Towards the Performance Analysis of IEEE in Multi-hop Ad-Hoc Networks

Transcription

1 Towards the Performance Analysis of IEEE in Multi-hop Ad-Hoc Networks Yawen D. Barowski Saâd Biaz Computer Science and Software Engineering Department Auburn University Auburn, AL , USA Technical Report # CSSE4-3 April 6, 24 Abstract IEEE standard protocol is widely used in wireless ad-hoc network. The performance of IEEE with different network densities and protocol configurations is of interest, particularly in distributed coordination function (DCF) mode. A two dimensional markov chain based mathematic model for one hop network IEEE protocol was introduced by to analytically derive the saturated throughput. Our ultimate goal is to extend this model to multihop ad hoc networks. Our work in this paper has two tiers of contribution. First, we revisit s model, modify the transition probabilities in order to more accurately match the IEEE standard. This modified model takes into consideration the frequency probability and the residence time proportion in each state, and provides detailed analytic derivation of transmission probability, collision probability, energy consumption, and saturated throughput. ns-2 simulations validate the modified model, and demonstrate an improvement in accuracy over s model. s model has up to 6% deviation on the transmission probability and up to 2% deviation on the collision probability. Second, queue length is considered as a third dimension to the modified two-dimensional model. The average queue length and average one hop message delay can then accurately be derived. By combining the two dimension model and the three dimension model, our paper proposes a method to analyze the performance of a network under different traffic loads and network densities, and opens the door to the analysis of IEEE on a multihop ad hoc network. 1 Introduction IEEE [1] medium access control (MAC) protocol is currently the most popular random access MAC layer protocol used in wireless ad-hoc network. It uses a distributed coordination function (DCF) as primary mechanism to access the medium. DCF has two modes: the basic broadcast mode, and the MACAW [8, 2] based RTS/CTS mode. The efficiency of IEEE protocol directly affects the utilization of the channel capacity and the system performance. When a collision occurs, IEEE protocol uses a binary exponential back off algorithm to randomize new access attempts based on the level of contention. Several parameters of this algorithm such as initial backoff window size and maximum backoff window size have a critical impact on IEEE performance. Performance analysis of IEEE has been done experimentally and analytically. Throughput of IEEE has been investigated by many researchers. Cali, Conti, and Gregori [4] established a theoretical upperbound for IEEE protocol capacity, showed that the standard may poorly perform, and proposed an appropriate tuning of the backoff algorithm to approximate the theoretical upperbound. [3] proposed a two-dimensional markov chain model to analyze the performance of the IEEE exponential backoff scheme, and evaluated the saturated throughput, as each station always has a packet to send. The IEEE exponential backoff scheme maintains a counter: a station cannot send unless this counter reaches the value. This counter is decremented after each σ if the medium is sensed idle. σ is normally a constant system time slot set by the physical layer. The main simplification of s model is to assume that the counter is decremented with probability 1. after each time slot, which violates IEEE standard (the counter is decremented only if the medium is sensed idle and the time it takes to decrement the backoff counter is system time slot. [1]). 1

