Log-Convexity of Rate Region in 802.11e WLANs

1 Log-Con v e xity of Rate Re gion in 80 2.11e WLAN s Douglas J. Leith, V ijay G. Subraman ian and Ken R. Du f fy Hamilton Institute, National Universit y of Ireland May nooth Abstract —In this paper we establish the log-con v exity of the rate r egion in 802.11 WLANs. This generalises pre vious results fo r Aloha networks an d h as immediate impl ications for optimisation based a pproaches to the analysis and design of 802.11 wireless networks. I . I N T RO D U C T I O N In this paper we consider the log-co n vexity of the rate re gion in 802.1 1 WLANs. The r ate region is defined as the set of achiev able throug hputs and we begin by notin g that the 80 2.11 rate r egion is well known to b e non-co n vex. This is illustrated, for exam ple, in Figu re 1 for a simple two-station WLAN (where σ , T c , T s are described in Section II). The shaded region indicates the set of achiev able rate pairs ( s 1 , s 2 ) where s i is the throughp ut of station i , i ∈ { 1 , 2 } . It c an be seen from this figure that the max imum throu ghpu t achiev able by the network when on ly a single station transmits (th e extrem e point alo ng the x- or y-axes) is g reater than th at whe n both stations are activ e (e.g. the extreme point along the y = x lin e). This n on-convex b ehaviour occurs b ecause in 802.1 1 there is a po siti ve pro bability of co lliding tran smissions when multiple stations are activ e, leading to lost transmission oppor tunities. In Figure 2 the same data is shown b ut now r eplotted as the log rate region, i.e. the set of pairs ( log s 1 , lo g s 2 ). Evidently , the log rate region is con vex. Our main result in this paper is to establish th at th is beh aviour is tru e in g eneral, not just in this particular example. Th at is, although the 802.11 rate regio n is non-co n vex, it is nevertheless log-co n vex. T he im plications of this f or optimisation -based approaches to the design a nd analysis of f air through put alloca tion sch emes are d iscussed after the result. In a WLAN context, rate region pr operties ha ve mainly been studied for Aloh a networks. The log -conve xity of the Aloha rate region in general mesh network settings has been established by several authors [7], [2], [3], [ 1], [8] in the context of utility optimisation . All of these results make the standar d Aloha assumption o f equ al idle and b usy slot duration s, wh ereas in 8 02.11 WLANs highly un equal slot duration s are the norm e. g. it is n ot u ncommo n to have busy slot dura tions that are 100 times larger than the PHY idle slot duration . This is key to improving thro ughpu t efficiency but also fundam entally alters o ther thro ughpu t properties since the mean MAC slot d uration and achieved rate are n ow strong ly coupled . W e note th at a num ber of recent papers have con - sidered alg orithms th at seek to achieve certain fair solutio ns (prop ortionally fair, max-m in fair) in 802. 11 networks, e. g see [6] an d r eferences there in. For the WLAN scenario in this paper we show how existence and u niquene ss of fair so lutions follows fr om log-co n vexity . W ork s upported by Scie nce Foundation Ireland grant 07/IN.1/I901. