MAC Layer Performance Analysis of the IEEE 1901 Standard for Power Line Communications Christina Vlachou I&C, EPFL

EDIC RESEARCH PROPOSAL 1 MAC Layer Performance Analysis of the IEEE 1901 Standard for Power Line Communications Christina Vlachou I&C, EPFL

Abstract—Power line communications are one of the fastest out special wiring needed for the installation of the physical growing technologies in home networking. The IEEE 1901 network. The power line channel varies widely with frequency standard specifies the physical and MAC layers of high data and time. Connected appliances have a complex impedance rate power line communications. We study analytical models of the MAC layer of the IEEE 802.11 standard which uses a similar, causing reflections of the transmitted signal and, as a result, but much simpler CSMA/CA mechanism. The analysis is under multipath propagation. Switching the electrical appliances saturated assumptions. The analytical models usually assume changes the characteristic impedance of the transmission line independence between the stations (decoupling assumption) [8]. causing channel characteristics to change over large time- We discuss whether the decoupling assumption holds in the scales. The physical layer of the 1901 standard provides stationary regime of a system [3]. We also study a discrete time Markov chain model of the IEEE 1901 standard [7]. It is yet an modulation techniques with robust forward error correction open question whether the decoupling assumption holds in the which offer the possibility of high speed communications. stationary regime of the 1901 standard. Both wireless and power line transceivers cannot sense the As future research directions, we propose mean field methods medium while transmitting. Hence, both MAC layer protocols to analyze the performance of the IEEE 1901 standard. We also use Carrier Sense Multiple Access with Collision Avoidance discuss future research plans in home networking technologies. (CSMA/CA) mechanisms. The MAC layer of the 1901 stan- Index Terms—HomePlug, Medium Access Control (MAC), dard uses both Time Division Multiple Access (TDMA) and CSMA/CA, decoupling assumption, mean field analysis CSMA/CA methods. The network is synchronized with the AC line cycle by a central coordinator that transmits beacons I.INTRODUCTION in order to synchronize the two different access methods. The IEEE 1901 standard specifies the physical and MAC The central coordinator is the first device activated, but can layers of high data rate power line communications. The change depending on the physical channel conditions and on advantage of power line communications over wireless tech- the availability of other devices. We are interested in studying nologies is that they provide wider communication range with- the CSMA/CA method which is similar to the CSMA/CA method of the 802.11 standard. Proposal submitted to committee: July 5th, 2012; Candidacy In this proposal we study analytical models of the two exam date: July 12th, 2012; Candidacy exam committee: Jean- CSMA/CA methods. These CSMA/CA methods have some Yves Le Boudec, Patrick Thiran, Christina Fragouli. differences which are explained in the following paragraphs. This research plan has been approved: We study analytical models of the 802.11 standard in order to model similarly the CSMA/CA method of the 1901 standard. In order to resolve contention, the 802.11 MAC layer uses Date: ———————————— a time-slotted random backoff procedure [1] with a backoff counter and a contention window (CW ). At each transmission attempt the station senses the medium. If the medium is sensed Doctoral candidate: ———————————— idle, the station selects a backoff counter uniformly at random (name and signature) over the range [0,CWmin 1], where CWmin denotes the minimum contention window− (equal to 32 for the 802.11b standard). The station decrements its backoff counter by 1 at Thesis director: ———————————— each time slot if the medium is sensed idle. If the medium is (name and signature) sensed busy, the station freezes its backoff procedure. When the medium becomes idle again, the station unfreezes its Thesis co-director: ———————————— backoff counter and starts decrementing it again. (if applicable) (name and signature) If the backoff counter is 0, the station transmits a frame. If the transmission is successful, the same procedure is followed for the next packet. If the transmission fails, the station Doct. prog. director:———————————— chooses a new backoff counter uniformly at random over (R. Urbanke) (signature) the range [0, 2CWmin 1] and retransmits when the backoff counter is 0. The contention− window is doubled whenever a EDIC-ru/05.05.2009 transmission attempt fails and does not go above a maximum EDIC RESEARCH PROPOSAL 2

