Distributed Resource Allocation for Proportional Fairness in Multi-Band Wireless Systems

Transcription

1 Distributed Resource Allocation for Proportional Fairness in Multi-Band Wireless Systes I-Hong Hou CSL and Departent of CS University of Illinois Urbana, IL 61801, USA Piyush Gupta Bell Labs, Alcatel-Lucent 600 Mountain Ave., 2C-374 Murray Hill, NJ Abstract A challenging proble in ulti-band ulti-cell self-organied wireless systes, such as ulti-channel Wi- Fi networks, feto/pico cells in 3G/4G cellular networks, and ore recent wireless networks over TV white spaces, is of distributed resource allocation. This involves four coponents: channel selection, client association, channel access, and client scheduling. In this paper, we present a unified fraework for jointly addressing the four coponents with the global syste objective of axiiing the clients throughput in a proportionally fair anner. Our forulation allows a natural dissociation of the proble into two subparts. We show that the first part, involving channel access and client scheduling, is convex and derive a distributed adaptation procedure for achieving Pareto-optial solution. For the second part, involving channel selection and client association, we develop a Gibbs-sapler based approach for local adaptation to achieve the global objective, as well as derive fast greedy algoriths fro it that achieve good solutions. I. INTRODUCTION Many of the existing and evolving wireless systes operate over spectru that spans ultiple bands. These bands ay be contiguous, as in OFDM-based systes, such as current IEEE based WLANs (a.k.a. Wi- Fi networks) and evolving fourth-generation LTE cellular wireless systes; or they ay be spread far apart, as in ulti-channel systes and in recently proposed wireless broadband networks over TV white spaces (discussed in the sequel). A coon issue in these ulti-band systes is how to perfor resource allocation aong different clients, possibly being served by different access points (APs). This needs to be done so as to efficiently utilie wireless resources spectru, transission opportunities and power while being fair to different clients. Furtherore, unlike traditional enterprise Wi-Fi networks and cellular wireless networks, where the placeent of APs and their operating bands are arrived at after careful capacity/coverage planning, ore and ore of the evolving wireless systes are going to be self-organied networks. There is extensive literature on copletely selforganied wireless networks, also referred to as ad hoc networks [10], which are often based on Even the eerging 4G cellular wireless networks, such as those based on LTE, are going to have a significant deployent of self-organied subsystes: naely, pico cells and feto cells [2]. These self-organied (sub)-systes will require that resource allocation is perfored dynaically in a distributed anner and with inial coordination between different APs and/or clients. In this paper, we consider the proble of joint resource allocation across different APs and their clients so as to achieve a global objective of axiiing the syste throughput while being fair to users. We focus on achieving proportional fairness, which has becoe essentially standard across current 3G cellular systes, as well as in eerging 4G systes based on LTE and WiMAX. Thus, the syste objective will be to allocate wireless resources, spectru and transission opportunities, so as to axiie the (weighted) su of the log of throughputs of different clients, which is known to achieve (weighted) proportional fairness. The cobined resource allocation in a ulti-band ulti-cell wireless syste involves four coponents: Channel Selection, Client Association, Channel Access, and Client Scheduling. The first coponent, Channel Selection, decides on how different APs share different bands of the spectru available to the wireless syste. The second coponent, Client Association, allows a client to decide on an AP to associate within its neighborhood that is likely to provide the best perforance. Once an AP has chosen a channel/band to operate in and a bunch of clients have associated with it, the third coponent, Channel Access, decides when it should access the channel so as to serve its clients while being fair to other access points in its neighborhood operating in the sae channel. The final coponent, Client Scheduling, decides which of its clients an AP should serve whenever it successfully accesses the channel. Our approach addresses the four coponents in a unified fraework, where the solutions to different coponents are arrived at through separation of tie scales of adaptation. More specifically, our forulation