2 In order to accommodate this fact, considers slot time as an average time between consecutive counter decrements. The saturated throughput predicted by s model matches the throughput measured through simulations despite the inaccuracy in transmission probability and collision probability. In fact, the inaccuracies on transmission probability and collision probability are such that they cancel each other in the estimation of the saturated throughput. Marcelo [5] derived the average delay based on s model and results. Marcelo used a probabilistic approach to analyze the average service time for each packet, then derived the one hop delay of the packet. The packet delay applies only to a packet at the head of the queue. In other words, Marcelo did not take into consideration the queueing delay of packets at the MAC layer. The inaccuracies on transmission probability and collision probability in s results directly impact Marcelo s results. [1] summarizes s work, provides an estimation of the average backoff window size during each transmission, thus derives the average delay. [11] considers another approach: the station is considered as a server and a G/G/1 queue model is used to analyze the average queue length and delay under unsaturated traffic. The author derives an abstract formula on G/G/1 queue model analysis based on an M/M/1 approximation of the probability of the server being idle. In this paper, we focus on the performance of IEEE DCF RTS/CTS scheme. We revisit s model to more accurately represent IEEE protocol. In order to analyze the queue length, we extend the two dimension model into a 3 dimension model. The 3D model allows the modeling of not only data sources (as in s model) but also relay stations that forward traffic. In our models, the residence time in each state and the energy consumption of each transition are analyzed as well as the probability of each state. The access delay and energy consumption rate are then derived from the model as well as the throughput. Our paper is outlined as follows. Section 2 briefly describes the IEEE DCF scheme and stresses on key elements related to this paper. We discuss in Section 3 work related to this paper and particularly a key contribution to this subject by [3]. In Section 4 we explain why s model does not accurately model an IEEE station. This model is extended in Section 5 into three dimensional model that could be used to model IEEE in multi-hop networks. 2 IEEE DCF scheme IEEE is a contention based MAC protocol. It has two working modes. The point coordination function(pcf) mode is a centralized scheme designed for an infrastructure network. This access method uses a point coordinator that operates at the base station to select the next wireless station to transmit. The distributed coordination function(dcf) makes use of carrier sense multiple access with collision avoidance (CSMA/CA [8, 2]. This paper focuses on DCF. DCF allows automatic and adaptive medium sharing between compatible physical layers (PHYs) through the use of the CSMA/CA and a random backoff procedure following a busy medium condition. Carrier sense is performed both through a physical mechanism at the physical layer and a virtual mechanism at the MAC layer. The virtual carrier sense mechanism is achieved by distributing reservation information announcing the impending use of the medium. The exchange of RTS and CTS frames prior to the actual data frame is one mean of distribution of this medium reservation information. A wireless station that needs to send a data frame should invoke the carrier sense mechanism to determine the state of the medium. If the medium is idle for a specific length of time, DCF interframe space (DIF S), the wireless station should generate a random time period for an additional deferral before transmission. Figure 1 illustrates the access procedure of DCF. The backoff procedure is invoked when the medium is sensed busy. The MAC sets its backoff timer to a random backoff time using formula, Count = Random() σ where Random() returns a random integer within [CW min, CW max] where CW min and CW max respectively are the minimal contention window size and the maximum contention window size. σ is the system time slot set by the physical layer. The binary exponential back off meachanism increases the range of [CW min, CW max] as contention increases with the objective of staggering the conflicting transmissions. A station performing the backoff procedure uses the carrier sense mechanism to determine whether there is activity during each backoff slot. If no medium activity is detected, the backoff procedure decrements Count by σ. Otherwise Count is not decremented for that slot. Figure 2 illustrates a successful transmission between a source and a destination. The minimum time between transmission of interactive packets (RTS or CTS) is the short interframe space (SIF S). With setting DIF S > SIF S, the protocol provides higher access priority to RTS and CTS frames. If a transmission succeeds, the wireless station will follow the same procedure for the next transmission. If a transmission fails, the DCF will redo the procedure with an exponential backoff mechanism: at the first transmission, the range of the Random() is from zero to W where W is the maximum contention window at stage. At the i th failure, the range of the Random() is extended from zero to W i, where W i = 2 i 1 W. i is the backoff stage. The first time transmission is called stage, the second time transmission is called stage 1, and so 2

3 DIFS Contention Window DIFS Busy Medium SIFS Backoff Window Next Frame Defer Access Slot time Select Slot and Decrement Backoff as long as the medium is idle DCF: SIFS DCF interframe space : short interframe space DIFS Figure 1: basic Access Method of DCF Source RTS DATA Destination SIFS CTS SIFS SIFS ACK Figure 2: Data transmission of DCF on. So, as the stage increases, the range of the contention window sizes increases. After a successful transmission, the stage is reset to. 3 Related Work [3] proposed a two-dimensional markov chain model to analyze the performance of the IEEE protocol. The model is shown in Figure 3. Two parameters, backoff stage and backoff counter value are used as a pair to describe the state of an IEEE station. The pair (backoff stage, backoff counter value), referred as (b, c), describes a given state of a station where c can take any value between and W b. Back off stage b varies from to a maximum back off stage MBS. assumes that a station will transfer from state (b, c) to state (b, c 1) with probability 1.. This assumption implies that the backoff counter value is decremented at each time slot under all conditions. This violates the IEEE standard 82.11[1]. IEEE specifies that the back off counter value is decremented only if the medium is sensed idle. accommodates nicely this violation by considering an average slot time, rather than the constant slot time σ set by the physical layer. If a station reaches state (b, ) (i.e., the backoff counter value gets null), the station will send out a packet. If the packet collides (with probability P coll ) then the station will transfer with probability P coll into some state W b+1 (b + 1, c) with a higher backoff stage. If the packet does not collide, the station will return to some state (, c) with backoff stage (recall that in such a state, c can be any value between and W ) with probability 1. P coll W. Derived from the model, the probability that a station transmits a packet at any time slot is The probability that a sent packet collides is τ = MBS b= π(b; ) P coll = 1 (1 τ) n 1 If n is the total number of stations, the expression above reflects the fact that a packet collides when at least another station is also sending. We denote π(b, ) as the probability that the station stays in state (b,). 3

4 (1 Pcoll)/W ; 1. ;1 1. ;2 ;W 2 1. Pcoll/W1 ;W 1 i 1; Pcoll/Wi i; i;1 i;2 i,wi 2 i;wi 1 Pcoll/Wm m; m;1 m;2 m,wm 2 m,wm 1 Pcoll/Wm Figure 3: s Model for IEEE Instead of the system time slot σ, [3] rather uses the average time slot T slot that he defines as the average time duration for a station to transfer from a state to another. T slot can be derived as follows. If there is no medium activity on the channel, the time slot would be the system time slot σ. If the medium is busy, the slot time would be either the time to complete a successful transmission or the time to perform a failed transmission. The average time slot is where T slot = (1 P tr) σ + P trp st s + P tr(1 P s)t c P tr = 1 (1 τ) n P s = nτ(1 τ)n 1 1 (1 τ) n T s = RT S + DIF S + CT S + SIF S + T [P ] + SIF S + ACK + DIF S T c = RT S + DIF S P tr is the probability that there is at least one transmission in the considered time slot, and P s is the probability that a transmission occurring in the considered slot is successful given there is a transmission in that slot. T s is the time needed to complete a successful transmission. T s includes the time to send RTS, CTS, ACK and data packet plus the defer and interframe time. T c is the time for a failed transmission. σ is the system time slot duration and T [P ] is the time needed to transmit the payload. The propagation delay is ignored. Thus, the saturated throughput is derived by as, where E[P ] is the average payload length. S = PtrE[P ] T slot 4 Modified IEEE Model s model structure is excellent and provides a good estimate of the saturated throughput. However, we pointed out that s model does not accurately model IEEE In this section, we will first explain the difference between our model and s, and second, we derive the expressions for the probability of transmission τ, the probability of collision P coll, the average access delay D access, and the average energy 4

