Analytical Models for Understanding Misbehavior and MAC Friendliness in CSMA Networks

Transcription

1 Analytical Models for Understanding Misbehavior and MAC Friendliness in CSMA Networks Zhefu Shi 1,CoryBeard 1,KenMitchell 1 University of Missouri-Kansas City Abstract Nodes using contention-based CSMA protocols are susceptible to the misbehavior of other nodes and also have little flexibility in controlling quality of service. To address the misbehavior problem, continuous-time protocols are proposed. The effects of misbehavior on the QoS of all nodes in the system caused by changing the cheater s arrival rate and backoff rate are studied. The problem of flexibility in controlling QoS is addressed by introducing the concept of MAC friendliness where nodes can adjust arrival rates and backoff rates while maintaining a fixed share of the channel. The continuous-time system is modeled using an iterative method and matrix exponential distributions. Collision probabilities are determined both by the channel utilization of the entire system, as well as the actual stage within the backoff process. The model itself is a key contribution because it is accurate over all ranges of traffic loads and models both queueing within nodes and contention for the channel. Key words: CSMA, QoS, matrix exponential, MAC Layer 1. Introduction Carrier-sense multiple-access with collision avoidance (CSMA/CA) protocols rely on the random deferment of frame transmissions, e.g., binary exponential backoff, in a shared wireless channel network. These types of MAC protocols are widely used and are being proposed in many new protocols, such as 4G wireless systems, ad hoc networks, etc. Generally, it is assumed that all nodes obey the protocol rules. CSMA provides cooperation among nodes, and is enforced by creating a standard approach that all nodes are expected to use or by having a base station that controls how nodes access a channel. Since the network adapter is programmable, however, as shown in [1], users can easily misbehave (not follow the rules) if the network adapters are reprogrammed. All types of users will utilize CSMA networks, including emergency users. But emergency users may experience two potential problems. First of all, emergency users might not be able to obtain acceptable quality of service (QoS) if there are too many nodes vying for a channel. Secondly, if several nodes do not follow the rules of the protocol, the QoS of all users, even emergency users, could be degraded [2]. This paper does not intend to solve the problem of priority access, but first addresses fairness and seeks to understand how a misbehaving node can hurt system performance. To this end, there are 4 major contributions to this paper Use of continuous-time protocols First, this work proposes the use of continuous-time protocols because of their benefits over slotted-time protocols regarding reduced vulnerability to misbehavior. The model is presented in Section 2. Slottedtime protocols do have advantages over continuous-time protocols in some respects, but in continuous-time protocols channel utilization does not go to zero when two or more nodes are misbehaving [2]. This work is supported by the National Science Foundation under CAREER Award ANI addresses: zskq4@umkc.edu (Zhefu Shi), beardc@umkc.edu (Cory Beard), mitchellke@umkc.edu (Ken Mitchell) Preprint submitted to Elsevier January 6, 29

2 In [2], it was shown for a slotted-time systems that a misbehaving node (cheater) can gain full channel utilization. By setting the cheater s contention window size to W = 1, it can cause the throughput of all other nodes to go to zero. If there are two misbehaving nodes, the throughput of everyone goes to zero, including the rogue nodes. This result is known as the tragedy of commons in economics. This is the case for slotted time systems, like IEEE We propose, therefore, that continuous-time protocols be used instead by taking advantage of the fact that no two events happen at the same time in a continuous-time protocol. Even though the current IEEE protocol uses slotted time, continuous-time protocols could be considered for ad hoc, sensor, or even future versions of networks for the contention-based CSMA point of view. In such cases, backoff times and channel sensing occur at any time, not just at time slot boundaries. But is there a loss of throughput when using continuous time? Kleinrock and Tobagi [3] have proven for a nonpersistent CSMA network (similar to the work here), both for continuous and slotted time, that both G 1+G have the same throughput S = when the propagation delay, a. Note that G is the offered channel traffic rate. If the propagation delay is included, then there is a difference depending on the propagation delay, but in our case the propagation delay as compared to average service time is very small and can be considered negligible. Figure 4 later shows simulation results that also show that the difference is very small between slotted and continuous time systems. Also note that the continuous-time approach is also less costly otherwise, because synchronization of time slots is not required Accurate over all offered loads Next, a new model of continuous-time CSMA systems is provided that is accurate over all ranges of potential offered loads. Equations and figures are presented in Sections 2.1 and2.2 to show this. In this paper, we use matrix exponential distributions to model the complete binary exponential backoff and queueing process, not only with respect to throughput, but also other QoS metrics by using an M/ME/1 model for analysis. We have formulated a continuous-time CSMA model that uses an iterative method and matrix exponential distributions [4] to model both queueing locally at each node and the channel contention. The backoff process is modeled as an exponential random variable with multiple backoff stages. The accuracy of the model is confirmed by comparing with simulation results. An important attribute of this model is the ability to study CSMA performance at all traffic loads, instead of commonly assumed saturated conditions like those in [5]. That means we allow the transmission queues of some nodes to be empty. Regarding the assumptions we use, Boorstyn [6] similarly used a continuous-time Markov model to develop a product form solution to analyze the throughput of arbitrary topology multihop packet radio networks that use the CSMA protocol with perfect capture and zero propagation delay. In this paper, we use the same assumptions. In [6], CSMA is modeled as a Markov process. We extend this to multiple backoff stages and exponentially changing backoff rates. Our work assumes exponential backoff and arrival rates. In [3], it has been shown only the first moment of the retransmission delay distribution has a noticeable effect on the average throughput-delay performance. So in this paper, an exponential distribution is used for the backoff process. Matrix exponential distributions also allow for modeling of different arrival processes. In our ongoing work [7], we have studied GI/GI/1 models and have shown there to be little difference from the M/G/1 results here. Regarding the accuracy of our model, Bianchi [5] first proposed a two dimensional discrete-time Markov Chain to model the IEEE protocol, with many follow-up papers. The key approximation in that model was that, at each transmission attempt, and regardless of the number of retransmissions suffered, each frame collides with constant and independent probability. One of the reasons for the accuracy of our model for continuous-time CSMA/CA is that we go beyond Bianchi s assumptions. We find the collision probability for node i at each stage g, δ i,g. We show that this is dependent on 1) the other nodes channel utilization, and 2) the backoff stage, g, for node i. For other work that is similar to ours, consider [8], which uses a fixed point method similar to our approach to determine packet loss, link utilization and TCP throughput across the network. Other previous modeling work has focused on slotted time protocols. In [9], Tickoo and Sikdar propose a G/G/1 model 2

