Distributed Interference Management Policies for Heterogeneous Small Cell Networks
11 Distributed Interference Management Policiesfor Heterogeneous Small Cell Networks
Kartik Ahuja, Yuanzhang Xiao and Mihaela van der SchaarDepartment of Electrical Engineering, UCLA, Los Angeles, CA, 90095Email: [email protected], [email protected] and [email protected]
Abstract
We study the problem of distributed interference management in a network of heterogeneoussmall cells with different cell sizes, different numbers of user equipments (UEs) served, and differentthroughput requirements by UEs. We consider the uplink transmission, where each UE determines whenand at what power level it should transmit to its serving small cell base station (SBS). We propose ageneral framework for designing distributed interference management policies, which exploits weakinterference among non-neighboring UEs by letting them transmit simultaneously (i.e., spatial reuse),while eliminating strong interference among neighboring UEs by letting them transmit in different timeslots. The design of optimal interference management policies has two key steps. Ideally, we need tofind all the subsets of non-interfering UEs, i.e., the maximal independent sets (MISs) of the interferencegraph, but this is NP-hard (non-deterministic polynomial time) even when solved in a centralized manner.Then, in order to maximize some given network performance criterion subject to UEs’ minimumthroughput requirements, we need to determine the optimal fraction of time occupied by each MIS,which requires global information (e.g., all the UEs’ throughput requirements and channel gains). Inour framework, we first propose a distributed algorithm for the UE-SBS pairs to find a subset of
MISs in logarithmic time (with respect to the number of UEs). Then we propose a novel problemreformulation which enables UE-SBS pairs to determine the optimal fraction of time occupied by eachMIS with only local message exchange among the neighbors in the interference graph. Despite thefact that our interference management policies are distributed and utilize only local information, wecan analytically bound their performance under a wide range of heterogeneous deployment scenariosin terms of the competitive ratio with respect to the optimal network performance, which can onlybe obtained in a centralized manner with NP complexity. Remarkably, we prove that the competitiveratio is independent of the network size. Through extensive simulations, we show that our proposedpolicies achieve significant performance improvements (ranging from 150% to 700%) over state-of-the-art policies. a r X i v : . [ c s . I T ] M a r I. I
NTRODUCTION
Dense deployment of low-cost heterogeneous small cells (e.g. picocells, femtocells) has be-come one of the most effective solutions to accommodate the exploding demand for wirelessspectrum [1] [2] [3]. On one hand, dense deployment of small cells significantly shortens thedistances between small cell base stations (SBSs) and their corresponding user equipments (UEs),thereby boosting the network capacity. On the other hand, dense deployment also shortens thedistances between neighboring SBSs, thereby potentially increasing the inter-cell interference.Hence, while the solution provided by the dense deployment of small cells is promising, itssuccess depends crucially on interference management by the small cells. Efficient interferencemanagement is even more challenging in heterogeneous small cell networks, due to the lack ofcentral coordinators, compared to that in traditional cellular networks.In this paper, we propose a novel framework for designing interference management policiesin the uplink of small cell networks, which specify when and at what power level each UEshould transmit . Our proposed design framework and the resulting interference managementpolicies fulfill all the following important requirements: • Deployment of heterogeneous small cell networks : Existing deployments of small cell net-works exhibit significant heterogeneity such as different types of small cells (picocells andfemtocells), different cell sizes, different number of UEs served, different UEs’ throughputrequirements etc. • Interference avoidance and spatial reuse : Effective interference management policies shouldtake into account the strong interference among neighboring UEs, as well as the weakinterference among non-neighboring UEs. Hence, the policies should effectively avoid in-terference among neighboring UEs and use spatial reuse to take advantage of the weakinterference among non-neighboring UEs. • Distributed implementation with local information and message exchange : Since there is nocentral coordinator in small cell networks, interference management policies need to becomputed and implemented by the UEs in a distributed manner, by exchanging only localinformation through local message exchanges among neighboring UE-SBS pairs. • Scalability to large networks : Small cells are often deployed over a large scale (e.g., in a Although we focus on uplink transmissions in this paper, our framework can be easily applied to downlink transmissions. city). Effective interference management policies should scale in large networks, namelyachieve efficient network performance while maintaining low computational complexity. • Ability to optimize different network performance criteria : Under different deployment sce-narios the small cell networks may have different performance criteria, e.g., weightedsum throughput or max-min fairness. The design framework should be general and shouldprescribe different policies to optimize different network performance criteria. • Performance guarantees for individual UEs : Effective interference management should pro-vide performance guarantees (e.g., minimum throughput guarantees) for individual UEs.As we will discuss in detail in Section II, existing state-of-the-art policies for interferencemanagement cannot simultaneously fulfill all of the above requirements.Next, we describe our key results and major contributions:1. We propose a general framework for designing distributed interference management poli-cies that maximizes the given network performance criterion subject to each UE’s minimumthroughput requirements. The proposed policies schedule maximal independent sets (MISs) ofthe interference graph to transmit in each time slot. In this way, they avoid strong interferenceamong neighboring UEs (since neighboring UEs cannot be in the same MIS), and efficientlyexploit the weak interference among UEs in a MIS by letting them to transmit at the same time.2. We propose a distributed algorithm for the UEs to determine a subset of MISs. The subsetof MISs generated ensures that each UE belongs to at least one MIS in this subset. Moreover,the subset of MISs can be generated in a distributed manner in logarithmic time (logarithmicin the number of UEs in the network) for bounded-degree interference graphs . The logarithmicconvergence time is significantly faster than the time (linear or quadratic in the number of UEs)required by the distributed algorithms for generating subsets of MISs in [4]–[6].3. Given the computed subsets of MISs, we propose a distributed algorithm in which eachUE determines the optimal fractions of time occupied by the MISs with only local message Consider the interference graph of the network, where each vertex is a UE-SBS pair and each edge indicates strong interferencebetween the two vertices. An independent set (IS) is a set of vertices in which no pair is connected by an edge. An IS is a MISif it is not a proper subset of another IS. Bounded-degree graphs are the graphs whose maximum degree can be bounded by a constant independent of the size of thegraph, i.e., ∆ = O (1) . As we will show in Theorem 5, for the interference graphs that are not bounded-degree graphs, eventhe centralized solution, given all the MISs, cannot satisfy the minimum throughput requirements. exchange. The message is exchanged only among the UE-SBS pairs that strongly interfere witheach other, i.e. among neighbors in the interference graph. The distributed algorithm will outputthe optimal fractions of time for each MIS such that the given network performance criterion ismaximized subject to the minimum throughput requirements.4. Under a wide range of conditions, we analytically characterize the competitive ratio ofthe proposed distributed policy with respect to the optimal network performance. Importantly,we prove that the competitive ratio is independent of the network size, which demonstratesthe scalability of our proposed policy in large networks. Remarkably, the constant competitiveratio is achieved even though our proposed policy requires only local information, is distributed,and can be computed fast, while the optimal network performance can only be obtained in acentralized manner with global information (e.g., all the UEs’ channel gains, maximum transmitpower levels, minimum throughput requirements) and NP (non-deterministic polynomial time)complexity.5. Through simulations, we demonstrate significant (from 160% to 700 %) performance gainsover state-of-the-art policies. Moreover, we show that our proposed policies can be easily adaptedto a variety of heterogeneous deployment scenarios, with dynamic entry and exit of UEs.The rest of the paper is organized as follows. In Section II we discuss the related worksand their limitations. We describe the system model in Section III. Then we formulate theinterference management problem and give a motivating example in Section IV. We propose thedesign framework in Section ?? , and demonstrate the performance gain of our proposed policiesin Section VI. Finally, we conclude the paper in Section VII.II. R ELATED W ORKS
State-of-the-art interference management policies can be divided into three main categories:policies based on power control, policies based on spatial reuse, and policies based on jointspatial reuse and power control.
A. Distributed Interference Management Based on Power Control
Policies based on distributed power control, with representative references [7]–[14] have beenused for interference management in both cellular and ad-hoc networks. In these policies, all theUEs in the network transmit at a constant power all the time (provided that the system parameters remain the same) . The major limitation of policies based on power control is the difficulty inproviding minimum throughput guarantees for each UE, especially in the presence of stronginterference. Some works [7], [8], [10] use pricing to mitigate the strong interference. However,they [7], [8], [10] cannot strictly guarantee the UEs’ minimum throughput requirements. Indeed,the low throughput experienced by some users, caused by strong interference, is the fundamentallimitation of such power control approaches - even the optimal power control policy obtainedby a central controller [15], [16] can be inefficient . Since strong interference is very commonin dense small cell deployments (e.g. in offices and apartments where SBSs are installed closeto each other [18]), more efficient policies are required which can guarantee the individualUEs’ throughput requirements. Also, there exist a different strand of work based on [19] whichproposes a distributed algorithm to achieve the desired minimum throughput requirement for eachUE. However, these works cannot optimize network performance criterion such as weighted sumthroughput, max-min fairness etc. and hence are suboptimal. B. Distributed Spatial Reuse Based on Maximal Independent Sets
An efficient solution to mitigate strong interference is spatial reuse, in which only a subset ofUEs (which do not significantly interfere with each other) transmit at the same time. Spatial Timereuse based Time Division Multiple Access (STDMA) has been widely used in existing workson broadcast scheduling in multi-hop networks [4]–[6] . Specifically, these policies construct acyclic schedule such that in each time slot an MIS of the interference graph is scheduled. Theconstructed schedule ensures that each UE is scheduled at least once in the cycle.In terms of performance, STDMA policies [4]–[6] cannot guarantee the minimum throughputrequirement of each UE, and usually adopt a fixed scheduling (i.e. follow a fixed order inwhich the MISs are scheduled), which may be very inefficient depending on the given networkperformance criteria. For example, the policies in [6] are inefficient in terms of fairness. In terms Although some power control policies [7], [8], [10] go through a transient period of adjusting the power levels before theconvergence to the optimal power levels, the users maintain constant power levels after the convergence. In the case of average sum throughput maximization given the minimum average throughput constraints of the UEs, thepower control policies are inefficient if the feasible rate region is non-convex [17] . These works [4]–[6] do not have the exactly same model as in our setting. However, these works can be adapted to ourmodel. Hence, we also compare with these works to have a comprehensive literature review. of complexity, for the distributed generation of the subsets of MISs, the STDMA policies in [4]–[6] require an ordering of all the UEs, and have a computational complexity (in terms of thenumber of steps executed by the algorithm) that scales as O ( | V | )) (in [5], [6]) or O ( | V || E | )) (in [4]), where | V | and | E | are the number of vertices/UEs and the number of edges in theinterference graph, respectively. Hence, in large-scale dense deployments, the complexity growssuperlinearly with the number of UEs, making the policies difficult to compute. By contrast, ourproposed distributed algorithm for generating subsets of MISs does not require the ordering ofall the UEs, and has a complexity that scales as O (log | V | ) , namely sublinearly with the numberof the UEs, for bounded-degree graphs. Finally, the STDMA policies in [4]–[6] are designed for the MAC layer and assume that allthe UEs are homogeneous at the physical layer. In practice, different UEs are heterogeneous dueto their different distances from their SBSs, their different maximum transmit power levels, etc.This heterogeneity is important, and will be considered in our design framework.
