Performance Model for TCP over Link Layer on Correlated Channels for Wireless Data Communication

Transcription

1 Performance Model for TCP over Link Layer on Correlated Channels for Wireless Data Communication Yi Wu, Zhisheng Niu, Junli Zheng State Key Lab on Microwave and Digital Communications Department of Electronic Engineering, Tsinghua University, Beijing, PR China Abstract - The focus of this paper is to propose an analytical model for the operation of TCP over multi-copy SR ARQ systems on wireless fading channels A major contribution of this paper is a competent analytical approach which promotes better behavior understanding of the wireless link layer protocol with correlated errors and the interaction between end-to-end TCP and local link layer protocol The analytical TCP model suggested by Padhye [1] is also reinvestigated in the wireless portion of the network The accuracy of the analytical model is validated by simulation I INTRODUCTION Recently, research work has begun to focus on performance enhancing mechanisms at all layers of the network in order to deliver high performance at the end-user level Residing between physical channel and transport layer, the link layer plays an important role in protocol stacks of the wireless networks A better understanding of link layer behavior on fading channel would significantly contribute to the design and evaluation of the higher layer protocols such as TCP [2] In particular, the objective of this paper is to analytically investigate the behavior of ARQ retransmission scheme over the correlated fading channels and to quantify the joint impact of the protocol stacks on the end-to-end TCP performance In this paper we focus on all three layers of the networking stacks: the physical layer with correlated channels, the wireless link layer with the NAK-based multi-copy Selective Repeat (SR) ARQ scheme, and the transport layer with TCP Reno In the network we consider, TCP is present at the wired portion as well as the mobile station and ARQ is active at the wireless portion of the end-to-end path By developing an accurate analytical model of link layer protocol over correlated fading channel, the performance of TCP/ARQ systems in the environment of correlated errors is investigated in depth II MODEL DESCRIPTION A Correlated Channel Model of the Physical Layer The error process at the frame level of wireless channel has been modeled as a first-order, discrete-time, two-state This paper has been supported by NSFC( ) & the Excellent Young Teachers Program of MOE, P R C Markov chain, which is shown to be accurate for correlated fading [3] in spite of its simplicity Let G and B denote good and bad states in which frame transmissions would be successful (without frame errors) and unsuccessful (with frame errors), respectively Then the one-step transition probability matrix of the Markov chain is given by [ ] 1 γgb γ Γ = GB (1) γ BG 1 γ BG where γ XY denotes the probability that the current frame transmission is successful (if Y = G) or unsuccessful (if Y = B), given that the previous frame transmission was unsuccessful (if X = B) or successful (if X = G) Denoting by P E the frame error rate of physical layer and by T the normalized Doppler bandwidth which describes the correlation of the fading process, the Markov parameters can be derived as shown in [3]: where γ GB γ BG = Q(θ, ρθ) Q(ρθ, θ), (Q(θ, ρθ) Q(ρθ, θ)), = 1 P E P E θ = (2) 2 ln(1 P E ) 1 ρ 2 (3) Here ρ = J 0 (2π T ) is the Gaussian correlation coefficient of two successive frames with T seconds interval over a fading channel with Doppler frequency J 0 ( ) is the first kind Bessel function of zero order, and Q(, ) is the Marcum Q function [4] In this way the Markov parameters are totally expressed in terms of the physical parameters P E and T In our analysis we apply this Markov model at frame level of the link layer Although motivated by the behavior of wireless fading process, the error model considered here also applies to any environment where the frame loss process exhibits correlation, eg, due to trellis coding B System Model of Radio Link Protocol In this paper we concern the NAK-based multi-copy Selective Repeat ARQ scheme which is widely applied by the radio link protocol of wireless systems The receiver does not acknowledge received data frames, but only requests the retransmission of the particular frames that were not received correctly In analysis, the basic unit we consider is a NAK round, defined as the interval that begins when