allows for the optiiation proble of axiiing the clients throughput with weighted proportional fairness to naturally dissociate into two sub-parts, which are adapted at different tie scales. Assuing that the channel selection and client association have been perfored, we show that the sub-proble of channel access and client scheduling becoes convex, which is also aenable to a distributed adaptation for achieving Pareto-optial weighed proportional fairness. At a slower tie scale, we adapt the channel selection and client association to varying deands and interference. This part is a non-convex proble in general, and thus, difficult to solve for a globally optial solution. We develop a Gibbs-sapler based approach to perfor local adaptation while iproving the global syste objective. The adaptation is randoied, and if done slowly enough, can achieve a globally optial solution. In practice, however, that ay not be always feasible; hence, we derive greedy heuristics fro it for channel selection and client association, which though not glob-

2 ally optial, provide fast and good distributed solutions with liited exchange of inforation, as the siulation results indicate. Siulation results also indicate that our policies achieve better perforance than state-of-the-art techniques, such as those proposed in [8] and [9]. The paper is organied as follows. Section II describes the ulti-cell ulti-band wireless syste odel and the joint resource allocation proble that we study. Section III gives an overview of our approach and discusses the separation of probles. The details of our approach and its desirable properties, including convergence to Paretooptial proportionally-fair allocation, are given in Section IV and Section V. The results of siulations are provided in Section VI. Concluding rearks are discussed in Section VII. II. SYSTEM MODEL We consider a syste with several access points (APs) and clients that can operate in a nuber of channels. We denote the set of APs by N, the set of clients by I, and the set of channels by C. Each client i is associated with an AP, which we denote by n(i), and is served by that AP. Each AP n is equipped with u n radios that can operate in different channels siultaneously and each client i is equipped with only one radio. When an AP n has ore than one radios, i.e, u n > 1, we can siplify the odel by assuing that there are u n APs, each with one radio, that are placed at the sae place as AP n. Each of these u n APs corresponds to one radio of AP n. This procedure allows us to only consider the odel in which each AP has one radio and siplifies notations. Throughout the rest of the paper, we assue that each AP only has one radio and operates in one channel unless otherwise specified. The channel that an AP n is operating in is denoted by c(n). APs can switch the channels that they operate in, although such switches can only be done infrequently due to the large overheads. When an AP switches channels, all its clients also switch channels accordingly. We focus on a server-centric schee where each AP schedules all transissions between itself and all clients that are associated with it. This schee is applicable to a wide varieties of wireless systes that include LTE, WiMax, and IEEE PCF 1. We assue that tie is slotted, with the duration of a tie slot equals the tie needed for a transission. If an AP n akes a successful transission toward client i in channel c, client i receives data at rate of B i,n,c in this slot. Since characteristics of different channels ay be different, B i,n,c depends on c. On the other hand, we assue that the APs are not synchronied and ay interfere with each other. We consider the interference relations using the protocol odel [5]. When an AP n operates in channel c, it ay be interfered by a subset M n,c of APs, where n M n,c for notational siplicity. When the AP n schedules a transission between itself and one of its clients, the transission is successful if n is the only AP aong M n,c that transits in channel c during the tie of transission; otherwise, the transission suffers fro a collision and fails. We assue that the interference relations are syetric, i.e., M n,c if and only if n M,c. Note that, since the propagation characteristics of different channels ay be different, especially in the case of TV white space access, the subset M n,c depends 1 A distributed schee, which is applicable to IEEE DCF, is discussed in the technical report [7]. on the channel c. This dependency further distinguishes our work fro ost existing works on ulti-channel access where interference relations are assued to be identical for all channels. We also define N i to