PRS0 PRS1 Priority: CA0 and CA1 CA2 and CA3 CA3 BPC CW DC CW DC CA2 0 8 0 8 0 CA1 1 16 1 16 1 CA0 2 32 3 16 3 3 64 15 32 15 TABLE I: Busy tones transmitted during the two priority ≥ resolution slots (PRS0, PRS1) in the IEEE 1901 Standard. TABLE II: IEEE 1901 standard values for CW and DC. Different priority levels have different parameters. value CWmax (equal to 1024 for the 802.11b standard). The CSMA/CA random backoff procedure of the IEEE 1901 standard is more complex than the procedure described due to collisions. above [2]. First, there are four priority levels CA0, CA1, CA2, • Homogeneous network: All stations have the same mini- CA3 (priority from the lowest to the highest) and only the mum and maximum contention windows. stations with the highest priority contend for the medium in Each model may use further specific assumptions that we the contention period. The highest priority is determined by mention in the following sections. In Section II we present two priority resolution slots PRS0, PRS1. Stations transmit a different analytical models for the backoff procedure of the busy tone in these slots according to their priority as shown IEEE 802.11 standard. Section III presents a discrete time in Table I. If the stations sense a busy tone in one of these Markov chain model of the IEEE 1901 standard [7]. Finally, slots, they differ from transmission. concluding remarks and our research proposal are given in Second, the backoff procedure uses three counters: the Section IV. backoff procedure counter (BPC), the backoff counter and the deferral counter (DC). BPC is set to 0 at the first transmission attempt. The deferral counter and contention window values II.MODELING SATURATED 802.11 NETWORKS are set according to BPC as shown in Table II. After choosing these values, BPC is increased by 1. The backoff counter is A. Bianchi’s model chosen uniformly at random over the range [0,CWmin 1] and is decremented by 1 at every idle time slot, as in− the The backoff procedure of the IEEE 802.11 is modeled with 802.11 standard. If the backoff counter reaches 0, the station a discrete time Markov chain in [5]. In the 802.11 standard the i transmits. After a successful transmission BPC is set to 0 and contention window CWi at backoff stage i is 2 CWmin. The the same procedure is followed for the next packet. After a Bianchi model consists of the variables (s(t), b(t)), with s(t) transmission failure, the contention window is doubled (up to representing the backoff stage and b(t) denoting the backoff a maximum value as shown in Table II) and a new backoff counter of the station at time slot t N. The backoff counter ∈ counter is chosen. The deferral counter is updated according is chosen uniformly at random over the range [0,CWi 1], − to BPC and then BPC is increased by 1. as in the real protocol. The specific assumptions of this model If a station senses the medium busy, it freezes its backoff are the following: counter. The difference with the 802.11 standard is that after • For each station the collision probability p conditioned sensing the medium idle again, the station may resume its on the event that the station transmits is time-invariant backoff counter depending on the deferral counter value. More and is independent of the backoff stage. precisely, if the deferral counter is 0, the station selects a • The backoff processes of the stations are independent. new contention window (CW ) and a new deferral counter according to the backoff procedure counter (Table II). Then After processing the equations of the stationary distribution a new backoff counter is chosen uniformly at random over of the irreducible Markov chain, the probability of transmis- the range [0,CW 1] and BPC is increased by 1. If the sion τ is given by − deferral counter is not 0, the station decrements the deferral 2(1 2p) τ(p) = , (1) and backoff counters by 1. The station keeps decrementing − m (1 2p)(CWmin + 1) + pCWmin(1 (2p) ) the backoff counter by 1 during each idle time slot, as in the − − CW 802.11 case. where min is the minimum contention window of the m In this proposal we study analytical models of the station and is the backoff stage such that the contention win- CW = 2mCW CSMA/CA mechanisms described above. The general assump- dow is max min. After reaching backoff stage m tions that hold for all the models are the following: , the station changes backoff stage only after a successful transmission (the retry limit K is infinite). The conditional col- • Perfect sensing: There are N stations in a single con- lision probability and the transmission probability are related tention domain. by • Saturated conditions: All stations always have packets to p = 1 (1 τ(p))N−1, (2) transmit. − − • Perfect channel: There is no packet loss or error due to and τ, p can be computed from the fixed point equation (2), the physical layer. Thus, transmission failures are only which has a unique solution [5]. EDIC RESEARCH PROPOSAL 3