2 H start phase end phase intermediate phase an arbitrary frame correct frame erroneous frame F 0 S 0 # 1 # 2 # l f # l e # l e +1 # 1 # 1 # l f # l f F 1 S 1 S 2 error burst loss detection by the receiver N 1 copies retransmission N 1 copies Fig 1 F R-1 F R S R The state transition diagram of wireless link recovery process receiver supplies the last of one or more NAK frames for a missing data frame described by a NAK list entry and ends when the retransmission timer expires for that NAK list entry[5] For convenience of expression, among the link layer settings we denote by R the maximum number of NAK rounds limited by link layer, and by N i (i = 1 R) the number of NAKs to be sent in the i th NAK round which will trigger N i multiple copy retransmissions for the lost data frame The system model for the wireless link recovery is illustrated in Fig1 with the state definitions as follows: H: transmitter begins to supply the particular data frame; S 0 : the data frame is received successfully by the receiver; F 0 : the data frame is detected missing by the receiver; S k : the recovery is successfully accomplished in the k th NAK rounds, for k = 1, 2,, R; F k : the recovery process fails in the k th NAK rounds, for k = 1, 2,, R C TCP End-to-end Connection Fig2 shows the network topology considered in this paper We focus on the scenario where the TCP packets are sent from a fixed station to a mobile host passing through an intermediate base station, which is general in practice The wired part of network is assumed to have a bandwidth of 40Mbps with a long propagation delay of 200ms As the last hop of the network, the wireless link is assumed to be the bottleneck oata transfer with the limited bandwidth of 1Mbps The propagation delay of the local wireless link is assumed to be negligible compared with the frame transmission duration This is a representative situation where the interaction of link layer protocol and TCP is minimized Fixed Station Fig 2 TCP packet ACK 40Mbps 200ms Base Station 1Mbps << 1ms link recovery TCP connection End-to-end TCP connection under study Mobile Host Time Fig 3 Illustration of the error burst and the retransmission sequence at receiving side and the local recovery process could have a more efficient reaction and consume less time than the end-to-end error correction process of TCP III ANALYTICAL APPROACH A Analysis of SR ARQ over Correlated Channel Models Without loss of generality, we assume each link frame is carried by a single physical layer frame For the local wireless link, the propagation delay and the transmission time of a NAK frame are assumed to be negligible compared with the data frame transmission duration Also, we assume the backward channel to be error-free so that the NAKs will never be lost This is reasonable in practice due to the robust error protection for NAK control frames From the diagram of the phase transition as shown in Fig 1, it is reality that P (F k H) = P (F k F k 1 )P (F k 1 H), 0 < k R (4) where P (Y X) denotes the transition probability from phase X to phase Y Since the phase transition probability P (S 0 H) and P (F 0 H) are the successful and unsuccessful probabilities of the original data frame transmission respectively, we get P (S 0 H) = 1 P E, P (F 0 H) = P E (5) If the initial transmission of the data frame fails, the link recovery process goes on for retransmissions as illustrated in Fig 1 According to the NAK-based SR ARQ strategy, the detection oata frame loss is only performed when at least one of the subsequent data frame with a larger sequence number is received successfully Thus, one can conclude that the first NAK round for any missing data frame must follow a successful transmission of some subsequent frame Therefore, the initial channel state distribution at the instance of loss detection denoted by α is α = [1 0] (6) Fig3 illustrates the retransmission process in details Since the frame errors are correlated, the sequence of the arbitrary lost frame within the error burst should be derived first For the two-state Markov chain as in Eq(1), it is well known that the length of the error burst N has the geometric distribution with average value of 1/γ BG : P {N = n} = γ BG (1 γ BG ) n 1, n = 1, 2, 3, (7)