5 consumption E. Other characteristics of the station are derived too. Finally, we use ns-2 simulations to check and validate our model and analysis. 4.1 Modified IEEE Model As in s paper [3], we assume that all stations always have at least one packet to send. Let us denote the state space as R, R = {(b,c): MBS b, c } where b is the current backoff stage and c is the current backoff counter value. c takes any value from to W b where W b = W 2 b. MBS is the maximum backoff stage of the IEEE protocol. The behavior of a station can be described as a chain of states on the time axis. When the station is in some state, different actions such as receiving a packet, sending a packet, or decrementing the counter will lead the station to different next states with some specific probability. The next state it will get to depends only on the current state, not on any previous state (Markovian chain). s Markov chain assumes a constant time T slot duration from a state to another where T slot is defined as the average slot time. This does not reflect accurately the behavior of a IEEE station. Therefore, we consider for our model that the time duration for the station to transfer from one state to another is different for different transitions. This time duration is randomly distributed and makes the chain a semi-markov process. With the stationary distribution of this semi-markov process, we can derive the frequency probability and time proportion of each state for further analysis. Our model takes into account the transition time, the transition probability, and transition energy in order to derive expressions of the probability of collision P coll, the saturated throughput, the average access delay D access, and energy consumption E. Figure 4 outlines our model. In general, a station in state (b, c) with a non-zero c value will transfer to state (b, c 1) if the medium ; ;1 ;2 ;W 2 ;W 1 i 1; P5[i] P5[i] i; i;1 i;2 i,wi 2 i;wi 1 P5[i+1] m; m;1 m;2 m,wm 2 m,wm 1 Figure 4: Overview of the Station Model is idle; a station in state (b, ) will send out a packet. If the packet is successfully sent, the station will transfer from (b, ) into some state (, i) (in stage ). If the packet collides, the station will transfer into some state (b+1, i) (in stage b + 1). A station in state (MBS, ) (recall MBS is the maximum back off stage) will transfer to some state (MBS, i) if the sent packet collides. One key difference between s model and ours is the transition probability from state (b, c) to state (b, c 1). s model assumes that the probability of the transition from state (b, c) to state (b,c-1) is 1.. 5

6 In order to accommodate with the fact that the station would only make this transition when the medium is idle, s model is based on an average transition time T slot instead of the system time slot σ. For a semi-markov chain, the frequency probability of each state - i.e., the frequency that a station gets in a specific state,- does not depend on the transition time. The frequency only depends on the transition probabilities. By assuming that the transition probability is 1., s model results in an abnormally larger frequency probability in each state (b, ). This over estimation results in an over estimation of the transmission probability τ. The over estimation of τ is confirmed through ns-2 simulations. b 1; b 1; P5[b] TFT EFT σ σ EIdle EIdle b; c 1 b; c b; c+1 ; i i =, 1,..., CW TST EST P5[b] TFT EFT b; σ EIdle b; 1 if c<>w b 1 P5[b+1] TFT EFT TStay EStay b+1; i i=, 1,..., CW b+1 1 (a) state (b, c) (b) state (b,) Figure 5: Examples of Detailed information on a state Figure 5.(a) and 5.(b) focus on the details of typical states (b, c) and (b,). In figure 5.a), the backoff stage is neither zero nor the maximum backoff stage MBS and the backoff counter value is not zero. Each ellipse in the figure represents a state, each line with an arrow represents a transition from a state to another. Each line bears three parameters: the first parameter is the probability of the transition, the second parameter is the duration of the transition, and the third parameter is the energy consumed during the transition. For example, the probability of the transition from (b,c+1) to (b, c) is P 4 instead of 1. as in s model. P 4 represents the probability that the medium is idle. For each state, the sum of all outgoing transitions must be 1., i.e., the station must transfer into some state from the current state. If a station must stay in the current state, then we can say that the station transfers into the same state with probability P s, which is equal to (1 P 4). For the transition from state (b,c+1) to state (b,c), the transition probability is P 4. The transition duration is σ. σ represents the system time slot. The energy consumed during the transition is EIdle. Since during the transition, there is no activity in the medium, and the station does not need to consume energy on listening to the medium, EIdle represents the energy consumed by a station when it stays in the idle state for time duration σ. For transition from state (b-1,) to state (b, c), this transition occurs when a station sends a packet and the packet collides. The transition probability is P 5[b]. The transition duration is T F T. T F T represents the time a failed transmission takes. The energy consumed during the transition is EF T. EF T represents the energy consumed for a failed transmission. For the transition that the station goes back to the current state, there are two situations. Either a successful or a failed transmission on the channel will cause the station to stay in the current state. However, successful and failed transmissions have each different probabilities and different durations. We use T Stay to represent the average duration of both cases, and EStay to represent the average energy consumed in both cases. In Figure 5.(b), the backoff stage is neither zero nor MBS and the backoff counter is zero. In such a state, a station will send out a packet. If the packet collides, the station will transfer into some state (, i). The transition probability is P 3, the transition duration is T ST, which represents the time of a successful transmission, and the energy consumed during the transition is EST, which represents the energy consumed for a successful transmission. The other transitions in figure 5.(b) are similar to those in figure 5.(a). The details about all parameters in our model are described below. From the medium point of view, time is divided into unequal time slots. There are three kind of time slots. First, when a packet is sent successfully, the length of the time slot is the time needed to transmit a packet; second, when a packet is sent unsuccessfully, the length of the time slot is the time needed to transmit RT S/CT S handshake packets; and the last, when no packet is sent, the medium is idle, and the length of the time slot is the system slot time, which is set by the physical layer. In the following description, when we mention time slots, it could mean any of the above three kinds of slots. 1. τ 6