(i) The throughput of the network is computed by defining the backoff of the jth packet is Bj , where i is the backoff stage probabilities (i) and 0 i Rj 1. Note that E[Bj ] = bi. An example of ≤ ≤ − N−1 the evolution of the processes is shown in Figure 1. The total N Nτ(1 τ) P = 1 (1 τ) P = , PRj −1 (i) tr s − backoff time for the jth packet is given by Xj = B . − − Ptr i=0 j X is a sequence of independent identically distributed non- which are the probability that at least one station transmits j negative random variables and N(t) = max n : T t in a time slot and the probability that a station successfully n with T = 0, T = X + X + ... + X { is a renewal≤ } transmits, respectively. The throughput of the network is 0 n 1 2 n process. The number of attempts R can be viewed as a reward computed by j associated with the renewal cycle length of Xj and thus, from PtrPs [P ] the renewal-reward theorem the average attempt rate β is given S = E , (3) σ(1 Ptr) + Ptr(PsTs + (1 Ps)Tc) by − − E[R] where σ is the time slot duration, E[P ] is the average packet β = , E[X] payload size, and Ts, Tc are the average time the channel is sensed busy because of a successful transmission and a where E[R], E[X] are given by collision respectively. Analytical results are found to be a good [R] = 1 + γ + γ2 + ... + γK approximation of simulation results presented in [5]. E 2 K E[X] = b0 + γb1 + γ b2 + ... + γ bK . B. The fixed point equation’s insensitivity to the distribution of the backoff counters Thus, the average attempt rate β is given by A different approach was followed by Kumar et al. [8] to study the backoff procedure of the 802.11 standard. The 1 + γ + γ2 + ... + γK β = G(γ) = 2 K . (4) authors use renewal theory to study the protocol and conclude b0 + γb1 + γ b2 + ... + γ bK that the fixed point equation of the Bianchi model does not Note that setting the same system parameters in (4) as in [5] depend on the distribution of the backoff counters. Their (K , CW = 2iCW and CW = 2mCW ), (4) assumptions are the following: i min max min gives→ (1). ∞ Under the renewal theory analysis (1) and (4) are • Decoupling assumption: The aggregate attempt process insensitive to the distribution of the backoff counters. of one station is independent of the backoff processes of the other N 1 stations. Attempts by a station experience Now, given that a station transmits in a time slot with a collision probability− γ. probability β, the collision probability is the probability that • The stations are assumed to attempt in each time slot with at least one other station attempts in the same slot. Under the a constant transmission probability equal to the average decoupling assumption the number of attempts of the other stations is binomially distributed with parameters β, N 1 attempt rate β. β is independent of the state of the station. − • The backoff counters follow a distribution with mean and the probability of collision γ is given by backoff duration bk at the kth packet transmission at- γ = Γ(β) = 1 (1 β)N−1. (5) tempt. − − The packet is discarded if the transmission fails K +1 times Equations (4), (5) lead to a fixed point equation (the retry limit is K), as in the real protocol. Thus, after a γ = Γ(G(γ)). (6) transmission attempt at backoff stage K, the backoff stage changes to 0. The authors observe that all stations freeze their Theorem 1. [8] Γ(G(γ)) : [0, 1] [0, 1] has a unique fixed backoff counters during the channel activity and that the total → point if bk, k 0 is a nondecreasing sequence. time spent in backoff up to any time t is the same for every ≥ station. Thus, they remove the channel activity and analyze The sequence of bk in the real 802.11 standard is nonde- the backoff process conditioned on being in backoff. creasing, thus (6) has a unique solution. It is shown that the collision probability obtained from the solution of the fixed R1 = 2 R2 = 1 R3 = 4 R4 = 1 point equation approximates well the collision probability obtained from ns2 simulations of the real 802.11b protocol [8].

(0) (1) (0) (0) (1) (2) (3) (0) B1 B1 B2 B3 B3 B3 B3 B4 Backoff Backoff successes collisions success collision Fig. 1: An example of the evolution of the backoffs of a station over time after removing channel activity. Each attempted packet starts a new backoff “cycle.” renewal instants Fig. 2: An example of the evolution of the channel activity Define Rj as the number attempts until the success of the jth packet transmitted by a station. The sequence of the over time. EDIC RESEARCH PROPOSAL 4

N−1 β(1 β) pi,jLi,j θi,j(β) = − (7) N N−1 N PN N−1 Pmi Li,k (1 β) + Tc (1 Nβ(1 β) (1 β) ) + β(1 β) pi,k + To − − − − − i=1 − k=1 Ci,k

