Queueing Analysis of IEEE MAC Protocol in Wireless LAN

Transcription

1 Queueing Analysis of IEEE MAC Protocol in Wireless LAN Chul Geun Park and Ho Suk Jung Department of Information and Comm Eng Sunmoon University Chung Nam, , Korea Dong Hwan Han Department of Mathematics Sunmoon University Chung Nam, , Korea Abstract In this paper, we present the performance analysis of IEEE MAC(Medium Access Control) protocol by investigating the queue dynamics of a wireless station based on the general probability distribution of packet sizes We use the MMPP(Markov Modulated Poison Process) as the input traffic model well describing the bursty nature of Internet traffic Thus we have the MMPP/G/1/K queueing model with the MAC layer service time to analyze the throughput and the delay performance of IEEE MAC protocol in wireless LAN We take some numerical examples for the queue dynamics including the mean packet delay and the packet blocking probability 1 Introduction WLANs(Wireless Local Area Networks) are one of the major growth factors for the network industry in the upcoming years The IEEE has developed the widely accepted specification[1], which defines the medium access and physical layer functions of WLANs The IEEE standard for wireless networks incorporates two medium access methods: the mandatory DCF(Distributed Coordination Functions) method and the optional PCF(Point Coordination Functions) method[2] If contention free service is required, it can be provided by PCF, which best suits delay sensitive traffic The DCF is an asynchronous data transmission function, which suits delay insensitive traffic The DCF defines two techniques for packet transmissions The default is a two-way handshaking mechanism called basic access method Another optional technique is a fourway handshaking mechanism called RTS/CTS(Request To Send/Clear To Send) method[3] The CSMA/CA(Carrier Sense Multiple Access with Collision Avoidance) protocol requires stations to sense the channel idle for a specific time interval before attempting any packet transmission In case that the channel is sensed busy, stations defer their transmission attempts to a later time on the basis of a backoff algorithm Collisions can still occur in case of hidden stations[3] Moreover, packets may be dropped either due to the buffer overflow or serious MAC layer contentions and the packet delay increases dramatically when the number of active stations increase[4] A large amount of works on IEEE MAC protocols has been studied for the saturation throughput and delay analysis of CSMA/CA[2-7] The 2-dimensional MC (Markov Chain) model introduced by Bianchi[5] for the analysis of saturation throughput has become a common method to study the performance of the IEEE MAC protocol[3] Most of the literature start from the model proposed in [5] In the references [2,3,6,7], the delay analysis as well as the throughput analysis of the IEEE MAC protocols have been done Most of the previous studies [2-7] dealt with the throughput and delay analysis of IEEE MAC layer in saturation traffic scenarios The MAC delay analysis of the current works except [4] has been limited to the derivation of mean value while the packet delay distribution is untouched Fortunately, in [4], they had maid a comprehensive study on the queue dynamics of IEEE MAC protocol by using M/G/1/K queueing model with the MAC layer service time, when the packet size has a uniform probability distribution The MAC layer service time is the time interval between the time instant that a packet starts to contend for transmission and the time instant that the packet either is acknowledged for correct reception by the intended receiver or is dropped On the other hand, references[8] indicate that the trimodal packet size distribution has been demonstrated in subscriber access networks We know that the IP traffic to Internet access lines has bursty and unpredictable characteristics in nature and the MMPP describes well the bursty nature of Internet traffic Thus we have the MMPP/G/1/K queueing model to analyze the performance of IEEE MAC protocol The rest of this paper is organized as follows In Section 2, we provide an overview of queueing system including /06 $ IEEE

