,*

1.Institute of Information Science,Beijing Jiaotong University,Beijing 100044,China;2.Beijing Key Laboratory of Advanced Information Science and Network Technology,Beijing 100044,China

Stability analysis of wireless network with improved fluimodel

Zhichao Zhou1,2,Yang Xiao1,2,and Dong Wang1,2,*

1.Institute of Information Science,Beijing Jiaotong University,Beijing 100044,China;
2.Beijing Key Laboratory of Advanced Information Science and Network Technology,Beijing 100044,China

Wireless communication is easily disturbed by unfortunate factors which drive the wireless environment unstable and complicated.Therefore,it is essential to consider these factors in stability analysis of the wireless network.However,wireless channel characteristics and packets collisions are neglected in the classical fluimodel.A wireless TCP fluimodel(WTFM)for stability analysis of wireless network is proposed based on cross layers,which not only makes the congestion control based on random early detection(RED)available for wireless network,but also provides a more accurate model to analyze the stability of wireless system theoretically.In the proposed model,active queue management,abnormality of wireless channels and packets collisions are taken into consideration.The comparisons between evaluating results from the WTFM and the practical performance from NS2 simulations validate the accuracy of the proposed WTFM in the perspectives of delay,dropping probability,throughput,sliding window size and queue length.A set of comparisons among the proposed WTFM,the classical fluimodel and the convex optimization model are conducted.The results demonstrate that the proposed WTFM model performs better than other schemes in comprehensive aspects on capturing the characteristic of the wireless network and computing complexity.

stability analysis,improved fluimodel,cross layers, collision,wireless channel characteristic.

1.Introduction

Stability of wireless network is becoming more and more important with the number of wireless real-time applications increasing.Such applications include streaming media,TV conferencing and online gaming,which cost huge link capacity and require higher throughput,smaller jitter, lower dropping probability and delay[1,2].However,the massive and fast data transmission in the wireless network suffers from congestion and interference easily due to the limited link capacity of wireless routers and the instability of wireless channels.Therefore,stability analysis of wireless network becomes important but difficul[3].The past networkstability researchesin[3–9]mainlyfocus oncongestion control mechanisms that over-dependon pure sliding window control in the transport layer and they do not consider the severe collisions and the unstable character of the wireless network.Thus such schemes are unavailable or inaccurate for the stability analysis of the wireless network.Other relevant schemes regard the stable problem as an issue of convex optimization,which is really capable of analyzing the network stability and obtain the optimized global data rate,but the problem of convex optimization is always difficul and complicate to deal with.Reference [1]proposedthree algorithms to control congestionand allocate the power for multihop wireless networks based on cross layers,but the algorithms resort to complicated convexoptimizationprocedureintheend.Therealsoexiststhe problems of convergence rate,global optimal source rate and increased network overhead,which affect the performance of the real network severely.References[4,5]studied the congestion control also based on scheduling optimization of the queue-length that can alleviate congestion of wireless network,but they need to coordinate between sub-frames circularly and periodically,which also leads to higher control overhead,longer delay and sluggish correspondence.In[6],the author modifiethe algorithm by decreasing the capacity region and virtualizing the wireless network as a wired network,which makes the complicatedcongestioncontrolofwirelessnetworkbecomemuch simpler.However,some constraints of wireless network are not considered completely in the scheme so that the analysis is not accurate enough.Reference[7]proposed a scheme in view of end-to-end notification There are two problems in the scheme:one is that every routing node has to maintain a table to store the window size of all sourcesrespectively,which needs to update periodically.The other is that it takes some time for the feedback notification to transmit from the destination to the source.Therefore, the condition of the transmitting link and routers can not be realized by the upper layers of the source instantly, and the window size in transport layer will not respond immediately when congestion occurs in the core router. References[8,9]designed a distributed congestion control mechanism to drive the elastic fl ws rate towards an optimization of all the system.Consequently,the approach will end up to a global optimization problem,which leads to complicate iterations until the optimized objective.The absence of conjunction between different layers in[9]also made the sources respond to collision and congestion in the wireless router slothfully.Reference[10]estimated the network congestion based on packet time series,where the sender transfers two continuousprobepackets labeled by 1 and 2.The receiver estimates a congestion by the interval of the two probed packets.The scheme assumes that the two probed packets will not be separated by other labeled packets and the probed packets will not drop in the process of transmission.In wireless network,the assumption is imprecise because the two conjoint probed packets may be separated by other labeled probed packets in transmission,which will lead the estimation of linking state information to be inaccurate.If only one of the probed series packets drops,congestion informationwill not be captured by the receiver.A new active queue management(AQM) algorithm was proposed in[11]for 802.16(WiMax)by using queue-delay as the congestion indicator instead of packet dropping.They consider that the cause of packet drops could be attributed to not only congestion but also channel impairments in the wireless network.The packet will bemarked,if its round-triptime exceedsa thresholdin the scheme.However,the threshold has to update every T seconds to reflec changes in the network.In[12,13],stability analysis was from a new perspective of cooperation between nodes for the vehicular Ad Hoc network and cooperative wireless communication respectively.Reference [14]set up a contractive interference function based on continuous-timepower control and provedthe relationship between the convergence of the function and the exponentially stability of the wireless network.Power control constrains the convergency of the interference function so as to obtain a stable state.However these approaches in[12–14]also relied on optimization.