After solving the fixed point equation and computing the are presented in the next section. average attempt rate β, throughput can be computed using again the renewal-reward theorem. As Figure 2 shows, the C. Mean field techniques renewal points are the instants just after the end of a transmission. A station makes an attempt with probability β in each Mean field techniques are used to study the performance of a system of N bodies, as N . Bena¨ım and Le Boudec [3] idle time slot. Thus, an attempt in the network is made with → ∞ probability 1 (1 β)N in a given slot and the aggregate present mean field models with applications on computer and attempt process− follows− a geometric distribution. The mean communication systems. The decoupling assumption of the backoff period is taken equal to the mean time until an attempt models described in the previous paragraphs assumes that the (which is 1/(1 (1 β)N )) in [8] and includes the time slot backoff processes of the stations are independent and that in which the attempt− − is made, which is not in the backoff collisions form an independent identically distributed sequence period. Thus, the actual mean backoff period between two for each station. Mean field methods model the evolution of consecutive attempts is 1/(1 (1 β)N ) 1. When there is a system without any independence assumption. The authors an attempt, the channel is allocated− − to station−i with probability in [3] study only the cases in which the objects have a finite (β(1 β)N−1)/(1 (1 β)N ), thus the mean reward during state space. − − − Mean field techniques in [3] consider N interacting objects a renewal cycle is N and a resource, where each object n has a state Xn (t), N−1 N N E[R] = (β(1 β) )/(1 (1 β) )pi,jLi,j, and the resource has a state R (t). The state of the system − − − Y N (t) = XN (t),XN (t), ..., XN (t),RN (t) is a homoge- where L is the packet size of a flow destined for station j 1 2 N i,j neous Markov chain on N , where = 1, 2, ..., l is and p is the probability that the packet transmitted belongs i,j the finite state space of XS N (×t) Rand =S 1, 2,{ ..., J is} the to the flow destined for station j. The mean length of a renewal n finite state space of RN (t). The processR Y N{(t) is called} mean cycle is field interaction model with N objects. (1 β)N Nβ(1 β)N−1 The mean field interaction model assumes that only the state E[X] = − + Tc 1 − 1 (1 β)N − 1 (1 β)N of an object can be observed, and not the object’s label n. N N −N − m − − !! Thus, the transition kernel of Y (t) is invariant under any β(1 β)N−1 i L X X i,k permutation of the labelingK of the N objects. The occupancy + − N pi,k + To , 1 (1 β) Ci,k N i=1 − − k=1 measure M (t) is defined as the vector of the proportions of objects in states i at time t with elements where mi is the number of flows handled by station i, To and ∈ S Tc are fixed overheads in time slots for a packet transmission N N 1 X and a packet collision, respectively, and Ci,k is the data rate M (t) = 1{XN (t)=i}. i N n from station i to station k. After canceling the term 1 (1 n=1 N − − β) , the throughput of the flow from station i to station j (per The state space of M N (t) is time slot) is obtained by (7). l It is shown in [8] that when each station handles a single l X ∆ = ~m R , mi = 1 and mi 0, i . flow and all stations have equal packets sizes, the total network { ∈ ≥ ∀ ∈ S} i=1 throughput Θ(β) is bounded above by the harmonic mean of the physical bit rates of the N flows: The authors prove that under some assumptions (Sec- tion II-C1) the occupancy measure converges to a deterministic 1 Θ(β) N min C (8) process as N . 1 1 i ≤ PN ≤ 1≤i≤N The intensity→ ∞(N) is defined as the expected number of i=1 N Ci transitions per object per time unit. The mean field occurs 1) An asymptotic analysis: The performance of the MAC when time is re-scaled by (N). If the intensity vanishes as layer is studied with backoff counters with mean bk = pbk−1 N , then the mean field limit is in continuous time. at backoff stage k, where p denotes the backoff multiplier of Otherwise,→ ∞ M N (t) converges to a discrete time process. We k the protocol. Hence, bk = p b0. As N , the number of study time-slotted systems in which the intensity vanishes → ∞ attempts of N 1 stations in each time slot can be modeled as N . Thus, the mean field is in continuous time. − as a Poisson random variable with mean (N 1)β. Thus, This model→ ∞ is applied to the MAC layer of the 802.11 − standard [9], [10], [6]. γ = Γ(β) = 1 e−(N−1)β. (9) − The drift f~N (~m,j) is defined as the expected change to It is shown that as K and N the attempt proba- M N in one time slot: bility per station behaves→ ∞like O(1/N→) [Theorem ∞ 7.2 [8]]. f~N (~m,j) = The performance of the MAC layer of the 802.11 standard N N N N as N can be studied using mean field techniques, which E M (t + 1) M (t) M (t) = ~m and R (t) = j → ∞ − | EDIC RESEARCH PROPOSAL 5

1) Assumptions of convergence to the mean field with a ~µ(τ), then a property of independence called propagation of vanishing intensity: Under some assumptions the re-scaled chaos holds for any finite number of objects at fixed time τ occupancy measure under the assumptions of Theorem 3.