2 DIFS Data +ACK Successful transmission SIFS Backoff DIFS W 0 = CW min T S Backoff W 0 Collided Data Collision SIFS +ACK T C DIFS Backoff 2W 0 Figure 1 Basic access method based on the DCF of CSMA/CA the basic access method Section 3 describes the probability distribution of the MAC layer service time Section 4 describes the MMPP/G/1/K queueing model In Section 5, we take some numerical examples for the queue dynamics including the mean packet delay and packet blocking probability 2 System model The essential method used in DCF is called basic access method The RTS/CTS provides an alternative way of transmitting data packets[2] In this paper, we focus on the basic access method which is illustrated in Fig 1 When a packet arrives at the head of the transmission queue, it will first monitor the channel activity If the channel is scanned busy, the MAC waits until the medium become idle, then defers the packet transmission for an extra time interval DIFS(Distributed Inter-frame Space) If the channel stays idle during the DIFS, the MAC then starts the backoff process To begin the backoff process, each station maintains a contention window size CW, which takes CW min as an initial value and doubles its value before it reaches a maximum limit CW max The backoff counter is measured in terms of time slot and uniformly chosen in the range of [0,CW), where CW is current contention window A time slot is the time needed at any station to detect the transmission of a packet from any other station[5] If the channel becomes busy during a backoff process, the backoff is frozen When the channel becomes idle again and the backoff counter reaches zero, the station attempts to retransmit the packet If the maximum transmission failure limit is reached, the retransmission shall stop, CW max shall be reset to CW min and the packet shall be discarded We assume that the collision probability does not change and is independent during the transmission regardless of backoff stages The collision probability is defined by the probability that there is at least one of other stations which will transmit at the same backoff time slot We also assume that packet sizes are generally distributed and the size of data buffer at each station is finite Since Internet traffic has bursty and unpredictable characteristics in nature, we will use the MMPP as the input traffic model well describing the burstiness Thus the resulting queueing model for each station can be described as an MMPP/G/1/K queue 3 The MAC layer service time distribution 31 The MAC layer packet service time Since we assume that the packet size is generally distributed, we can obtain that the MAC layer service time is a non-negative random variable, which will be denoted by T B When the MAC layer transmits a packet, it experiences three basic processes: the decrement process of the backoff counter, the two frozen backoff counter processes that take time slots of T S and T C, respectively Hereafter we will use one time slot as a unit service time Thus T S (T C ) is the random variable representing amount of time slots that channel is sensed busy due to a successful transmission (collision) We define s i = P {T S = i} and c i = P {T C = i}, i =1, 2, as the probability distribution of T S and T C The MAC layer service time T B has a discrete probability distribution of b i = P {T B = i}, i =1, 2, Obviously, the probability distribution b i,i=1, 2, depends on the transmission rate, the length of the packet, and the specific medium access method[4,5] To find the PGF(Probability Generating Function) of T B, let Y p be a general discrete random variable of packet sizes and let a i = P {Y p = i},, 1, 2,,L p, where L p is the maximum packet length in time slot As shown in Fig 1, the period of successful transmission T S consists of DATA, ACK, SIFS and DIFS intervals Let h 1 =[ACK + SIFS + DIFS] +1in slot, where [ ] is the Gaussian integer Then we have the PGF of T S as follows L p S(z) = a i z i+h1, (1) where a i is the probability mass function for the packet size The collision probability T C is determined by the longest one of the collided packets Assume that the probability of three or more packets simultaneously colliding is neglected and let Y 1 and Y 2 be two random variables of packet sizes engaged in collision, then the probability distribution of T C can be calculated by the following equation P {T C = h 1 + i} = P {Y 1 = i, Y 2 i} + P {Y 2 = i, Y 1 i} P {Y 1 = i, Y 2 = i} /06 $ IEEE