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 s 1 s 2 Fig. 1. Illust rating non-co n ve xity of 802.11 rat e re gion. Plot sho ws through- put normalised by PHY rate for n = 2 sta tions and σ /T c = 1 / 10 an d T s = T c (i.e. for packet sizes where the pack et transmission duration is 10 times lar ger than the PHY idle slot duration). −5 −4.5 −4 −3.5 −3 −2.5 −2 −5 −4.5 −4 −3.5 −3 −2.5 −2 log s 1 log s 2 Fig. 2. Log rate reg ion corresponding to data shown in Figure 1. I I . N E T WO R K M O D E L The 802.11 e stand ard extends and sub sumes the standard 802.1 1 DCF (Distributed Coord inated Function) conten tion mechanism by allowing the adjustment of MA C parameters that w ere p reviously fixed. With 80 2.11, on detectin g th e wireless medium to be idle for a perio d D I F S , eac h station initializes a counter to a ran dom numb er selected u niform ly in the set { 0, ...,CW -1 } where CW is the co ntention window . T ime is slotted and this counter is decr emented once f or each slot th at the m edium is idle. An im portant fea ture is th at the cou ntdown ha lts when th e mediu m becomes busy and only resumes af ter the med ium is idle ag ain fo r a period D I F S . On the coun ter reachin g zero , the station transmits a packet. If a c ollision occ urs (two or mo re stations transm it simultaneou sly), CW is set to min(2 × C W , C W max ) and the process r epeated. On a successful transmission, CW is reset to the value C W min and a new coun tdown starts for the next packet. Again, each packet transmission in this phase 2 includes th e tim e sp ent waiting fo r an ack nowledgement from the r eceiv er . Th e 802. 11e MA C enables the values of D I F S (called AI F S in 8 02.11 e), C W min and C W max to be set on a per class basis for each station. Thro ughou t this paper we restrict atten tion to situation s wh ere AI F S ha s the legacy value D I F S . In addition , 8 02.11 e adds a TXOP mechanism that specifies the duration d uring which a station can keep transmitting withou t r eleasing the channel onc e it wins a transmission oppor tunity . In order not to release the channel, a SIFS interval is inserted between each p acket-A CK pair . A successful tr ansmission roun d consists of multiple packets and A CKs. By adjusting this time , the num ber of packets that may b e tr ansmitted b y a station a t each transmission oppor tunity can b e contr olled. A salient feature of the TXOP operation is th at, if a large T XOP is assigned and there are n ot enoug h packets to be tran smitted, the TXOP per iod is ended immediately to av oid wasting bandwidth. W e conside r an 802.11e WLAN with n station s. As d e- scribed in [4], [5], we d i vide time into MAC slots, wh ere ea ch MA C slot may con sist either of a PHY id le slot, a successful transmission or a colliding transm ission (wh ere m ore th an o ne station attempts to transmit simultaneo usly). Let τ i denote the probab ility that station i attempts a transmission. The mean throug hput of station i is then shown in [4] to be s i ( T ) = τ i Q k ∈ N \{ i } (1 − τ k ) L i σ P idle + T s P succ + T c (1 − P idle − P succ ) (1) where P idle = Q k ∈ N (1 − τ k ) and P succ = P i ∈ N τ i Q k ∈ N \{ i } (1 − τ k ) , T = [ τ 1 ... τ n ] T , L i is the mean frame paylo ad size at station i in b its and N = { 1 , .., n } , σ is th e PHY idle slot d uration, T s is the du ration of a successful tran smission (inclu ding time to transmit the data frame, receiv e the MA C ACK and wait for DIFS) an d T c the duration of a collision . In this paper we prove useful analytical p roperties of the thro ughp ut exp ression (1). It will prove usef ul to work in terms o f the qua ntity x i = τ i / (1 − τ i ) rather than τ i . W ith this transform ation we hav e that P idle = 1 / Q k ∈ N (1 + x k ) and P succ = P i ∈ N x i / Q k ∈ N (1 + x k ) and so s i ( T ) = x i L i /T c σ /T c − 1 + ( T s /T c − 1) P i ∈ N x i + Q k ∈ N (1 + x k ) Definition 1: Rate Region . The rate region is the set R ( ¯ τ ) of achiev able throu ghput vectors S ( T ) = [ s 1 ... s n ] T as the vector