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Performance Analysis of IEEE 802.11 DCF under Limited Load Yong Shyang Liaw, Arek Dadej, Aruna Jayasuriya Institute for Telecommunications Research, University of South Australia Yong.Liaw@postgrads.unisa.edu.au, Arek.Dadej@unisa.edu.au, Aruna.Jayasuriya@unisa.edu.au presents the model and its analysis. Section III presents The goal of effective distributed network control is that each numerical and simulation results. We conclude in Section IV. wireless node operates near its maximum capacity, with bounded queue delay. This “optimum” operating point, where maximum throughput is achieved, is usually located well below the II. LIMITED LOAD ANALYTICAL MODEL & ANALYSIS saturation point. The choice of optimum operating point requires knowledge of MAC performance below saturation, A. Analytical Model hence a limited load model of MAC operation needs to be studied We extend the discrete-time Markov model from [2] by for useful insights. In this paper, we extend the analytical model adding another state (i.e. E) to represent the node with empty of IEEE 802.11 DCF from [2] to study throughput performance queue, as shown in Fig. 1. The parameters used in the model under limited load, where the node’s transmit queue may be at are summarised in Table 1. Time is divided into slots of times empty. duration σ. A packet is discarded after m failed retransmissions. We assume that unsuccessful transmissions I. INTRODUCTION are solely due to collisions (perfect channel). The model does The performance of IEEE 802.11 DCF MAC has been well not account for packet capture effect. Similarly as in [2], a studied under the saturation conditions, where each node key assumption in this model is that collisions happen with the always has packets in its transmission queue [2, 3, 4, 7]. In [2], same probability p regardless of the number of retransmission an analytical Markov model is used to study the IEEE 802.11 attempts. DCF under saturation. Alternatively, [3, 4] use a model based When a packet arrives in state bE,, the system enters with on average values to provide a closed-form solution for transition probability pe0/W0 one of the back-off states b0,i. The saturation throughput. These works provide useful insights pe0 depends on the packet arrival distribution function. At ith into the IEEE 802.11 performance and in performance tuning back-off stage, the back-off timer assumes a random value k of the MAC protocol [7, 8]. However, all are based on the chosen from {0, 1, 2, …, Wi-1} with uniform distribution, assumption of saturation, and fail to capture the throughput where Wi is given by (1). Each time slot, the back-off timer is behaviour below saturation. decremented by 1, and when zero is reached, the packet is A robust analytical model is required to study the DCF transmitted. When the transmission collides with another throughput under a full range of traffic loads. The benefits of transmission (with probability p), the next (i+1)th back-off such studies include traffic admission policies suitable for use stage is entered with probability p/Wi+1. There is a maximum of at individual nodes, under fully distributed network control. It m back-offs. At mth failed retransmission, the packet is is with this in mind that our analytical model is built; we are discarded. After a successful transmission (with probability 1- particularly interested in expressing the throughput p) or after mth failed retransmission, the transmission queue is performance as a function of queue utilisation measurable empty with probability q and non-empty with probability 1-q. locally at every node. Furthermore, we extend our analysis to If not empty, the system enters state bi,k to serve the next packet account for the presence of hidden terminals [5, 10]. in the queue. When empty, the system enters state bE, and waits for a packet arrival. As soon as a packet arrives in state There are some works on finite load models for IEEE bE, the system enters b0,k to contend for the channel. 