The detailed process of the MAC layer has been researched in[15,16],in which the author proposed a two dimensional Markov model that provided an accurate approach to analyze the detailed process of MAC layer and deal with collisions,but it did not provide any ways to research the stability of the wireless network.A classical flui model was researched in[17–19]and the model was verifie to be effective to research congestion control of wired network with the RED scheme.However,the flui model is not suitable for the wireless network since it does not consider the problems of packets collision and the channel character,which are notable in the wireless network.Therefore,the classical fluimodel is inaccurate to analyze the stability of the wireless network due to the neglect of some considerable factors.Additionally, as the classical flui model,most of the network stability researches are based on the wired network rather than the wireless network in order to simplify the analysis process. Consequently,the drawback of the classical flui models and the complication of convex optimization issue in the previous researches of stability analysis for the wireless network motivates us to propose an available,accurate and simple model in this paper.

In this paper,we propose a dynamic wireless TCP flui model(WTFM)to analyze the stability of the wireless network in the perspective of cross layers,in which not only the active packets drops are taken into consideration but alsothe passivepacketsdropsduetothe abnormalityofthe unstable wireless channel and the severe packets collisions are considered.Therefore,the proposed model is more accurate than the classical flui model to capture the character of wireless network.Besides,the simulation demonstrates that the proposedWTFM is effectiveto analyze stability of the wireless network theoretically and much simpler than the scheme of convex optimization.

The rest of this paper is organized as follows.Section 2 describes the passive packets drops and the characteristic of the wireless channel.In Section 3,the proposed WTFM is described.In Section 4,the NS2 simulations verify the proposed WTFM.Then we compare the WTFM with the classical flui model and the convex optimization model in performance and processing complexity.Finally, we present our conclusions in Section 5.

2.Analysis of wireless channels and passive drops

Stability analysis of the wireless network is more difficul than the wired scenario due to the complexity of the wireless one.Therefore,the proposed WTFM,which is used for wireless network stability analysis,is based on cross layers involving the MAC layer and the TCP layer.So we analyze the details of the MAC layer in this part to make the proposed WTFM understood clearer.