be the set of APs that client i can be associated with, and M i := { N c C, n N i s.t. M n,c } for each client i. Since APs are not coordinated, we assue that they access the channel by rando access. Each AP n chooses a rando access probability, p n. In each slot, AP n accesses the channel c(n) with probability p n. The transission is successful if n is the only AP in M n,c(n) that transits over channel c(n). Thus, the probability that AP n successfully accesses the channel can be expressed as p n M n,c(n), (1 p ) = p n 1 p n M n,c(n), c()=c(n) (1 p ). (1) The AP is in charge of scheduling transissions for its clients. When AP n accesses the channel, it schedules the transission for client i, where n(i) = n, with probability φ i,n, φ i,n 0 and i:n(i)=n φ i,n = 1. Since the data rate of client i when it is served is B i,n(i),c(n(i)) and the probability that the AP n(i) akes a successful transission is as in Eq. (1), its throughput per tie slot is, assuing n(i) = n and c(n) = c, p n r i := B i,n,c φ i,n 1 p n M n,c,c()=c (1 p ). (2) Finally, we assue that each client is associated with a positive weight w i > 0. We also denote the total weights of clients associated with AP n by w n, i.e., w n := i:n(i)=n w i. The goal is to achieve weighted proportional fairness aong clients, that is, to axiie i I w i log r i. III. SOLUTION OVERVIEW AND TIME-SCALE SEPARATION We now give an overview of our approach to achieve weighted proportional fairness, which consists of separating the proble into four coponents and solving the. By Eq. (2), we can forulate the proble of achieving weighted proportional fairness as the following optiiation proble: Max i w i log r i = i w i[log B i,n(i),c(n(i)) + log φ i,n(i) + log p n(i) 1 p n(i) + log M n(i),c(n(i)),c()=c(n(i)) (1 p )], s.t. c(n) C, for all n; n(i) N i, for all i; 0 p n 1, for all n; φ i,n(i) 0, for all i; i:n(i)=n φ i,n = 1, for all n. Based on this forulation, the proble of achieving weighted proportional fairness consists of four iportant coponents, in increasing order of tie scales: First, whenever the AP accesses the channel, it needs to schedule one client for service. That is, the AP has to decide the values of φ i,n. Second, in each tie slot, the AP has to decide whether it should access the channel, which consists of deterining the values of p n. Third, each client needs to decide which AP it should be associated with, i.e., deciding n(i). Finally, each AP n needs to choose a channel, c(n), to operate in. We denote the four coponents as Scheduling Proble, Channel Access Proble, Client Association Proble, and Channel Selection

3 Proble, respectively. Weighted proportional fairness is achieved by jointly solving the four coponents. Since the overhead for a client to change the AP it is associated with and for an AP to change the channel it operates in are high, solutions to the Client Association Proble and the Channel Selection Proble are updated at a uch slower tie scale copared to solutions to the Scheduling Proble and the Channel Access Proble. Based on this tiescale separation, we first study the solutions to the Scheduling Proble and the Channel Access Proble, given fixed solutions to the Client Association Proble and the Channel Selection Proble. We then study the solutions to the Client Association Proble and the Channel Selection Proble, under the knowledge of how solutions to the Scheduling Proble and the Channel Access Proble react. Thus, solutions to the Client Association Proble and the Channel Selection Proble are indeed joint solutions to all the four coponents, and their optial solutions achieve Pareto-optial weighted proportional fairness. In addition to solving the four coponents, we will show that the solutions naturally turn into distributed algoriths where each client/ap akes decisions based on local knowledge. IV. THE SCHEDULING PROBLEM AND THE CHANNEL ACCESS PROBLEM In this section, we assue that solutions to the Client Association Proble and the Channel Selection Proble, i.e., n(i) and c(n), are fixed. Since n(i) and c(n) are fixed, values of B i,n(i),c(n(i)) are constant. The optiiation proble can be rewritten as Max i I w i[log φ i,n(i) + log p n(i) 1 p n(i) + log M n(i),c(n(i)),c()=c(n(i)) (1 p )] = i I w i log φ i,n(i) + n N [wn log p n + ( M n,c(n),c()=c(n) w w n ) log(1 p n )], s.t. 0 p n 1, for all n; φ i,n(i) 0, for all i; i:n(i)=n φ i,n = 1, for all n, where w n = i:n(i)=n w i, as defined in Section II. We also define n := M n,c(n),c()=c(n) w to be the total weights of clients that are associated with APs that interfere with n, including itself. This forulation naturally decoposes the optiiation proble into two