C. Distributed Power Control and Spatial Reuse For Multi-Cell Networks
As we have discussed, the works in the above two categories either focus on distributed powercontrol in the physical layer [7], [8], [10] or focus on distributed spatial reuse in the MAC layer[4]–[6]. Similar to our paper, some works (representative references [20]–[24] ) adopted a cross-layer approach and proposed distributed joint power control and spatial reuse for multi-cellnetworks. However, although these works schedule a subset of UEs to transmit at the same time,the subset is not the MIS of the interference graph [22], [23]. For example, the policies in [22],[23] schedule one UE from each small cell at the same time, even if some UEs are from smallcells very close to each other. In this case, the UEs will experience strong inter-cell interference.Hence, the works in [22], [23] cannot perfectly eliminate strong interference from neighboringcells and exploit weak interference from non-neighboring cells. Moreover, the works in [20]–[24]cannot provide minimum throughput guarantees for the UEs. As will be shown in Theorem 5, for graphs which are not bounded degree graphs, even a centralized solution based on allthe MISs cannot satisfy the minimum throughput requirements.
III. S
YSTEM M ODEL
A. Heterogeneous Network of Small Cells
We consider a heterogeneous network of K small cells operating in the same frequencyband (see Fig. 1), which represents a common deployment scenario considered in practice[10] [13] [25]. Note that the small cells can be of different types (e.g. picocells, femtocells,etc.) and thereby belong to different tiers in the heterogeneous network. Each small cell j hasone SBS, (SBS- j ), which serves a set of UEs under a closed access scenario [10]. Denotethe set of UEs by U = { , ..., N } . We write the association of UEs to SBSs as a mapping T : { , ..., N } → { , .., K } , where each UE- i is served by SBS- T ( i ) . We focus on the uplinktransmissions; the extension to downlink transmissions is straightforward when each SBS servesone UE at a time (e.g. TDMA among UEs connected to the same SBS).Each UE- i chooses its transmit power p i from a compact set P i ⊆ R + . We assume that ∈ P i , ∀ i ∈ { , ..., N } , namely any UE can choose not to transmit. The joint power profileof all the UEs is denoted by p = ( p , ...., p N ) ∈ P (cid:44) Π Ni =1 P i . Under the joint power profile p , the signal to interference and noise ratio (SINR) of UE- i ’s signal, experienced at its servingSBS- j = T ( i ) , can be calculated as γ i ( p ) = g ij p iN (cid:80) k =1 ,k (cid:54) = i g kj p k + σ j , where g ij is the channel gain fromUE- i to SBS- j , and σ j is the noise power at SBS j . The UEs do not cooperate to encode theirsignals to avoid interference, hence, each UE-SBS pair treats the interference from other UEsas white noise. Hence, each UE- i gets the following throughput [22], r i ( p ) = log (1 + γ i ( p )) . B. Interference Management Policies
The system is time slotted at t = 0,1,2..., and the UEs are assumed to be synchronized as in[22], [23] [26] [27].. At the beginning of each time slot t , each UE- i decides its transmit power p ti and obtains a throughput of r i ( p t ) . Each UE i ’s strategy, denoted by π i : Z + = { , , .. } → P i ,is a mapping from time t to a transmission power level p i ∈ P i . The interference managementpolicy is then the collection of all the UEs’ strategies, denoted by π = ( π , ..., π N ) . The average Our solutions will be based on spatial time reuse assuming every UE uses the same frequency. Our solutions can be extendedto spatial frequency reuse, where we let different MISs operate in non-overlapping frequency bands. We use the Shannon capacity here. However, our analysis is general and applies to the throughput models that consider themodulation scheme used.
FBS-1 FUE-2 PUE-1 PUE-2
PBS
FBS-2FUE-3
FUE
PUEFBSPBS Femtocell User
Equipment
Picocell User EquipmentFemtocell Base StationPicocell Base Station Direct channel gain
Cross channel gainLocal message exchange Local message exchange
Femto/Pico Cell
FUE-1
Figure 1. Illustration of a heterogeneous small cell network. throughput for UE i is given as R i ( π ) = lim T →∞ T +1 T (cid:80) t =0 r i ( p t ) , where p t = ( π ( t ) , ..., π N ( t )) is the power profile at time t . We assume the channel gain to be fixed over the consideredtime horizon as in [22] [28]–[31]. However, we will illustrate in Section VI that our frameworkperforms well under dynamic channel conditions (due to fading, time varying channel) as well.An interference management policy π const is a policy based on power control [7], [8], [10]if π const ( t ) = p for all t . As we have discussed before, our proposed policy is based on MISsof the interference graph. The interference graph G has N vertices, each of which is one ofthe N UE-SBS pairs. There is an edge between two pairs/vertices if their cross interferenceis high (rules for deciding if interference is high will be discussed in Section V) and let therebe M edges in the graph. Given an interference graph, we write I = { I , ..., I N MIS } as the setof all the MISs of the interference graph. Let p I j be a power profile in which the UEs in theMIS I j transmit at their maximum power levels and the other UEs do not transmit, namely p k = p maxk (cid:44) max P k if k ∈ I j and p k = 0 otherwise. Let P MIS = { p I , ..., p I NMIS } be theset of all such power profiles. Then π is a policy based on MIS if π ( t ) ∈ P MIS for all t . Wedenote the set of policies based on MISs by Π MIS = { π : Z + → P MIS } .IV. P ROBLEM F ORMULATION AND
A M
OTIVATING E XAMPLE
In this section, we formulate the interference management policy design problem and give amotivating example to highlight the advantages of the proposed policy over existing policies.
A. The Interference Management Policy Design Problem
We aim to optimize a chosen network performance criterion W ( R ( π ) , ...., R N ( π )) , definedas a function of the UEs’ average throughput. We can choose any performance criterion thatis concave in R ( π ) , ...., R N ( π ) . For instance, W can be the weighted sum of all the UEs’ Step 1.
Each UE identifies the interfering UE-
SBS pairs. Step 2.
Distributed generation of MISs: each UE executes Phase 1 and 2 to identify the MISs it belongs to. (Theorem 1)
Step 3.
Each UE executes the procedure in Table I, to arrive at the optimal fraction of time allocated to each
MIS. (Theorem 2 and 3)
Step 4.
Each UE computes the cycle length and the duration of each
MIS in the cycle.
Figure 2. Steps in the Design Framework. throughput, i.e. N (cid:80) i =1 w i R i ( π ) with N (cid:80) i =1 w i = 1 and w i ≥ . Alternatively, the network performancecan be max-min fairness (i.e. the worst UE’s throughput) and hence W can be defined as min i R i ( π ) . The policy design problem can be then formalized as follows: Policy Design Problem (PDP) max π W ( R ( π ) , ..., R N ( π )) subject to R i ( π ) ≥ R mini , ∀ i ∈ { , ..., N } The above design problem is very challenging to solve even in a centralized manner (it hasbeen shown to be NP-hard [32] even when we restrict to policies based on power control π const ).Denote the optimal value of the PDP as W opt . Our goal is to develop distributed, polynomial-timealgorithms to construct policies that achieve a constant competitive ratio with respect to W opt ,with the competitive ratio independent of the network size. We achieve our goal by focusingon policies based on MISs Π MIS , among other innovations that will be described in Section V.Next, we provide a motivating example to demonstrate the efficiency of our proposed policy.V. D
ESIGN F RAMEWORK FOR D ISTRIBUTED I NTERFERENCE M ANAGEMENT
A. Proposed Design Framework
Our proposed design framework (see Fig. 2) consists of the following four steps.
Step 1. Identification of the interfering neighbors:
In Step 1, each UE-SBS pair identifiesthe UE-SBS pairs that strongly interfere with it. Essentially, each pair obtains a local view (i.e.,its neighbors) of the interference graph. Note that an edge exists between two pairs if at leastone of them identifies the other as a strong interferer.Specifically, each UE-SBS pair is first informed of other pairs in the geographical proximityby managing servers (e.g., femtocell controllers/gateways) [33] [34] [29] [30]. Then each paircan decide whether another pair is strongly interfering based on various rules, such as rulesbased on Received Signal Strength (RSS) in the
Physical Interference Model [33] [29] [30], and ={R,Y,G} C1 = {R,Y,G}C1 = {R,Y,G}C1 ={R,Y,G} 1 234C1 = {} 1 3224C1 = {Y}C1 = {} C1 = {G} C1 = {}C1 = {}C1 = {} C1 = {} A). Before Phase 1 and Phase 2
B). After Phase 1, (Time = P1 time slots). C). After Phase 2, (Time=P1+P2 time slots). C1 Qi List of colors remaining for UE-i at time slot QY/G/R color acquired by a UESet of colors {R,G},{R,Y} acquired by a UE {R,Y,G} {Red,Yellow,Green}
1. Color classes corresponding to Y,G are ISs after Phase 1, R is MIS 2. Color classes corresponding to R,Y,G are MISs after Phase 2.3
Figure 3. Illustration of the distributed generation of MISs in Step 2. rules based on the locations in the
Protocol Model [28]. If one pair identifies another pair asstrongly interfering, its decision can be relayed by the managing servers to the latter, such thatany two pairs can reach consensus of whether there exists an edge between them.
Step 2. Distributed generation of MISs that span all the UEs:
In Step 2, the UE-SBSpairs generate a subset of MISs in a distributed fashion. It is important that the generated subsetspans all the UEs, namely every UE is contained in at least one MIS in the subset. Otherwise,some UEs will never be scheduled.The key idea is that from a given list of colors, each UE has to choose a set of colors suchthat the choice does not conflict with its neighbors. We should ensure that each UE has at leastone color. We call the set of UEs with the same color “a color class”. In addition, we should alsoensure that every color class is a MIS. This step is composed of two phases: first, distributedcoloring of the interference graph based on [35], and second, extension of color classes to MISs.All the UEs are synchronized and carry out their computation simultaneously. We now explainthe algorithm in detail. The pseudo-codes can be found in Table II and III in the Appendix.