7 τ = MBS b= π(b; ) τ is the probability that a station sends a packet at any time slot. Whenever a station gets in state (b, ), it will send a packet. So τ should be the sum of the frequency probabilities of all states (b, ). 2. P coll P coll is the probability that a packet collides given that the packet is sent. When one station is sending a packet, collisions occur if there is at least another station sending a packet in the same time slot. If no other station is sending packets, there will be no collision. So the probability that a sent packet gets collided should be 1. minus the probability that all other stations are not sending packet in the same time slot. So, given the total station number is n, P(a station will not send a packet within a time slot) = 1. τ P coll = 1 (1 τ) n 1 3. P succ, P idle, and P fail P succ is the probability that there is a successful packet in a time slot. There will be a successful packet only if there is only one station sending in a time slot. P idle is the probability that there is no activity in a time slot, which means no station is sending in a time slot. At any time slot, each station has probability τ to send a packet, and the probability that this packet is successful is (1 P coll ). So, P(one station sends a successful packet in a time slot) = τ (1 P coll ) then, the probability that there is a successful packet in a time slot is P succ = n τ (1 P coll ) And, the probability that no station is sending in a time slot is P idle = (1 τ) n Then, the probability that there are collided packets in a time slot should be P fail = 1. P succ P idle 4. P 3 P 3 is the probability that a station sends a packet and transfers into some state (, i). There are W states at backoff stage, and a station has even chances to get in any of these states after it successfully sends a packet. So, P 3 = (1 P coll )/W 5. P 4 P 4 is the probability that the medium is idle in a time slot. The medium is idle at a time slot when no station is sending packets. So, P 4 = P idle = (1 τ) n 6. P 5[b] P 5[b] is the probability that a station in state (b 1, ) sends a packet and transfers into some state (b, i). There are W b states at backoff stage b, and a station has even chances to get in any of these states after a sent packet collides. So, P 5[b] = (1 P coll )/W b 7. T ST, T F T, T SR, and T F R T ST, T F T, T SR, and T F R are time respectively to perform a successful transmission, to perform a failed transmission, to receive a successful packet and to receive a collided packet. Refer to Figure 2, TST = T rts+3.*sifs+t cts+t data +T ack +DIFS TSR = TST If a transmission fails, most likely, only RTS packets have been sent. In our model, we calculate T F T and T F R as 7

8 T F T = T rts + SIF S + T cts T F R = T rts + EIF S where EIF S is the extended interframe space. EIF S is used as a delay by a station when a collided RTS packet is heard on the channel. When a station hears a collided RTS packet, it will not get the NAV time indicated in the RTS packet which is supposed to be used as a channel reservation duration. It will only defer an EIF S to resume with the existing procedure. 8. EST, EF T, ESR, EF R, ESL, and EF L EST, EF T, ESR, EF R, ESL, and EF L are the energy consumed to transmit a successful packet, to transmit a collided packet, to receive a successful packet, to receive a collided packet, to listen to a successful packet and to listen to a collided packet, respectively. We adopted the energy model in [6] for the calculation of energy consumption. A linear model for energy consumption in IEEE network is derived, E = a size + b where the energy consumed in IEEE network is linear to the size of the packet involved. a and b are different for different situations, such as sending a packet, receiving a packet, listening to a packet when a station is in the range of both data source and data destination, listening to a packet when a station is only in the range of data source or listening to a packet when a station is only in the range of the data destination. 9. P s P s is the probability that a station stays in its current state. P s is equal to 1. minus the sum of probabilities of the station transferring to other states. P s = 1 P 4 1. T Stay, and EStay T Stay is the average time during which a station stays in the current state and does not do any transition. Both successful packets and corrupted packets on the channel will cause the station to stay in the current states. Therefore the mean time is T Stay = ucc T ST +P fail T F T P succ+p fail The energy consumed in waiting is similarly calculated. 4.2 Solution of the Models EStay = ucc ESL+P fail EF L P succ+p fail In the description of the models and the calculation of the parameters, we use unknowns such as π is and τ to recursively express themselves. In order to solve these unknowns and get the stationary distribution π, system equations are established for all stations. Let i denote any possible state (b, c). The range of i is from 1 to MaxStates, the number of all possible states (b, c). Equations for stations in IEEE 82.11, π i = MaxStates j=1 MaxStates τ = i=1 MBS b=1 π jp ji (1) π i = 1. (2) π(b; ) (3) for all i, j [1, MaxStates]. In all the equations above, j s are all possible states that can get in state i, and P ji s are the probabilities with which state j can get in state i. 8