Communication in the wireless channel is easily disturbed by some abnormal factors due to the unstable properties of the wireless channel.The most considerableones are collisions and error packets in the transmission. These packets can not be received or decoded correctly by the destinations,so we consider such packets as passive packets drops in the paper.We analyze the details of the wireless channels and passive packet drops in this part. The characteristics of the MAC layer were researched in [15,16],in which a two dimensional Markov model was used to analyze the details of the wireless network.The bidimensional Markov process(s(t),b(t))is a discretetime Markov chain in the precondition that the packet collision probabilitypcis independent on the states(t)of the station.The nonnullone-step transition probabilities of the Markov chain can be expressed as

whereb(t)is the size of the back-off window for a given station at slot timet,s(t)is the back-off stage(0,...,m) of the station at timet.mis the largest back-off stage.Widenotes the back-off window for simplicity,which can beWi=2iW0.iistheback-offstageandW0is theminimum back-off window size.

Because(s(t),b(t))is assumed as a discrete-time Markov chain,we can get the following Theorem 1 in the aspect of mathematic according to the property of Markov chain and the specifi problem of back-off window.

Theorem 1Ifi∈(0,m),k∈(0,Wi?1),mandWiare finit states,is a stationary distribution of the Markov chain.

ProofConsiderWi=2iW0,i∈(0,m),W0=32. The largest back-off window size is 1 024,so we obtainm=5.The whole states are finit and irreducible.The transferred probability between different stages is random but not periodical.Besides,the back-off window will return back to the initial state to send packet no mater what the stage and current window size are.Therefore,it is sufficien and necessary to derive thatis a stationary distribution.

From Theorem 1 and properties of the Markov chain, we can obtain

Then we can get

A transmission will happen when the back-off window decreases to 0,regardless of the back-off stage.τdenotes the transmission probability of one station in a given slot. Then we can obtain the expression of transmitting probabilityτ,which is shown in(5).

When at least one of the remaining stations transmits packet at the same slot with the current station,collision will occur.Therefore,we get the relation between the collision probability and the transmission probability of one station.

It is not difficul to prove that there is a unique crosspoint between(5)and(6).Now we prove the uniqueness of the result. From(6),we can obtain the expressionτ?(pc)=1?This is a monotone increasing and continuous function in the scope ofpc∈(0,1),which starts withand ends up with (1?(2pc)m)can be expanded as(1?2pc)(1+2pc+ (2pc)2+...+(2pc)m?1).Therefore,(5)can be rewrit-Note that ten asIt is clear that the expression is a monotone decreasing function that begins withand reduces toNow we can draw the conclusion that the result is unique noting thatAccordingly,we can get the results of the collision probability and the transmission probability from(5) and(6)with Matlab.

We cannot get the instantaneous packet error ratio due to the signal interference in the process of transmission and the abnormality of the wireless channel.Additionally, the problem is not the key researched issue of this paper. Therefore,we adopt the statistical results that are studied in[20],which provides that the packet error ratio is in order of 10?2from the perspective of statistics.

3.Proposed WTFM and stability analysis

3.1Proposed WTFM

The dynamic WTFM is established by a set of stochastic differential equations like the classical flui model.The classical flui modelwas developedin[17–19],which did not consider the passive packets drops and time-out in order to simplify the model.However,these factors play an important part in the wireless network.Consequently,the model is inaccurate and imprecise to analyze the stability of the wireless network theoretically and mathematically. In this paper,we take the collision probability,packet error probability and active dropping probability into consideration in WTFM.The rough graphic process is shown in Fig.1,both the active drop(the RED scheme in the router) and the passive drop(the part surrounded by the ellipse on the left)are taken into consideration.The WTFM is described by the following nonlinear equations:

Fig.1 Simple process of proposed WTFM

In the analysis,we assume that the number of fl ws and capacity of routerCare constantN(t)≡N,so the equilibrium point can be expressed as follows:

where(W0,q0,R0,P0)is the equilibrium point.The round-trip timeR(t)can be presented as

and the range ofR(t)is

whereTpdenotes the propagation time for each packet.qminandqmaxare the minimum and maximum queue length respectively.And the throughput of the router can be expressed as follows:

wherePmac(t)=pa(t)+pc+pe.pa(t)is define as the active dropping probability,andpeis the statistical packet error ratio.