802.11 [6, 9, 11]. However, these models are either based on simplifying assumptions that limit their use near the optimum Wi = 2 i W0 (1) throughput point, or use variables not measurable locally at a node, hence are not suitable for distributed network control. B. Transmission probability in a slot The main contribution of our work is the limited load To solve for p, we balance the equations from the Markov Markov model of IEEE 802.11, and its analysis based on chain in Fig. 1 and obtain the following equations for k є {0, variables observable locally at each node. We offer insight …, Wi -1}. It shall be noted that y in Fig.1 denotes not a state, into the throughput behaviour of 802.11 over a wide range of but the sum of the probabilities of successful transmission and traffic loads. The paper is organized as follows. Section II discarding a packet, as given in (4). 1 m W i − 1 i Wi − k 1 − ( 2 p ) m +1 1 − p m +1 ∑ ∑0 p W =W0 1 − 2 p + 1 − p TABLE I. PARAMETERS FOR THE MODEL β = p Probability a transmission is unsuccessful/collided. 2 i=0 k = i q Probability an empty queue is found after a packet is served. bi,k State being in ith back-off stage, with back-off timer of value k. for 0 ≤ p < 1. bE State being in queue empty state. Pi,k Stationary probability being in state bi,k. The node will only transmit in state bi,0, hence the probability PE Stationary probability being in state bE. ptx that a node will transmit in an arbitrary time slot is the pe0 Transition probability from state bE to a non-empty state. probability of being in any of the states bi,0. W0 The initial maximum contention window. m m 1 − p m +1 Wi m Maximum contention window at ith back-off stage. The maximum number of retransmissions allowed. p tx = ∑ i=0 Pi , 0 =ε P0 , 0 where ε = ∑ i=0 pi = 1− p (8) q 1-Pe0 E We note that (6), (7) and (8) are independent of q and pe0, (1-q)/W0 i.e. independent of packet arrival distribution. So far, (6) and y Pe0/W0 Pe0/W0 Pe0/W0 Pe0/W0 (8) are equivalent to expressions obtained (in terms of P0,0) in [2] for a saturated system. 1-p 0, 0 0,1 ... 0,W0-2 0,W0-1 C. Solving for probability of collision, p p/W1 p/W1 p/W1 p/W1 1) No Hidden Terminals 1-p 1,W1-2 In a neighborhood with N identical nodes where every node 1, 0 1,1 ... 1,W1-1 can sense other nodes’ transmissions (no hidden terminals), a transmission results in a collision when any of the (N-1) p/W2 p/W2 p/W2 p/W2 neighbors also transmits in the same time slot. Eq. (8) must 1-p satisfy the constraint given by (9), and the system may be 2,1 2,W2-2 2,W2-1 2, 0 ... solved numerically for a given value of PE. p = 1 − (1 − p tx ) N −1 ⇒ p tx = 1 − (1 − p)1 /( N −1) (9) . 1-p . 2) Hidden Terminals . Let us assume that in addition to the (N-1) direct neighbors, p/Wm p/Wm p/Wm p/Wm there are, on average, Nht hidden terminals in the region, as in 1 m, 0 m,1 ... m,Wm-2 m,Wm-1 Fig. 2. A collision with a neighboring node can occur only when the nodes transmit in the same slot, otherwise the nodes would be able to sense busy channel. However, a hidden Figure 1. IEEE 802.11 Markov model for limited load terminal cannot sense our node, hence can begin the colliding transmission in any slot. Therefore, for a successful W0 − k P0 , k = ⋅ [ p e 0 . P E + (1 − q ) y ] (2) transmission, the N-1 neighbors must not transmit in the same W0 slot, and the Nht hidden terminals must not transmit during the entire transmission period. Wi − k for 0 < i ≤ m (3) Pi , k = . p i . P0 , 0 Let Ltx be the average number of slots in a transmission Wi period (ttx), and ξ be the average slot duration (i.e. average time m spent in a state). We categorise states in the Markov model in y= ∑ (1 − i=0 p ). Pi , 0 + p . Pm , 0 = P0 , 0 (4) Fig. 1 into 3 types, empty-queue state (EQ), transmission state (TX) and backoff state (BO). The average time spent in each PE ⋅ p e 0 = q ⋅ y (5) state type before a transition occurs is different. Let teq, ttx and tbo be the average times spent in states of type EQ, TX and BO With (4) and (5), (2) and (3) can be rewritten as (6). respectively (they are functions of p (and PE), given in the later W −k i (6) section). Then Ltx and ξ are given by (10), and ptx by (11). Pi , k = i . p . P0 , 0 Eq. (8) must satisfy constraints given by (10) and (11). Wi for i є {0, …,m}, k є {0,…, Wi -1} From the normalising condition and (6), we obtain an expression for P0,0 in terms of p and PE. C W i −1 m 2 (1 − PE ) A B ∑∑P i=0 k =0 i ,k + PE = 1 ⇒ P0 , 0 = β (7) B is a neighbor C is a hidden terminal Figure 2. Transmission in the presence of hidden terminal t tx period is the transmission period ttx, hence the average time L tx = spent in a BO state, tbo, is given by (15). ξ m W i −1 (10) t bo = 1 − Pn −1 + Pn −1 t tx ξ = PE t eq + p tx t tx + t bo ∑ (15) ∑P i = 0 k =1 i ,k Pn −1 = 1 − (1 − p tx ) N −1 α The average time taken to transmit a packet, ts, is the total = PE t eq + ( ε t