3 which can account for arbitrary arrival patterns, packet size distributions and numbers of nodes for slotted time protocols; in [1], Alizadeh-Shabdiz and Subramaniam present an M/G/1 queueing model based on [5] for slotted time protocols. In [11], Medepalli and Tobagi found that the binary exponential backoff algorithm offers negligible gains once contention window size is appropriately set in IEEE 82.11; also in [12] and [13], Medepalli and Tobagi present a simple M/M/1 analytical model for IEEE which addresses network performance characteristics such as throughput, delay and fairness Investigation of Misbehavior The third contribution of this work is the analysis of misbehavior and consequences for QoS by 1) increasing the arrival rate, λ i, to the MAC layer, 2) adjusting the backoff rate, b i, and 3) adjusting both λ i and b i. Misbehavior analysis is the subject of Section 3 and Section 4. We demonstrate that in a continuous model, channel utilization is monotonically increasing to a value less than 1 with the increasing of arrival rate, λ i. When increasing backoff rate, the channel utilization first increases, then asymptotically decreases to approach a value greater than. Therefore, increasing backoff rate might not be helpful. The conclusions on the effect to delay, loss and jitter by changing λ i and b i are also presented. By using the model developed in this paper, we show that no nodes, either the misbehaving nodes or the regular nodes, will have their channel utilization decrease to zero in continuous-time systems, and no misbehaving nodes can occupy the whole channel either. We assume these nodes are rational, and not malicious (they misbehave only when they can benefit from it). Each node also has flexibility to adjust its arrival rate and backoff rate, so if one node misbehaves, the others could also misbehave in response. In a continuous-time system, the effect of misbehavior can vary and the response to misbehavior can vary. Regarding related work, to address problems of unfairness, Kar [14] presented a similar model as [6] and focused on fairness issues regarding throughput with the same assumption as [6]. In [15], Garetto, Salonidis and Knightly develop an analytical model that incorporates this lack of coordination (i.e., all nodes not following the same rules to access the channel), identifies dominating and starving flows and accurately predicts per-flow throughput in a large-scale network MAC Friendliness The fourth contribution of this paper is that it proposes a new method for sharing the channel in a flexible yet fair manner according to what we call MAC Friendliness. This is presented in Section 5. Analogous to the idea of TCP-friendliness where nodes are expected to provide the same impact to a network even if they behave differently, we introduce MAC friendliness to CSMA protocols where nodes can behave in different ways but still have the same impact on the channel. They can balance their requirements for loss and delay by changing backoff rates and arrival rates, but their impact on the channel remains the same because they can keep their channel utilization constant or below a required value. Regarding our approach of nodes adapting their backoff rates to channel conditions, in [16], Cantieni, Ni, Barakat, Turletti analyze the throughput degradation caused by automatic rate adaptation in CSMA/CA wireless networks. 2. Matrix Exponential Analytical Model In Wireless Local Area Networks (WLANs) and wireless ad hoc networks, in order to guarantee that each node which contends for the channel can gain access, Carrier Sense Multiple Access (CSMA) protocols are proposed. In this section, we present an analytic model combined with an iterative solution method. We specify the following notation first: 1) λ i. The frame arrival rate to the MAC layer is exponentially distributed with rate λ i at Node i. It is assumed that users have control over this arrival rate to the MAC layer either by modification of application parameters (e.g., compression) or with frame dropping policies before frames are to be sent, for example using active queue management. More advanced and realistic arrival distributions than the exponential distribution can easily be added to the model later using Matrix Exponential methods [4] that can approximate many arrival distributions. 3

4 b i λ i 1,b,1 δ i,1 b i 2 bi G 1 δ i,g 2,b,1.... N,b,1 r N S i 1,b,2 b i δi,g G ,b,2. 2 δ ) (1 i,2 b i (1 δi,1) 1,b,G 2,b,G N,b,G b i ( 1 δ i,g ) S i 2 G 1 1,S 2,S.... N,b,2. N,S (a) Markov chain with queueing of the multi-stage back off process. r 1 chl, chl,1... chl,n s s (b) Markov chain of channel. Figure 1: Markov Chains for Single Node and Channel 2) b i. The backoff rate is exponentially distributed with rate b i at the first stage, b i /2 at the next stage, and so on, where i is the node id. Random backoff in the MAC layer could be implemented according to any random variable, but it is advantageous here to use the exponential random variable to facilitate analysis. 3) s i. The frame service rate for Node i is exponential with rate s i. In the simulation, we set all s i equal to each other. 4) δ i,g. The probability that the channel is busy when one node finishes the current backoff stage and tries to access the channel is defined as the collision probability. Different backoff stages have different collision probabilities which are denoted as δ i,g for each node. Here i is the node id, g =1 G is the backoff stage. 5) Q i. We will discuss finite queue and infinite queue cases, each one separately. The queue length at the i th node is Q i. Now we describe the matrix exponential (ME) distributions we use. The matrix B is the service rate matrix of a matrix exponential distribution which is a family of distributions having rational Laplace- Stieltjes transforms [4]. The density function is f(t) =pbe Bt ε and the corresponding distribution function is F (t) =1 pe Bt ε,where,pis the starting vector and ε is a summation operator (usually a column vector of all 1 s). Fig. 1(a) illustrates the Markov chain for one node using the CSMA process modeled here. Queue fill increases horizontally to the right, and backoff stages increase downward. We use matrix exponential methods to characterize the general service process at each queue fill (each column in Fig. 1(a)), which makes the model M/ME/1. The operation of the model is illustrated, for example, by considering state q,b,g. In this state, q frames are in the system, the node is in backoff mode, and it is in the gth stage of backoff. The node transfers to the sending state q,s if when it finishes backoff the channel is idle. The 1) service matrix B i and the 2) leaving matrix L i for a G stage exponential backoff process for the i th node are B i = b i b i δ i1... b i (1 δ i1 ) b i 2 bi 1 2 δ 1 i2... bi 2 (1 δ 1 i2 ) , L i = b... i bi (1 δ 2 G 1 2 G 1 ig ) s (1) b i 2 δ G 1 i,g s In this paper, all the nodes are in the transmission range of the others. Follow-up work to this paper will present results for multi-hop scenarios. Similar to [6] and [14], the propagation delay is assumed to be 4