Phase 1. Distributed coloring of the interference graph : Let H be the maximum numberof colors given to all SBSs at the installation and d i be the degree (number of neighbors in theinterference graph) of the i th pair. The goal of this phase is to let each UE-SBS pair i choose one color from C i (cid:44) { , ...H } ∩ { , .., d i + 1 } , such that no neighbors choose the same color.The distributed coloring works as follows. The maximum number of colors H should be set to be larger than the maximum number of UE-SBS pairs interfering withany UE-SBS pair. The SBSs can determine H according to the deployment scenario. H in general will also include the numberof UEs that use the same SBS who interfere with each other along with the other neighboring UEs. For example, H can be10-15 in an office building with dense deployment of SBSs, and can be 3-5 in a residential area. i) At the beginning of each time slot t , each UE i chooses a color from the set of remainingcolors C ti uniformly randomly, and informs its neighbors of its tentative choice. This informationcan be transmitted through the back-haul network/X2 interface that is used for ICIC [34].ii) If the tentative choice of a UE does not conflict with any of its neighbor, then it fixes itscolor choice and informs the neighbors of its choice. This UE does not contend for colors anyfurther in Phase 1. The neighbors delete the color chosen by i from their lists C t +1 j , ∀ j ∈ N ( i ) ,where N ( i ) is the set of i ’s neighbors.iii) Otherwise, if there is a conflict, then the UE does not choose that color and repeats i) andii) in the next time slot.There are (cid:100) c log N (cid:101) + 1 time slots in Phase 1, where c is the parameter given by theprotocol. The number of time slots is known to the SBSs at installation. Phase 1 is successful ifall the UEs acquire a color, which implies that the set of color classes (i.e., the set of UE-SBSpairs with the same color) spans all the UEs. Phase 2. Extending color classes to the MISs:
Each color class obtained at the end of Phase1 is an independent set (IS) of the graph. In Phase 2, we extend each of these ISs to MISs andpossibly generate additional MISs. After Phase 1, each UE has chosen one color and deletedsome colors from its list. But there may still be remaining colors in its list that are not acquiredby any of its neighbors. If the UEs can acquire these remaining colors without conflicting withits neighbors, then each color class will be a MIS. Phase 2 works as follows.i) At each time slot in Phase 2, UE i chooses each color from the remaining colors in itslist independently with probability c . Each UE i then sends the set of its tentative choices to itsneighboring UEs, and receives their neighbors’ choices.ii) For any tentative choice of color, if there is a conflict with at least one neighbor, then thatcolor is not fixed; otherwise, it is fixed.iii) At the end of each time slot, each UE deletes its set of fixed colors from its list, andtransmits this set of fixed colors to its neighbors, who will delete these fixed colors from theirlists as well. Note that a UE deletes a particular color if and only if the UE itself or some ofits neighbors have chosen this color. Based on this key observation, we can see that if a coloris not in any UE’s list, the set of UEs with this color is a MIS. If all the UEs have an emptylist, then for any color in the set { , ..., H } , the set of UEs with this color is a MIS.There are (cid:100) c log x N (cid:101) + 1 time slots in Phase 2, where x = − ( c ) H (1 − c ) H , and c is the parameter given by the protocol. The number of time slots is known to the SBSs at installation.We say that Phase 2 is successful, if it finds H MISs, or equivalently if all the UEs have anempty list.
Example:
We illustrate Step 2 in a network of 4 UE-SBS pairs, whose interference graph isshown in Fig. 3. At the start, each UE-SBS pair has a list of 3 colors { Red, Yellow, Green } .Phase 1 is run for P (cid:100) c log (cid:101) time slots. At the end of Phase 1, UE 1 and UE 2 acquireGreen and Yellow respectively, while UEs 3-4 acquire Red. Hence, UE 1 (UE 2) has an emptylist, as Green (Yellow) is acquired by itself and Red, Yellow (Green) by its neighbors. UE 3(UE 4) has Green (Yellow) color in its list of remaining colors. At the end of Phase 1, theRed color class is a MIS, while the Yellow and Green color classes are not. Phase 2 is run for P (cid:100) c log x (cid:101) + 1 time slots. UE 3 (UE 4) acquires the remaining color Green (Yellow). Atthe end of Phase 2, the Green and Yellow color classes become MISs too.The next theorem establishes the high success probability of Step 2. Theorem 1.
For any interference graph with the maximum degree ∆ ≤ H − , the proposedalgorithm in Table II and III outputs a set of H MISs that span all the UEs in ( (cid:100) c log N (cid:101) + (cid:100) c log x N (cid:101) + 2) time slots with a probability no smaller than (1 − N c − )(1 − N c − ) , where c and c are design parameters that trade-off the run time and the success probability.See the Appendix for detailed proofs.Theorem 1 characterizes the performance of our proposed algorithm, in terms of the run timeof the algorithm and the lower bound of the success probability. When the parameters c and c are larger, the lower bound of the success probability increases at the expense of a longer runtime. When the maximum degree of the interference graph is larger, we need to set a higher H , which results in a longer run time. This is reasonable, because it is harder to find coloringand MISs when the number of interfering neighbors is higher. Finally, we can see that the lowerbound of the successful probability is very high even under smaller c and c , especially ifthe number of UEs is large. Note that the exact successful probability should depend on theprobability c in Phase 2, while the lower bound in Theorem 1 does not. Hence, our lower boundis robust to different system parameters. Note also that the interference graph here is a bounded-degree graph since the maximum degree is bounded by a given constant, H − . The algorithmsin [4] [6] (require ordering of the vertices, work sequentially and have a higher complexity) canbe used to output the MISs spanning all the UEs for arbitrary graphs. However, we will show in Theorem 5, that the restriction to bounded-degree graphs is a must to ensure that the minimumthroughput requirement of each UE is satisfied for any MIS based policy.
Step 3. Distributed computation of the optimal fractions of time for each MIS:
Let the setof MISs generated in Step 2 be { I (cid:48) , ..., I (cid:48) H } . In Step 3, the UE-SBS pairs compute the fractionsof time allocated to each MIS in a distributed manner.When an MIS is scheduled, the UEs in this MIS transmit at their maximum power levels,and the other UEs do not transmit. Define R ki as the instantaneous throughput obtained by UE i in the MIS I (cid:48) k , which can be calculated as log (1 + g iT ( i ) p I (cid:48) ki (cid:80) Nr =1 ,r (cid:54) = i g rT ( i ) p I (cid:48) kr + σ T ( i ) ) , where p I (cid:48) k i = p maxi if i ∈ I (cid:48) k and p I (cid:48) k i = 0 otherwise. To determine R ki , the UE needs to know the total interferenceit experiences when transmitting in I (cid:48) k . This can be measured by having an initial cycle oftransmissions of UEs in each MIS in the order of the indices of MISs/colors.From now on, we assume that the network performance criterion W ( y ) is concave in y and is separable, namely W ( y , ...y N ) = (cid:80) Ni =1 W i ( y i ) . Examples of separable criteria includeweighted sum throughput and proportional fairness. Our framework can also deal with max-min fairness min i R i ( π ) , although it is not separable (see the discussion in the Appendix) Theproblem of computing the optimal fractions of time for the MISs is given as follows: Coupled Problem (CP) max α N (cid:88) i =1 W i (cid:32) H (cid:88) k =1 α k R ki (cid:33) subject to H (cid:88) k =1 α k R ki ≥ R mini , ∀ i ∈ { , .., N } H (cid:88) k =1 α k = 1 , α k ≥ , ∀ k ∈ { , .., H } Each UE i knows only its own utility function W i and minimum throughput requirement R mini . Hence, it cannot solve the above problem by itself. We will first reformulate the aboveproblem into a decoupled problem and then show that the reformulated problem can be solvedin a distributed manner. Let each UE i have a local estimate β ki of the fractions of time allocatedto each MIS I (cid:48) k (including those MISs that UE i does not belong to). We impose an additionalconstraint that all the UEs’ local estimates are the same. Note that this constraint will be satisfiedby our solution, and is not an assumption. Such a constraint is still global, because any two UEs,even if they are not neighbors, need to have the same local estimate. Hence, global message exchange among any pair of UEs is still needed to solve this problem with local estimates andglobal constraints . To avoid global message exchange, we reformulate the CP into a decoupledproblem (DP) that involves only local coupling among the neighbors and can be solved withlocal message exchange using Alternating Direction Method of Multipliers (ADMM) [37].Now we reformulate the CP into a decoupled problem (DP) that involves only local couplingamong the neighbors and that can be solved by Alternating Direction Method of Multipliers(ADMM) [37]. If UE i and l are connected by an edge ( i, l ) then for each set I (cid:48) k define θ k ( i,l ) i = β ki and θ k ( i,l ) l = − β kl , note that these auxiliary variables are introduced to formulate the probleminto the ADMM framework [37]. Define a polyhedron for each i , T i = { β i | s.t. t β i = 1 , β i ≥ , R (cid:48) i β i ≥ R mini } , here β i = ( β i , ..., β Hi ) and R i = ( R i , ..., R Hi ) and () (cid:48) corresponds to thetranspose. Let β = ( β , ..., β N ) ∈ T , where T = (cid:81) Ni =1 T i and (cid:81) corresponds to the Cartesianproduct of the sets. Also, let β k = ( β k , ..., β kN ) , ∀ k ∈ { , .., H } . Define another polyhedron Θ k ( i,l ) = { ( θ k ( i,l ) i , θ k ( i,l ) l ) : θ k ( i,l ) i + θ k ( i,l ) l = 0 , − ≤ θ k ( i,l ) s ≤ , ∀ s ∈ { i, l }} , Θ k = (cid:81) ( i,l ) ∈ E Θ k ( i,l ) here E = ( e , ..e M ) is the set of all the M edges in the interference graph. A vector θ k ∈ Θ k is written as θ k = ( θ ke ,z ( e ) , θ ke ,t ( e ) , .., θ ke M ,z ( e M ) , θ ke M ,t ( e M ) ) , here z ( e i ) , t ( e i ) correspond to thevertices in the edge, e i . Similarly define, θ = ( θ , ..., θ H ) ∈ Θ , where Θ = (cid:81) Hk =1 Θ k . Decoupled Problem (DP) min β ∈T , θ ∈ Θ − (cid:80) Ni =1 W i ( R i (cid:48) β i ) subject to D k β k − θ k = 0 , ∀ k ∈ { , .., H } Here, D k ∈ R M × N , is a matrix in which each row has exactly one non-zero element whichis or − . Each element of the matrix, D kvj is evaluated as follows, the index v can be uniquelyexpressed in terms of quotient q and the remainder w as v = 2 q + w , and if j (cid:54) = z ( e q +1 ) , j (cid:54) = t ( e q +1 ) then D kvj = 0 . If w = 1 , j = z ( e q +1 ) , then D kvj = 1 else if w = 0 , j = z ( e q +1 ) then D kvj = 0 . Also, if w = 0 , j = t ( e q +1 ) , then D kvj = − else if w = 1 , j = t ( e q +1 ) then D kvj = 0 . Theorem 2:
For any connected interference graph, the coupled problem (CP) is equivalent tothe decoupled problem (DP). If the UEs could exchange messages globally, i.e. broadcast messages to all the UEs in the network, and if the networkperformance criterion is strictly concave, we could use standard dual decomposition with augmented Lagrangian in [36] toderive a distributed algorithm. However, in large networks, the UEs cannot exchange messages globally with other UEs, andthe network performance criterion may not be strictly concave (e.g., the weighted sum throughput is linear). FBS-1
FBS- FUE-
10 m gain
PBS- PBS-
FUEFBS
Picocell
Base Station Picocell User Equipment
Femtocell
Base Station
Femtocell
User EquipmentDirect channelgain
Figure 4. A heterogeneous network of 2 PBS and 2 FBS and their corresponding UEs.