T o f a ttempt proba bilities range s over domain D ( ¯ τ ) = [0 , ¯ τ 1 ] × · · · × [0 , ¯ τ n ] , wh ere ¯ τ i denotes th e i ’th elem ent of vector ¯ τ and 0 ≤ ¯ τ i ≤ 1 , ∀ i ∈ { 1 , ..., n } . In this pap er we assume that the value of τ i can be fr eely selected in the interval [0 , ¯ τ i ] . Th is is a mild assumption . For example, suppo se C W max is set equal to C W min . T hen 1 τ = 2 q / C W min where q is the p robability that the re is a packet av ailable fo r tr ansmission wh en the station wins a transmission opp ortunity an d so is related to the p acket ar riv al rate. When a station is saturated we have q = 1 . W e n ote that the value q here is similar to the q uantity in [4] also referred to as q . By adjusting q (via the packet arr i val p rocess) and/or 1 Ignoring post back off for simplicity C W min , it can be seen that the value of τ i can be controlled as req uired. Definition 2: Log-conve xity . Recall that a set C ∈ R n is conv ex if for a ny s 1 , s 2 ∈ C and 0 ≤ α ≤ 1 , there exists a n s ∗ ∈ C such that s ∗ = αs 1 + (1 − α ) s 2 . A set C is log-convex if th e set lo g C := { log s : s ∈ C } is co n vex. I I I . L O G - C O N V E X I T Y A. Log-Con ve xity W e begin in th is section by assuming that ¯ τ = 1 , where 1 den otes the all 1’ s vector . This assump tion is relaxed later on. For co n venience we set a := σ / T c with a ∈ [0 , 1] and K := T s /T c − 1 with K ≥ 0 . Th e thro ughpu t expre ssion can now b e written as s i ( T ) = x i L i /T c X ( T ) (2) where X ( T ) := a + K X i ∈ N x i + Y i ∈ N (1 + x i ) − 1 = a + ( K + 1 ) X i ∈ N x i + n X k =2 X A ⊆ N : | A | = k Y j ∈ A x j . (3) W e kn ow that the rate region R ( 1 ) may be non- conv ex, but ask whether it is log-conv ex. Let log S ( T ) = [log s 1 ... log s n ] T . The rate region R ( 1 ) is log-conv ex if ∀ T 1 , T 2 ∈ (0 , 1) n and ∀ α ∈ [0 , 1] , ∃T ∗ ∈ (0 , 1) n such that α log S ( T 1 ) + (1 − α ) log S ( T 2 ) = log S ( T ∗ ) . (4) Rearrangin g terms we g et for every i = 1 , . . . , n , x ∗ i X ( T ∗ ) =  x 1 i X ( T 1 )  α  x 2 i X ( T 2 )  (1 − α ) , or ( x 1 i ) α ( x 2 i ) (1 − α ) x ∗ i = X ( T 1 ) α X ( T 2 ) (1 − α ) X ( T ∗ ) . (5) Note that here we restrict T to (0 , 1 ) n rather th an [0 , 1] n . This inv olves no loss of ge nerality since S ( T ) is a con tinuous function of T . Note th at the L i /T c term in (2) canc els on both sides of (4) so the log-co n vexity result is independ ent of this term. W e proc eed by po stulating that x ∗ is of the fo rm x ∗ i = ( x 1 i ) α ( x 2 i ) (1 − α ) δ (6) as the right side of (5) does not depend on any particular i . The log-convexity qu estion is wh ether we can find δ > 0 satisfyin g δ = X ( T 1 ) α X ( T 2 ) (1 − α ) X ( T ∗ ) (7) Substituting from (6) into (7), then using th e first expression in (3), and definin g y k = ( x 1 k ) α ( x 2 k ) (1 − α ) , we will need to 3 solve for a δ > 0 such that δ = X ( T 1 ) α X ( T 2 ) (1 − α ) a + K P i ∈ N y i δ + Q i ∈ N  1 + y i δ  − 1 , i.e. δ a + K X i ∈ N y i δ + Y i ∈ N  1 + y i δ  − 1 ! = X ( T 1 ) α X ( T 2 ) (1 − α ) . (8) Recalling H ¨ olders inequality for tw o non-negative vectors u and v , X k u k ! α X k v k ! (1 − α ) ≥ X k u α k v (1 − α ) k ∀ α ∈ [0 , 1] , we have using the secon d expression in (3) that the rig ht-hand side o f (8) is positive and lower bo unded by a + K X i ∈ N y i + Y i ∈ N (1 + y i ) − 1 . Choosing δ = 1 it can be seen that this lo wer bo und lies within the ran ge of the left-hand side of (8). Con sidering th e left-hand side of (8) in more detail, its second deriv ati ve is giv en by 1 δ 3 X i,j ∈ N : j 6 = i y i y j Y k ∈ N : k 6 = i,j  1 + y k