independent parts: axiiing i w i log φ i,n(i) over φ i,n, which is the Scheduling Proble, and axiiing n [wn log p n +( n w n ) log(1 p n )] over p n, which is the Channel Access Proble. Thus, we can solve these two probles independently. We first solve the Scheduling Proble by the following theore, whose proof is included in [7]. Theore 1: Given n(i), c(n), and p n, for all i and n, i w i log r i is axiied by setting φ i,n(i) w i /w n(i). We solve the Channel Access Proble next. The following theore is the direct result of Theore 1 in [6]. Theore 2: Given n(i), c(n), and φ i,n, for all i and n, i w i log r i is axiied by setting p n w n / n. In suary, when the solutions to the Client Association Proble and the Channel Selection Proble, i.e., n(i) and c(n), are fixed, the AP n should access the channel with probability p n = w n / n in each tie slot and should schedule the transission for its client i with probability φ i,n = w i /w n whenever it accesses the channel. In addition to achieving the optial solution to both the Scheduling Proble and the Channel Access Proble, this solution only requires n to know the local inforation of w and c() for all AP that ay interfere with itself. Thus, this solution can be easily ipleented in a distributed anner. V. THE CLIENT ASSOCIATION PROBLEM AND THE CHANNEL SELECTION PROBLEM We now propose a distributed algorith that solves the Client Association Proble and the Channel Selection Proble based on the knowledge of optial solutions to the Scheduling Proble and the Channel Access Proble. These two probles are non-convex and a local optial solution to the two probles ay not be globally optiu, which we will also illustrate by siulations in Section VI. Thus, coon techniques for solving convex probles are not suitable for these probles. Instead, the proposed algorith uses a siulated annealing technique that is based on the Gibbs Sapler [4], which is proven to converge to the global optiu point alost surely. In the sequel, we also derive a greedy heuristic that is easier to ipleent and converges faster. We call a joint solution to both the Client Association Proble and the Channel Selection Proble as a configuration of the syste. A configuration is thus fully specified by the AP each client is associated with, and the channel each AP operates in. Define ψ t as the configuration of the syste at tie t. We define the utility of the syste under configuration ψ t, which we denote by U(ψ t ), as the value of i w i log r i when APs and clients choose their channels to operate in and APs to be associated with according to ψ t, and apply the optial solution to the Scheduling Proble and the Channel Access Proble under ψ t. We then have U(ψ t ) = i I w i[log B i,n(i),c(n(i)) + log w i + n N [wn log wn + ( n w n ) log n w n n ]. n (3) Finding the joint solution that achieves Pareto-optial proportional fairness is equivalent to finding the configuration ψ that axiies U(ψ). We apply the Gibbs sapler to solve the Client Association Proble and the Channel Selection Proble jointly. At each tie t, either a client or an AP is selected in a rando or round-robin fashion. The selected client, or AP, then changes the AP it is associated with, or the channel it operates in, randoly, while all other clients and APs ake no changes. The solutions to the Scheduling Proble and the Channel Access Proble are then updated according to the new configuration. We now discuss how the selected client, or AP, changes the AP it is associated with, or the channel it operates in. Let ψ t (n(i) = n) be the configuration where client i is associated with AP n, and the reaining of the syste is the sae as in configuration ψ t. We can define ψ t (c(n) = c) for AP n siilarly. If client i is selected at tie t, it changes the AP it is associated with to n with probability e U(ψ t(n(i)=n))/t (t) / eu(ψ t(n(i)=))/t (t), where T (t) is a positive decreasing function. On the other hand, if AP n is selected at tie t, it changes the channel it operates in to c with probability e U(ψ t(c(n)=c))/t (t) / d eu(ψ t(c(n)=d))/t (t). It reains to copute the values of U(ψ t (n(i) = n)) for client i and U(ψ t (c(n) = c)) for AP n. We first discuss how to copute U(ψ t (n(i) = n)). Let w i n := j:n(j)=n,j i w j w n(i) ]