The above theorem shows that the original problem (CP), which requires global informationand global message exchange to solve, is transformed into an equivalent problem (DP), whichas we will show, can be solved in a distributed manner with local message exchangeWe denote the optimal solution to the DP by W G distributed . We associate with each constraint D keq β kq = θ keq a dual variable λ keq . The augmented Lagrangian for DP is L y (cid:0) { β i } i , { θ keq } k,e,q , { λ keq } k,e,q (cid:1) = − (cid:80) Ni =1 W i ( β Ti R i ) + (cid:80) Hk =1 (cid:80) e ∈ E (cid:80) q ∈ e (cid:104) λ keq (cid:0) D keq β kq − θ keq (cid:1) + y (cid:0) D keq β kq − θ keq (cid:1) (cid:105) . In the ADMMprocedure (see Table IV in the Appendix), each UE i solves for its optimal local estimates β i ( t ) that maximizes the augmented Lagrangian given the previous dual variables λ kei ( t − andauxiliary variables θ kei ( t − . Then it updates its dual variable λ kei ( t ) and auxiliary variable θ kei ( t ) based on its local estimate β ki ( t ) and its neighbor j ’s local estimate β kj ( t ) . This iterationof updating local estimates, dual variables, and auxiliary variables is repeated P times. Next, itis shown that this procedure will indeed converge. Theorem 3:
If DP is feasible , then the ADMM algorithm in Table IV converges to theoptimal value W G distributed with a rate of convergence O ( P ) . Step 4. Determining the cycle length and transmission times:
At the end of Step 3, allthe UEs have a consensus about the optimal fractions of time allocated to each MIS, namely β ∗ i = γ ∗ = ( γ ∗ , ..., γ ∗ H ) , ∀ i ∈ { , .., N } . The MISs transmit in the order of their indices (i.e., { , .., H } ) in cycles. In each cycle of transmission, MIS I (cid:48) k transmits for (cid:108) γ ∗ k min i ∈ ,...,N γ ∗ i × d (cid:109) slots, where we multiply by d such that the rounding error is reduced or eliminated in casethat γ ∗ k min i ∈ ,...,N γ ∗ i is not an integer. DP is feasible, if the feasible region resulting from the constraints in DP is non-empty. B. A Motivation Example
Consider a network of 2 picocell base stations (PBS) and 2 femtocell base stations (FBS),each serving one UE. The network topology is shown in Fig. 4. We assume a path loss modelfor channel gains, with path loss exponent . The maximum transmit power of each UE is mW, and the noise power at each SBS is . × − mW. UEs in different tiers have differentminimum throughput requirements: FUE (femtocell UE) 1 and FUE 2 in the femtocells require aminimum throughput . bits/s/Hz, and PUE (picocell UE) 1 and PUE 2 in the picocells require . bits/s/Hz. The interference graph is constructed according to a distance based threshold rulesimilar to [28]. Specifically, an edge exists between two UE-BS pairs if the distance betweenany pair of SBSs is less than a threshold, which is set to be . m here. There are two MISs.MIS 1 consists of FUE 1 and FUE 2, and MIS 2 consists of PUE 1 and PUE 2. We considertwo performance criteria: the max-min fairness and the sum throughput. We will compare withthe following state-or-the-art policies:
1. Distributed Constant Power Control Policies [7], [8], [10] : In these policies, all the UEschoose constant power levels determined by distributed algorithms utilizing information (e.g.,power levels used by neighbors) made available through local/global message exchange.
2. Optimal Centralized Constant Power Policies:
In these policies, all the UEs choose constant power levels determined by a central controller utilizing global information.
3. Distributed MIS STDMA -1 [6] and
STDMA -2 [4] : These policies construct a subset ofthe MISs of the interference graph in a distributed manner and propose fixed schedules of theMISs. Different works adopt different schedules, and we differentiate them by referring to themas MIS STDMA-1 [6] and STDMA-2 [4].
4. Distributed Joint Power Control and Spatial Reuse [22] [23] : These policies choose oneUE from each cell to form a subset, and schedule these subsets of UEs based on their channelgains to maximize the sum throughput. The policies are named power matched scheduling (PMS).In Table 1, we compare the performance of our proposed policy with state-of-the-art policiesfor the same setup as in Fig. 4. We compute the optimal centralized constant power control policyby exhaustive search, which serves as the performance upper bound of the distributed constantpower control policies [7], [8], [10] centralized constant power control policies [15]. In PMSpolicies [22] [23], UEs within the same cell are scheduled in a time-division multiple access Table IC
OMPARISONS IN TERMS OF MAX - MIN FAIRNESS & SUM THROUGHPUT CRITERION
Policies Max-min Performance Sum Performancethroughput (bits/s/Hz) Gain % throughput (bits/s/Hz) Gain %Distributed constant power control [7], [8], [10] <0.28 >375 % 6.1 32.8 %Distributed PMS [22], [23] <0.28 >375% 6.1 32.8 %Optimal centralized constant power control 0.28 375% 6.1 32.8 %Distributed MIS STDMA-2/1 [4], [6] ) - - Benchmark Problem (BP) (Section- VI) 1.33 - - (TDMA) fashion, and the active UEs in different cells transmit simultaneously. In this motivatingexample, there is one UE in each cell, which will be scheduled to transmit all the time. Therefore,the PMS policy reduces to a constant power control policy, and is worse than the optimalcentralized constant power control policy. We can see that our proposed policy outperformsall constant power control policies and distributed PMS policies by at least 375% and 32.8%,in terms of max-min fairness and sum throughput, respectively. The significant performanceimprovement over the constant power control policies results from the elimination of the highinterference among the users through scheduling MISs. Our proposed policy also outperformsdistributed STDMA policies by 30%-40%. As we will see in Section VI, the performance gain iseven higher (160%-700%) in realistic deployment scenarios. Finally, in this motivating example,the proposed policy achieves the optimal performance of the benchmark problem defined inSection VI, which is a close approximation of the original problem (CP). C. Performance Guarantees for Large Networks and Properties of Interference Graphs
In this subsection, we provide performance guarantees for our proposed framework describedin Section V-A. Specifically, we prove that the network performance W G distributed achieved by theproposed distributed algorithms has a constant competitive ratio with respect to the optimal value W opt of the PDP. Moreover, we prove that the competitive ratio does not depend on the networksize. Our result is strong, because the solution to PDP needs to be computed by a centralizedcontroller with global information and with NP complexity, while our proposed framework allowsthe UEs to compute the policy fast in a distributed manner with local information and local message exchange.Before characterizing the competitive ratio analytically, we define some auxiliary variables.Define the upper and lower bounds on the UEs’ maximum transmit power levels and throughputrequirements as, < p maxlb ≤ p maxi ≤ p maxub , ∀ i ∈ { , ..., N } and, < R minlb ≤ R mini ≤ R minub , ∀ i ∈{ , ..., N } respectively. Let D ij is the distance between UE i and SBS j . Define upper and lowerbounds on the distance between any UE and its serving SBS and the noise power at the SBSsas, < D lb ≤ D iT ( i ) ≤ D ub , ∀ i ∈ { , ..., N } and, σ lb ≤ σ j ≤ σ ub , ∀ j ∈ { , ..., K } respectively.We assume that the channel gain is g ij = D ij ) np , where np is the path loss exponent. Definition 1 (Weak Non-neighboring Interference):
The interference graph G exhibits ζ Weak Non-neighboring Interference ( ζ -WNI) if for each UE i the maximum interference fromits non-neighbors is bounded, namely (cid:80) j (cid:54)∈N ( i ) ,j (cid:54) = i g jT ( i ) p maxj ≤ (2 ζ − σ ub , ∀ i ∈ { , ..., N } .Define ∆ max = log (1+ pmaxlb ( Dub ) np ζσ ub ) R minub − . Then we have the following theorem for the networkperformance criterion, sum throughput . Theorem 4:
For any connected interference graph, if the maximum degree ∆ ≤ ∆ max and itexhibits ζ -WNI then, our proposed framework of interference management described in SectionV-A achieves a performance W G distributed ≥ Γ · W opt with a probability no smaller than (1 − N c − )(1 − N c − ) . Moreover, the competitive ratio Γ = R minub log (1+ pmaxub ( Dlb ) npσ lb ) is independent of thenetwork size.Note that the analytical expression of competitive ratio, Γ = R minub log (1+ pmaxub ( Dlb ) npσ lb ) , does not dependon the size of the network. Our results are derived under the conditions that the interferencegraph has a maximum degree bounded by ∆ max , and that the interference from non-neighborsis bounded (i.e. ζ − WNI). These conditions do not restrict the size of the network, next exampleillustrates this. In addition, our results hold for any interference graph that satisfy the conditionsin Theorem 4, regardless of how the graph is constructed.