9 4.3 Performance Metrics from the Model Solutions to the system equations are the frequency probabilities that the system stays in each state, π i, τ, and T i where T i is average residence time in state i. Starting from this point, we can derive several system performance metrics: average energy consumption rate, average throughput, and average access delay Time proportion of each state As we mentioned before, for IEEE 82.11, the time proportion of each state can be calculated as: P i = π i µ i MaxStates π j µ j j=1 where the µ i s are the mean time for the station to stay in state i. µ i = P ij t ij for j, such that P ij j where P ij is the probability for the station to transfer from state i to state j and t ij is the transition time for the station to transfer from state i into state j. For convenience, we use T state to denote, T state = i πiµi T state represents the average time a station will stay in state state Average energy Consumption Rate The average energy consumption of a station can be calculated as E = Pi εi i where ε i is the mean energy consumption rate for the station to remain in state i. ε i = Pij ɛij j µ i for all j, such that P ij <> where ɛ ij is the energy consumption for the station to transfer from state i to state j Average Throughput As in s work, the throughput we consider here is the saturated throughput. By saturated throughput, we mean that a station has always a packet to send. From the MAC layer point of the view, the time gap between finishing sending a packet and getting another packet from upper layer is zero. The average system throughput should be the sum of throughputs of all stations Average Access Delay T hr = 8 P AY LOAD { τ(1 P coll) n T state } Let us denote D access the average access delay of a packet, i.e., the time between when a packet gets to the MAC layer and when it is successfully sent. Suppose that a packet is successfully sent on the first try, and the time it takes is T s. Otherwise, if it fails on the first try, which takes time T f, then it will have to wait for the station to get in the next sending state (b, ) before it is sent again. In order to derive D access, we need to express the average time between two sending states. Let us denote D the average time between two sending states. In practice, D is also the time that a station takes to complete a backoff procedure after a failed transmission. Consider that each transmission starts with a backoff procedure. We have, Let R τ be the set of sending states, i.e., T f = T F T + D T s = T ST + D R τ ={(b; c) : c = } The probability that a station is in R τ is τ. Suppose τ i R and τ j R are two consecutive states (in R τ ) that the station goes through. Then between two consecutive visits to R τ (τ i and τ j), the expected number of visits to any state k / R τ is π k τ. Since the average time a station will stay in state k after it gets in state k is µ k, we can derive the time D between two consecutive sending state as 9

10 D = k;k R, / R τ D is also the time between two consecutive transmissions of any packet. The probability that a packet would be successfully sent at the first try is (1 P coll ), at the second try is P coll (1 P coll ), and so on. The probability that a packet would be sent successfully at i th try is P i 1 coll (1 P coll). if a packet is sent successfully on the first try, it takes T s. If on the second try, it takes (T f + D + T ST ), which is (T f + T s), and so on. If a packet is sent successfully on the i th try, it takes ((i 1) T f + T s). The average access delay can be derived as N P n 1 n=1 coll (1 P coll)(t s + (n 1) T f ). where N is the number of retransmissions time minus one. When N goes to infinity, the access delay will be 4.4 Experiments and Results π k µ k τ D access = T s + P coll T f 1 P coll. In this section, we explain how to collect data measurements in ns-2 simulations. Then we present the simulations results. Extensive simulation results validate our model and match the analytical results Experiment Scenarios Numerical analysis methods are used to implement s model and ours. For analytical results and simulations, we used the same configuration parameters. The maximum backoff stage is kept at 3. Varied bandwidth, initial backoff window size, network size and packet size are used. ns-2 simulations are carried with the same configuration parameters to collect data on transmission probability and access delay. In this section, we will describe how we collect the measurements. We define five types of time slot for each station in our simulation. The first two types of slots are: (1) slots in which the station detects a successful packet, and (2) slots in which the station detects collided packets. In these two kinds of slots, we assume the station is not sending any packet. The three last types of slots are: (3) slots on which the station sends out a successful packet, (4) slots on which the station sends a packet and this packet collides, and (5) slots in which there is no medium activity. All five types of slots are exclusive. We use N rs, N rc, N ts, N tc and N idle to respectively represent the number of each type of slot that occur during our experiments. According to the description of how ns-2 implements the IEEE protocol, we know that each successful ACK packet a station gets means a successful transmission of a data packet. Based on this observation, N ts is incremented each time an ACK packet is received at the station. Only RT S packet is sent during each unsuccessful transmission and every data packet must have a preceding RT S packet. Therefore, N tc is incremented each time an RT S packet is sent and is decremented each time an ACK packet is received. N rs is incremented each time a station detects an ACK packet that is sent to some other station or receives a data packet for itself. N rs is decremented each time the station s outgoing ACK packet is collided. N rc is incremented when the station detects a collision while it is not sending packets. Consecutive collisions which overlap each other are counted only once. N idle increments each time the backoff timer decrements. Based one the five numbers (N rs, N rc, N ts, N tc and N idle ) we collect, we can calculate the probability of a sending packet within any time slot as τ = N tc + N ts N tc + N ts + N rc + N rs + N idle (4) Access delay is collected as the simulation counts the average time from when a packet gets to its outgoing MAC layer to when a packet is finished receiving by the incoming MAC layer. For the ns-2 simulation, every station is within all other stations range. Each station has a saturated traffic source connected, and also serves as a sink for some other station. For each performance metrics, probability of transmission, probability of collision, access delay and average throughput, we plot four sets of results. The first is the performance versus network size, in which the bandwidth is 2 Mbps, the packet size is 15 bytes, the initial window size is 32 and the network size varies from 5 to 4. The second is the performance versus bandwidth, in which the packet size is 15 bytes, the initial window size is 32, the network size is 2 and the bandwidth varies from 1 Mbps to 11 Mbps. The third is the performance versus initial window size, in which the network size is 2, the packet size is 15 bytes, the bandwidth is 2 Mbps and the initial window size varies from 32 to 256. The last is the performance versus packet size, in which the network size is 2, the initial window size is 32, the bandwidth is 2 Mbps and the packet size varies from 512 to 8192 bytes. 1