3.2Stability analysis with the proposed WTFM

The proposedWTFM considersthe RED algorithm[21]as the queue management in the wireless access point(AP). The active dropping expression of RED is shown as

whereqave(t)represents the average queue length in the AP.The averagequeuelengthis calculated with a low-pass filte with an exponential weighted moving average parameterw.We adopt the smoothing average queue weight (w=0.05)from the underlying parameters of NS2 to smooth the instant queue length.Pmaxis the maximumactive dropping probability.

There are no simple and accurate approaches to directly analyze nonlinear equations like the expression in (7).Thus we transfer the nonlinear equation(7)into a linear time-delay model with the approach in[22].The author of[22]transferred the nonlinear equation into a set of linear time-delay system,which was verifie simple and effectiveto deal with such nonlinear problem.Then we get

In order to analyze the problem easily,we transform the linear equation(12)into frequency domain with the approach proposed in[23].The approach aims at processing complicated continuous-discrete signals and systems with 2-D Laplace-Ztransformation,which converts the continuous-discrete system from time-domain into frequency-domain.First,we transfer(12)into frequency domain with 2-D Laplace.We obtain the presentation as follows:

The characteristic polynomial of(13)is

Letz=es,we obtain the 2-D Laplace-Zsystem of(12)

Then we can get the 2-D characteristic polynomial of the system(12)

There are two theorems on the characteristic polynomials (14)and(16)as follows.

Theorem 2Equation(14)is stable when

Theorem 3The polynomialB(s,z)is stable if and only if

ProofAccordingto Theorem2,the stability ofB(s,z) is sufficien for the polynomialB(s).Based on Theorem3,B(s,z)is Hurwitz-Schur stable,if and only if(18)and (19)are satisfie[24].

According to the theorems above,it is easy to get such proposition about the stability of WTFM as follows.

PropositionThe system is stable if and only if the parameters satisfy the following conditions.

The proposition is used to verify the stability of the wireless system analyzed with the proposed WTFM under given parameters.The validity of the proposed scheme in capturing the characteristic of the wireless network will be demonstrated by the system simulations with NS2 in Section 4.

4.Experiments and results analysis

This section is to analyze the stability of the wireless network numerically and verify the accuracy of the proposed WTFM with the NS2 system simulator.In the experiment, we consider 10 FTP fl ws.The link between the senders and the AP1(access point 1)is wireless channel.The other linksarewireline.Table1showsthecorrespondingsimulation parameters.We adopt a dumbbell simulation topology which is shown in Fig.2.The active queue managementin AP1 is the RED scheme.

Table 1 Simulation parameters

Fig.2 Frame of simulation topology

4.1Stability verificatio and experiments

First,we analyze the stability of the wireless network with theproposedWTFMunderthegivenparameterscondition. Substituting the parameters in Table 1 into(5)and(6),we get the collision probabilitypc=0.02 with 10 terminals in the wireless system.

Therefore,we consider that the maximum boundary of the active dropping probability in the quasi-equilibrium point ofPmacisP′0=0.02+0.03+0.02=0.07,which onlyacts as a beginningto analyzethe stabilityof the wireless system.From(8),we obtain

Equation(22)clearly shows thatwhen

Furthermore,we substitute the quasi-equilibriuminto(21).Then we get the expression

It is not difficul to realize that the symmetry axis of the quadratic in(23)isandAdditionally,the coefficien of the quadratic terms2is positive.ThereforeB(s,e?jw)＞0 is always right whenw∈Rand Re(s)≥0 according to the parabola.The analytical result is corresponding with proposition(21).

Based on the analysis with the proposed WTFM above, we can conclude that the wireless system is stable under the given parameters from the theoretical analysis.