N N M (t(N)) = M (t) for all t N Theorem 3. Assume that the initial values of the state of N ∈ the system XN (0),XN (0), ..., XN (0) are exchangeable and M (τ) is affine on τ [t(N); (t + 1)(N)] 1 2 N ∈ that the initial occupancy measures are such the assumptions converges to a deterministic process ~µ(τ). The re-scaled ver- of Corollary 1 hold. Then for a mean field interaction model N N sion Xn (τ) of Xn (t) is defined similarly. The assumptions with vanishing intensity are the following [3]: N N N N lim P X1 (τ) = i1, X2 (τ) = i2, ..., Xk (τ) = ik H1 The resource does not scale with N. If Kj,j0 (~m) N→∞

are the elements of the marginal transition matrix = µi1 (τ)µi2 (τ) µik (τ), N N ··· K (~m) for the resource, then limN→∞ Kj,j0 (~m) = 0 2 for any fixed k, τ. Kj,j0 (~m) exists for all ~m ∆, (j, j ) . The ∈ ∈ R matrix K(~m) is indecomposable for all ~m ∆ The theorem can be used for the following approximation: ∈ H2 Intensity vanishes at rate (N) such that N N N f~N ( ~m,j) lim P X1 (t) = i1,X2 (t) = i2, ..., Xk (t) = ik limN→∞ (N) = 0, and limN→∞ = N→∞ (N) f~(~m,j) exists for all ~m ∆, j . t t t µi1 µi2 µik . H3 If W N (t) is an upper bound∈ on the∈ numberR of objects ≈ N N ··· N that do a transition in time slot t, i.e. Theorem 3 states that if the coupled system is started such that N the initial values of the state of the system are exchangeable, X N 1{XN (t)6=XN (t+1)} W (t + 1), then as the system becomes large, any finite number of n n ≤ n=1 objects are almost independent. Thus, the mean field is the N 2 2 2 approximation of both the occupancy measure and the state then E W (t) c1N (N) , where c1 is a constant independent≤ of t, N. probability for one object at time τ. N 1 3) Decoupling assumption in the stationary regime: In H4 Kj,j0 (~m) is a smooth function of N and ~m for all (j, j0) 2. this paragraph we discuss whether the decoupling assumption H5 f~N (~m,j∈) Ris a smooth function of 1 and ~m, j . holds in the stationary regime of the system, as t . N ∀ ∈ R Generally, the decoupling assumption might not hold→ in ∞the The deterministic process ~µ(τ) satisfies the ODE stationary regime, because d~µ = F~ (~µ), (10) lim lim (M N (t)) = lim lim (M N (t)). dτ N→∞ t→∞ L 6 t→∞ N→∞ L PJ ~ with F~ (~m) = πj(~m)f(~m,j), where π(~m) is the invari- This occurs, for example, when the stationary distribution of j=1 N ant probability of the transition matrix K(~m), and f~(~m,j) is the process M (t) is unique due to irreducibility, but the deterministic process µ(τ) has a limit cycle as τ . defined in H2. Let Φτ (~m) be the solution of the ODE (10) If the process Y N (t) is an irreducible Markov chain,→ ∞ it has with initial condition ~m at time τ : N a unique stationary distribution η . We denote with PηN the dΦτ (~m) N N = F~ (Φτ (~m)) probability obtained when we initialize Y with η . In this dτ case, Y N (t) is a stationary sequence. Φ0(~m) = ~m. t∈N Corollary 2. [3] Assume that the ODE (10) has a unique Theorem 2. [3] For all T > 0 there exist constants C1(T ), ∗ N stationary point ~m to which all trajectories converge. Then C2(T ) and a random variable B (T ) such that under the stationary distribution ηN ,M N (0) converges in N N N ∗ N sup kM (τ) − Φτ (~m)k ≤ C1(T ) B (T ) + kM (0) − ~mk distribution and in probability to ~m . Furthermore, if η is 0≤τ≤T exchangeable, then and kBN (T )k2 ≤ C (T )(N). E 2 N N N lim PηN X1 (t) = i1,X2 (t) = i2, ..., Xk (t) = ik N→∞ = m∗ m∗ m∗ . i1 i2 ik Corollary 1. [3] If M N (0) ~m in probability [resp. in ··· → N The decoupling assumption holds in the stationary regime mean square] as N , then sup0≤τ≤T M (τ) ~µ(τ) 0 in probability [resp.→ ∞ in mean square], wherek ~µ(−τ) satisfiesk → if the ODE has a unique global stable point to which all the ODE (10) and ~µ(0) = ~m. trajectories converge. Thus, in order to ensure the validity of the decoupling assumption in the stationary regime, it is not Thus, as N the re-scaled occupancy measure con- sufficient to show that the fixed point equation F~ (~µ) = 0 has → ∞ verges to a deterministic process, which is the solution of an a unique solution, but it must also be shown that the ODE has ODE defined by the drift. a unique global attractor. 