tx + t bo ) p 0 , 0 2 of the transmission time and MAC access time, as given in W i −1 Wi − k (16), and the average transmission rate (µ) is 1/ts. However, 1 m 1 − ( 2 p ) m +1 1 − p m +1 α = 2 ∑∑ pi Wi =W 0 1− 2p − 1− p only (1-p) proportion of packets transmitted are successful, pdisc i=0 k =1 proportion are discarded after maximum number of retransmissions (m), and the rest (i.e. p-pdisc) result in a p = 1 − (1 − p tx ) N − 1 + L tx N ht (11) collision after which the packet is re-inserted into the queue. ⇒ p tx = 1 − (1 − p ) 1 /( N − 1 + L tx N ht ) Hence the effective packet service rate (µeff) is a sum of the rate of successful transmissions (µsucc) and the rate at which packets For uniform node distribution over a large area with node are discarded (µdisc), as in (17). For small p and m > 1, pdisc is density D, it can be shown that Nht is given by (12). Eq. (8), negligible, hence we may approximate µeff with µsucc. (10), (11) and (12) may be solved for p iteratively. We note that Nht = 0 is a special case without hidden terminals. t s = taccess + ttx (16) 3 3 2 3 3 (12) µ eff = µ succ + µ disc (17) N ht = R D = N 4 4π where µ succ = 1 − p , µ disc = p disc D. Packet Service Rate ts ts Next, we find an expression for µ, the packet service rate (rate at which packets are removed from the queue), in terms of p . Pm , 0 (1 − p ) p m + 1 PE and p. Let taccess be the average time spent backing off p disc = = P0 , 0 m 1 − p m +1 before a packet is transmitted (i.e. MAC access time), and ttx be ∑P i ,0 the packet transmission time (i.e. the slot time for TX state). i=0 Since at ith back-off stage the average number of back-off slots is Wi/2, and the system enters the next back-off stage with E. Throughput probability p, the average access time taccess is given by (13). For limited load where the arrival rate λ is less than µeff, the node’s throughput (Tnode) is the portion of traffic that arrived m Wi ∑ pi( 2 t bo ) φ W 0 t bo (13) minus the portion that is discarded i.e. λ*(1-Pdisc). To derive t access = i=0 = the throughput of a node, it is necessary to know the m 2ε distribution of packet service time. Here, we assume an ∑ i=0 pi idealized M/M/1/kq model for the transmission queue, hence the packet arrival rate (λ), throughput at a node (Tnode), and the m 1 − ( 2 p ) m +1 where φ = ∑ i=0 p iW i = 1− 2p total normalized throughput for a neighborhood of N identical nodes, Ttotal, are given by (18), where ρ is the traffic intensity. The transmission time ttx depends on the payload size and λ = ρ . µ eff the access mechanism (i.e. basic access or RTS/CTS). Note ⎧ ρ ⋅ µ succ for ρ < 1 (18) that the RTS/CTS mechanism combats the hidden-terminal T node = ⎨ problem, but amplifies the exposed-terminal problem, with a ⎩ µ succ for ρ ≥ 1 potentially greater performance degradation effect than hidden N ⋅ T node ⋅ PL terminal [10]. For this reason, we limit our analysis to the T total = basic access mechanism. For basic access mechanism, ttx is ChannelRat e given by (14), where H is the packet header, PL the payload For M/M/1/kq queue, ρ is related to the queue utilization as size, τ propagation delay, and tsucc and tcoll the times taken by in (19), where kq is the size of the queue buffer. It is noted that successful and unsuccessful transmissions respectively. the model in Fig. 1 uses time slot of varying duration, and the probabilities PE and Pi,k are relative frequencies of visiting the t tx = (1 − p ) t succ + pt coll corresponding states. Let SE and Si,k be the equivalent t succ = H + PL + SIFS + DIFS + ACK + 2τ (14) probabilities with respect to arbitrary time. They are given by (20), and specify the distribution of time the node spends in the t coll = H + PL + DIFS + τ corresponding states; in particular SE is the proportion of time At the beginning of a BO-slot, the node senses either an the queue is empty, hence (1- SE) is the queue utilization. idle channel with probability Pn-1, or a busy channel followed 1− ρ (19) by an idle slot with probability (1- Pn-1). The average busy SE = k +1 = 1−Uq 1− ρ q PEteq PEteq 0.9 SE = = m Wi α ⋅ tbo 0.8 PEteq + Ptxttx + ∑∑ Pi, k tbo PEteq + (ε ⋅ ttx + ) P0,0 i = 0 k =1 2 0.7 0.6 (20) Throughput Pi ,0 t tx Pi ,0 t tx 0.5 S i ,0 = = m Wi α ⋅ t bo 0.4 PE t eq + Ptx t tx + ∑∑ Pi ,k t bo N=4 PE t eq + (ε ⋅ t tx + ) P0,0 N=10 i =0 k =1 2 0.3 N=20 0.2 N=30 Pi ,k t bo Pi ,k t bo for i ≥ 1 N=40 0.1 S i ,k = = m Wi α ⋅ t bo N=50 PE t eq + Ptx t tx + ∑∑ Pi ,k t bo PE t eq + (ε ⋅ t tx + 0 ) P0,0 i =0 k =1 2 0 0.1 0.2 0.3 0.4 0.5 0.6 Queue Utilization 0.7 0.8 0.9 1 PE.teq is the proportion of time the queue is empty in an Figure 3. System throughput versus queue utilization average time slot. Eq. (20) gives a unique solution for any given value of λ, for 0 < teq < 1/λ. Then, for simplicity, a 0.9 solution is obtained when PE = SE, and teq is given by (21). 