5 zero in our model. The RTS and CTS transmission times can be ignored. Acknowledgements are obtained instantaneously. Transmission interference is negligible. When one node is sending a frame, it is not affected by another node. If another node finishes its backoff process and intends to send a frame, it only listens to the channel, but does not transmit, hence it does not interfere with the active transmission. In this case, the second node starts a new backoff stage or drops the current frame L i,s = , L i,d = (2) b i 2 δ G 1 i,g s Analytical queueing model and solution We describe an analytical queueing model using an iterative solution method. 1) The standard Matrix Exponential M/ME/1 solution for the steady state probability at queue fill q for Node i is π i,q = π i, p(λ i (λ i I + B i λ i εp) 1 ) l, (3) [ ] [ ] T p = 1, ε = ) Define π i,s, the probability that Node i is in a sending state, as that node s channel utilization, Q i [ ] T π i,s = π i,l ε 1, ε 1 = 1. (4) l=1 This is the steady state probability of being in a sending state for the i th node. Assume all s i are equal. For a backoff process with only a single stage, we have δ i,1 = N j=1,j i π j,s. (5) For a multi-stage backoff process, the collision probability for stage g>1 takes into account the likelihood of colliding again with the same frame as before, and is approximately The average collision probability is s δ i,g = δ i,1 s + b/2 g 1 + b/2g 1. (6) s + b/2g 1 δ i,avg = λ i(δ i,1 + δ i,1 δ i,2 + + δ i,1 δ i,g ) λ i (1 + δ i,1 + δ i,1 δ i,g 1 ) = δ i,1 + δ i,1 δ i,2 + + δ i,1 δ i,g 1+δ i,1 + δ i,1 δ i,g 1. (7) This equation is found by taking the ratio of total number of collisions to total number of attempts to use the channel per unit time. For stage 1, there are λ i attempts and λ i δ i,1 collisions. For stage 2, there are λ i δ i,1 attempts and λ i δ i,1 δ i,2 collisions. For the last stage, stage G, thereareλ i (δ i,1 δ i,g 1 ) attempts and λ i (δ i,1 δ i,g ) collisions. 3) Use step 2 to define δ i,g for every node so the collision probability for every node can be evaluated. 4) The δ i,g for each node is dependent on the π j,s of other nodes. This creates a set of mutually dependent equations. These equations can be solved exactly for a few nodes or with numerical methods for more nodes. 5

6 By reconstructing the B i matrix each time using the collision probability evaluated from the previous iteration step using the above equations, the queue fill distribution can be evaluated using the following equations. π i,q = π i p(λ i (λ i I + B i λ i εp) 1 ) q, <q<q i, (8) π Qi = π i pλ i B 1 i (λ i (λ i I + B i λ i εp) 1 ) Qi 1, (9) Q i π i + π i,q ε =1. (1) q=1 Fig. 2(a) through Fig. 5(a) show the channel utilization, mean system delay, loss ratio and collision probability evaluated by the theoretical model and simulation for the case of six nodes with a 3-stage exponential backoff process with changes in arrival rate and backoff rate. Simulation was performed using a custom built model in CSIM using the same assumptions as the analytical model. For the case of changing arrival rates, the arrival rates of all the nodes, λ i=1 6, are from 5 to 85 frames/sec, the backoff rates b i=1 6 = 4 attempts/sec, the sending rates s i=1 6 = 2 frames/sec (same for all nodes). For the case of changing backoff rates, the arrival rates are λ i=1 6 = 45, the backoff rates b i=2 6 = 4 attempts/sec, b 1 is from 1 to 17 occurences/sec, and the sending rates are s i=1 6 = 2 frames/sec Node1 Utilization, 3 level backoff, Simulation Node1 Utilization, 3 level backoff, Analysis Node1 Utilization, 1 level backoff, Simulation Node1 Utilization, 1 level backoff, Analytisis 3 25 Node1 Delay, 3 level backoff, Simulation Node1 Delay, 3 level backoff, Analysis, Infinite Node1 Delay, 3 level backoff, Analysis, Finite Node1 Delay, 1 level backoff, Simulation Node1 Delay, 1 level backoff, Analysis, Infinite Node1 Delay, 1 level backoff, Analysis, Finite 1.8 Node1 Loss, 3 level backoff, Simulation Node1 Loss, 3 level backoff, Analysis Node1 Loss, 1 level backoff, Simulation Node1 Loss, 1 level backoff, Analysis Channel Utilization Mean System Delay (ms) Loss Arrival rate (frames/second) (a) Channel utilization with change of arrival rate λ Arrival rate (frames/second) (b) Mean system delay with change of arrival rate λ Arrival rate (frames/second) (c) Loss ratio with change of arrival rate λ 1. Figure 2: Change of channel utilization, mean system delay and loss ratio with change of arrival rate λ 1. The simulation results match the analytical results very closely for all performance metrics. A key benefit of this model is also seen in that the analytical results match with the simulation results for all ranges of traffic loads. This is in contrast to the throughput model in [5] which only addresses constant backlog conditions (i.e., high loads). Fig. 2(a) and Fig. 2(c) show the advantage gained by increasing the number of backoff stages, but Fig. 2(b) shows the cost incurred through increased delay. Fig. 2(a) also shows how channel utilization increases with increasing arrival rate, which is an important topic of consideration for our misbehavior analysis later in the paper. Fig. 3(a) and Fig. 3(c) show the benefits of increasing backoff rate, but that if backoff rate becomes large beyond a certain point, then performance degrades. Fig. 3(b), however, shows that delay keeps decreasing no matter how large the backoff rate becomes. To show the impact of the model assumptions, Fig. 4 shows plots the same as Fig. 2(a) but with different probability distributions for the backoff process and also comparing discrete versus continuous-time processes. In all cases, the results are very close. And finally, Fig. 5(a) shows how the collision probability at each stage changes, which will also be important in our misbehavior discussion. 6