Example:
Consider a layout of SBSs in a K × K square grid, i.e. K SBSs with a distanceof m between the nearest SBSs. Assume that each UE is located vertically below its SBS ata distance of m. Fix the parameters p maxi = 100 mW, σ i = 3 mW, R mini = 0 . bits/s/Hz, ∀ i ∈ { , .., K } , np = 4 . We construct the interference graph based on the distance rule [28], We can extend this result for weighted sum throughput, with weights w i = Θ( N ) , it is not done to avoid complex notations. namely there is an edge between two pairs if the distance between their SBSs exceeds 6m,which gives us the maximum degree ∆ = 4 . We can also verify that the interference graphsunder any number K of SBSs exhibit ζ -WNI with ζ = 0 . and ∆ < ∆ max , where ∆ max = 48 .Given ∆ = 4 and ζ = 0 . , from Theorem 4, we get the performance guarantee of . forany network size K . Note that the number . is a performance guarantee, and that the actualperformance is much higher compared to the performance guarantee as well as those achievedby state-of-the-art policies (see Section VI).Both Theorem 1 and 4 required the maximum degree of the interference graph to be boundedby a given constant. Here, we show that constraint on the degree is natural and is a must to ensurefeasibility, i.e. to satisfy the minimum throughput requirements of every UE. Specifically, weprove that if the maximum degree exceeds some threshold, then no policy based on schedulingMISs in Π MIS (a large space of policies, see Section III) is feasible. Let the construction ofinterference graph be based on a distance based threshold rule similar to [28]. An edge existsbetween two UE-SBS pairs if and only if, the distance between two SBS is no greater than D th .We define the threshold of the maximum degree as ∆ ∗ (See the Appendix for the expression). Theorem 5:
If the maximum degree of the interference graph ∆ ≥ ∆ ∗ , then any policy basedon scheduling MISs in Π MIS fails to satisfy the minimum throughput requirements of the UEs.The intuition behind Theorem 5 is that, if the degree of the interference graph is large thenthere must be a large number of UE-SBS pairs which interfere with each other strongly (mutuallyconnected) which makes it impossible to allocate each UE enough transmission time to satisfyits minimum throughput requirement.
D. Self-Adjusting Mechanism for Dynamic Entry/Exit of UEs
We now describe how the proposed framework can adjust to dynamic entry/exit by the UEsin the network without recomputing all the four steps. We allow the UEs to enter and exit, butnumber of SBSs is fixed. We only allow let one UE enter or leave the network in any time slot.1.
UE leaves the network:
Suppose a UE i which was transmitting to SBS T ( i ) leaves thenetwork. If the UE i was transmitting in a set of colors C i , then as soon as it leaves, thesecolors can be potentially used by some neighbors, N ( i ) . The SBS T ( i ) which was serving theUE i can have other UEs which are still in the network and transmitting to it. Then for eachcolor c (cid:48) ∈ C i it first searches among the UEs which it serves that are not already transmitting in c (cid:48) and who also do not have a neighboring UE-SBS pair which is already transmitting in c (cid:48) . Let the set of such UEs be U E c (cid:48) i,left . SBS T ( i ) allocates color c (cid:48) to the UE whose index isarg max j ∈ UE c (cid:48) i,left R c (cid:48) j . In case U E c (cid:48) i,left is empty then that color, c (cid:48) is left unused.2. UE enters the network:
Suppose a UE i registered with SBS T ( i ) enters the network.i). Given the minimum throughput requirement of the UE i the SBS T ( i ) first creates a list ofUEs, U E i,enter , which consists of the UEs it is serving and who are transmitting at more thantheir minimum throughput requirement.ii). SBS T ( i ) creates the list of colors, C i,enter in which UEs in U E i,enter are transmitting, italso consists of the colors that are not being used by any UE served by T ( i ) . Next, it createsvalid colors list i.e. C validi,enter from C i,enter , where a color c ∈ C validi,enter if c ∈ C i,enter and if noneof the neighbors of i in N ( i ) that are not in U E i,enter are already using that color.iii). Next, the SBS T ( i ) has to allocate some portions from the fractions of time allocated tothe colors in C validi,enter , such that UE- i can transmit and its minimum throughput requirement issatisfied to the best possible extent. The allocation is done as follows, let C validi,enter = { c (cid:48) , ...., c (cid:48) s } .Proceeding sequentially, for each color c (cid:48) i , SBS T ( i ) selects the maximum possible portionto satisfy the minimum throughput requirement of UE- i , such that the minimum throughputrequirements of UEs in U E i,enter , who are using this color, c (cid:48) i are not violated.iv). If the requirement of UE- i is not satisfied then, SBS T ( i ) requests the neighboring UE-SBSs (apart from the UEs that are served by T ( i ) ) to announce the set of colors which are eithernot being used or in which their corresponding UEs are operating at more than the minimumthroughput requirement. From the set of colors that are received, the SBS- T ( i ) chooses eachcolor from the list if it is not being used by any other neighboring UE apart from the ones whosent the announcement. The resulting list of colors is Cl validi,enter = { c (cid:48) , ..., c (cid:48) l } .v). Proceeding sequentially with the colors in Cl validi,enter , for each color, SBS- T ( i ) requests aportion from the fraction of time allocated to that color, to the neighboring UE-SBSs allocatedthat color, such that the throughput requirement of UE- i is satisfied. The neighboring UE-SBSseither allow the requested portion or send the portion which is acceptable to them, i.e. theirthroughput requirements are not violated. SBS- T ( i ) allocates the minimum acceptable portionto UE- i and proceeds to the next color in the list if the throughput requirements are not satisfied. E. Extensions
In our model, UEs operate in the same frequency band. However, our methodology can beextended to scenarios where UEs operate in different frequency channels (frequency reuse) andtransmit at the same time. In this case, the problem is to find the optimal frequency allocationwith the same objective function and constraints as in PDP. To solve this problem, the first twosteps of the framework remain the same. In Step 3, the UEs compute distributedly the optimalfractions of bandwidth to be allocated to each MIS. This step is equivalent to computing theoptimal fraction of time allocated to each MIS as in our current formulation. In Step 4, theUEs compute the number of frequency channels allocated to each MIS based on the bandwidthallocation.Note that we do not implement beamforming, although beamforming can be used in conjunc-tion with our policy. If the UEs transmitting to the same SBS cooperate to do beamforming, wecan delete the edge between them in the interference graph, and use the new interference graphin the scenario with beamforming.VI. I
LLUSTRATIVE R ESULTS
In this section, we evaluate our proposed policy under a variety of scenarios with differentlevels of interference, large numbers of UEs, different performance criteria, time-varying channelconditions, and dynamic entry and exit of UEs.We compare our policy with the optimal centralized constant power control policy, the dis-tributed MIS STDMA-1 [6] and STDMA-2 [4], distributed PMS [22] [23], in terms of sumthroughput and max-min fairness. We do not separately compare with distributed/centralizedconstant power control policies in [7], [8], [10] [15], because their performance is upper boundedby the optimal centralized power control. Since it is difficult to compute the solution to the NP-hard PDP, we define a benchmark problem, where we restrict our search to policies in which aUE either transmits at its maximum power level or does not transmit.The space of such policiescan be writtenas Π BC = { π = ( π , ..., π N ) : π i : Z + → { , p maxi } ∀ i ∈ { , .., N }} . The policyspace Π BC is a subset of all policies Π and is a superset of MIS based policies Π MIS . Inother words, the benchmark problem has the same objective and constraints as PDP; the onlydifference is the policy space to search . Hence, the benchmark problem is a close approximationof the PDP. Note that the benchmark problem is also NP-hard (see the appendix). Fading parameter β M i n i m u m t h r oughpu t ac h i e v e d by a ny u s e r ( b it s / s / H z ) Proposed PolicyOptimal Centralized Constant PowerDistributed MIS STDMA−1Distributed MIS STDMA−2Benchmark Problem 18 % Gap40 %Increase
Fig. 5 a)
Fading parameter β A v e r a g e t h r oughpu t p e r U E ( b it s / s / H z ) Proposed PolicyOptimal Centralized Constant PowerDistributed MIS STDMA−2Benchmark Problem 9 % Gap88 %Increase
Fig. 5 b)
Figure 5. Comparison of the proposed policy with state of the art under different interference strength and time-varying channelconditions
A. Performance under time-varying channel conditions
Consider a x square grid of 9 SBSs with the minimum distance between any two SBSsbeing d = 4 . m. Each SBS serves one UE, who has a maximum power of mW and aminimum throughput requirement of . bits/s/Hz. The UEs and the SBSs are in two parallelhorizontal hyperplanes, and each SBS is vertically above its UE with a distance of √ m .Then the distance from UE i to another SBS j is D ij = (cid:113)