δ  where p roduct over an em pty set is defined to be 1 . Since the second-d eriv ati ve is p ositiv e for δ ≥ 0 , it implies the (strict) convexity of the left-han d side of (8). Th is quantity is unboun ded an d has range that in cludes [ a + K P i ∈ N y i + Q i ∈ N (1 + y i ) − 1 , ∞ ) . I t follows that there exists a positive δ satisfying (8), as required . I ndeed, in genera l there m ay exist two values of δ solving ( 8). T o see this observe that the left -hand side is unbo unded b oth as δ → 0 a nd as δ → ∞ . The first-deriv ativ e is n egati ve as δ → 0 and positive as δ → ∞ , so we hav e a turning po int δ ∗ , which due to the co n vexity of the f unction is uniqu e. This turning po int p artitions the real line and two solutio ns to (8) then exist, one lying in (0 , δ ∗ ) and the other in ( δ ∗ , ∞ ) . Ad ditionally , this argume nt also say s that th ere exists at least on e solution o f (8) where δ ≥ 1 . W e h av e theref ore established the following theor em. Theor em 1: The r ate region R ( 1 ) is log-co n vex. B. Constraints on τ W e can extend the foregoing analysis to situations where the station attempt pr obability is constrained, i.e. the vector T of attempt probab ilities range s over D ( ¯ τ ) = [0 , ¯ τ 1 ] × · · · × [0 , ¯ τ n ] , where 0 ≤ ¯ τ i ≤ 1 , ∀ i ∈ { 1 , ..., n } . Note that an uppe r b ound on τ i of ¯ τ i results in an upp er bou nd ¯ x i = ¯ τ i / (1 − ¯ τ i ) on x i . Therefo re if T 1 , T 2 ∈ ¯ τ , then x 1 , x 2 ∈ D ( ¯ x ) = [0 , ¯ τ 1 / (1 − ¯ τ ) 1 ] × · · · × [0 , ¯ τ n / (1 − ¯ τ ) 1 ] an d f or every α ∈ [0 , 1] we also have y ∈ D ( ¯ x ) . From the proo f of Theo rem 1 we know that there exists at least o ne δ ≥ 1 th at solves (8 ). Using that solution we find that x ∗ = y /δ ≤ y so that x ∗ ∈ D ( ¯ x ) . Note that we can h av e different values of ¯ τ i for every i . Th erefore we h av e the f ollowing cor ollary to Theorem 1. Cor ollary 1: The rate region R ( ¯ τ ) is log-conve x for every ¯ τ ∈ [0 , 1] n . I V . D I S CU S S I O N These log-co n vexity results allow us to im mediately apply powerful o ptimisation results to th e analysis and design of fair thr oughp ut allocations for 802.1 1 WLANs. First, using [9, Th eorem 1] , the existence of a max-m in fair solution immediately follows. W e also ha ve that any optimisation o f the for m max S f ( S ) s.t. S ∈ R ( ¯ τ ) , h i ( S ) ≤ 0 , i = 1 , .., m can be con verted into an o ptimisation max S ˜ f (log S ) s.t. log S ∈ log R ( ¯ τ ) , ˜ h i (log S ) ≤ 0 , i = 1 , .., m where ˜ f ( z ) = f (ex p( z )) ( so, in particular, ˜ f (log S ) = f ( S ) ), log S ( T ) = [log s 1 ... lo g s n ] T , log R = { log s : s ∈ R } and ˜ h i ( z ) = h (exp( z )) . Provided − ˜ f ( · ) and the ˜ h i ( · ) are con vex function s, the optimisation is a convex proble m to which standard tools can the n b e ap plied. From this poin t of view it now follows that we can naturally extend th e con gestion an d contention co ntrol id eas of [3] to the more general scenario considered in [4], [ 5]. In par ticular , for the standard family of utility fairness function s given fo r w > 0 , α ≥ 1 and z > 0 by f w, α ( z ) = ( wz 1 − α / (1 − α ) if α 6 = 1 , w log ( z ) if α = 1 , we have ˜ f w, α ( z ) = f w, α (exp( z )) is conca ve fo r all α ≥ 1 . In the α > 1 case we also get strict con cavity o f f , and the existence an d u niquen ess of utility fair so lutions immediately follows fr om our log-convexity r esult. For ¯ τ = 1 an an alysis of the bou ndary of th e log ra te-region also allows one to show uniquen ess of the solu tion in the case o f α = 1 . V . 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