7 .3.25 Node1 Utilization, 3 level backoff, Simulation Node1 Utilization, 3 level backoff, Analysis Node1 Utilization, 1 level backoff, Simulation Node1 Utilization, 1 level backoff, Analytisis 1 8 Node1 Delay, 3 level backoff, Simulation Node1 Delay, 3 level backoff, Analysis, Infinite Node1 Delay, 3 level backoff, Analysis, Finite Node1 Delay, 1 level backoff, Simulation Node1 Delay, 1 level backoff, Analysis, Infinite Node1 Delay, 1 level backoff, Analysis, Finite 1.8 Node1 Loss, 3 level backoff, Simulation Node1 Loss, 3 level backoff, Analysis Node1 Loss, 1 level backoff, Simulation Node1 Loss, 1 level backoff, Analysis Channel Utilization Mean System Delay (ms) 6 4 Loss Backoff Rate (occurence to access the channel/second)) (a) Channel utilization with change of backoff rate b Backoff Rate (occurence to access the channel/second)) (b) Mean system delay with change of backoff rate b Backoff Rate (occurence to access the channel/second)) (c) Loss ratio with change of backoff rate b 1. Figure 3: Change of channel utilization, mean system delay and loss ratio with change of arrival rate b 1. Throughput in a 7 node 6 path network Throughput in a 7 node 6 path network Throughput in a 7 node 6 path network level backoff, exponential 3 level backoff, gamma 3 level backoff, uniform level backoff, exponential 3 level backoff, gamma 3 level backoff, uniform level backoff, exponential, continous system 3 level backoff, exponential, discrete system End to end throughput End to end throughput End to end throughput Arrival rate (frames/second) (a) Channel utilization in continuous model with change of arrival rate λ Arrival rate (frames/second) (b) Channel utilization in discrete model with change of arrival rate λ Arrival rate (frames/second) (c) Comparison with continuous and discrete model with change of arrival rate λ 1. Figure 4: Comparison of channel utilization in continuous and discrete model with change of arrival rate λ 1. Collision Probability Node1 1st Backoff Collision Probability, 3 level Backoff, Simulation Node1 2nd Backoff Collision Probability, 3 level Backoff, Simulation Node1 3rd Backoff Collision Probability, 3 level Backoff, Simulation Node1 1st Backoff Collision Probability, 3 level Backoff, Analysis Node1 2nd Backoff Collision Probability, 3 level Backoff, Analysis Node1 3rd Backoff Collision Probability, 3 level Backoff, Analysis Node1 1st Backoff Collision Probability, 1 level Backoff, Simulation Node1 1st Backoff Collision Probability, 1 level Backoff, Analysis Channel Utilization of Cheater level backoff, arrival rate of other nodes 3 1 level backoff, arrival rate of other nodes 45 3 level backoff, arrival rate of other nodes 3 3 level backoff, arrival rate of other nodes Backoff Rate (occurence to access the channel/second) (a) Collision probability with change of backoff rate b Backoff rate (frames/second) (b) Cheater s Channel utilization with change of backoff rate b 1. Figure 5: Collision probability and Cheater s Channel utilization 7