0.85 α ⋅ t bo 0.8 (ε ⋅ t tx + ) P0 , 0 S E = PE ⇒ t eq = 2 (21) 0.75 Throughput 1 − PE 0.7 0.65 F. Queue Delay 0.6 For M/M/1/kq queue, the queue delay is given by 0.55 Saturation throughput Max. throughput 0.5 ⎛ ρ ( k + 1) ρ k q +1 ⎞1 0 10 20 30 40 50 t queue = ⎜ − q ⎟ (22) No. of Nodes (N) ⎜1− ρ k +1 1− ρ q ⎟λ ⎝ ⎠ Figure 4. Maximum and saturation throughput 0.3 III. SIMULATIONS & RESULTS N=10 N=20 0.25 Table II shows the default parameters used for both N=30 simulations and numerical evaluation. Uq is approximated by Queue Delay (seconds) N=40 0.2 ρ, since a buffer size (kq) of 60 packets is assumed, and Tnode is N=10 (simulation) N=20 (simulatoion) taken as equivalent to λ since the pdisc is negligible in (18). 0.15 N=30 (simulation) 0.1 A. Without Hidden Terminals Fig. 3 shows the total system throughput (Ttotal) in respect to 0.05 queue utilization (Uq) at the node. The point of maximum throughput is at the saturation point (i.e. Uq = 1) only for less 0 0 1 2 3 4 5 6 7 8 9 10 than 10 nodes in the neighborhood. For N ≥ 10, the maximum Node Throughput (pps) throughput is achieved below the saturation point (i.e. Uq < 1). Figure 5. Queue delay The maximum throughput point shifts towards lower Uq with increasing N. This result agrees with the results in [11]. 0.6 N=4 Fig. 4 shows that maximum throughput degrades slower 0.5 N=10 than the saturation throughput, and approaches an asymptotic N=20 value with large N. Significant benefits can be obtained 0.4 N=30 prob of collission N=40 N=50 0.3 TABLE II. VALUES FOR SIMULATION & NUMERICAL EVALUATION Payload size (PL) 8184 bits 0.2 MAC header 272 bits PHY header 128 bits 0.1 ACK 112 bits + PHY header Channel Bit Rate 1 Mbit/s 0 Propagation delay (τ) 1 µs 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Node Throughput (pps) Slot time (σ) 20 µs SIFS 10 µs Figure 6. Transmission collission probability DIFS 50 µs Initial contention window (W0-1) 31 through distributed control of queue utilization at each node. Maximum number of retransmissions (m) 5 Uq is very sensitive to the traffic arrival rate near the conditions, in respect to queue utilization before saturation. maximum throughput point. A small increase in λ will cause a We have shown that for large numbers of nodes the maximum significant increase in Uq, and consequently a dramatic increase throughput is achieved well before the saturation, and have in queue delay as shown in Fig. 5. This can be explained by validated our model via simulations. Our model offers a useful sharp increase in collision probability when the throughput method of estimating the optimum operating point for the exceeds certain threshold, as illustrated in Fig. 6. As a result, purpose of distributed network control, based on Uq. operating at saturation point is not desirable as it dramatically m =5 increases queue delay and reduces throughput for larger 0.9 numbers of nodes. For maximum throughput with low delay, 0.8 the “optimum” operating point for IEEE 802.11 MAC is at the 0.7 “knees” of the delay curves shown in Fig. 5. Normalised Throughput 0.6 0.5 N=4 (Simulation) B. Hidden Terminals N=10 (Simulation) 0.4 N=20 (Simulation) 0.6 N=30 (simulation) 0.3 N=4 (Analytical) 0.5 0.2 N=10 (Analytical) N=20 (Analytical) 0.1 0.4 N=30 (Analytical) Throughput 0.0 0 5 10 15 20 25 30 35 0.3 Arrival Rates (pps) N=4 0.2 N=10 Figure 8. Throughput performance of IEEE 802.11 N=20 N=30 0.1 N=40 TABLE III. ESTIMATED QUEUE ULTILIZATION N=50 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 N Uq (Analytical) Uq (Simulation) Queue Utilization 10 0.33±0.04 0.31±0.09 20 0.17±0.04 0.18±0.05 Figure 7. Throughput performance in the presence of hidden terminals 30 0.09±0.04 0.082±0.029 Fig. 7 depicts throughput performance of the IEEE 802.11 in the presence of hidden terminals. 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[11] C. L. Fullmer, et al, "Solutions to Hidden Terminal Problems in IV. CONCLUSIONS Wireless Networks", Proc. of ACM SIGCOMM 97, Sept 1997. We have presented an analytical model to study the performance of IEEE 802.11 DCF under a full range of load