2) Mean field independence or Propagation of chaos: If 4) Application of the mean field method on the 802.11 the occupancy measure converges to the deterministic process standard: In order to investigate the performance of a fully- EDIC RESEARCH PROPOSAL 6 coupled system, Sharma et al. [9] used mean field techniques conclude whether the processes of the objects of the system to study the performance of the 802.11 MAC layer as the become independent in the stationary regime. Thus, mean field number of stations N tends to infinity. The state of the stations techniques can shed light on existing analytical models which represents the backoff stage from 0 to the maximum backoff assume independence on the objects of a system. stage m. There is no resource in this model. The specific assumptions of this model are: III.MODELING SATURATED 1901 NETWORKS • The backoff counters are assumed to be geometrically A. Performace analysis of the IEEE 1901 Backoff Procedure distributed with mean equal to CW/2, same as the mean Chung et al. [7] use the same assumptions mentioned in of the uniform distribution in the real protocol. Section II-A to analyze the performance of the HomePlug 1.0 • The backoff stage is reset to 0 only after a successful MAC layer, which is compliant with the 1901 standard. The transmission attempt. Thus, the retry limit K is infinite. probability of transmission is denoted by τ and the conditional This study does not make any independence assumption, in collision probability by p. Moreover, Chung et al. define a comparison with the Bianchi model which replaces the N- new conditional probability pb, which is the probability that dimensional chain of the system by N independent one- the medium is sensed busy in a time slot conditioned on the dimensional Markov chains for each station in the system. Due event that the station is not transmitting. The model consists to the geometric distribution of the backoff counters, stations of a tridimensional process (BPC(t),DC(t),BC(t)), where attempt to transmit with probability pi = 2/CWi in a given BPC(t),DC(t),BC(t) denote the station’s backoff proce- slot, where i denotes the backoff stage of the station. dure counter, deferral counter and backoff counter at time slot The transmission probability is scaled to pi/N and the t N, respectively. Since the backoff procedure counter is model has an intensity (N) = 1/N. The vanishing intensity increased∈ by 1 immediately after the selection of the backoff prevents the saturation of the aggregate transmission probabil- counter, BPC takes the values 1, 2, 3, 4 in this model. The ities and the throughput. This model satisfies the assumptions contention window and maximum deferral counter values are of Section II-C1. It is shown in [9] that the re-scaled process denoted by CWi−1 and Mi−1 when BPC is i. N M (τ) converges to a deterministic limit ~µ(τ) which is the The Markov chain is shown in Figure 3. The transition from solution of the equations: state i, j, k to state i, j, k 1 occurs with probability 1 { } { − } − pb. After having sensed the medium busy, the station must dµ0 = µ0p0 + β(~µ)(1 γ(~µ)) select a new backoff counter if the deferral counter is 0. Thus, dτ − − if the station senses the medium busy, then the transition from dµi = µipi + µi−1pi−1γ(~µ), for 1 i m 1 (11) state i, j1, k1 , to either the state i, j1 1, k1 1 or the dτ − ≤ ≤ − { } { − − } state i + 1, j2, k2 occurs depending on whether j1 = 0 or dµm { } 6 = µm−1pm−1γ(~µ) µmpm(1 γ(~µ)), j = 0. The new values j of the deferral counter and k of dτ − − 1 2 2 the backoff counter depend on i according to Table II. Pm −β(~µ) where β(~µ) = µipi and γ(~µ) = 1 e . It is proven i=0 − The probability pb is the probability that a station senses in [9] that the fixed point equation F~ (~µ) = 0 has a unique the medium busy conditioned on the event that the station is ∗ solution ~µ and that for maximum backoff stage m = 1, not transmitting. Hence, M N (t) converges to ~µ∗ from all possible initial values. N−1 pb = 1 (1 τ) . (13) In [10] the authors study the same mean field model with − − retry limit K equal to the maximum backoff stage, as in [8]. The Markov chain is homogeneous and ergodic, thus there N ∗ It is shown that M (t) converges to the solution ~µ of the is a unique stationary distribution P . The authors do not ~ i,j,k fixed point equation F (~µ) = 0 from all possible initial values, present a fixed point equation for the collision probability. if pi 1, for 0 i K. In this case, the stationary points P , τ, p and p are computed using a numerical method ∗ ≤ ≤ ≤ i,j,k b ~µ of the ODE are from the equations of the stationary distribution given in [7], γi 1 (13), (2), µ∗ = , 0 i K i K γi for pi P ≤ ≤ 4 Mi−1 4 Mi−1 CWi−1−1−j i=0 pi X X X X X τ = Pi,j,0, and Pi,j,k = 1. where γ is the solution of i=1 j=0 i=1 j=0 k=0 −β γ = 1 e p τ − Throughput is computed as in (3) after obtaining and . PK γi (12) Although analytical results are accurate when compared to β = i=0 . PK γi simulation results, it is not shown whether the system of i=0 pi equations has a unique solution. Finally, when the number of backoff stages m is infinite 1) Mean MAC Delay: The mean MAC delay is computed −i and pi = 2 p0, it is shown in [6] that if p0 < ln 2, then the recursively by computing the mean remaining MAC delay after ODE equation is globally stable as m . a station changes its backoff stage. At backoff stage i, the → ∞ Mean field techniques are used to study the evolution of station chooses a new backoff counter. Depending on Mi−1, a stochastic system through a deterministic process. If the on the backoff counter value and on the number of time slots intensity vanishes as N , the deterministic process during which the station senses the medium idle, there are is the solution of an ODE.