10 + ( D BSij ) , where D BSij is thedistance between SBSs i and j . The channel gain from UE i to SBS j is a product of pathloss and Rayleigh fading f ij ∼ Rayleigh ( β ) , namely g ij = D ij ) f ij . The density function of Rayleigh ( β ) is v ( z ) = zβ e − z β for z ≥ , and v ( z ) = 0 for z < . The SBSs identify neighborsusing a distance based rule with the threshold distance as in Section V-C with D th = 7 m. Notethat different thresholds lead to different interference graphs, and hence different performance,which will be discussed next. Although, we use a distance based threshold rule, our frameworkis general and does not rely on a particular rule. The resulting interference graph for this settingis graph 3 shown in Fig. 7 a).At the beginning, the UE-SBS pairs generate the set of MISs (Step 2 of the design frameworkin Section V), and compute the optimal fractions of time allocated to each MIS (Step 3). Inour simulation, we assume a block fading model [38] and the fading changes every timeslots independently. To reduce complexity, the UEs do not recompute the interference graph and M i n i m u m t h r oughpu t ac h i e v e d by a ny u s e r ( b it s / s / H z ) d=4.74 md=3.70 md= 2.52 mGraph 3 isoptimal for d=4.74mGraph 2 isoptimal ford=3.70mGraph 1 isoptimal ford=2.52m Fig. 6 a)
Time instance at which number of UEs change S u m t h r oughpu t ( b it s / s / H z ) Rmin tol =0.23Rmin tol =0.25Rmin tol =0.25 (1,2,2) (1,3,1) (1,3,1) (1,2,1) (1,2,2) (0,2,2) (0,2,2)(3,3,2) (3,2,2) (3,2,1) (3,3,1) (2,3,1) (2,3,0) (1,3,0)(0,1,2)(1,3,1)(1,2,1)(2,3,1)(2,3,2)(2,1,2) (1,2,1)(2,1,1) (1,1,1) (1,1,1) (1,1,2) (1,2,2) (0,2,3) (0,2,2)(1,1,2) (a,b,c)−number of UEsin room 1,2 and 3 (1,2,3)
Fig. 6 b)
Figure 6. a) Comparison of max-min fairness under different grid sizes, b) Sample paths of sum throughput under dynamicentry/exit of UEs in the network the MISs, but will recompute the optimal fractions of time under the new channel gains every100 time slots. In Fig. 5, we compare the performance of the proposed policy with state ofthe art policies under different variances β of Rayleigh fading. We do not plot the performanceof distributed PMS for this scenario since it is upper bounded by optimal centralized constantpower control (because there is one UE per cell). We do not plot the distributed MIS STDMA -1either, when the performance criterion is average throughput per UE (i.e., sum throughput N ), becauseit cannot satisfy the minimum throughput constraints. From Fig. 5, we can see that in terms ofboth average throughput and max-min fairness, our proposed policy achieves large performancegain (up to 88%) over existing policies, and achieves performance close to the benchmark (asclose as 9%). Selecting the Optimal Interference Graph : For different values of d , there can be five possibleinterference graphs, which are shown in Fig. 7 a). In Fig. 6 a) we show that as the grid size d decreases ( d = 4 . m, d = 3 . m and d = 2 . m ), the levels of interference from the adjacentUEs increases, and as a result, the interference graph with higher degrees perform better (as d decreases, the optimal graph changes from graph 3 to graph 1) . B. Performance scaling in large networks
Consider the uplink of a femtocell network in a building with 12 rooms adjacent to each other.Fig. 7 b) illustrates 3 of the 12 rooms with 5 UEs in each room. For simplicity, we consider Fig. 7 a)
Length = 20 mFBS FBS FBS Height = 2 m 5 m 5 m
Fig. 7 b)
Figure 7. a) Different interference graphs for the 3 x 3 BS grid, b) Illustration of setup with 3 rooms. a 2-dimensional geometry. Each room has a length of 20 meters. In each room, there are P uniformly spaced UEs, and one SBS installed on the left wall of the room at a height of 2m.The distance from the left wall to the first UE, as well as the distance between two adjacent UEsin a room, is P ) meters. Based on the path loss model in [39], the channel gain from eachSBS i to a UE j is D ij ) ∆ nij , where ∆ = 10 . is the coefficient representing the loss from thewall, and n ij is the number of walls between UE i and SBS j . Each UE has a maximum transmitpower level of mW, a minimum throughput requirement of R mini = 0 . bits/s/Hz, and anoise power level of − mW at its receiver. Here, we consider that the UEs use a distancebased threshold rule as in Section V-B with D th = 30 m. This results in interference graphswhich connects all the UE-SBS pairs within the room and in the adjacent rooms. We vary thenumber P of UEs in each room from to 9 and compare the performance in Fig. 8. Note thatthe optimal centralized constant power policy cannot satisfy the feasibility conditions for anynumber of UEs in each room. Hence, only the performance of distributed MIS STDMA-1,2 anddistributed PMS is shown in Fig. 8. We can see that under both criteria, the performance gainof our proposed policy is significant (from 160% to 700%). Note that since the number of UEsis large, it is impossible to solve the benchmark problem (which is NP-hard) is not possible. C. Self-adjusting mechanism for dynamic entry/exit of the UEs
The self-adjusting mechanism proposed in Section V-D is aimed to provide incoming UEswith the maximum possible throughput without affecting the incumbent UEs, and to reuse thetime slots left vacant by exiting UEs efficiently. Consider the same setup as in Section VI-Bwith 3 rooms and a maximum of P = 3 UEs in each room. Each UE has a maximum transmitpower of mW and a minimum throughput requirement of . bits/s/Hz.We assume that at a given time only one UE either enters or leaves the network. In Fig. 6 b)we show different sample paths of the sum throughput under different entry and exit processes.
60 65 70 75 80 85 90 95 100 10500.010.020.030.040.050.060.070.08
Number of UEs M i n i m u m a v e r a g e t h r oughpu t ac h i e v e d by a U E ( b it s / s / H z ) Proposed PolicyDistributed PMSDistributed MIS STDMA−2Distributed MIS STDMA−1161 %Gain 180 %Gain
Fig. 8 a)
60 65 70 75 80 85 90 95 100 10500.10.20.30.40.50.60.7
Number of UEs A v e r a g e t h r oughpu t p e r U E ( b it s / s / H z ) Proposed PolicyDistributed PMSDistributed MIS STMDA−2Distributed MIS STDMA−17 x (times) gain
Fig. 8 b)
Figure 8. Comparison of max-min fairness and average throughput per UE against state of the art for large networks
In the legends (i.e.,
Rmin tol ), we show the minimum throughput achieved at any point in thesample path. We repeated the same procedure 100 times. We can see that the self-adjustingmechanism works well by guaranteeing a worst-case minimum throughput requirement of . bits/s/Hz, which is just . bits/s/Hz below the original requirement more than % of the time.VII. C ONCLUSION
We proposed a design framework for distributed interference management in large-scale,heterogeneous networks, which are composed of different types of cells (e.g. femtocell, picocell),have different number of UEs in each cell, and have UEs with different minimum throughputrequirements and channel conditions. Our framework allows each UE to have only local knowl-edge about the network and communicate only with its interfering neighbors. There are two keysteps in our framework. First, we propose a novel distributed algorithm for the UEs to generatea set of MISs that span all the UEs. The distributed algorithm for generating MISs requires O (log N ) steps (which is much faster than state-of-the-art) before it converges to the set ofMISs with a high probability. Second, we reformulate the problem of determining the optimalfractions of time allocated to the MISs in a novel manner such that the optimal solution can bedetermined by a distributed algorithm based on ADMM. Importantly, we prove that under widerange of conditions, the proposed policy can achieve a constant competitive ratio with respect tothe policy design problem which is NP-hard. Moreover, we show that our framework can adjustto UEs entering or leaving the network. Our simulation results show that the proposed policycan achieve large performance gains (up to %). Table IIG
ENERATING
MIS
S IN A DISTRIBUTED MANNER , ALGORITHM FOR UE i Phase 1 - Initialization: Tx i tent = φ , Tx i final = φ , tentative and final choice of UE i , Rx N ( i ) tent = φ ,Rx N ( i ) final = φ tentativeand final choice made by the neighbors, C i = { , ..., H } ∩ { , .., d i + 1 } the current list of subset of available colors, C i = φ, list of colors used by i , F i colored = φ , C i = { , ..., H } , the current list of all available colorsfor n = 0 to (cid:100) c log N (cid:101) Tx i tent = φ , Tx i final = φ if(F i colored = φ )Tx i tent = rand{ C ni } , rand represents randomly selecting a color and informing the neighbors about it.Rx N ( i ) tent = { Tx k tent , ∀ k ∈ N ( i ) } If(Tx i tent (cid:54) = Rx N ( i ) tent ( j ) , ∀ j ∈ N ( i ) ), here UE- i checks if there is a conflict with any of the neighbor’s choiceTx i final = Tx i tent , C i = { Tx i final } ,if there is no conflict then UE- i transmits its final color choice to the neighbors,elseTx i final = φ endendRx N ( i ) final = { Tx k final , ∀ k ∈ N ( i ) } C n +1 i = C ni ∩ {Rx N ( i ) final ∪ Tx i final } c , C n +1 i = C ni ∩ {Rx N ( i ) final ∪ Tx i final } c if(Tx i final (cid:54) = φ )F i colored = 1 endend A PPENDIX
Discussion on max-min fairness:
We now discuss as to how the proposed framework can beextended to incorporate inseparable function like max-min fairness. The coupled problem withmax-min fairness objective is restated below:
Coupled Problem (CP) max α min i ∈{ ,..,N } W i ( H (cid:88) k =1 α k R ki ) subject to H (cid:88) k =1 α k R ki ≥ R mini , ∀ i ∈ { , ...N } H (cid:88) k =1 α k = 1 , α k ≥ , ∀ k ∈ { , ..., H } Table IIIP
HASE OF THE DISTRIBUTED
MIS
GENERATION