4.2Performance analysis from the simulation results

We conduct a set of experiments to verify the theoretical conclusion under the same scenario.We obtain a set of graphic results from the simulation as follows.According to Fig.3,the queue length computed from the proposed WTFM fluctuate at the beginning and then quickly converges to a stable value 65 packets as expected.The result from the WTFM is in good agreement with the practical simulation results with NS2 when the system goes into stable state.While the queue length computed from the classical fluimodel is much bigger than the practical result from NS2 due to the negligence of the passive packets drops.The result of the joint congestion control with convexoptimization[4]is also in accordancewith the practical NS2 result as the proposed WTFM.However,wenote that the computing time complexity of the convexoptimizationis muchhigherthantheproposedWTFM,which is analyzed in Section 4.3.The comparison of the computing complexity and the variance of the three schemes is shown in the Table 2.

Fig.3 Comparison of queue length among the classical model

Fig.4 shows the evolution of the window size with different models under the same parameters condition.It suggests that the results from the proposed WTFM and the convex optimization scheme are almost anastomotic.The NS2 practical result fluctuate around the two curves with only small errors,which demonstrates that both the proposed WTFM and the optimization scheme are accurate enough in capturing the characters of the wireless network from the aspect of window size.While,the classical flui model result is about three packets bigger than the practical window size.The observation suggests the proposed WTFM performance and the optimization scheme are better than the classical flui model.

Fig.4 Comparison of window size among the classical model

Fig.5 suggests the throughput computed from the theoretical analysis and the practical result obtained from the NS2.The figur shows that the throughput computed from the proposed WTFM is a little bigger than the result that is from NS2 simulation.Because the packets error probability adopted in the proposedWTFM is a statistical value, which may lead to a small error with its peer in the practical wireless environment.The optimization result is consistent with the NS2 result.However the classical flui model result is much bigger than the practical result because it does not consider the collision drops and packet error drops.Such packets are included in the result when the throughput is computed by the classical flui model. Which leads to the difference between the computed result and the practical NS2 result.By contrast,the proposed WTFM is more accurate than the classical model,because the passive packets drops are considered in the WTFM.

Fig.5 Comparison of throughput among the classical model

Fig.6 is the comparison of delay among the four cases. The round-trip time which is computed from the proposed WTFM and the optimization scheme is much closer to the practical result than the delay computed from the classical flui model.To simplify the TCP behavior in the classical model,only active dropping probability is taken into account in the theoretical analysis.In fact,both the active packets drops and the passive packets drops are needed to resend in the practical network,so the average packet round-trip time from the classical fluimodel is shorter than the practical packet delay.We consider the passive packets drops and active packets drops in the proposed WTFM,so the delay from which converges to the prac-tical packet delay in a short response time and performs better than the classical flui model.The reason also accounts for the phenomenon that illustrates in Fig.7.From the perspectiveof packets drops,the proposedWTFM is in betteragreementwith the practicalwireless networkdueto the complementary packet drop information than the classical flui model.The optimization scheme is also able to reflec the accurate dropping probability when it goes to stable state according to Fig.7.We are aware that it takes a longer period of fluctuation for the convex optimization scheme to get to the stable state because of its iterations. However,thefluctuation forWTFMandtheclassical flui model are smaller than the former.

Fig.6 Comparison of delay among the classical model

Fig.7 Comparison of drop among the classical model

We also note that the results come from the practical wireless network begin with a period of fluctuation and then converges to a stable state.The convex optimization scheme coincides with the practical result better than the two other models at the beginning period.The reason is thatadjustmentandcontinuousiterationsareproceededfor thepracticalwireless networkandtheoptimizationscheme respectively before the stable state.While,both the proposed WTFM and the classical flui model aim at analysis atthestablestatewiththequasi-equilibriumparameters,so the results that are from the two flui models converge to the stable state quicker.Because the classical flui model and the proposedWTFM concentrateon the analysis at the stable state,they can not reflec the fluctuate process at the beginning as apparently as the optimization scheme.