3 transmit Let p s be the probability that there is one successful transmission among other (n 1) stations in the considered slot given that the current station does not transmit Then, by the equation (3), we have p s =(n 1)[(1 ) n 2 n 1 + pc 1] To find the packet transmission probability, we let W 0 = CW min and m be the maximum backoff stage such that 2 m W 0 = CW max and let W i =2 i W 0, where i, 0 i m, is called backoff stage Let s(t), b(t) and k(t) be the random processes representing the backoff stage, the backoff counter and the frozen period of the station at a slot t respectively Then the 3-dimensional process {(s(t),b(t),k(t))} forms a discrete time MC depicted in Fig 2 At each transmission, the backoff counter is uniformly chosen in the range (0,W i ), where i is the current backoff stage, ie the number of transmission failed for the considered packet We denote the one-step transition probability of {(s(t),b(t),k(t))} by P {(i 1,j 1,k 1 ) (i 0,j 0,k 0 )} = P {(s(t +1)=i 1,b(t +1) = j 1,k(t +1)=k 1 ) (s(t) =i 0,b(t) =j 0,k(t) =k 0 )} Figure 2 State transition diagram of the discrete time Markov chain Thus we can obtain the PGF of the random variable T C as L p C(z) = [2a i F Y (i) a 2 i ]z i+h1, (2) where F Y (i) is the cumulative distribution function of a i,, 1, 2,,L p 32 Packet transmission probability and Markov chain The backoff process decreases its counter by one for every idle slot detected We let be the collision probability seen by a packet being transmitted on the medium Assume that there are n active stations in wireless LAN and packet arrival processes at all stations are independently and identically distributed Then we have =1 [1 (1 p 0 )τ] n 1, (3) where p 0 is the idle probability that there are no packets to transmit at the MAC layer of the considered station and τ is the packet transmission probability that the station transmits in a random time slot given that the station has packets to Then the one-step transition probabilities for the backoff stages and backoff counters at the original state of the frozen period are summarized as follows, for i =0, 1,,m, P {(i, j, 0) (i, j +1, 0)} =1, 0 j W i 2, P {(i, j, 0) (i 1, 0, 0)} = /W i,i 0, 0 j W i 1, and for 0 j W i 1, P {(0,j,0) (i, 0, 0)} = { (1 )/W 0, i m, 1/W 0, i = m The one-step transition probabilities of the frozen period at the backoff stages i and the backoff counter j are given by, for i =0, 1,,m, 0 j W i 1, P {(i, j, k 1) (i, j, k)} =1, 0 <k L, P {(i, j, k) (i, j, 0)} = q k, 0 <h 1 <k L, where L = h 1 + L p Thus we have s k = a k h1, c k = 2a k h1 F Y (k h 1 ) a 2 k h 1, k = h 1,,L, by the definitions of T S and T C Hence q k, is a distribution of the discrete random variable representing sum of packet size and some inter-frame spaces as follows q k = p s s k + p s c k Let b i,j,k = lim P {s(t) =i, b(t) =j, k(t) =k}, 0 i t m, 0 j W i 1, 0 k L be the stationary distribution of the MC Then, in steady state, we can derive following relations through the state transition diagram First, /06 $ IEEE

4 we have, for i =0, 1,,m, j =1,,W i 1, b i,j,0, 1 k h 1, b i,j,k = b i,j,0 q l, h 1 +1 k L l=k By the recursive relation, we have b i,0,0 =( ) i b i,0,0, 1 i m, b i,0,0 =(1 )b i,1,0 + b i 1,0,0, 1 i m, W i b i,j,0 =(1 )b i,j+1,0 + b i 1,0,0 + b i,j,1 W i = W i j b i,0,0, 1 i m, 1 j W i 1 (1 )W i By the normalization condition, we have 1= m W i 1 j=1 b i,j,0 + m For simplicity, let η =1+ b 0,0,0 = [ ηw0 (1 (2 ) m+1 ) 2(1 )(1 2 ) W i 1 j=1 m b i,j,k + b i,0,0 k=1 k=h 1+1 η(1 pm+1 c ) kq k, then we have 2(1 ) ] 1 pm+1 c 1 When the backoff counter reaches zero, a station will attempt to transmit packet regardless of backoff stage So we can find the probability τ that the station which has a packet in its buffer transmits in a randomly chosen time slot Thus we have[ ηw0 (1 (2 ) m+1 ] 1 ) τ = 2(1 2 )(1 p m+1 c ) η 2(1 ) The PGF of MAC layer service time In the backoff process, if the medium is idle, the backoff counter has the probability 1 to decrement by 1 during a time slot and the probability to stay at the original state (i, j, 0) during the frozen period Moreover the frozen period has the probability p s to stay at