Stochastic Delay Analysis for Protocols with Self-similar Traffic

Transcription

1 22 7th International ICST Conference on Communications and Networking in China (CHINACOM) Stochastic Delay Analysis for 82 Protocols with Self-similar Traffic Jie Zhang, Li Yu, Lie Jiang, Jingjing Luo Department of Electronics and Information Engineering, Huazhong University of Science and Technology, PRC Wuhan National Laboratory for Optoelectronics, Wuhan, Hubei, PRC Abstract IEEE 82 wireless local area network (WLAN) is one of the most deployed wireless technologies all over the world An accurate and comprehensive performance evaluation is crucial for the quality of service (QoS) guarantee in 82 networks In this paper, we use the method of stochastic network calculus to obtain the stochastic delay bounds of 82 protocols In particular, we take the self-similar property of wireless traffic into consideration and fractional Brownian motion (fbm) traffic model is adopted to describe the arrival traffic We derive the stochastic arrival curve of fbm traffic and the stochastic service curve of existing model of 82, then obtain the stochastic delay and backlog bounds Finally, we use matlab to calculate theoretical results and ns-2 for simulations The results show that the delay of the 82 wireless node increases with the increasing of Hurst parameter H and the number of competition stations n And the results also indicate that the theoretical results well bound simulation results I INTRODUCTION Wireless networking is one of the exciting developments in the world of networking technology IEEE 82 protocols have been proposed as the standard protocols for wireless local area networks In order to provide QoS guarantees, it is necessary to characterize the delay and other performance metrics in 82 networks Traditional method for performance analysis of 82 protocols is the method of queuing theory In [], Zhai et al assumed Poisson traffic arrival and proposed an M/G/ queueing model of 82 In [2], [3], Tickoo modeled each node as a discrete time G/G/ queue and derived the service time distribution while accounting for a number of factors However, recent researches [4], [5] show that the wireless LAN traffic processes have the property of long-range independence (LRD) and statistical self-similarity, which means that the traditional traffic model such as Poisson arrival model and CBR arrival model can not fulfill these properties FBm [6], [7], [8] is a traffic model which captures self-similarity FBm is characterized by the Hurst parameter H WhenH ( 2, ), fbm is LRD Due to the complexity and the burst property of self-similar traffic, queuing theory is not suitable for the performance analysis of wireless network In this paper, we adopt the method of stochastic network calculus to analyze this problem Network calculus which is proposed in [9], [], [] is a recently developed approach to take a deep insight into flow problems encountered in networking, especially on QoS guarantee analysis Due to the stochastic property of arrival and service process, network calculus has developed from deterministic to stochastic in [2], [3], [4] The stochastic network calculus is more suitable for the networks which only provide stochastic guarantees Unlike queuing theory, network calculus provides a more general way to describe the arrival and service process It is convenient to use network calculus to get the delay and backlog bounds of network Though in [5], the author derived the backlog and delay bounds of an 82 wireless node based on an existing model of 82, however, the traffic model adopted was Poisson arrival model and Constants Bit Rate (CBR) arrival model, and it still did not consider the property of long-range dependence and selfsimilarity of the traffic In light of such need, we develop a stochastic network calculus model for performance analysis of 82 protocols We derive the stochastic delay bound of an 82 wireless node using stochastic network calculus and take the property of long-range dependence and self-similarity of the traffic into consideration, where fbm model is chosen as the traffic model In this paper, we first derive the form of stochastic arrival curve of fbm model and the stochastic service curve of an 82 wireless node Secondly, based on the theory of stochastic network calculus, we get the stochastic delay and backlog bounds of an 82 wireless node Through the results, we can analyze how the parameters of both fbm traffic andthe setting of 82 protocols impact the delay and backlog of an 82 wireless node Then, we use matlab to calculate the theoretical stochastic delay bounds and ns-2 for simulations The resulting delay and backlog distribution own practical significance to guide the parameters configuration of network applications, eg, the impact of the number of competition nodes can help design the network topology and the theoretical stochastic delay and backlog can help design 82 systems The rest of this paper is organized as follows In Section II, we give a brief introduction of the definitions and theory about stochastic network calculus In Section III, the corresponding stochastic arrival curve and service curve are derived, and we obtain the stochastic delay and backlog bounds conducted by stochastic network calculus In Section IV, we analyze the derived bounds and compare the bounds with simulation results Conclusions are drawn in Section V /2/$3 22 IEEE