→ By ∞ studying the ODE, we can three cases of transmission: EDIC RESEARCH PROPOSAL 7

1 p − Case 1 CW0 1) D (l) = lσ +(k l)Tb +(1 p)Ts +p(Tc +Di+1), i − − 1 i m, 0 l k Mi−1, 1-p 1-p b 1-pb b ≤Case≤2 ≤ ≤ ≤ 1,0,0 1,0,1 ... 1,0,CW0-2 1,0,CW0-1 2) D (l) = lσ + (M + 1)T + D , 1 i m, p i i−1 b i+1 b pb CW ≤ ≤ p 1 CW1 pb 0 l k Mi−1 1, CW1 CW1 ≤Case≤3 − − 1-p 1-p 3) D (l) = lσ +(k l)Tb +(1 p)Ts +p(Tc +Di+1), b 1-pb b i 2,1,0 2,1,1 ... 2,1,CW1-2 2,1,CW1-1 − − 1 i m, k Mi−1 l k, pb ≤ ≤ − ≤ ≤ pb pb where σ is the duration of a time slot, Ts, Tc are the duration of 1-p 1-p b 1-pb b a successful packet transmission and a collision respectively, 2,0,0 2,0,1 ... 2,0,CW1-3 2,0,CW1-2 pb pb and Tb is the mean duration of a time interval in which the CW2 CW2 p p b CW medium is sensed busy. The medium is sensed busy due to CW 2 2 1-p 1-p a successful transmission or a collision of the other N 1 b 1-...pb b 3,3,0 3,3,1 3,3,CW2-2 3,3,CW2-1 stations. Thus, − pb pb pb (N − 1)τ(1 − τ)N−2 (N − 1)τ(1 − τ)N−2 1-p 1-p T = T +T 1 − . b 1-pb b b s c 3,2,0 3,2,1 ...... 3,2,CW2-3 3,2,CW2-2 pb pb p pb CW3 pb pb The mean MAC delay after the station changes the backoff 1-p 1-p b 1-pb b stage to i is 3,1,0 3,1,1 ... 3,1,CW2-4 3,1,CW2-3

pb Mi−1 k pb pb 1 X X Case 1 Case 1 pb 1-pb 1-pb Di = P (l)D (l) 1-pb CW3 i i 3,0,0 3,0,1 ... 3,0,CW2-5 3,0,CW2-4 CWi−1 p pb k=0 l=0 CW CW3 3 CWi−1−1 k−Mi−1−1 1 X X Case 2 Case 2 1-pb 1-p 1-pb 4,15,0 4,15,1 ... b 4,15,CW -2 4,15,CW -1 + Pi (l)Di (l) 3 3 CWi−1 p k=M +1 l=0 CW pb i−1 3 pb pb CWi−1−1 k 1-pb 1-p 1-pb 1 4,14,0 4,14,1 ... b 4,14,CW -3 4,14,CW -2 X X Case 3 Case 3 3 3 + Pi (l)Di (l) CWi−1 p . . k=Mi−1+1 l=k−Mi−1 CW pb pb 3 pb pb ··· = f(Di+1). 1-p 1-p b 1-pb b 4,1,0 4,1,1 ... 4,1,CW3-16 4,1,CW3-15 p The mean MAC delay when a packet is at the head-of- pb CW3 p p b b the-line for transmission is equal to D1. D1 is computed 1-p 1-p b 1-pb b 4,0,0 4,0,1 ... 4,0,CW3-17 4,0,CW3-16 recursively since Di = f(Di+1) and Dm+1 = Dm, where pb pb pb m CW3 CW3 CW3 is the backoff stage such that the contention window is m−1 CWmax = 2 CWmin. Analytical formulas of the results Fig. 3: Dicrete Markov chain model of the 1901 standard can be found in [7]. MAC layer. The transitions due to collisions are shown with IV. CONCLUSIONANDRESEARCHPROPOSAL dashed lines. The minimum contention window is CW0 and i CWi = 2 CW0. The Markov chain models the process Both MAC layers of the IEEE 1901 and 802.11 standards (BPC(t),DC(t),BC(t)). for power line and wireless communications, respectively, use CSMA/CA mechanisms. We studied analytical models of the 802.11 standard in order to apply similar techniques on the 1) The station chooses a backoff counter k such that k MAC layer of the 1901 standard. The first two methods (the ≤ Mi−1, and attempts to transmit at backoff stage i. Bianchi model and the fixed point analysis by Kumar et al.) 2) The station chooses a backoff counter k such that consider independence of the stations and a time-invariant k > Mi−1 and changes a backoff stage before making collision probability. a transmission attempt due to Mi−1 + 1 transmissions Mean field techniques do not rely on any independence by other stations. assumptions and can be used to study the performance of 3) The station chooses a backoff counter k such that k > the system as the number of stations N gets large. The Mi−1 and makes a transmission attempt at backoff stage intractable Markov chain model of the system is studied i. through a determinisic process, which is the solution of an ODE. By studying the ODE we can conclude whether the The probability that l slots among k slots are sensed idle for stations become independent in the stationary regime of the each case is system (decoupling assumption). Case 1 k l k−l 1) P (l) = (1 pb) p , 1 i m, i l − b ≤ ≤ The analytical model [7] studied for the 1901 standard 0 l k Mi−1, approximates well the MAC throughput and delay, but whether ≤Case≤2 ≤ Mi−1+l l Mi−1+1 2) P (l) = (1 pb) p , 1 i m, i l − b ≤ ≤ the decoupling assumption holds in the stationary regime is not 0 l k Mi−1 1, studied yet. The authors do not present a fixed point equation ≤Case≤3 − k − l k−l 3) P (l) = (1 pb) p , 1 i m, i l − b ≤ ≤ for the collision probability, thus neither the uniqueness of the k Mi−1 l k. − ≤ ≤ fixed point nor the validity of the decoupling assumption is The MAC delay for each case is computed as studied for this model. EDIC RESEARCH PROPOSAL 8