11 Probability of transmission.3.2 Probability of transmission packet size=15 bytes Node Number=2 MBS=3 W=32.1 Node Number=2 MBS=3 W=32 packet size= 15 bytes Network Size BandWidth (Mbps) (a).vs. Network Size (b) Vs. Bandwidth Probability of transmission.3.2 Probability of transmission packet size=15 bytes MBS=3 Node Number=2 Bandwidth=2Mbps.1 MBS=3 W=32 Node Number=2 Bandwidth=2Mbps Initial Window Size Packet Size (c) Vs. Initial Window Size (d) Vs. Packet Size Figure 6: Average transmission Probability Probability of transmission τ Figures 6(a), (b), (c) and (d) plot τ, the probability of a station sending a packet at any time slot, on y-axis, versus network size, bandwidth, W and packet size, respectively. τ is derived for each station using Formula (4) and average is done over all stations. According to the analytic model, τ will decrease as the network size increases, because more stations cause more collisions, larger defer time and smaller transmission probability. The bandwidth and packet size do not affect the calculation of τ. This is confirmed in Figure 6 (b) and (d). As W increases, each station will defer for a longer time before the transmission of the packet, the probability of the channel being idle, P idle, will increase and P 4 will approach 1.. Thus the difference between the result of s model and our modified model will decrease. This is confirmed by Figure 6 (c). Figure 6(a), 6(b) and 6(d) confirm that (1) s model overestimates τ by up to 6%, and (2) simulation results fit quite well with analytical results from our model Probability of Collision P coll P coll is calculated for a station i as (1 τj) where τj is the probability of transmission of station j. We j=1,j i compute P coll for the overall network by averaging P coll over all stations. Figure 7 plots the average collision probability P coll on the y-axis. Figure 7 (a) shows quite clearly that as the network size increases, the collision increases. Figure 7 (c) shows that the collision probability decreases as W increases. We know from Figure 6 that τ decreases as W increases. Since the network size does not change in this set of results, a lower τ will naturally lead to a less collision probability. Figure 7 shows that (1) s model overestimates P coll by up to 2%, and (2) our model fits quite well with the simulation results. 11

12 Probability of collision Probability of collision packet size=15 bytes Node Number=2 MBS=3 W=32.1 Node Number=2 MBS=3 W=32 packet size= 15 bytes Network Size BandWidth (Mbps) (a).vs. Network Size (b) Vs. Bandwidth Probability of collision.3.2 Probability of collision packet size=15 bytes MBS=3 Node Number=2 Bandwidth=2Mbps.2 MBS=3 W=32 Node Number=2 Bandwidth=2Mbps Initial Window Size Packet Size (c) Vs. Initial Window Size (d) Vs. Packet Size Figure 7: Average Collision Probability Average Access Delay Figure 8 shows the access delay averaged on all packets of all stations. Since s model did not give out the estimation of the access delay, we only present the results from our model and the simulation results. The analytic results fits well with the simulation results. Figure 8(a) shows that the average access delay increases as the network size increases. This is because the collision probability increases as the network size increases. Thus, it will take more retransmissions to successfully send a packet. Figure 8(b) and 8(d) shows that the acess delay decreases as the bandwidth increases, and increases as the packet size increases. This is because the transmission time of each packet will decrease as the bandwidth increases, and increase as the packet size increases. Figure 8(c) shows that the initial window size has little effect on the delay. As the initial window size increases, the defer time before each station sends a packet will increase. This will positively affect access delay, but the collision probability decreases as the window size increases, which it takes fewer retransmissions to be successfully send a packet. This will negatively affect access delay. These two effects counteract each other. Somehow, Figure 8(b) and 8(d) show a relatively larger difference between the analytic results and the simulation results as the bandwidth and packet size increase. More work needs to be done to investigate this Average Throughput Every station in the ns-2 simulations are in the exact same condition, so they should have same access delay and throughput as well. In the simulation, stations get different access delays and throughputs. In order to get the characteristic of the system considering the difference between stations, it is important to find a proper way to derive the system parameters. For saturated throughput, every packet is sent to the MAC layer immediately after 12