8 2 b i δi,g π G 1 i, π i, s i λ i s i ( 1 π i, ) b δ i i,g 2 G 1 ( 1 π i, ) b i 2 G 1 b,1 δ i,g b (1 δ i,1 ) i δ i,1 b i bi δ i,2 (1 b,2 b. i δ 2 i,2 b i δi,g 1 b,g 2 G 2 (1 (a) The stage analytical model of the multi-stage backoff process. s ) 2 ) s i b δ i i,g 2 G 1 b i 2 G 1 b,1 δ i,g b (1 δ i,1 ) i δ i,1 b i bi δ i,2 (1 b,2 b. i δ 2 i,2 b i δi,g 1 b,g 2 G 2 (1 (b) The stage analytical model of the multi-stage backoff process when π i,. s ) 2 ) Figure 6: The stage analytical model 2.2. Stage analysis model Consider what could be called the stage analysis model in Fig. 6. This is a collapsed version of Fig. 1(a), used to compute average service rates, depending on the number of backoff stages. For π i, <ɛ, ɛ is a small positive value, Fig. 6(a) can collapse to an even simpler Fig. 6(b). First define the mean service time T i,srv and ρ as follows. From Fig. 6, μ i can also be found. T i,srv = 1 μ i = pb i 1 ε, ρ i = λ i μ i, b i s i s i+2s iδ i1+ +2 G 1 s G 1 i μ i = G. (11) g=1 δi,g+bi bi g=1 δi,g Also from this Markov Chain, we derive the following results, π i,s = λ i (1 π i,q )(1 G g=1 δ i,g) ; (12) s i The term (1 π i,q ) means the ratio of the frames left after the dropping because the queue is full. The term (1 G g=1 δ i,g) means the ratio of the frames left after collision. This result is applicable for cases of 1) infinite queue and 2) finite queue because the frame loss comes from blocking when queues are full and from dropping after collisions through all backoff stages. When π i,q approaches, formula (12) can be approximated as π i,s = λ i(1 G g=1 δ i,g). (13) s i Formula (13) is applicable for 1) infinite queues and 2) finite queues when π i,q is sufficiently close to. Here π i, is the probability that the system is empty and π i,q is the probability the queue is full. For π i, ɛ, whichmeansπ i,, the stage analysis model changes from Fig. 6(a) to Fig. 6(b) and the probability of being in a sending state, π i,s, can be found as follows. b G i(1 g=1 π i,s = δi,g) s i+2s iδ i1+ +2 G 1 s G 1 G. (14) i g=1 δi,g+bi bi g=1 δi,g This formula is only applicable for a finite queue in overloaded conditions (i.e., π i, ). In this situation, one can also use an M/M/1 approximation, which yields the following derivation. π i,q = ρ Q i 1+ρ i + ρ 2 i ρq i 8 ρ Q i i 1 ; (15) = ρq+1 i ρ Q+1

9 Based on the assumption that Q is sufficiently large, π i,q = ɛ ; 1 π i,q =1 ɛ 1 when ρ i < 1; (16) π i,q 1 1 ρ i ; 1 π i,q 1 ρ i when ρ i > 1; (17) π i,q = 1 Q +1 ; 1 π i,q = Q Q +1 when ρ i =1. (18) We will use this result in later sections. 3. Analysis of throughput and quality of service In this section, we begin by analyzing the effects of the arrival rate λ i and backoff rate b i from the perspective of Node i s channel utilization and quality of service. Compared with the saturation model in [5], we use this model that works over all traffic loads to understand the performance effects due to variation of the arrival rate. When π i, >ɛ,i=1...n, combining formulas (12), (5) and (6), the collision probability for each node is δ i,1 = N j=2 λ j (1 π j,q )(1 G g=1 δ j,g) s j. (19) δ i,g = N λ j j=2 (1 π j,q)(1 G g=1 δj,g) s+b 1/2 g 1 + b1/2g 1 s+b 1/2 g 1. (2) From formulas (12), (19) and (2), there are two findings: 1) if λ 1 (λ 1 increases), this causes π 1,s, then by symmetry δ j,1 and π j,s (π j,s decreases) for the other nodes, 2) as π j,s,thenδ 1,1 and δ 1,g. From our analytical model and simulation results, δ 1,1 decreases with increasing λ 1 and π 1,s approaches a constant value with increasing λ 1. Now, finding 1) implies Node 1 s behavior affects the channel utilization of all of the other nodes. It follows from finding 2) that π 1,s as λ 1, but there exists a balance relationship between λ 1 and π 1,s due to the values of b and s; Node 1 cannot increase its channel utilization by increasing its arrival rate without any limitation. We will explain the reason for this phenomenon in more detail in a later section when misbehavior is considered. We have applied the formulas for channel utilization, delay, jitter and loss based on LAQT [4] in [17]. From all of the above formulas, channel utilization, delay, jitter and loss are functions of arrival rate λ i and backoff rate b i. The two parameters each node controls, the arrival rate λ i and backoff rate b i, not only affect the channel utilization and quality of service for Node i, but also change these metrics for the other nodes. 4. Misbehaving nodes In this section, we analyze the effect of misbehaving nodes on the wireless network. A misbehaving node is one that seeks only to maximize its own performance, and might violate any rules of cooperation with other nodes. It is not necessarily malicious, but it is simply focused on its own benefit gained from the wireless network. We consider the effects as these nodes increase their arrival rates or backoff rates. For each parameter change: 1) Is the channel utilization bounded by a value less than 1? 2) Is the channel utilization monotonically increasing and does it approach the optimal solution as the arrival rate or backoff rate keeps increasing? 9