3 Then it is reasonable to assume that the particular lost frame is uniformly distributed within the N-frame error burst Denoting by L the sequence of the lost frame,its probability distribution is given by P {L = l} = n=l 1 np {N = n} = γ BG 1 γ BG lnγ BG l 1 n=1 Fixing the random variable L as l, we have (1 γ BG ) n 1 γ BG n (8) P (l) (F 1 F 0 ) = αγ (l 1)N 1 Φ N 1 e (9) where e denotes the all 1 s column vector and Φ denotes the transition probability matrix of an erroneous frame transmission, which represents that the correlated channel falls in the bad state: [ ] [ ] γgb Φ = Γ = (10) γ BG In Eq(9), Γ (l 1)N 1 denotes the transition probability of the channel state during the retransmissions of the previous l 1 lost frames of the current error burst, and Φ N 1 represents the probability matrix for the event that all N 1 retransmitted copies of frame l are corrupted Denote by τ LL the retransmission timeout interval normalized by the frame transmission time Using Bayes formula, from Eq(9) we have τ LL P (F 1 F 0 ) = P {L = l}p (l) (F 1 F 0 ) (11) l=1 where the summation is truncated at τ LL because of the assumption that τ LL is set sufficiently large compared to the transmission time of frame Recalling Eq(4), the residual error probability after the first retransmission is given by P (F 1 H) = P (F 1 F 0 )P (F 0 H) τ LL = P {L = l}p (l) (F 1 F 0 )P E (12) l=1 From the second NAK round the retransmissions are all triggered by timeout events, and therefore the residual error probability P (F k H) after k NAK rounds is derived recursively by [ k ] P (F k H) = β Γ τ LL Φ N i e P (F 1 H), i=2 k = 2,, R (13) where β = [0 1] represents the distribution of the channel states after the first failed retransmission (F 1 ), Γ τ LL represents the transition probability of the channel states during the timeout interval τ LL in each NAK round, and Φ Ni is the probability matrix that the N i retransmitted packets lost in the i th NAK round Since the number of retransmission times at the link layer is limited by R, the residual frame error rate denoted by P LL being provided to TCP is given by P LL = P (F R H) (14) To be consistent with the general definition of throughput at TCP layer, we denote by η LL the normalized throughput of the link layer which is defined as the average number of data frames transmitted during a frame transmission time, independent of whether the delivery is successful or not As every failure of link recovery in the i th NAK round results in the (i + 1) th retransmission of N i+1 multiple retransmitted copies, the normalized throughput is derived as 1 η LL = 1 + Σ R k=1 N kp (F k 1 H) (15) B Analysis of TCP over link layer To keep track of the analytical model developed in [1], let d RT T denote the round trip time of TCP connection, T 0 denote the value of TCP time-out and b denote the number of segments that are acknowledged by a received ACK (b is typically 2 due to the mechanism oelayed ACK) Because this paper concentrates on performance influence of the wireless fading, we suppose that all the packet errors are due to wireless loss for convenience of analysis And the assumption in [1] that the segment losses are bursty in a window due to network congestion does not work anymore As derived in [7], with a large advertised TCP window, the expression for the steady state TCP throughput denoted by, in a wireless context only, as a function of wireless loss probability p is: where = E[W ] + (2 + 1 p ) + 3 E[W ](1 p) ( b 2 E[W ] + 1)d, (16) f(p) RT T + 3T 0 1 p E[W ] = 4 3b [( b4 + 1) + ( b 4 + 1)2 + 3b 2p ], (17) f(p) = 1 + p + 2p 2 + 4p 3 + 8p p p 6 (18) Because the maximum number of local retransmission times has been strictly limited which is usually set as small as 3, link layer retransmissions are indeed much faster than end-to-end ones driven by TCP Furthermore, when we focus on a heterogeneous network with the topology as in Fig 2, where the delay time of the wired portion is much larger than that of the local wired link due to multiple routing and queueing, the impact of the local link recovery delay on the TCP end-to-end round trip time can be ignored Consequently, the probability of the interference between TCP and local recovery process that may lead to the conflicting retransmissions is kept small enough Therefore in this sense, the residual error probability and the normalized throughput provided by the link layer perform as the major factors that would influence the end-to-end TCP performance The so-called goodput defined as the normalized rate of successful data transmission at the receiver yields η LL (1 P LL )