13 Average Access Delay(s) Average Access Delay(s) packet size=15 bytes Node Number=2 MBS=3 W= Node Number=2 MBS=3 W=32 packet size= 15 bytes Network Size BandWidth (Mbps) (a).vs. Network Size (b) Vs. Bandwidth Average Access Delay(s) Average Access Delay(s) packet size=15 bytes MBS=3 Node Number=2 Bandwidth=2Mbps.2.1 MBS=3 W=32 Node Number=2 Bandwidth=2Mbps Initial Window Size Packet Size (c) Vs. Initial Window Size (d) Vs. Packet Size Figure 8: Average Access Delay the previous packet is sent out, which means there is no time gap between packets. Given the system s average access time, D access, for each packet, the throughput is calculated as T hroughput = P AY LOAD 8 D access Figure 9 plots the average throughput on the y-axis and shows that s model and ours closely match the simulation results. s model achieves a close estimation on throughput despite the overestimation of both transmission probability and collision probability. This is due to the fact that τ and P coll have opposite effects on throughput. A larger probability of transmission τ results in a station sending out packets more often, so system throughput is higher. On the contrary, a larger collision probability P coll implies more collisions and less successful transmissions. Therefore, a larger P coll negatively affects the system throughput. In conclusion, overestimates of τ and P coll cancel each other in the estimate of the system throughput. For the same reason as in the results in Figure 8(b) and 8(d), Figure 9(b) and 9(d) shows a relatively larger difference between simulation results and analytic results. 5 Extended IEEE model In Section 4, we discussed the model for the station that generates saturated traffic. We propose in this section to extend the model to describe the behavior of a network that may include source stations that generate traffic (not necessarily saturated) and relay stations that only forward traffic. By choosing different numbers of source and relay stations, we can analyze different network densities and traffic loads. To model relay stations, we add link layer queue length into the model to construct a three-dimensional markov model. This model allows to estimate the average queue length of relay stations and derive average one hop delay in ad hoc networks. Let us denote state space R, 13

14 Average Throughput(Mbps) 1.5 packet size=15 bytes Node Number=2 MBS=3 W=32 Average Throughput(Mbps) Node Number=2 MBS=3 W=32 packet size= 15 bytes Network Size BandWidth (Mbps) (a).vs. Network Size (b) Vs. Bandwidth Average Throughput(Mbps) 1 Average Throughput(Mbps) 1.5 packet size=15 bytes MBS=3 Node Number=2 Bandwidth=2Mbps.5 MBS=3 W=32 Node Number=2 Bandwidth=2Mbps Initial Window Size Packet Size (c) Vs. Initial Window Size (d) Vs. Packet Size Figure 9: Average Throughput R = {(q, b, c) : MaxQueuelength q, MBS b, c } where q is the current queue length, b is the current backoff stage and c is the current backoff counter value. A relay station listens to the medium, get packets from it and forward the packets it receives. The number of packets a relay station receives and accepts to forward depends on the upper layer routing protocol. For example, in the flooding protocol, a station broadcasts all its own packets and forwards packets from/to all its neighbors. A station will accept 1% of the traffic within its range. In diffusion routing protocol [7], a station will forward most of its traffic to the neighbor station on its estimated shortest path to the destination, and very little traffic to other neighbor stations. So the station on the shorted path will accept the packet with probability P a which is much higher than those of the other stations. In gossip based routing [9], the probability P a will be much smaller. In this mathematical model, we assume that the proability a station will accept to forward a successful packet is P in, which means this station accepts 1 P in percent of the successful packets in the network. Figure 1 outlines the model. The foreground plane in Figure 1 merely represents the two dimensional Markov model with a queue length q = i The model is extended in depth towards the background with increasing queue length q. The background plane is the two dimensional Markov model with queue length q = i + 1. Within the (b, c) plane with a fixed value q, the model works similarly to two dimensional model. A station in states on the (b, c) plane with queue value i (i.e., in state (i, b, c)) will transfer into states on the (b, c) plane with queue length i + 1 if it accepts a packet. A station in state on the (b, c) plane with queue value i + 1 (i.e., in state (i + 1, b, c)) will transfer some state (i,, c) if it completes a successful transmission. If a station stays in some state (, b, c) that has queue length of zero then the station has no packet to send. Therefore, we assume that states (, b, c) must have backoff stage b = and will not transfer into any other state unless the station gets a successful packet from the channel. This feature of the relay stations is different from that of the source stations since we assume that the source stations always have packets to send. 14