Phase 2-Initialization: Tx settent ,i = φ ,Tx setfinal ,i = φ , the set of tentative and final colors chosen by i ,Rx settent ,i = φ ,Rx setfinal ,i = φ , the set of tentative and final colors chosen that are received from the neighbors, x = − ( c ) H (1 − c ) H for n = (cid:100) c log N (cid:101) + 1 to (cid:100) c log N (cid:101) + (cid:100) c log x N (cid:101) + 1 Tx settent ,i = φ ,Tx setfinal ,i = φ ,for m = 1 to | C ni | with probability c , Tx settent,i ( m ) = C ni ( m ) , randomly selecting and informing the neighbors about tentative choicewith probability − c , Tx settent,i ( m ) = φ endRx settent ,i = ∪ k ∈N ( i ) Tx settent,k , set of tentative color choices of the neighbors of i for r = 1 to | Tx settent,i | If(Tx settent ,i ( r ) (cid:54) = Rx settent ,i ( j ) ∀ j ∈ N ( i ) )Tx setfinal ,i ( r ) = Tx settent ,i ( r ) elseTx setfinal ,i ( r ) = φ end C i = C i ∪ Tx setfinal ,i Rx setfinal ,i = ∪ k ∈N ( i ) Tx setfinal,k , set of final color choices of the neighbors of iC n +1 i = C ni ∩ {Rx setfinal ,i ∪ Tx setfinal,i } c end Table IVADMM UPDATE ALGORITHM FOR UE i Initialization: arbitrary β i (0) ∈ B i , θ kei (0) such that θ k ∈ Θ k , and λ kei (0) = 0 , ∀ k ∈ { , ..., H } , ∀ e such that i ∈ e For t = 0 to P − β i ( t + 1) = arg min β i ∈B i − (cid:80) Ni =1 W i ( β Ti R i ) + (cid:80) Hk =1 (cid:80) e ∈ E (cid:80) q ∈ e (cid:104) λ keq (cid:0) D keq β kq − θ keq (cid:1) + y (cid:0) D keq β kq − θ keq (cid:1) (cid:105) β i ( t + 1) is transmitted to all of its neighbors in N ( i ) . λ kei ( t ) is transmitted to its neighbor connected with edge e , ∀ k ∈ { , ..., H } and ∀ e such that i ∈ e Update ∀ k ∈ { , ..., H } and ∀ e such that i ∈ eλ kei ( t + 1) = ( λ kei ( t ) + λ kej ( t )) − y ( D kei β ki ( t + 1) + D kej β kj ( t + 1)) , where j is the other endpoint of e . θ kei ( t + 1) = y ( λ kei ( t + 1) − λ ke,i ( t )) + D kei β ki ( t + 1) end Transforming the above problem into an equivalent problem with auxiliary variable t is givenas max α, t t subject to W i ( H (cid:88) k =1 α k R ki ) ≥ t, ∀ i ∈ { , ..., N } H (cid:88) k =1 α k R ki ≥ R mini , ∀ i ∈ { , ...N } H (cid:88) k =1 α k = 1 , α k ≥ , ∀ k ∈ { , ..., H } To decouple the above problem, we introduce local variables for each UE i given as, { β i , ..., β H +1 i } .Now we state a problem which we claim is equivalent to CP,(the proof to this claim is verysimilar to the proof of Theorem 2 and we will highlight this fact in the proof clearly). P1 max β N (cid:88) i =1 β H +1 i subject to W i ( H (cid:88) k =1 β ki R ki ) ≥ β H +1 i , ∀ i ∈ { , ..., N } H (cid:88) k =1 β ki R ki ≥ R mini , ∀ i ∈ { , ...N } H (cid:88) k =1 β ki = 1 , β ki ≥ , ∀ k ∈ { , ..., H } , ∀ i ∈ { , ..., N } β ki = β kj , ∀ j ∈ N ( i ) , ∀ k ∈ { , ..., H + 1 } Here, β = ( β , .., β N ) , with β i = ( β i , ..., β H +1 i ) , ∀ i ∈ { , ..., N } . Now, given the twoproblems CP and the problem P1 are equivalent, we focus on solving P1. P1 can be changedto a problem similar to DP. To do that we introduce some additional variables similar tothe ones introduced for DP. If UE i and l are connected by an edge ( i, l ) then for each set I (cid:48) k define θ k ( i,l ) i = β ki and θ k ( i,l ) l = − β kl , note that these auxiliary variables are introducedto formulate the problem into the ADMM framework [37]. Define a polyhedron for each i , T (cid:48) i = { ( β i | s.t. t ( β (cid:48)(cid:48) i ) = 1 , ( β i ≥ , R (cid:48) i ( β (cid:48)(cid:48) i ) ≥ R mini , W i ( R (cid:48) i ( β (cid:48)(cid:48) i ) ) − β H +1 i ≥ } , here β (cid:48)(cid:48) i = ( β i , ..., β Hi ) and R i = ( R i , ..., R Hi ) and () (cid:48) corresponds to the transpose. Let β =( β , ..., β N ) ∈ T (cid:48) , where T (cid:48) = (cid:81) Ni =1 T (cid:48) i and (cid:81) corresponds to the Cartesian product ofthe sets. Also, let β k = ( β k , ..., β kN ) , ∀ k ∈ { , .., H } . Define another polyhedron Θ k ( i,l ) = { ( θ k ( i,l ) i , θ k ( i,l ) l ) : θ k ( i,l ) i + θ k ( i,l ) l = 0 , − ≤ θ k ( i,l ) s ≤ , ∀ s ∈ { i, l }} , Θ k = (cid:81) ( i,l ) ∈ E Θ k ( i,l ) here E = ( e , ..e M ) is the set of all the M edges in the interference graph. A vector θ k ∈ Θ k is written as θ k = ( θ ke ,z ( e ) , θ ke ,t ( e ) , .., θ ke M ,z ( e M ) , θ ke M ,t ( e M ) ) , here z ( e i ) , t ( e i ) correspond to thevertices in the edge, e i . Similarly define, θ = ( θ , ..., θ H +1 ) ∈ Θ (cid:48) , where Θ (cid:48) = (cid:81) H +1 k =1 Θ k .The reformulated problem is stated as follows: DP1 min β ∈T (cid:48) , θ ∈ Θ (cid:48) − (cid:80) Ni =1 W i ( R i (cid:48) β i ) subject to D k β k − θ k = 0 , ∀ k ∈ { , .., H + 1 } Then, DP1 can be solved using the ADMM procedure similar to the one described for DP.
Discussion on Benchmark Problem’s complexity:
Benchmark Problem is restated here forconvenience:
Benchmark Problem (BP) max π ∈ Π BC W ( R ( π ) , ..., R N ( π )) subject to. R i ( π ) ≥ R mini , ∀ i ∈ { , ..., N } Let the power set of U be S U , where S U consists of N subsets of UEs. Let S U ( j ) denote the j th element of S U . Define a set of power profiles, P S U , where the P S U ( j ) corresponds to the j th element in the set and it corresponds to the power profile when the UEs in set S U ( j ) transmit attheir maximum power levels and the rest of the UEs do not transmit. Note that for π ∈ Π BC , π ( t ) corresponds to a power profile in P S U . Therefore, the average throughput achieved byUE i , R i ( π ) , where π ∈ Π BC , can also be expressed as R i ( π ) = (cid:80) N j =1 α j r i ( P S U ( j )) , with α j ≥ , ∀ j ∈ { , .., N } and (cid:80) N j =1 α j = 1 . Here the fraction α j associated with each profile P S U ( j ) corresponds to the fraction of transmission time associated with that power profile. Consider the following problem:
BP1 max y, α W ( y , ..., y N ) subject to. y i ≥ R mini , ∀ i ∈ { , ..., N } y i = N (cid:88) i =1 α i r i ( P S U ( j )) , ∀ i ∈ { , ..., N } α j ≥ , ∀ j ∈ { , .., N } , N (cid:88) j =1 α j = 1 Next, in order to show that the above problem is NP-hard we will show intuitively why is itso, but the detailed proof follows from proof of Theorem 1 in [40]. Consider W ( y , .., y N ) = (cid:80) Ni =1 y i ,to be a linear function, R mini = 0 , ∀ i ∈ { , ..., N } and the cross channel gains amongstsome users who do not share an edge in the interference graph to be and the cross channelgains amongst the interfering neighbors to be ∞ . This implies that in any optimal solutionwill correspond to the transmission by a MIS of the interference graph. This can be justified asfollows. Consider an optimal solution in which two neighboring UEs are transmitting, making oneof the UEs not transmit will definitely increase the sum throughput contradicting the optimality.Specifically, this problem reduces to finding the maximum weighted maximum indpendenet setwhich is NP hard. Here the weight of each MIS corresponds to (cid:80) Ni =1 r i ( p I j ) . Proof of Theorem 1:
The success probability of Phase 1 is high, (1 − N c − ) (lower bound),(see [35] for detail), here we analyze Phase 2.We first show that, if the list of remaining colors given as, C ni is empty at n ≥ (cid:100) c log N (cid:101) + (cid:100) c log x N (cid:101) + 2 and if this holds ∀ i ∈ { , ..., N } then the Phase 2 has converged to a set of H MISs which span all the UEs. Let us assume otherwise, i.e. C ni is empty ∀ i ∈ { , ..., N } however, the set corresponding to some color h ∈ { , ..., H } , I (cid:48) h is not a MIS. I (cid:48) h has to be anIS. Assume otherwise, i.e. I (cid:48) h is not an IS, which implies that there must exist a pair of UEs, i and j , which are neighbors and are a part of I (cid:48) h . If this is true then both then both acquiredthe color h either in the same time slot or in different time slots, in Phase 1 or 2. In case thecolor is acquired in different time slots, then after the first time slot when either of the UEs inthe pair acquires the color it will transmit the final color choice, h to the neighbors (see TableII and III) who in turn delete that color. However, if the color is deleted by the neighbor thenit cannot acquire it in the future thus, ruling out the case that the colors were acquired in two different time slots. If the color was acquired by the UEs in the same time slot, then it impliesthat despite the conflict in tentative choice the UEs acquire the color which is not possible (seeTable II and III). This shows that I (cid:48) h is an IS.Since I (cid:48) h is not maximal then ∃ at least one UE- j (cid:54)∈ I (cid:48) k which can be added to this set withoutviolating independence. From the assumption, we have C nj = φ which implies that the color h was deleted at some stage from the original list of all the colors either in Phase 1 or 2. Thedeletion of h was a result of that color being acquired finally by at least one of the neighbors k ∈ N ( j ) since j (cid:54)∈ I (cid:48) k . In that case, j cannot acquire h as it will violate the independenceproperty.Next, we show that indeed the list of all colors available C ni is empty at the end of Phase 2with a high probability. Let U n correspond to the number of UEs which have a non-empty listat the beginning of time slot n and, let T n ( U n ) correspond to the total time needed before allthe UEs have an empty list. The probability that a UE at time slot n with a non-empty list willhave an empty list in next time slot is always greater than c H (1 − c ) H . This can be explainedas, if the UE chooses all the colors in the list assuming (worst case H number of colors remain)and all the neighbors (worst case H neighbors) do not choose any color, then all the colorsin the UE’s list will be deleted. From this, we get E ( U n +1 ) ≤ (1 − c H (1 − c ) H ) U n = x U n and T n ( U n ) = 1 + T n ( U n +1 ) . Assuming that the Phase 2 will start with N UEs whose listare non-empty (worst case) and from [41] we get P ( T n ( N ) ≥ (cid:100) c log x N (cid:101) ) ≤ N c − . Thisgives the lower bound on success probability of Phase 2 and thereby the result in the Theorem.(Q.E.D) Proof of Theorem 2:
The two problems which are introduced to transit from CP to DP are,