10 4.1. The misbehavior strategy of variation of arrival rate The organization for this subsection is related to answering the following questions: 1) Is π i,s bounded for a value less than 1? The answer is Yes, and is shown for the 1.1) infinite queue, and 1.2) finite queue. 2) Is π i,s monotonically increasing as λ i increases? Also, does π i,s approach the optimal value as λ i increases? The answer is Yes to both questions. For the situation where queue length Q is sufficiently large, we show analytically that π i,s increases monotonically with the increasing of λ i. Therefore, it automatically follows that π i,s approaches the optimal value. 3) Based on experimental results, we show that π i,s is maximized with the increasing of λ i using the assumption that queue length Q is NOT sufficiently large. 4) Finally, we consider a strategy where a cheater, node i, canhavemoreπ i,s. We also look at the corresponding changes of the QoS metrics of all the nodes in the network. Question 1: Is π i,s bounded with increasing λ i? Now we show that π i,s is indeed bounded to a value less than 1.. Assume: 1) a cheating node maximizes its arrival rate to increase its channel utilization, which means it doesn t maliciously try to hurt the performance of others, it just wants to maximize its own utilization and 2) each node has an infinite queue and a single stage backoff process. From these assumptions, formulas (13) and (19) can be simplified as follows. π i,s = λ N i (1 δ i,1 ), δ i,1 = π j,s i =1...N. s i The corresponding solution is δ i,1 = Using the definitions of μ i and ρ i in formula (11), j=1,j i (1 s i )( N j=1,j i 1+(1 s i )( N j=1,j i λ max = μ i = λ j s j λ j ) λ j s j λ j ). (21) b i s i s i + b i b i δ i,1. (22) Notice that λ i < μ i is necessary because this is an infinite queue. Formula (22) defines the upper bound of λ i. Now we substitute the value of λ i from (22) into (21) to calculate the lower bound δ i,1 because λ i is inversely proportional to δ i,1. Since δ i,1 is also inversely proportional to π i,s, we can substitute the newly found lower bound of δ i,1 into formula (13) to obtain the upper bound of π i,s. The same conclusion that λ i, δ i,1,andπ i,s are bounded holds for the infinite queue with a multi-stage backoff process because because there is an upper bound for λ i. For a finite queue, λ i can. π i,s is certainly bounded, but is the upper bound less than 1? Will node i occupy the whole channel, which means π i,s = 1? The answer is No. 1. If π i,s =1,thenπ j,s =,j i, and based on formula (5), δ i,1 =. 2. If δ 1,1 =, then from formula (14), all of the terms with product of δ i,g will equal zero, so π i,s = b i (s i+b i) 1= contradiction. 3. By contradiction, max π i,s < 1. Which means a node must always spend some proportion of time in backoff. Now we have shown that the channel utilization, π i,s is bounded with increasing λ i. Next we consider how channel utilization is maximized, and break the discussion into two parts, for large queue length and finite queue length. Question 2: Is the channel utilization maximized with increasing λ i when Q is sufficiently large? The Answer is Yes. Since we have shown that max π i,s < 1, what is the maximum value? Does it occur as λ i? This is indeed true, and we will analyze this by showing that π i,s is a monotonically increasing function of λ i. The general behavior of π i,s with the increasing of λ i is as follows, shown with arrows to indicate how values 1

11 are increasing or decreasing. λ i π i,s δ j,g π j,s (j i) δ i,1 δ i,g (g =1..G) 1 G g=1 δ i,g π i,s. This results in a double effect to increase π i,s caused by both the increase in λ i and by the decrease in π j,s. First we define the notations: 1)λ i,p is the arrival rate of node i at step p. 2)μ i,p is the mean service rate of node i at step p. 3)ρ i,p is the system utilization of node i at step p. 4)π i,q,p is the steady state probability of node i at the last position of the queue at step p. 5)π i,s,p is the sending steady state probability of node i at step p. Assume there are two nodes, each with a very large queue. Formula (16), (17) and (18) can be applied and we consider node 1 is the cheater. By introducing dynamic game methods, the sequence of action and the consequences are now of concern. At the beginning, all parameters have initial values, ρ 1, < 1and ρ 2, < 1. Then at the first step, p = 1, node 1 increases its arrival rate by a factor of k, k>1, and π 1,s.At the second step node 2 doesn t change any of its parameters, but it s channel utilization is changed because π 1,s δ 2,g. Sequentially at step 3, node 1 s channel utilization changes also because the change of node 2 s channel utilization, π 2,s, has an effect on node 1 s collision probability. The detailed analysis is given in Appendix A, in which we show that an increase in λ 1 always causes an increase in π 1,s, hence the function is monotonically increasing. The conclusion holds for any number of nodes. Based on the assumption that the queue length Q is sufficiently large, it is guaranteed that π 1,s as λ 1,soπ 1,s is monotonically increasing. Question 3: Is π i,s maximized with increasing of λ i when Q is not sufficiently large? The answer again is Yes. We have used different sets of values for N, λ i, λ j,(j i), b i, b j (j i), s and used numerical analysis. All of the results show that π i,s, the cheater s channel utilization, is maximized with increasing λ i. In fact, λ i π i,s δ j,g and π j,q,j i. From formula (12) or (13), π j,s δ i,g π i,s. When λ i, π i,q 1, the value of π i,s π i,q,s. This is the sending probability for a full queue, and this parameter approaches a constant value. Consequently, this is the limitation of the effect of node i on all the other nodes in the system. Again, δ j,g constant (goes to constant) and π j,q constant, making π j,s constant and δ i,g constant. Then by formula (12), π i,s constant. We conclude that the channel utilization approaches a constant value for all nodes with increasing λ i and the result is shown in Fig. 7. At every value of b 1, the curve flattens with larger λ 1. Therefore, π i,s is asymptotic. Also as shown, the optimal solution for channel utilization π i,s, which is related to the arrival rate λ i, is the same as the asymptotic limit, because the function is monotonically increasing. Now the three subquestions related to the arrival rate λ i have been answered. Question 4: Response of other nodes to cheating nodes Given below in Table 1 is a summary of the effects of increasing arrival rate by the cheater node, Node 1. The table applies for both cases of when the Q is and is not sufficiently large. This includes the effects on both on the cheater node and the other nodes. Table 1: QoS Metrics with Respect to Increasing λ 1 Channel Utilization Loss Delay Jitter Cheater ρ 1 < 1 increasing increasing increasing increasing ρ 1 > 1 increasing, increasing, increasing, increasing, asymptotic asymptotic asymptotic asymptotic The Other Nodes ρ 1 < 1 decreasing increasing increasing increasing ρ 1 > 1 decreasing, increasing, increasing, increasing, asymptotic asymptotic asymptotic asymptotic 11