4 be the total weights of clients, excluding i, associated with AP n. Let n i := M n,c(n),c()=c(n) w i. Define U 0 i (ψ t) = j I,j i w j[log B j,n(j),c(n(j)) + log + n N [wn i log wn i n i w j w n(j) i + ( i n wn i ) log n i wn i ], i n which can be thought of as the utility of the syste as if the weight of client i were ero. We then define Ui n(ψ t) := U(ψ t (n(i) = n)) Ui 0. Since in the configuration ψ t (n(i) = n), w = w i for all n; w n = w i n + w i; = i + w i if M n,c(n), c() = c(n), and n; and = i, otherwise, we have U n i (ψ t ) = w i [log B i,n,c(n) + log w i w n ] + j:n(j)=n w j log +w i log wn n ] w n i w n i +wi + w n i log n i (wn i +wi) w n i (n i +w i) + (n i wn i ) log + M n,c(n), [w i log i i +wi ( i w i ) log ( i w i +wi) i ( i +w i)( i w i ) + w i log w ] = w i [log B i,n,c(n)w i + log w n M n,c(n), ] n i n i +w i + log[(1 + wi M n,c(n), ) i i w i /(1 + w i w i ) i i ] + log(1 wi n i +w i )n i w i [log( B i,n,c(n)w i M w n n,c(n), )] + α, where α is a constant. Since (1 + w i A )A e wi and (1 w i A )A e w i for all A >> w i, the last approxiation holds when i >> w i, which is true in a dense network where the weights of all clients are within the sae order. Suppose a client i is selected to change its state at tie t, at which tie the configuration of the syste is ψ t. The probability that i chooses AP n to be associated with is e [U 0 i (ψ t)+ U n i (ψ t)]/t (t) / e[u 0 i (ψ t)+ U i (ψ t)]/t (t) w ) w i/t (t) /γ, where γ is ( B i,n,c(n)w i n M n,c(n), the noralier. To copute the probability of choosing AP n to be associated with, client i only needs the values of B i,n,c(n) for all n N i, w and for all M i. Thus, this probability can be coputed by client i using its local inforation. We also note that this probability has the following properties: First, it increases with B i,n,c(n), eaning that client i tends to choose the AP that has higher data rate; Second, it decreases with n, which is the total weights of clients that interfere with n; Finally, it increases with M n,c(n), w, which is the probability that none of the APs that interfere with n access the channel in a tie slot. Thus, this probability jointly considers the three iportant factors for the Client Association Proble: data rate, interference, and channel congestion. Next we discuss the coputation of the probability that an AP n should choose channel c to operate in, if it is selected. Let n := o M,c(),c(o)=c(),o n wo w. Let U 0 n(ψ t ) be the utility of the syste under configuration ψ t, if the weights of all its clients were ero. That is, U 0 n(ψ t ) = j I,n(j) n w j[log B j,n(j),c(n(j)) + log + N, n [w log w n w j w n(j) ] + ( n w ) log n w ]. n We then define U c n(ψ t ) := U(ψ t (c(n) = c)) U 0 n. Since in the configuration ψ t (c(n) = c), = n + w n if M n,c(n), c() = c, and n; and = n, otherwise, we have U c n(ψ t ) = i:n(i)=n w i[log B i,n,c + log wi w n ] + w n log wn n + ( n w n ) log n w n n + M n,c(n), [w log n n +wn ( n w ) log ( n w +w n ) n ( n +wn )( n w ) + wn log n w +w n n +wn ]. When an AP n is selected by the Gibbs sapler at tie t, it changes the channel it operates randoly, with the probability of changing to channel c proportional to e U(ψ t(c(n)=c))/t (t) = e [U 0 n (ψ t)+ U c n (ψ t)]/t (t) e U c n (ψ t)/t (t). We note that, to copute Un(ψ c t ), AP n only needs the values of B i,n,c, w i, for each client i that is associated with n, and n, w for all c M n,c. Thus, Un(ψ c t ) can also be coputed using only local inforation. Based on the above discussion, it is straightforward to design a distributed protocol (DP) using the Gibbs sapler. In DP, all clients and APs in the syste only need to exchange inforation within their local areas, as they only need local inforation to copute the probability of choosing an AP to be associated with or a channel to operate in. Thus, DP is easily scalable. Further, DP achieves the Pareto-optial proportional fairness as t alost surely by the following theore derived fro [4]. Theore 3: If T (t) satisfies the following conditions: 1) T (t) 0, as t ; 2) T (t) log t, as t ; then li t U(ψ t ) = ax ψ U(ψ) with probability 1, for any initial configuration ψ 1. In addition to DP, we can also consider a greedy policy (Greedy) that is easier to ipleent and converges faster. Greedy works siilar to DP, except that when a client i, or an AP n, is selected by the sapler, it chooses the AP that axiies U(ψ t (n(i) = n)) to be associated with, or the channel that axiies U(ψ t (c(n) = c)) to operate in, respectively. It is essentially a steepest descent direction approach and is guaranteed to converge to a local optial point. In addition to siple ipleentation, Greedy is also consistent with the selfish behavior of clients. Each client i chooses the AP n that axiies B i,n,c(n) o M o n n,c(n),c(o)=c(n),o n o +w, which is indeed o the value of r i when i is associated with n. Thus, in Greedy, every client always chooses to associate with the AP that offers the highest throughput. The pseudo-codes of DP and Greedy are included in the technical report [7]. VI. SIMULATION RESULTS We have ipleented both DP and Greedy algoriths and copared the against policies that use state-of-the-