the original state during T S and has the probability p s to stay at the original state during T C LetH b (z) be the PGF of time interval needed to decrement the backoff counter by 1 Then we have (1 )z H b (z) = 1 z q k z k k=h 1+1 By using the above equation, we can obtain the PGF of the MAC layer service time T B, m 1 B(z) =(1 )S(z) ( C(z)) i H i (z) +( C(z)) m S(z)H m (z), (4) where S(z) and C(z) are given in the equations (1) and (2) and H i (z) is defined by i 2 j W H i (z) = 2 j (H b (z)) k W 0 j=0 4 Queueing analysis In this section we present the considered MMPP/G/1/K queue with the MAC layer service time A data buffer with finite capacity K including a single server is fed by an d- state MMPP process The MMPP is an elementary and useful doubly stochastic Poisson process in which the arrival rate is given by λ[j(t)], where J(t), t 0 is an d-state irreducible MC The d-state MMPP is characterized by the infinitesimal generator Q of the underlying MC and by an arrival rate matrix Λ=diag(λ 1,λ 2,,λ d )Weuseadstate MMPP to model the input traffic Let A(t) be the number of arrival packets during the time interval [0,t) and let J(t) be the state of the underlying MC at time slot t We denote the transition probabilities of the process {A(t),J(t),t 0} by P ij (n, t) =P {A(t) =n, J(t) =j A(0) = 0,J(0) = i} The d d matrix P (n, t) of the transition probabilities has the PGF as follows P (z,t) =e [Q+(z 1)Λ]t, z 1 Let X(t) be the number of packets in the data buffer at time slot t Let t n be the n-th packet departure instant and let X n = X(t n +) and J n = J(t n +) be the state of the queueing system and the state of the underlying MC just after t n respectively Then {(X n,j n ),n 0} forms a finite MC with state space {0, 1,,K 1} {1, 2,,d} The onestep transition probability matrix P is given by A 0 A 1 A 2 A K 2 Ā K 1 A 0 A 1 A 2 A K 2 Ā K 1 P = 0 A 0 A 1 A K 3 Ā K 2, A 0 Ā 1 where A k = P (k, t)p (T B = t), Ā k = A i, and T B t=1 i=k is the MAC layer service time in Section 3 To find A k,let U be the matrix which accounts for the evolution of J(t) during server s idle periods and whose (i, j) entry denotes the conditional probability of reaching phase j at the end of an idle period, starting from phase i, then we have U = 0 e (Q Λ)t Λdt =(Λ Q) 1 Λ By using the matrices U and A k, we can obtain A k and Ā K 1 as follows A k = UA k =(Λ Q) 1 ΛA k, Ā K 1 = A i i=k /06 $ IEEE

5 Let π k,j be the limiting probability of (X n,j n ), ie, π k,j = lim n P {X n = k, J n = j}, and let π = (π 0,π 1,,π K 1 ) with π k = (π k,1,π k,2,,π k,d ), 0 k K 1 Then we have the steady-state probability vector π by the equation π P = π with πe =1, where e is a column vector with all ones Next we will find the limiting distribution of (X(t), J(t)) at an arbitrary time Let x k,j be the limiting probability as x k,j = lim t P {X(t) =k, J(t) =j X(0) = 0,J(0) = i}, and let x =(x 0,x 1,,x K ) with x k =(x k,1,,x k,d ) By supplementary variable method, we can find the steady-state probability vector x Let the random variable Γ be the steady-state of server, namely Γ=1if the server is busy and Γ=0if it is idle Then the probability that the system is busy is given by γ 1 P {Γ =1} = E[T B ] E[T B ]+π 0 (Λ Q) 1 e, where E[T B ] is the MAC layer mean service time Thus the relation between the state probabilities at an embedded time point and an arbitrary time point is given by x 0 = γ 1 π 0 (Λ Q) 1 E[T B ] 1, x n = γ 1 [π n + n 1 π k U n 1 k (U I)](Λ Q) 1 E[T B ] 1, x K = θ K 1 n=0 x n, where θ is the steady-state probability vector of the underlying MC, ie, θq =0[9] Note that x 0 gives the idle probability p 0 = x 0 e appeared in the equation (3) The packet blocking probability P B for an arbitrary packet is given by K P B = x K Λe/( x k Λe) We can use Little s law with the effective arrival rate λ = θλe of in order to find the mean packet delay W, K λ (1 P B )W = kx k e By using P B and the packet discard probability due to transmission failures, we have the throughput S at each station by S = λ (1 P B )(1 p m+1 c ) 5 Numerical results In this section, we present some numerical results to show the property of the discrete probability distribution of B(z) for the MAC layer service time, the packet delay and the packet loss performance and the queue dynamics of