20.210.220.230.24 Number of Users F a i r ne ss Greedy ApproximationSequential FixingRandomized AlgorithmSelfish UserRounding Approximation (c) Fairness vs. number of usersFig. 2. Performance evaluation of the open access strategy.TABLE IIE
XECUTION T IMES OF THE P ROPOSED A LGORITHMS UNDER THE O PEN A CCESS S TRATEGY ( S )No. users 30 40 50 60 70 80Greedy 0.024 0.034 0.024 0.030 0.026 0.038Approx.Sequential 16.532 24.020 30.809 48.713 47.842 50.654FixingRandomized 0.030 0.048 0.077 0.136 0.132 0.151AlgorithmSelfish User 0.035 0.035 0.035 0.035 0.036 0.026SchemeRounding 0.133 0.148 0.160 0.168 0.176 0.213Approx. with step size . They are randomly located in network area.Each user can connect to one of the BS’s.We first examine the impact of the number of users ontotal service time. In Fig. 2(a), we plot the maximum totalservice time for the five algorithms along with the lowerbound found by solving the relaxed LP. As expected, the moreusers, the more total service time on BS’s. Except for the lowbound, the sequential fixing algorithm achieves the smallesttotal service time. The rounding approximation algorithm hasa slightly better performance than the greedy approximationalgorithm and the result justifies the approximation ratioproven in Section IV-C. Both approximation algorithms alwaysachieve lower load than both the randomized algorithm andthe selfish scheme. We also observe that beyond users,all the proposed algorithms have lower service times than thesimple selfish scheme. When number of users becomes larger,the simple selfish scheme becomes less competitive and therounding approximation algorithm achieves almost lesstotal service time in the case of users.After cell association, users should be properly scheduledto get service in BS’s to minimize average waiting time. InFig. 2(b), we investigate the impact of the number of users onaverage waiting time. In the scheme of greedy approximation,randomized algorithm and sequential fixing, we use the servicescheduling policy in Section V to schedule users in BS’sand obtain the corresponding waiting time. For comparison,we randomly schedule users in BS’s in the selfish schemeand rounding approximation scheme. Intuitively, the larger thenumber of users, the larger the average waiting time. We cansee from the figure that, the average waiting time obtained by the greedy approximation algorithm is very close to thatby the sequential fixing algorithm, while without appropriatescheduling, the rounding approximation algorithm achieves thelargest waiting time, which is almost twice as large as thewaiting time achieved by greedy approximation algorithm.To evaluate the fairness performance, we adopt Raj Jain’sfairness index given by J ( C , C , · · · , C N ) = ( P Nn =1 C n ) N × P Nn =1 C n ,where C n is the network throughput for user n [8]. The valueof the index ranges from /N (worst case) to (best case).It can be seen from Fig. 2(c) that fairness indexes decreasewhen the number of users is increased. We notice that, theselfish scheme and the randomized algorithm achieve betterfairness than the other three schemes. Figs. 2(a) and 2(c) showthat from operator’s viewpoint, the selfish and the randomizedschemes are not preferred since they produce less balancedload on BS’s. From users’s viewpoint, these two schemes maybe appealing due to their fairness performance.We list the execution times of the five schemes in Ta-ble VI-A. We find the execution time increases as the numberof users is increased. The selfish scheme always has thesmallest execution time, while sequential fixing has the largestexecution time. Although the rounding approximation algo-rithm can achieve smaller load on the BS’s, its execution timeis greater than that of the greedy approximation algorithm.This result also justifies the complexity analysis for the pro-posed schemes. The running time of the greedy approximationalgorithm and the selfish scheme is always much smallerthan other schemes and does not increase obviously with thenumber of users. For the closed access simulations shown inSection VI-B, the execution times of the proposed algorithmsare all much smaller than that shown in Table VI-A, sincethe user list include fewer users in the closed access case. Weomit these results for brevity. B. Closed Access Strategy
We next investigate the second scenario with closed accessfemtocells. Now each FBS maintains a user list and only servesthe listed users. Note that the MBS will always serve all theusers inside its coverage.In Fig. 3(a), we evaluate the impact of the number ofusers on total service time. Intuitively, the total service timeincreases as the number of users. However, we find that italso depends on the user list at each FBS. In the simulation,8 M a x i m u m T o t a l S e r v i c e T i m e ( m s ) Greedy ApproximationSequential FixingRandomized AlgorithmSelfish UserRounding ApproximationLow Bound (a) Total service time vs. number of users
30 40 50 60 70 805001000150020002500 Number of Users A v e r age W a i t i ng t i m e ( u s ) Greedy ApproximationSequential FixingRandomized AlgorithmSelfish UserRounding Approximation (b) Average waiting time vs. number of users
30 40 50 60 70 800.160.170.180.190.20.21 Number of Users F a i r ne ss Greedy ApproximationSequential FixingRandomized AlgorithmSelfish UserRounding Approximation (c) Fariness vs. number of usersFig. 3. Performance evaluation of the closed access strategy. we randomly choose the user set A m for BS m . Moreover,the user list at each FBS is further reduced due to the SINRthreshold. Consequently, all the proposed algorithms achieveclose performance in the closed access scenario. The totalservice time of the proposed algorithms is close to the lowbound in closed access scenario. However, the performanceof all the proposed algorithms is better than that of the selfishscheme, as we can see in Fig. 3(a).We next show the impact of the number of users on averagewaiting time in Fig. 3(b). The scheduling policy setting is thesame as that in the open access scenario. The result thus is alsosimilar to the open access case that, the selfish scheme andthe rounding approximation scheme achieve the largest waitingtime. Actually with proposed optimal service scheduling, theapproximation algorithms will achieve as less waiting time asthat of the sequential fixing scheme.Finally, we plot the fairness indices in Fig 3(c). Therandomized algorithm, although not better than the selfishscheme, achieves the best performance in fairness than theother proposed schemes. Despite of its good performance inminimizing the maximum service time, the rounding approx-imation algorithm, is not competitive with respect to fairness.Due to the randomness of user lists at BS’s, the confidentialintervals are larger than those in the open access scenario.VII. C ONCLUSION
In this paper, we investigated the problem of cell associationand service scheduling in two-tier femtocell networks. Wedeveloped several algorithms and analyzed their performance.The sequential fixing algorithm achieves the best performancein total service time but it has a relatively high complexity.Then we presented two approximation algorithms with lowercomplexity and proven approximation ratios. We also proposeda randomized algorithm with a proven performance boundthat requires the least information exchange among users. Inaddition, we addressed the service scheduling problem withan optimal solution. The proposed algorithms were validatedwith simulations in both open and closed access scenarios.R
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