5 id frequency bandwidth id frequency bandwith A 524 MH 12 MH E 659 MH 6 MH B 593 MH 6 MH F 671 MH 6 MH C 608 MH 12 MH G 683 MH 6 MH D 641 MH 6 MH TABLE I: List of white spaces in New York City. art techniques for solving the Client Association Proble and the Channel Selection Proble. We copare with [8], which proposes a distributed algorith for achieving iniu total interference aong APs, for the Channel Selection Proble. For the Client Association Proble and the Scheduling Proble, we copare with two techniques. The first technique uses a Wifi-like approach where clients are associated with the closest AP and the AP schedules clients so that the throughput of each client is the sae. The protocol that applies both [8] and the Wifi-like approach is called MinInt-Wifi. The other technique is one that is proposed in [9], which, under a fixed solution of the Channel Selection Proble, is a centralied algorith that ais to find the joint optial solution to the Client Association Proble and the Scheduling Proble that achieves weighted proportional fairness. This technique first relaxes the Client Association Proble by assuing that each client can be associated with ore than one APs and forulates the proble as a convex prograing proble. It then rounds up the solution to the convex prograing proble and finds a solution to the Client Association Proble where each client is associated with only one AP. For ease of coparison, we use the solutions to the relaxed convex prograing proble, which is indeed an upper-bound on the perforance of [9]. The protocol that applies both [8] and [9] is called MinInt-PF. The Channel Access Proble is then solved by the optial solutions based on the resulting solutions of MinInt-Wifi and MinInt-PF, respectively. We copare the policies on two etrics: the weighted su of the logariths of throughput for clients, i I w i log r i, and the total weighted throughput i I w ir i. All reported data are the average over 20 runs. We consider a syste consisting of 16 APs that are placed on a 4 by 4 grid. Each AP has 2 radios and adjacent APs are separated by 300 eters. There are 16 clients uniforly distributed in each of the two sectors [0, 300] [0, 300] and [600, 900] [600, 900]; There are 9 clients uniforly distributed in each of the two sectors [0, 300] [600, 900] and [600, 900] [0, 300]. We consider the TV white spaces available in New York City [3]. The list of available channels is shown in Table I. The data rates between APs and clients, and interference relations aong APs, for each channel are derived fro the ITU path loss odel [1] and the siulation settings in [9]. We consider two settings: an unweighted setting where all clients have weights 1.0, and a weighted setting where clients within the region [0, 300] [0, 900] have weights 1.5 and clients outside this region have weights 0.5. Siulation results are shown in Fig. 1. For both the unweighted and weighted settings, MinInt-Wifi and MinInt- PF are far fro optiu. The total weighted throughputs achieved by the two policies are less than half of those achieved by DP under both settings. These results show that policies that optiie the four coponents independently can result in poor perforance. The perforance Fig. 1: Perforance coparison. of Greedy is close to optiu, whose weighted total throughputs are about 88% of those by DP for both the unweighted and weighted settings. VII. CONCLUSION We have studied the proble of achieving weighted proportional fairness in ulti-band wireless networks. We have considered a syste that consists of several APs and clients operating in a nuber of available channels, accounting for interference aong APs and heterogeneous characteristics of different channels. We have identified that the proble of achieving weighted proportional fairness in such a syste involves four iportant coponents: client scheduling, channel access, client association, and channel selection. We have proposed a distributed protocol that jointly considers the four coponents and achieves weighted proportional fairness. We have also derived a greedy policy based on the distributed protocol that is easier to ipleent. Siulation results have shown that the distributed protocol outperfors state-of-the-art techniques. The total weighted throughputs achieved by the distributed protocol can be twice as large as state-of-the-art techniques. Siulation results have also shown that, while being suboptial, the perforance of the greedy policy is actually close to optiu quite often. 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