the MMPP/G/1/K queue with B(z) as a service time model We use the system parameters for FHSS(Frequency Hopping Spread Spectrum) PHY-specification and DCF access Probability distribution n=10, m=5, =03, basic Real data Exponentail Log normal Erlang(E2) MAC Service Time [x5ms] Figure 3 PDFs of MAC layer service time Packet blocking probability =01 =013 =02 MMPP/G/1/K queue, = 01, 013, Effective arrival rate Figure 4 Packet blocking probability wrtλ method[4] In addition, the other parameters such as the channel bit rate 2Mbps, the length of a time slot 50μs, the number of active stations n =10, the maximum backoff stage m =5and the initial value of a contention window W 0 =32are fixed We assume that the propagation delay is neglected and the channel is error-free[2] We assume that three modes correspond to the most frequent packet sizes, 64 bytes(47%), 594 bytes(23%) and 1518 bytes(30%)[8] From Fig 3, we can see that the 2-stage Erlangian distribution is a good approximation to the real distribution of the MAC layer service time for the basic access method Now we present some numerical result of MMPP/G/1/K queueing model with the MAC layer service time In this part, we deal with also the basic access method We use a simple 2-state MMPP (Q, Λ) as an arrival process, which has the following representation with the effective arrival rate λ =(r 2 λ 1 + r 1 λ 2 )/(r 1 + r 2 ), ( r1, r Q = 1 r 2, r 2 ), Λ= ( λ1, 0 0, λ 2 ) /06 $ IEEE

6 Packet delay MMPP/G/1/K queue, = 01, 013, 02 =01 =013 =02 tensity varies From this figure, we can see that the system throughput depends on the collision probability when the traffic intensity varies We need further studies to give the detail comparison of the queue dynamics between the basic access and the RTS/CTS access and to investigate the effect of the traffic burstiness on the queue dynamics Acknowledgement This research was supported by the MIC, Korea, under the ITRC support program supervised by the IITA Thouughput Effective arrival rate Figure 5 Mean packet delay wrt λ =01 =013 =02 MMPP/G/1/K queue, = 01, 013, Traffic intensity Figure 6 System throughput wrt ρ Figs 4 and 5 show the packet blocking probabilities and the mean packet delays of the basic access method when the effective arrival rate varies in three cases of collision probabilities =01, 013, 02, respectively Moreover, from the equation (4), we can obtain the corresponding mean MAC service times E[T B ]=11037, and 21387, respectively We take the parameters r 1 =01, r 2 =02 and λ 2 =004 fixed But the effective arrival rate varies, since the arrival rate λ 1 varies From Fig 4, we can see that the packet blocking probabilities have sharhanges around effective arrival rates λ =0036, 006 and 074 in cases of =01, 013 and 02, respectively This is because the collisions increase significantly around this traffic load and increase rapidly as large as the collision probabilities are From Fig 5, we the similar results on the mean packet delays when the effective arrival rate varies in three cases of collision probabilities =01, 013, 02, respectively Fig 6 shows the system throughput when the traffic in- References [1] IEEE Standard ParT II: Wireless LAN Medium Access Control (MAC) and Physical Layer(PHY) Specifications (1999) [2] 2 P Chatzimisios, V Vitas and AC Boucouvalas, Throughput and delay analysis of IEEE protocol, IEEE INFOCOM 02 1 (2002) [3] YT Lee, DH Han and CG Park, Saturation Throughput and Dealy Analysis of DCF in the IEEE Wireless LAN, WSEAS Trans on Comm 4 (2005) [4] H Zhai, YG Kwon and Y Fang, Performance analysis of IEEE MAC protocols in wireless LANs, Wireless Comm and Mobile Computing 4 (2004) [5] G Bianchi, Performance Analysis of the IEEE Distributed Coordination Function, IEEE J on Selected Areas in Comm 18 (2000) [6] P Chatzimisios, AC Boucouvalas and V Vitas, Packet Dealy Analysis of IEEE MAC Protocol, Electronics Letters 39 (2003) [7] H Chen and Y Li, Analytical Analysis of Hybrid Access Mechanism of IEEE DCF, IEICE Trans on Comm E87-B(2004) [8] CG Park, DH Han and B Kim, Packet Dealy Analysis of Dynamic Bandwidth Allocation Scheme in an Ethernet PON, Networking-ICN05, LNCS 3420 (2005) [9] A Baiocchi and N Blefari-Malzzi, Steady-State Analysis of the MMPP/G/1/K queue, IEEE Trans on Comm 41 (1993) /06 $ IEEE