12 The collision probability of Node j (j i) is an increasing function of λ i, which causes the channel utilization probability π j,s to be a decreasing function of λ i. It s natural for the other nodes to increase their arrival rates, as a response to the cheater. The other node, say j, either increases λ j until π j,s approaches a constant value, or sets λ j = λ j,max if π j,s is not at the asymptotic limit yet. Once all of the nodes maximize their channel utilization combined with QoS constraints, no node needs to change its arrival rate any more. This results in an equilibrium for all nodes. This causes some of the effects from the cheater to be counteracted by the other nodes. The end result is that the cheating node has not increased its channel utilization substantially, but it has increased its frame loss probability and/or delay because it increased arrival rate. Therefore, one would expect the cheater to reduce arrival rates back to original levels The misbehavior strategy of varying back off rate When the arrival rate stays constant, one might assume that an increase in b i would increase π i,s. In reality it may increase or may decrease π i,s, depending on the loading conditions, number of backoff stages, etc. We answer four subquestions. 1) Is π i,s bounded by a value less than 1. with the increasing of b i? We show that it is indeed bounded and consider three scenarios. 1.1) π 1,Q = (low load), 1.2) π 1, <ɛand π 1,Q > (high load), 1.3) π 1, >ɛand π 1,Q > (moderate load). 2) Does π i,s Asymptotically Approach a Constant with Increasing b i? In this case, Yes. 3) Is π i,s maximized with the increasing of b i? Using the assumption that queue length Q is sufficiently large, does π i,s reach its maximum as b i? We answer this by dividing the discussion into two parts: 3.1) a single stage backoff process where the answer is Yes, and 3.2) a multi stage backoff process where the answer is No. In general, the value of π i,s for multiple stages will increase then decrease as b i increases. For each division we use four conditions based on ρ i in a sequence of steps. 4) Then we discuss Is π i,s maximized with the increasing of b i? using the assumption that queue length Q is NOT sufficiently large based on experimental results. Question 1: Does π i,s have a bound < 1 with increasing b i? The Answer is Yes. Assume Node 1 is the cheater and it increases the backoff rate b 1. We divide the analysis into three parts based on corresponding conditions: 1) Condition A, for light load traffic, when π 1,Q =, 2) Condition B, for heavy load traffic, when π 1, <ɛand π 1,Q >, 3) Condition C, formoderate traffic load, when C, for middle load traffic, when π 1, >ɛand π 1,Q >. The detailed analysis is given in Appendix B. From the above analysis, π i,s does have a bound < 1with increasing b i. Our model is a continuous time model, so as long as there is a competitor, say node j, the competitor always has some probability to seize the channel even if b 1,aslongasλ 1. This means no node can occupy the whole channel, that is <π i,s < 1. For example, the probability for node j seizing the b channel is j b 1+b j if node j has a frame to transmit and both node 1 and j have a single backoff process. When b 1, and new frames arrive, if the channel is busy, μ 1 as in the definition of formula (11); node 1 will immediately backoff and collide G times, then drop the frame. If the channel is not busy, node 1 will obtain the channel and keep it until it has no more frames to send, since its backoff times are so short that no other nodes can get a chance to transmit. Other nodes will only collide since there will be 1 b 1 seconds between the transmission of node 1 s frames. Question 2: Does π i,s asymptotically approach a constant with increasing b i? The Answer is Yes. Until now we have shown that the channel utilization, π i,s is bounded with the change of b i, but we need to know the asymptotic behavior of π i,s with the change of b i. Still we assume node 1 is the cheater. 1) As b 1, from formula (6), δ 1,g1 1, (g 1 > 1), so the system simplifies approximately to a single backoff process system. 2) From formula (11), μ 1 = s1 1 δ 1,1 = s 1 1 N j=2 πj,s. 3) Once the values of δ 1,g and μ 1 have stabilized, using formula (12), we can calculate π 1,s. 12

13 4) We can calculate all the other node s π j,s, (j 1). From the above steps, the asymptotic behavior of π 1,s has been shown. Question 3: Is π i,s maximized with increasing of b i when Q is sufficiently large? Next we will answer the third subquestion: Is channel utilization maximized as b 1?Theansweris Yes for a single backoff stage and No when there are multiple backoff stages. Fig. 5(b) shows the channel utilization of the cheater, node 1, when λ 1 = 1, λ j =3(j 1)and λ j =45(j 1),b 1 = 4...2, b j = 4 (j 1). In Fig. 5(b), the channel utilization, π 1,s monotonically increases with increasing of backoff rate b 1 for a single-stage backoff process. Also π 1,s first increases, then decreases with increasing of backoff rate b 1 for a multi-stage backoff process. We divide the answer into two parts. First we will show the analysis of the single backoff process case in Appendix C1. From the analysis in Appendix C1, an increasing in b 1 causes 1) no change to π 1,s or 2) an increase in π 1,s, hence the function is monotonically increasing. The same conclusion holds for N>2 number of nodes. Then we analyze of the multi backoff process case in Appendix C2. It is not guaranteed that π 1,s as b 1, as can been seen in Fig. 5(b). Question 4: Is π i,s maximized with increasing of b i when Q is not sufficiently large? Now how about the case when Q is not sufficiently large? Here is an example. 1) At step 1, b 1,1 >b 1, μ 1,1 > μ 1, (1 π 1,Q,1 ) > (1 π 1,Q, ). 2) Also, b 1,1 >b 1, δ 1,g1,1 >δ 1,g1,, (g 1 > 1), (1 G g=1 δ 1,g,1) < (1 G g=1 δ 1,g,). 3) In fact, because the changes in the terms (1 π 1,Q ) and (1 G g=1 δ 1,g,1) for the change of b 1,the value of π 1,s could or, which depends on the values of λ 1, λ 2, b 1, b 2 and s. By using numerical analysis, we have found many sets of these values to increase or decrease the value of π 1,s with the increasing of b 1. Until now we have shown π 1,s is not monotonic with the increasing of b 1. Also by using numerical analysis, we have found π 1,s is not maximized as b 1. Based on the above analysis, the cheater cannot always increase its backoff rate to make π i,s. Merely increasing backoff rate b i doesn t guarantee π i,s can be increased. Optimization methods to find the maximal channel utilization are a subject of future work. The relationship between the collision probability, δ i,g the and backoff rate, b i is presented in Table 2 with the assumption that the queue length, Q is sufficiently large. Note that increasing b 1 moves the scenario from ρ 1 > 1toρ 1 < 1 since frames are served faster. Table 2: Collision Probability with Respect to Increasing b 1 in a Multi-Stage Back Off Process δ 1,1 δ 1,g, g 1 δ j 1,q, q =1 G ρ 1 < 1 decreasing, asymptotic increasing, asymptotic increasing, asymptotic ρ 1 > 1 decreasing increasing increasing 4.3. The misbehavior strategy of varying both arrival and back off rates It has been shown that channel utilization is bounded less than 1. for increasing λ i and increasing b i individually. What if both are increased together in a coordinated fashion? Would the cheater then be able to obtain a utilization approaching 1.? The answer is Yes, but if another node does the same, they share the channel. Fig. 1(b) shows a simplified channel model, where each node (one node at a time) goes from idle to busy state with a general rate r i and finishes service with rate s. The rate r i from the channel idle state, state (chl, ), to the state (chl, i) whennodei is using the channel, is a function of 1) the probability that node i is in a backoff stage, 2) the arrival rate λ i, 3) backoff rate b i, and 4) the other nodes channel utilization. s From Fig. 1(b), the channel idle probability, π chl,,canbederivedasπ chl, = s+ N. Node i s i=1 r ri channel utilization, π chl,i,isπ chl,i = i s+ N. These equations show that π chl, asanyr i. If i=1 ri 13