2 A Network Model II SYSTEM MODEL In 82 networks, the situation that several stations compete for the same access point always occurs We consider how to derive stochastic delay and backlog bounds of an 82 wireless node under this condition Fig shows the situation where several stations compete for the same access point And by F the set of non-negative wide-sense decreasing function, or F = {f( ) : x y, f(y) f(x)} (2) The following operator under the (min, +) algebra will be used frequently in the following sections The (min, +) convolution of function f and g is (f g)(x) [f(y)+g(x y)] (3) y x For service guarantee analysis of a system, we are interested in two quantities: backlog and delay, which are defined in [6], [7], [8]: The backlog B(t) in the system at time t is The delay D(t) at time t is B(t) =A(t) A (t) (4) D(t) {d :A(t) A (t + d)} (5) Fig The network model For simplicity, our analysis is based on the following assumptions: (i) There are n stations transmitting to an access point; (ii) There are no hidden terminals and capture in the channel condition, which is ideal; (iii) The saturation condition is fulfilled Each station always has a packet available for transmission In other words, the transmission queue of each station is assumed to be nonempty; (iv) Each node is fed with fbm flow Based on the assumptions above, the stochastic arrival curve and stochastic service curve are derived in the following analysis B Stochastic Network Calculus Basics We first introduce the fundamental notations of stochastic network calculus We consider a lossless communication system which is modeled by various processes A process is defined by a function of time t, (t=,, 2) It could count cumulative amount of traffic arriving at some network element, the cumulative amount of traffic departing from the network element, the amount of service provided by the network element In this case, we name the process as arrival process, departure process and service process, and denoted them by A(t), A (t), and S(t) respectively In this paper, we assume all process are defined on t and by convention, have zero value at t= For any s (,t), we denote A(s, t) =A(t) A(s), A (s, t) =A (t) A (s), S(s, t) =S(t) S(s) We denote the set of non-negative wide-sense increasing functions by F F = {f( ) : x y, f(x) f(y)} () For traffic model, the virtual-backlog-centric (vbc) stochastic arrival curve is mainly used in this paper The vbc stochastic traffic model explores the virtual backlog property of the deterministic arrival curve, which is that the queue length of a virtual single-server queue fed with the same flow with a deterministic arrival curve is upper-bounded [4] The definition of the vbc stochastic arrival curve is defined as follow: Definition A flow is said to have a virtual-backlog-centric (vbc) stochastic arrival curve α F with bounding function f F, denoted by A vb <f,α>,ifforallt and x, there holds P { sup [A(s, t) α(t s)] >x} f(x) (6) Definition 2 AServerS is said to prove a stochastic service curve β F with bounding function g F, denoted by S sc <g,β>,ifforallt and x, there holds P { sup [A β(s) A (s)] >x} g(x) (7) Theorem Consider a server fed with flow A Iftheserver provides stochastic service curve to the flow and the flow has a vbc stochastic arrival curve, then, (i) The backlog B(t) of the flow in the server at time t satisfies: for all t and x, P {B(t) >x} f g(x +inf[β(s) α(s)]) (8) s (ii) The delay D(t) of the flow in the server at time t satisfies: for all t and x, P {D(t) >x} f g(inf[β(s) α(s x)]) (9) s III PERFORMANCE ANALYSIS In this section, we obtain the stochastic arrival curve and stochastic service curve based on the system model and derive the stochastic delay and backlog bounds conducted by stochastic network calculus 25

3 A Stochastic Service Curve of 82 Node In the 82 protocols, distributed coordination function (DCF) is the fundamental mechanism to access the medium This random access scheme is based on the carrier sense multiple access with collision avoidance (CSMA/CA) protocol, and an exponential backoff scheme is adopted The backoff time is uniformly chosen in the range (, w-) The contention window w is set equal to the value of minimum contention window CW min After each unsuccessful transmission, w is doubled, up to the maximum value CW max = 2 m CW min we assume the channel conditions are ideal, which means there are no hidden terminals and capture And we operate in saturation conditions, which means the transmission queue of each station is assumed to be always nonempty The saturated throughput of the channel is denoted by R Then, we have R = C τ ( γ) () where C is the capacity of the link, τ is the stationary probability that the station transmits a packet in a generic (ie, randomly chosen) slot time, and γ is the conditional collision probability, meaning that this is the probability of a collision seen by a packet being transmitted on the channel The value of τ and γ is determined by the following equations [9] τ = 2( 2γ) ( 2γ)(CW min +)+γcw min [ (2γ) m ], () n= Eq() Eq(2) γ = ( τ) n (2) n= Fig 2 n=3 n=4 The Plots of Eq() and Eq(2) Fig 2 shows the curves of Eq() and Eq(2) when n =, 2, 3, 4 and 5 For different n, the points of intersection are the solution of τ and γ Whenn increases, γ increases and τ decreases According to Definition 2, we have the service curve S sc <g,β>,whereβ(t) =Rt and g(t) = n=5 B Stochastic Arrival Curve of fbm Traffic Fractional Brownian motion (fbm) is a widely accepted model for self-similar traffic [6], [7], [8] FBm is characterized by the Hurst parameter H Depending on its Hurst parameter H, it can be short-range dependent or long-range dependent When H ( 2, ), fbm is LRD, which is the traffic model we consider Definition 3 An arrival process is said to be fraction Brownian motion arrival if it satisfies A(t) =λt + aλz H (t),t, (3) where λ > is the mean rate of fbm traffic and a is variation coefficient Z H (t) is standard fraction Brownian motion process Z H (t) has the following basic properties: (i) Z H (t) has stationary Gaussian increment; (ii) For all t, Z H (t)=, and E[Z H (t)]=; (iii) For all t, E[Z 2 H (t)] = t 2H ; (iv) Z H (t) is continuous and its finite dimension distribution is Gaussian process Since α(t) =rt is the basic form of α(t), we can get the bounding function f F of the virtual-backlog-centric (vbc) stochastic arrival curve with α(t) =rt: P {A(s, t) α(t s) >x} P {A(s, t) α(s, t) >x} sup P {A(s, t) r(t s) >x} P {A(t) A(s) r(t s) >x} P {A(t) A(s) >r(t s)+x} P {λt + aλz H (t) λs aλz H (s) >r(t s)+x} P {Z H (t) Z H (s) > (r λ)s+(r λ)t+x aλ } P {N(, t 2H s 2H ) > (r λ)s+(r λ)t+b+x aλ )} exp( (4) Thus, according to Definition, we have the stochastic arrival curve A vb <f,α>,where { α(t) =rt f(x) =exp( C Performance Bounds From the above analysis, we can get that the arrival process A vb <f,α>,where { α(t) =rt f(x) =exp( The service S fulfills S sc <g,β>,where: { β(t) =C τ ( γ) t g(t) = Substituting the above results obtained in Theorem, the Stochastic backlog and delay bounds are derived In order to simplify the calculation, r = R is adopted 26

4 For backlog B(t): P {B(t) >x} f g(x +inf(β(s) α(s))) s = f g(x) For delay D(t): y x y x (f(y)) = f(x) (f(y)+g(x y)) =exp( 2aλ ( R λ P {D(t) >x} f g(inf(β(s) α(s x))) s = f g(rx) y x (f(ry)) y x = f(rx) =exp( (f(ry)+g(rx Ry)) 2aλ ( R λ H )2H ( Rx (5) (6) Through the above inequality, we can analyze the impact of the self-similar Hurst parameter H to the backlogand delay of the network, and the impact of the number of stations which are transmitting to the same access point to the network delay and the backlog IV PERFORMANCE EVALUATION In this section, numerically simulation is conducted by matlab to analyze the relationship between self-similar Hurst parameter H, the number of stations which contents the same channel n and stochastic delay bound And we use ns-2 to verify our delay bound for a certain scenario A Theoretical Calculation In each simulation, the value of minimum contention window is 32 (w=32), the mean rate of fbm traffic is 4kb/s (λ=4kb/s) and the variation is 2kb/s The following simulation is conducted by matlab and based on the analytical result of (6) ) Impact of Hurst Parameter H: In this simulation, we set three stations transmitting to the same access point (n=3), the backoff index is five (m=5) We set the value of H 6, 65, 7, 75, 8 and 85 to see the impact of Hurst parameter on stochastic delay of the node Fig3 shows the results Fig3 shows that, as H increases, the violation probability of the same delay increases, so does the delay of the same violation probability Which is to say, as Hurst parameter increases, the delay of the 82 wireless node increases This is because the larger the value of H, the stronger the burst of fbm flow, which leads to the increasing of the delay of the node 2) Impact of the Number of Stations n: In this simulation, we set Hurst parameter H=75, the backoff index m=5 We change the number of stations which transmit to the same access point from one to five to see how it impacts stochastic delay of the node Fig4 shows the results Fig4 shows that, the violation probability of the same delay increases with the number of competing stations, so does the H=6 H=65 H=7 H=75 H=8 H= Dealy(ms) Fig 3 Impact of Hurst parameter H n= n=2 n=3 n=4 n= Delay(ms) Fig 4 Impact of competition station number n delay of the same violation probability Which is to say, as the number of competing station increases, the delay of node generally increase The reason is that the more the competing stations, the less the service each station gets, which leads to the increase of backlog, and the increase of delay B Comparison between Simulation Results