15 i+1;, i+1;,1 i+1;,2 i+1;,w 2 i+1;,w 1 i+1;j 1, P2 P5[j] P5[i] i+1;j, i+1;j,1 i+1;j,2 i+1;j,wi 2 i+1;j,wi 1 P5[j+1] P2 i+1;m, i+1;m,1 i+1;m,2 i+1;m,.. i+1;m,wm 1 P2 i;, i;,1 i;,2 i;,w 2 i;,w 1 P2 P5[j] i;j 1, P5[i] P2 P2 P2 i;j, i;j,1 i;j,2 i;j,wj 2 i;j,wj 1 P5[j+1] i;m, i;m,1 i;m,2 i;m,wm 2 i;m,wm 1 Figure 1: Overview of the Station Model As an example of how these three parameters describe the behavior, let us suppose that at any moment, the state of the station is (q, b, c). As time progresses, the backoff counter may go to zero, and one packet is sent out. If the packet is sent out successfully, the station transfers into state (q 1,, c); otherwise the station transfers into state (q, b + 1, c). If a station accepts a successful packet, then it transfers into state (q + 1, b, c). Parameters such as P 3, P 5[i], and P 4 can be derived the same way as they were in two dimensional model. The additional parameter introduced in this model is P 2. P 2 is the probability that a station transfers from state (q, b, c) to state (q + 1, b, c). In order to do such a transition, there must be a successful packet on the channel and the station is willing to accept the packet. From the derivation in two dimensional model, we know that the probability that there is a successful packet on the channel is P succ, and a station has probability P in to accept a packet, thus, P 2 = P succ P in P s in the three dimensional model is different P s in the two dimensional model. Since a station will not always stay in the current state when there is a successful packet on the channel, P s is modified as P s = 1 P 4 P 2 And, T Stay and EStay will be adjusted as well. Besides the performance metrics mentioned above, we can also derive the estimated average queue length for the relay stations. 15

16 Q = b c q Πi(q; b; c) q With the estimation of average queue length and average access delay, we can therefore estimate the one-hop message delay of a relay station. By one-hop message delay, we mean the time between when a packet gets into queue and when the packet is successfully sent. One-hop message delay is the accumulated access delay of all packets in the queue. Let D hop denote the average one-hop delay, we can calculate it as D hop = Q D access With the same methodology as we used in the previous section, we can derive the performance metrics for a network that has both source and relay stations. We already derived these performance metrics that we cannot include here for lack of space or refer to due to blind review requirements. 6 Conclusion and Further Work We discussed in this work s model, modified it, and proposed how to extend it to model and analyze a network with source and relay stations. We also derived some performance metrics of particular interest in ad hoc networks: one hop delay (queuing and access delay) and energy cost. s model is accurate in predicting the saturated throughput. However, this saturated throughput is based on overestimated probability of transmission τ and probability of collision P coll. Since the extra τ and extra P coll have opposite effects on the saturated throughput, they cancel each other and result in an accurate saturated throughput. However, τ and P coll are critical for the access delay: accuracy is critical. ns-2 simulations validate our model and yield τ and P coll results that fit quite well our model predictions. We propose a three dimensional model that take into consideration the queue length. We also introduce the probability P in to reflect the upper layer routing protocol. This enriched model extends the scope of from one single hop to multi-hop networks. A key part of the networking is the routing protocol, and routing protocols are all about how to distribute the load on every neighbor. If we understand the characteristics of the upper layer routing protocol and get the idea of how to simulate the routing protocol with a specified P in, which does represent the average load on each neighbor station, then we can describe the specified multi-hop network and analyze it. By combining our two dimensional model and three dimensional model, we can simulate a network with both source stations and relay stations. Different choices of parameters such as number of source stations, number of relay stations and P in, can result in different network scenarios. It provides a method to estimate the impact of network load on network performance. Although we limit our discussion to saturated traffic in our two dimensional model, it is easy to extend it to analyze the exponential traffic. For example, with saturated traffic, after a station completes a successful transmission, it will transfer into one of the states at stage. Then it will start to decrement the backoff counter to prepare for the next transmission. It is not always true if there is no saturated traffic since it could be possible that the station has no packet in the queue at all. If we consider an exponential traffic load with rate λ, then within any time interval of length t, the probability there is at least on packet in the queue is P λ (t) = 1. e λ t Then, if we simplify the probability function assuming λ t is small, let np denote the number of packets generated within time interval t, we have, P {np = } = e λt 1 λt P {np = 1} = λt For a station in any state such as (, c), it will transfer into state (, c 1) with probability P = P idle (λ D access) P s = 1 P With this modification, we can analyze the station with an exponential traffic load. If we do not make this simplification, we can add queue length into the model for source stations for further analysis. For example, a station in state (q,b,c) will transfer into state (q+np,b,c) with probability P = P idle e λ σ (λ σ)np np! + P succ e λ T ST (λ T ST )np np! + P fail e λ T F T (λ T F T )np np! In which, np (number of packets) varies from to infinity or queue limit. Further work is undergoing. We did not include this work here for lack of space. 16