4 Residual Error Rate after Local Recovery 1E-6 1E-7 T= (analysis) T= (simulation) Fig 4 Residual FER of RLP on correlated channels (P LL ) Normalized Throughput of Link Layer 95% 90% 85% 80% 75% 70% 65% T= (simulation) T= (analysis) Fig 5 Normalized throughput of RLP on correlated channels (η LL ) In the case of TCP over SR ARQ, the error process of the physical channel is shielded by the link layer so that the loss probability provided to TCP as p in Eq(16) equals the residual error probability of the TCP segment after the local link recovery Consider the residual error process after the link recovery process 1)When the wireless channel suffers fast fading, ie the normalized Doppler bandwidth T is large enough (eg T = 1), the correlation of errors is limited within a link frame Thus the residual error probability of TCP segment is determined by p = 1 (1 P LL ) l num, where P LL is the residual frame error rate of link recovery and l num represents the number of link frames that a TCP segment is divided in 2)When the wireless channel suffers slow fading, ie T is sufficiently small (eg T = ), the correlation degree is high that errors always occur in burst The residual error probability of TCP segment is approximated by p = P LL 3)When the fading speed of the correlated channel is moderate, the residual error probability of TCP segment is bounded by P LL p 1 (1 P LL ) l num Actually the TCP throughput will be limited by the data throughput of the underlying link layer [6] Assume that the channel capacity of the wireless link as C l and together with the expression derived in Eq(16), finally the TCP throughput is determined by { E[W ] + (2 + 1 p = min η LL C l, ) + } 3 E[W ](1 p) ( b 2 E[W ] + 1)d, f(p) RT T + 3T 0 1 p (19) where η LL is the normalized data throughput of link layer in Eq(15) IV NUMERICAL RESULTS AND DISCUSSIONS To study the interactions between link layer schemes, wireless fading channels and TCP at transport layer, we examine the numerical examples based on analysis and simulation In our simulation, we employ ns-2(version 21b9) Additional source codes for wireless error modules and radio link module of the NAK-based multi-copy SR ARQ schemes are also developed and implemented Numerical results of the theoretical analysis on link layer performance are exhibited in Fig 4-5 Besides the analytical curve, simulation points are also given simultaneously showing the accuracy of the analytical approach Fig 4 compares the residual frame error rate after local wireless link recovery over fading channels with different normalized Doppler bandwidth T The residual frame error rate of link layer increases with correlation degree of the error process Explanatorily, the bursty characteristic of error process increases the failing probability of all retransmission trails which leads to the degradation of reliability performance Fig 5 shows the normalized throughput performance of link layer over the correlated channels The results show that the data throughput of link layer benefits from correlation of the underlying error process What makes the throughput performance of link layer benefit from the error correlation? The relationship between throughput and residual frame error rate of link layer in the environment of burst errors need to be thought over To gain an insight into the behavior of link layer protocol, the evolution of the local link recovery process along with increasing NAK rounds is illustrated in Fig 6 Considering the NAK-based SR retransmission process, the first NAK round is not triggered at the receiver until that a successful delivery event of the future data frame occurs, ie, the initial channel state at the beginning of the first NAK round is good as expressed in Eq(6) Therefore the retransmission of the first NAK round unconsciously utilizes the correlated information of the physical channels and certainly benefits from the correlation of the error process Unlike the first NAK round, subsequent NAK rounds are all triggered by timeout events of retransmission timer, so that the benefit from the initial state distribution does not work anymore and the degradation influence of the burst error process dominates the residual error rate This finally makes the residual error rate P (F 3 H) smaller in the case of less correlated channels Then recalling Eq(15), the value of throughput η LL directly depends on weighted aggregation of the residual frame error rates of all NAK rounds As the dominating element with the largest value (above 10 times of P (F 2 H)), P (F 1 H) absolutely determines the relative