14 one r i,itsπ chl,i 1. If more than one r i, they share the channel depending the relationship between their rates. From experimental results, by increasing λ i and and setting b i appropriately, node i is indeed able to increase r i for higher channel utilization. Note once again that large b i does not necessarily produce the highest channel utilization Response of other nodes to cheaters From the analysis in the previous two subsections, a cheater can adjust its arrival rate and backoff rate to achieve its maximum channel utilization and satisfy other QoS requirements, such as loss ratio or delay. However, the channel is shared by all of the nodes and the maximum channel utilization probability of a cheater is asymptotic. Let s describe a typical scenario in the following steps: 1) suppose Node 1, the cheater, keeps on increasing its arrival rate and backoff rates together to get a desired high channel utilization and high QoS. 2) Node 1 s action will cause corresponding actions of another node (e.g., Node 2) to guarantee π j,s. In essence, Node 2 must cheat to counteract Node 1. Here we show the results for two competing nodes. In this scenario, there are 6 total nodes in the system. The service rate of the channel is 2 MAC frames per second. The range of arrival rates for the two cheaters, Node 1 and Node 2, is from 2 to 2 frames per second; the range of the backoff rate is from 5 to 1 backoffs per second. λ k, (k =3..6) = 45 and b k, (k =3..6) = 4 will be the constant values for the other regular nodes, Node 3 to Node 6. Fig. 7 shows the changing of π 1,s with respect to only changing λ 1 and b 1 and shows the changing of π 2,s with respect to only changing λ 1 and b 1. Initially, Node 1 increases π 1,s which makes π j 1,s decrease. Consequently, the mean system delay and loss ratio will also change for the benefit of Node 1, the cheater. These results are shown in Fig. 8 and 9. Next, Node 2 takes corresponding actions to increase π 2,s, and has the same effect on the other nodes, like the cheater, Node 1. This is shown in Fig. 1. The corresponding changes for the mean system delay and loss ratio of the current cheater, Node 2, and the previous cheater, Node 1, are shown in Fig. 11 and Fig. 12. It s easy to extend the same steps to all nodes until they reach the point that all the nodes have maximum arrival rate (within QoS constraints) and corresponding large backoff rate. Note that the effect of changes to λ i and b i on π i,s and π j,s are asymptotic, which is consistent with the analysis of the last three subsections. Our research also shows that total channel utilization also increases with respect to increasing λ 1, b 1, λ 2, b 2. Once each node approaches its asymptotic value, no node takes more action since it will not gain any benefit. This equilibrium point is where misbehaving nodes share the channel capacity. Note that this is a benefit for the continuous model compared to a discrete model. In the discrete model, the channel utilization decreases to zero if there are at least two cheaters [2] when both of them set their contention window size to W =1. Channel Utilization Of Node Id #1 Channel Utilization Of Node Id # Channel Utilization Channel Utilization λ1 b λ1 b1 (a) Node 1 s channel utilization. (b) Node 2 s channel utilization. Figure 7: Channel utilization of the cheater (node 1) and the other node vs. λ 1 vs. b 1. 14

15 Delay Of Node Id #1 Delay Of Node Id #2 Mean System Delay (ms) Mean System Delay (ms) λ b λ b (a) Node 1 s mean system delay. (b) Node 2 s mean system delay. Figure 8: Mean system delay of the cheater (node 1) and the other node vs. λ 1 vs. b 1. Loss Ratio Of Node Id #1 Loss Ratio Of Node Id # Loss Ratio Loss Ratio λ b λ b (a) Node 1 s loss ratio. (b) Node 2 s loss ratio. Figure 9: Loss ratio of the cheater (node 1) and the other node vs. λ 1 vs. b 1. Channel Utilization Of Node Id #1 Channel Utilization Of Node Id # Channel Utilization Channel Utilization λ b λ b (a) Node 1 s channel utilization. (b) Node 2 s channel utilization. Figure 1: Channel utilization of the cheater (node 2) and the other node vs. λ 2 vs. b MAC friendliness In the previous section, we discussed the effect of a misbehaving node on other nodes. Now we present a new definition of what we call MAC friendliness. All of the nodes agree on their predefined respective π i,s and comply with them. In the previous analysis for cheating nodes, if Node i increases its λ i and b i,this changes its channel utilization ratio π i,s, and affects the other nodes. Conversely, if a node is to be MAC friendly, a node 1) keeps the same π i,s ; 2) adjusts λ i and b i correspondingly to adjust its delay, service 15