4.3Comparisons and evaluations

In this section,we compare the differences among the four cases numerically from the aspects of queue length,window size,throughput,packet delay and packets drops in the stable state.Furthermore,we also compare the computing time complexity among the three schemes.The result is shown in the following Table 2(NS2:the practical result from NS2;CFM:the result is from the classical flui model;WTFM:the result is from the proposed model;COS:the result is from the convex optimization scheme.).

Table 2 Comparison result

Table 2 shows the comparisons of the four cases in the experiment.It shows that the difference is three packets between the proposed WTFM and the NS2 result,and its peersare9and2.3respectivelyfortheclassicalflui model and convex optimization scheme in the aspect of queue length.

Correspondingly,the window size differences are 0.7, 1.4 and 0.3 for the three schemes.The performance of the proposed WTFM is a little worse than the convex optimization but is much better than the classical flui model for some the parameter indexes.The convex optimization scheme performs a little better than the proposed WTFM in the aspect of queue length,window size,throughputand packets drops,while there is a bigger difference between the classical flui model and the practical results due to its neglect of passive packets drops.However,we note that both the classical fluimodel and the proposed WTFM (O(x2))(the WTFM is a quadratic term,so we get suchconclusion according to the definitio of the computing time complexity)are simpler than the convex optimization (O(x8))(at least eight times basic computations iterations to get the stable state with the preset parameters)in the computingtime complexity when the parameters are set as Table 1.In fact,the bigger the number of terminals is and the larger the link capacity is,the higher the computing order of convex optimization would be.Because the optimized data rate is obtained from continuous iterations and it has little chance to get to the expected state with one or two iterations in a wireless system with ten or more terminals.While the order of analysis complexity does not increase with the number of the terminals for the proposed WTFM.Therefore,the proposed WTFM is better in the comprehensive perspectives of performance and computing time complexity.

5.Conclusions

In this paper,we propose a WTFM based on cross layers to analyze stability of the wireless system theoretically.Not only are active drops considered according to the RED scheme,but also severe collisions and error packets drops caused by abnormality of the wireless channel are taken into account in the proposed WTFM.From the comparison of the simulation,the proposed WTFM is more veracious and accurate than the classical flui model in capturing the properties of wireless network.Both the theoretical analysis and simulation results verify that the proposed WTFM is accurate and effective enough to analyze the stability of the wireless network theoretically.Although the performance of the optimization scheme is a little better than the proposed WTFM,the analysis process of the WTFM is less complicate than the former.In conclusion,the WTFM proposed in this paper performs better from the comprehensive aspects of accurate performance and low computation complexity.

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Biographies

Zhichao Zhouwas born in 1989.He received his B.S.degree in Langfang Normal College in 2012. He made research from 2012 to 2013 as a M.S. candidate in Beijing Jiaotong University.He is now pursuing his Ph.D degree in the Institute of Information Science,Beijing Jiaotong University. His research interests include congestion control of network,stability analysis of wireless network, communication signal processing.

E-mail:zczhou@bjtu.edu.cn

Yang Xiaowas born in 1955.He received his B.S. degree from Beijing Posts and Teleconununications College in 1983 and M.S.degree and Ph.D.degree in 1989 and 1991 respectively from Beijing Jiaotong University.He is now a full professor in Beijing Jiaotong University.His main research interests include congestion control of the network,stability analysis of systems,communication signal processing and MIMO.

E-mail:yxiao@bjtu.edu.cn

Dong Wangwas born in 1981.He received his B.S. and Ph.D.degrees in electronic engineering in 2006 and 2010 from Xi’an Jiaotong University.He was a visiting student in the Computer Science Department of the University of California,Los Angeles, during 2009-2010.He is now working at Beijing Jiaotong University.His research interests include computer arithmetic forreconfigurabl and high performance computing architectures for embedded applications.

E-mail:wangdong@bjtu.edu.cn

10.1109/JSEE.2015.00125

Manuscript received December 10,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61106022)and the Beijing Natural Science Foundation (4143066).