The differences between the two protocols raise questions probability to access the channel than the other stations. such as which one performs better and under which conditions. In order to study the performance of the system as N gets We built two simulators in Matlab in order to evaluate the large, we propose to use mean field techniques and a similar performance of the protocols. Analytical results of [5] and [7] method as the method presented in [6], [9] for the 802.11 were compared with our simulation results and were found standard. We propose to study the model in discrete time since, to be quite accurate. Performance metrics such as throughput, as discussed above, collisions have a great impact on the short- delay, collision probability and fairness were studied for both term fairness of the protocols. The state of a station of an IEEE protocols. However, we focused our analysis on fairness which 1901 standard model consists of two processes: the backoff depends only on the backoff process and not on the system stage and the deferral counter. It is an open question whether parameters such as the duration of time slots. the decoupling assumption holds in the stationary regime of Since the stations with deferral counter 0 increase their the 1901 system. This question can be answered by studying window and select a new backoff counter whenever they the ODE of the mean field model. sense the medium busy, these stations have lower probabil- After evaluating analytically the performance of the 1901 ity to access the channel than the stations that transmitted network as described in this proposal, we intend to study successfully a packet in the 1901 standard. In the 802.11 the performance of the standard in multi-hop topologies. standard, the stations just resume their backoff counters after Depending on the results of our performance evaluation, we the end of a sensed transmission in the channel and thus, have may propose enhancements of the MAC layer protocol by larger probability to access the channel than the station that proving their superiority over the existing protocol analytically, just transmitted. The probability to access the channel should in simulation and by real network experiments. be less for the station that just transmitted successfully to Finally, after studying IEEE 1901 networks we intend to ensure short-term fairness in the system. Short-term fairness study hybrid wireless and power line technologies, since is important, since unfair systems may hurt delay sensitive combining both technologies increases capacity and coverage applications. in home networks. IEEE is going to publish a new standard We analyzed the backoff process of both protocols for called P1905.1 for multiple home networking technologies, N = 2 using continuous-time backoff counters uniformly among which are wireless and power line communications. distributed over the range [0,CW ]. The authors in [4] define The IEEE P1905.1 standard introduces a software layer be- the number of intertransmissions as a short-term fairness tween layers 2 and 3 that abstracts the individual details of metric. Suppose that we tag a station M. The number of in- each interface, aggregates available bandwidth, and specifies tertransmissions is the number of other stations transmissions end-to-end quality of service (QoS). between two transmissions of M. We proved that the 802.11 is fairer in the short-term than the 1901 for a network with 2 REFERENCES stations by showing that the distribution of intertransmissions [1] IEEE Standard for Information Technology-Telecommunications and for the 1901 standard has a longer tail, and much higher Information Exchange Between Systems-Local and Metropolitan Area mean and variance than the distribution for the 802.11 standard Networks-Specific Requirements - Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications. IEEE in [4]. We plan to validate our analytical and simulation results Std 802.11-2007 (Revision of IEEE Std 802.11-1999), pages C1–1184, for N = 2 with a HomePlug AV test-bed, which is compliant 12 2007. with the 1901 standard. [2] IEEE Standard for Broadband over Power Line Networks: Medium Access Control and Physical Layer Specifications. 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