5 Residual Error Probability after 1st NAK Round Residual Error Probability after 2nd NAK Round Residual Error Probability after 3rd NAK Round 01 1E-6 1E-7 1E-8 1E-9 Fig 6 P(F 1 H) T= T=1 P(F 2 H) T= T=1 P(F 3 H) T= T=1 Evolution of residual FER for all 3 NAK rounds difference of link layer throughput over different fading channels Next the performance issue of end-to-end TCP connection at user level is studied The values of T adopted are and 1 which correspond to the different correlated channels with slow fading and fast fading, respectively The analysis results from the refined TCP throughput model are also validated by computer simulations as shown in Fig 7 Although it has been investigated that TCP could perform better due to the high degree of correlation in the error process, the research there was engaged in the case of wireless TCP without local recovery of radio link protocols Our work reveals that with the cooperation of link layer, the TCP performance mainly lies on three factors: the bandwidth limit of wireless link, the residual error probability after local recovery and the correlation degree of the residual errors As shown in Fig 7, with a low physical frame error rate the TCP end-to-end throughput is chiefly limited by the capacity of wireless link C l which is supposed as 1Mbps in this paper When the channel condition turns worsen, the failure probability of local link recovery increases as in Fig 4 Incremental loss of TCP segments frequently shrinks the TCP congestion window and declines the performance sharply In this error-prone situation, the TCP throughput becomes under the joint effect of the residual error probability and the residual error correlation The higher probability of residual errors leads to more degradation of TCP performance, whereas the higher correlation degree of residual errors leads to the performance improvement due TCP end-to-end throughput Kbps T= (analysis) T= (simulation) Fig 7 TCP end-to-end throughput with local link recovery over fading channels to the TCP AIMD (Additional Increase Multiple Decrease) mechanism However, the residual error probability and the correlation degree of the residual packet errors are always a pair of contraries, both increasing with the error correlation of physical layer Accordingly, in the case of T = 1 the comparatively low residual error rate causes the distinct advantage of the performance over the T = case V CONCLUSION In this paper, we have developed a new analytical approach of TCP over link layer protocols for wireless systems in the two-state Markov channels Results show that the burstiness in the error process of physical layer caused by wireless fading significantly affects the system performance Although the data throughput efficiency of link layer benefits from the channel correlation, the degradation of the residual error probability after local recovery due to the error burstiness dominates the behavior of link layer and finally degrades the end-to-end TCP throughput The analytical model is suggested to be helpful for the designing and configuration of the mobile networks to finally improve the end-to-end TCP performance at user level REFERENCES [1] JPadhye, VFiroiu, DTowsley, JKurose, Modeling TCP Reno performance: a simple model and its empirical validation, IEEE/ACM Trans Networking, Vol 8, pp , April 2000 [2] C Hoene, I Carreras, T Chen, A wolisz, Design and deployment of link-layer boosters for per-flow improvement of QOS in wireless Internet access, in European Wireless 2002, Florence, Italy, Feb 2002 [3] M Zorzi, RR Rao, LB Milstein, On the accuracy of a first-order Markov model for data transmission on fading channels, in Proc IEEE ICUPC 95, pp , Nov 1995 [4] Marvin K Simon, Mohamed-Slim Alouini, Digital Communication over Fading Channels: A Unified Approach to Performance Analysis, Wiley, New York, 2000 [5] TIA/EIA/IS-707-A-2, Data services options for spread spectrum systems: Radio link protocol Type 3, Addendum No 2, Washington: Telecommunication Industry Association, Mar 2001 [6] F Khafizov, M Yavuz, Analytical Model of RLP in IS-2000 CDMA Networks, in Proc IEEE VTC 2002 Fall, vol1, pp , Oct 2002 [7] F Zheng, M Li, C Gao, An analytic throughput model for TCP Reno over wireless networks, in Proc IEEE ICCNMC 2001, pp , Oct 2001