and Numerical Results We adopt ns-2 to verify the theoretical results Firstly, an approximate fbm generation method known as the random midpoint displacement (RMD) algorithm [2] is adopted to generate fbm traffic The parameters of fbm traffic areset as follows: the mean rate of fbm traffic is 4kb/s (λ=4kb/s), the variation is 2kb/s We show in Fig5 fbm trace with input H=75 We set three stations transmitting to the same access point (n=3) The value of minimum contention window is 32 (w=32), and the backoff index is five (m=5) The statistical data of traffic delay is obtained from the trace file of ns-2 simulation The comparison between simulation results and analytical results is shown in Fig6 The blue bar represents the analytical results of (6) and the red bar represents the simulation results from ns-2 Fig6 shows that theoretical results are well upper- 27

5 Generated Traffic (kb) Fig Time(s) x Fig 6 FBM trace based on the RMD algorithm (H=75) Analytical Simulation Delay(ms) Compare simulation results with analytical results bounded and quite close to the results of simulations It indicates that our analysis about the stochastic delay of 82 protocols with self-similar traffic input is precise Thus it can be a practical reference when analyzing the QoS of 82 protocols and helping the design of 82 systems V CONCLUSIONS In this paper, we firstly take the self-similar property of WLAN traffic into consideration to analysis the delay performance of 82 protocols With the method of stochastic network calculus, we have derived the stochastic arrival curve and stochastic service curve of an 82 wireless node Based on derivation above, the delay and backlog bounds of the node have been obtained Matlab is used to calculate the theoretical bounds and predict the impact of the Hurst parameter of fbm traffic and the number of competition nodes on the performance of an 82 wireless node The delay of the 82 wireless node increases with the increasing of Hurst parameter H and the number of competition stations n Finally, we carried out simulations to verify these bounds We believe that these results will shed light on deeper understanding of 82 protocols, the impact of the property of self-similarity of wireless traffic and the design of 82 networks ACKNOWLEDGMENT This work was supported in part by National Natural Science Fundation of China (No 69726), Funds of Distinguished Young Scientists (No 29CDA5), China-Finish cooperation Project (No 2DFB57) REFERENCES [] Zhai, and Y Fang, Performance of Wireless LANs Based on IEEE 82 MAC Protocols, Procs, IEEE PIMRC, vol 3, pp , 23 [2] O Tickoo, and B Sikdar, Queuing Analysis and Delay Mitigation in IEEE 82 random access MAC based Wireless networks, Procs, IEEE INFOCOM, vol 2, pp 44-43, 24 [3] O Tickoo, and B Sikdar, A queueing model for finite load IEEE 82 random access MAC, Procs, IEEE ICC, vol, pp 75-79, 24 [4] C Oliveira, J B Kim, and T Suda, Long-range dependence in IEEE 82b wireless LAN traffic: An empirical study, Proceedings of IEEE 8th Annual Workshop on Computer Communications, pp 7-23, 23 [5] Y Qin, M Yuming, W Taijun, and W Fan, Hurst parameter estimation and characteristics analysis of aggregate wireless LAN traffic, Prcos, IEEE ICCCAS, vol, pp , 25 [6] W Leland, M Taqqu, W Willinger, and D Wilson, On the self-similar nature of Ethernet traffic (extended version), IEEE/ACM Trans Netw, 2(), pp -5, Feb 994 [7] I Norros, A storage model with self-similar input, Queueing Systems, 6(3), pp , Sep 994 [8] I Norros, On the use of fractional Brownian motion in the theory of connectionless networks, IEEE J Sel Areas Commun, 3(6), pp , Aug 995 [9] R L Cruz, A Calculus for Network Delay, Part I: Network Elements in Isolation, IEEE Trans Information Theory, 37(), pp 4-3, Jan 99 [] R L Cruz, A Calculus for Network Delay, Part II: Network Analysis, IEEE Trans Information Theory, 37(), pp 32-4, Jan 99 [] J-Y Le Boudec, and P Thiran, Network Calculus: A Theory of Deterministic Queueing Systems for the Internet, (Springer-Verlag), 2 [2] Y Jiang, and P J Emstad, Analysis of Stochastic Service Guarantees in Communication Networks: A Server Model, Procs IEEE IWQoS, 25 [3] Y Jiang, A Basic Stochastic Network Calculus, Procs SIGCOMM, 36(4), pp 23-34, Oct 26 [4] Y Jiang, and Y Liu, Stochastic Network Calculus, Springer, 28 [5] Y Wang, T Wang, Applying Stochastic Network Calculus to 82 Backlog and Delay Analysis, Quality of Service (IWQoS), San Jose, CA, Jun 2 [6] R L Cruz, Quality of service guarantees in virtual circuit switched networks, IEEE JSAC, 3(6), pp 48-56, Aug 995 [7] C-S Chang, Performance Guarantees in Communication Networks, Springer-Verlag, 2 [8] J-Y Le Boudec and P Thiran, Network Calculus: A Theory of Deterministic Queueing Systems for the Internet, Springer-Verlag, 2 [9] G Bianchi, Performance analysis of the IEEE 82 distributed coordination function, IEEE J Sel Areas Commun, vol 8, pp , Mar 2 [2] Lau, Wing-Cheong et al, Self-similar traffic generation: the random midpoint displacement algorithm and its properties, Procs, IEEE ICC, vol, pp ,