Global Primal Problem (GPP) max { β ki } i,k (cid:80) Hk =1 W i ( (cid:80) Ni =1 β ki R ki ) subject to (cid:80) Hk =1 β ki R ki ≥ R mini , (cid:80) Hk =1 β ki = 1 , ∀ i ∈ { , ..., N } β ki = β kl , ∀ i (cid:54) = l, ∀ k ∈ { , ..., H } , β ki ≥ , ∀ i ∈ { , ..., N } , ∀ k ∈ { , ...H } The second problem, Local Primal Problem (LPP) is the same as GPP except we choose asubset of the constraints from the above problem. Basically, instead of an equality constraintbetween the UE’s estimate and every other UE in the network, we only keep the equality constraints between the UE and its neighbors, i.e. β ki = β kl , ∀ k ∈ { , ..., H } , ∀ l ∈ N ( i ) . This isformally stated below: Local Primal Problem (LPP) max { β ki } i,k (cid:80) Hk =1 W i ( (cid:80) Ni =1 β ki R ki ) subject to (cid:80) Hk =1 β ki R ki ≥ R mini , (cid:80) Hk =1 β ki = 1 , ∀ i ∈ { , ..., N } β ki = β kl , ∀ l (cid:54)∈ N ( i ) , ∀ k ∈ { , ..., H } , β ki ≥ , ∀ i ∈ { , ..., N } , ∀ k ∈ { , ...H } To show that problems CP and GPP are equivalent, we need to show that from β ∗ =( β ∗ , .., β ∗ N ) , an optimal argument of GPP, we can obtain an optimal argument of CP, i.e. α ∗ and vice versa. Since β ∗ is the optimal value (assuming feasibility) we know that β ∗ i = β ∗ j (component-wise) holds ∀ i, j ∈ { , ..., N } .a).Let α (cid:48) = β ∗ i . α (cid:48) satisfies the constraints in CP. The objective of CP at α (cid:48) attains the optimalvalue of GPP. We need to establish that α (cid:48) is indeed the optimal argument of CP. Assume that α (cid:48) is not the optimal value, then there exists another α ∗ which is indeed the optimal. Next, using α ∗ , we can obtain another β (cid:48) as follows, β (cid:48) = α ∗ and β (cid:48) i = β (cid:48) , ∀ i ∈ { , ..., N } . The objectiveof GPP at β (cid:48) should be higher than β ∗ which contradicts β ∗ being the optimal argument. Notethat if either of CP or GPP is infeasible then the other problem can be shown to be infeasibleas well. On the same lines we can show that from an α ∗ we can obtain β ∗ as well.b). Let α ∗ be the optimal solution to CP, and define β (cid:48)(cid:48) a solution to GPP as follows. Let β (cid:48)(cid:48) = α ∗ and β (cid:48)(cid:48) i = β (cid:48)(cid:48) j , ∀ j (cid:54) = i and since α ∗ satisfies the constraints of CP, i.e. it is feasible,implies that β (cid:48)(cid:48) as well satisfies constraints of GPP. We want to show that β (cid:48)(cid:48) is the optimal valueas well, assume that it is not and there exists an argument β ∗ for which the objective takes ahigher value. If this is the case then, from β ∗ we can construct a α (cid:48) as in part a). which, if β ∗ takes a higher value than β (cid:48)(cid:48) , takes a higher value than α ∗ thus, contradicting optimality.To show that GPP and LPP are equivalent, we use the following fact, since LPP consists of asubset of the constraints then the solution of LPP is an upper bound of the solution to GPP. Weneed to show that the gap between the solution of LPP and GPP is always 0. Note that for anoptimal solution of LPP, γ ∗ = ( γ ∗ , .., γ ∗ N ) we know that γ ∗ i = γ ∗ j ∀ j ∈ N ( i ) (component-wise).If we can show that γ ∗ i = γ ∗ j ∀ j ∈ { , ..., N } then LPP and GPP will be equivalent, since itwill also satisfy all the constraints of GPP. Assume that this does not hold then ∃ i, j such that LPP DP Problems given in Appendix
CP- Coupled Problem
GPP-Global Primal ProblemLPP-Local Primal ProblemDP-Decoupled Problem
Figure 9. Problems used to transit from the Coupled Problem (CP) to Decoupled Problem (DP). γ ∗ i (cid:54) = γ ∗ j . Since, the interference graph is connected ∃ a path i → j = { i , ..., i s } which implies, γ ∗ i = γ ∗ i ... = γ ∗ j . This leads to a contradiction, thereby establishing the claim.Lastly, to show that DP is equivalent LPP. Given γ ∗ , define κ = γ ∗ and a θ = ( θ , ..., θ H ) to satisfy D k κ k − θ k = 0 , ∀ k ∈ { , .., H } , where κ k = ( γ ∗ ,k , .., γ ∗ ,kN ) . It can be shown usingthe same approach as we did for GPP and CP that ( κ , θ ) is indeed optimal argument for DP.Assume that ( κ , θ ) is not the optimal solution then we know that there exists ( κ ∗ , θ ∗ ) for whichthe objective in DP takes a higher value. If this is the case, let us define γ (cid:48) = κ ∗ , here γ (cid:48) satisfies the constraints in LPP. Also, since the objective in DP at ( κ ∗ , θ ∗ ) takes a higher valuethan that at ( κ , θ ) , this yields that the objective in LPP at γ (cid:48) should take a higher value than thatat γ ∗ , which contradicts optimality of γ ∗ . On the same lines, it can be easily shown that from ( κ ∗ , θ ∗ ) we can construct the optimal solution γ ∗ of the LPP. This, will establish equivalencebetween LPP and DP. Hence, all the four problems are equivalent. This is shown in Fig. 10.(Q.E.D) Proof of Theorem 3:
According to [37], the ADMM algorithm converges with rate O (1 /P ) if the DP is feasible and if the feasible set is compact. Since B i and Θ k are all closed andbounded polyhedrons, the feasible set is compact. (Q.E.D) Proof of Theorem 4:
Here, we need to show three things,i). if ∆ ≤ ∆ max then the distributed policy yields a feasible solution,ii). the size of any MIS is ≥ N ∆+1 , thereby using this to show that the distributed policy, iffeasible will yield a network performance of at least N ∆+1 log (1 + p maxlb ( D ub ) np ζ σ ) andiii). the upper bound on the network performance, sum throughput here is N log (1+ p maxub ( D lb ) np σ ) .i). In the Phase 1 of the algorithm the maximum number of colors used is ∆+1 , since each UEselects colors from subset of { , ..., H }∩{ , ..., d i +1 } . The first ∆+1 output MISs, { I (cid:48) , ..., I (cid:48) ∆+1 } span all the UEs in the network. If the fraction of time assigned to each of these ∆ + 1 MISs is, α (cid:48) k = R minub log (1+ pmaxlb ( Dub ) np ζσ ub ) , ∀ k ∈ { , .., ∆ + 1 } then such an assignment satisfies the constraintthat sum of fractions assigned to all the colors cannot be more than 1, i.e. since ∆ ≤ ∆ max = ⇒ (∆ + 1) R minub log (1+ pmaxlb ( Dub ) np ζσ ) ≤ . Using the fact that network exhibits ζ − WNI we can write theminimum instantaneous throughput that can be obtained by UE- i as, log (1 + p maxi ( D iT ( i ) ) np ζ σ ub ) , andminimum instantaneous throughput of any UE as, log (1+ p maxlb ( D ub ) np ζ σ ub ) . Thus, given the fractionsassigned to the MISs, α (cid:48) k = R minub log (1+ pmaxlb ( Dub ) np ζσ ub ) , ∀ k ∈ { , .., ∆ + 1 } , which span all the UEs. eachUE i ’s throughput requirement is satisfied, , R minub log (1+ pmaxlb ( Dub ) np ζσ ub ) log (1 + p maxi ( D iT ( i ) ) np ζ σ ub ) ≥ R minub .ii). Assume that ∃ an MIS whose size is S < N ∆+1 . Each UE in the MIS can exclude amaximum of ∆ UEs from being included in the MIS. This implies that S (∆ + 1) , represents thetotal number of UEs excluded and the UEs in the MIS which put together should exceed N . Sincethis is not the case here, the contradiction implies that S ≥ N ∆+1 . This combined with minimuminstantaneous throughput of any UE, we get the lower bound N ∆+1 log (1 + p maxlb ( D ub ) np ζ σ ub ) , for ourpolicy.iii). The upper bound on the optimal network performance is obtained by summing maximuminstantaneous throughput of any UE log (1 + p maxub ( D lb ) np σ lb ) for all UEs, N log (1 + p maxub ( D lb ) np σ lb ) .Computing the ratio of the lower bound of proposed scheme N ∆+1 log (1 + p maxlb ( D ub ) np ζ σ ) and N log (1 + p maxub ( D lb ) np σ ) , we get log (1+ pmaxlb ( Dub ) np ζσ )(∆+1) log (1+ pmaxub ( Dlb ) npσ ) which is no less than, Γ = R minub log (1+ pmaxub ( Dlb ) npσ ) since ∆ ≤ ∆ max . (Q.E.D) Proof of Theorem 5:
Let ∆ ∗ = 6 η with η = (cid:100) log (1+ Dlb ) npσ lb p maxub ) R minlb (cid:101) . We assume that theinterference graph is constructed using a distance threshold rule (Subsection V-B). Note thateach UE’s minimum throughput requirement is at least R minlb , this combined with maximuminstantaneous throughput of any UE log (1 + p maxub ( D lb ) np σ lb ) yields that each UE needs at least R minlb log (1+ pmaxub ( Dlb ) npσ lb ) fraction of time slots. First, we need to show that if there exists a clique (asubset of vertices in the graph which are mutually connected) in the interference graph of size, X greater than η then the minimum throughput constraints cannot be satisfied. Assume thatthere does exist such a clique, then any MIS based scheduling policy will allocate separate timeslots to each UE in the clique. This is true because no two UEs in the clique will belong to thesame MIS. This implies that X R minlb log (1+ pmaxub ( Dlb ) npσ lb ) is the total fraction separate time slots neededwhich has to be less than 1. But as X ≥ η , this leads to infeasibility. Next, if ∆ ≥ ∆ ∗ , we claim that we will have at least one clique in the graph satisfying this condition. Then ∃ UE- i with adegree d i ≥ η , this implies that within a radius of D th around SBS- T ( i ) ∃ η SBSs. Also, thiscircle around SBS- T ( i ) can be partitioned into 6 sectors subtending π at the center.The distancebetween any two points located in the sector is ≤ D th , which we justify next. Hence, all thepoints in a sector are mutually connected, thus forming a clique.Let the 2-D polar coordinates of two points i, j in a sector be ( r i , and ( r j , θ ) , where ≤ r i ≤ D th , ≤ r j ≤ D th and ≤ θ ≤ π . Hence, the square of the distance between the twopoints is expressed as f ( r i , r j , θ ) = r i + r j − r i r j cosθ and our claim is that the maximum value f ( r i , r j , θ ) , in the set of constraints above is no greater than ( D th ) . We formally state this asan optimization problem below: max r i ,r j ,θ f ( r i , r j , θ )0 ≤ r i ≤ D th , ≤ r j ≤ D th ≤ θ ≤ π Since, both r i , r j are non-negative, this implies that in the above optimization problem, θ = π has to be satisfied in the optimal argument. Substituting θ = π in f ( r i , r j , θ ) we get, f ( r i , r j , π ) = r i + r j − r i r j . Next, we show that r i + r j − r i r j ≤ ( D th ) for ≤ r i ≤ D th , ≤ r j ≤ D th .Fix a ≤ r j ≤ D th ,then r i + r j − r i r j takes its maximum value at r i = D th , which gives ( D th ) + r j − D th r j . Since ≤ r j ≤ D th , this yields ( D th ) + r j − D th r j ≤ ( D th ) whichestablishes the claim.If we have a total of η SBSs in the circle then at least one sector has to have more than η SBSs (Pigeonhole principle), which implies that a clique of size X ≥ η will exist. (Q.E.D)R EFERENCES [1] E. Hossain, L. B. Le, and D. Niyato, “Self-organizing small cell networks,” in
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