A New Feedback Control Mechanism for Error Correction in Packet-Switched Networks

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1 Proceedngs of the th IEEE Conference on Decson and Control, and the European Control Conference 005 Sevlle, Span, December -15, 005 MoA1. A New Control Mechansm for Error Correcton n Packet-Swtched Networks Oscar Flärdh, Karl H. Johansson and Mkael Johansson Department of Sgnals, Sensors and Systems KTH, SE-0, Stockholm, Sweden Emal: {oscar.flardh, kallej, mkael.johansson}@s3.kth.se Abstract Error correcton mechansms enable control and other real-tme applcatons to be executed over unrelable packet-swtched networks. By addng carefully adjusted redundancy to transmtted data at the sender, t s possble to recover lost data at the recever and thereby mprove effectve throughput. We descrbe smple models for packet loss, whch allow us to fnd the optmal redundancy as a functon of packet loss probablty. Two feedforward control mechansms based on the packet loss probablty are presented: one that s computed off-lne and another one usng an on-lne algorthm. A drawback wth these approaches s ther dependency on accurate network state nformaton and precse loss models. To cope wth these ssues, we propose a new feedback soluton that tracks the optmum usng gradent estmaton. Smulatons n ns- llustrate the behavor of the error correcton schemes, demonstratng that the feedback soluton outperforms the feedforward soluton n presence of model errors. I. INTRODUCTION The emergng technology of network embedded control opens up vast possbltes for flexble automaton solutons n applcaton areas such as smart buldngs, ndustral automaton and power dstrbuton. At the same tme, the desre to engneer vable networked systems rases many fundamental research challenges on predctablty, relablty and scalablty [1]. One mportant subproblem s on how to execute feedback control applcatons over unrelable communcaton channels. There are two natural approaches to address ths problem. One s to modfy each control algorthm to cope wth the varatons and uncertantes posed by the network []. Another approach s to develop generc nterfaces between control applcatons and transport protocols that are sutable for a large class of control applcatons and network condtons. The latter approach s less studed n the lterature, but consdered n ths paper. In general, these nterfaces are aware of features of both control applcatons (e.g., control qualty, stablty) and networks (e.g., delay, jtter, packet loss, dsconnecton) [3], []. Here, the focus s on adaptaton mechansm for packet losses. The man contrbuton of the paper s a comparson of a set of end-to-end error correcton algorthms that support control applcatons on top of an unrelable packet-swtched network. By adaptng the amount of transmtted redundancy, t s possble to cope wth large changes n the packet loss probablty. In partcular, nterestng features of feedforward and feedback control of error correcton are dscussed. The presented schemes are nspred by an adaptve forward error Ths work was partally supported by the European Commsson through the Integrated Project RUNES and the Network of Excellence HYCON, by the Swedsh Research Councl, and by the Swedsh Foundaton for Strategc Research through an Indvdual Grant for the Advancement of Research Leaders. correcton (AFEC) method proposed by Park and Wang [5] for end-to-end transport of real-tme traffc. Smlar AFEC schemes have been appled to Internet telephony [] and MPEG streamng [7]. We show that error correcton control can be posed as an extremum control problem [], n whch the statc nonlnear map corresponds to the normalzed effectve throughput. However, snce the measurements are extremely nosy n our applcaton, t seems hard to utlze more advanced extremum seekng methods. A smple scheme based drectly on fltered dervatves of the redundancy and the effectve throughput s shown to work well. In partcular, t s llustrated how proper tunng of a sngle parameter makes the feedback loop generate suffcent exctaton for approprate estmaton to stay close to the optmum. Smulatons n ns- (the network smulator [9]) wth dfferent loss models llustrate the results. The outlne of the paper s as follows. Secton II presents the network model and the control problem. A feedforward control strategy for error correcton based on estmates of loss probablty s presented n Secton III, whereas a more robust feedback control strategy based on a block recovery ndcator s ntroduced n Secton IV. The error correcton schemes are evaluated through smulatons descrbed n Secton V and the paper s concluded n Secton VI. II. PROBLEM FORMULATION The consdered error correcton problem s defned next, by frst ntroducng a smple network model followed by the control objectve for error correcton. A. Network Model Consder the followng model for unrelable end-to-end communcaton over a packet-swtched network. A sequence of data packets are transferred across the network usng an unrelable transport protocol, such as UDP or RTP. Congeston and lossy lnks cause some packets to be dropped by the network. Let v be an ndcator for successful transfer of data packet,.e., { 1, f packet was receved successfully v = 0, otherwse We consder varous models for packet loss. The smplest one assumes a fxed probablty p for losng a packet, and that the packet losses are ndependent. To evaluate the scheme for correlated losses, we consder a two-state Markov model, also known as the Glbert model []. The two states are loss and no loss, wth the probablty q 1 of gong from no loss to loss, and q of gong back to no loss. The average packet loss probablty s then equal to q 1 /(q 1 +q ) /05/$ IEEE

2 1 3 Codng Block 1 3 N 1 N Network 1 3 N 1 N? Decodng 1 3 Fg.. Objectve functon J as a functon of fxed redundancy u and ndependent packet loss wth fxed probablty p. Fg. 1. Error correcton. If at most u out of N sent packets n a block are lost, then t s possble to recover the orgnal N u data packets of the block. Consecutve packets are organzed nto blocks of fxed sze N packets, and error correcton s appled on block level as llustrated n Fgure 1. Each block b contans a varable number 0 N of redundancy packets and N data packets (applcaton nformaton). It s assumed that the data n a block can be recovered f the number of packet losses s less than or equal to. Ths s true for maxmum dstance separable codes (such as Reed Solomon codes), snce the mnmum dstance for the code n ths case s +1 [11]. Let I b be the ndex set for data packets belongng to block b and ntroduce x b = I b v as the total number of receved data packets n block b. Let z b be the block recovery ndcator for block b: { 1 f xb N u z b = b 0 otherwse For the mplementaton of the error correcton mechansms dscussed below, t s mportant to properly choose the observaton nstances of the packet loss process. In order to decde when a packet s lost or not, a tme-out nterval between consecutve packets can be ntroduced smlarly to what s done n TCP. Ths ssue s not further dscussed n the paper. B. Control Objectve We focus on maxmzng the normalzed effectve throughput, whch we defne as the average number of transmtted data packets per block over a horzon of B blocks 1 B (N )z b BN b=1 The control objectve s to maxmze the expected value of ths quantty, whch leads us to the followng objectve functon J(u) = 1 BN B E {(N )z b } (1) b=1 It s nterestng to qualtatvely study how J s nfluenced by a fxed control = u. A partcular case s when the packet losses are ndependent and equal to p b, say, for each packet n block b. Then, the probablty of block recovery can be expressed as Prob(z b =1)= p b(1 p b ) N () Fgure shows J as a functon of a fxed redundancy u for ndependent packet loss wth fxed probablty p. For large p and small u, the objectve functon s close to zero snce the probablty for block recovery s very low n ths case. There s a threshold where the probablty for block recovery ncreases rapdly wth u to a value close to one, whch corresponds to the rdge n J. Increasng eyond ths threshold ncreases the probablty for block recovery very lttle, but decreases the number of data packets per block. In ths stuaton, J thus decreases approxmately lnearly wth u. For a gven loss probablty, there s a unque maxmum and hence a unque optmal control. It s shown below that even f packet losses are correlated, t s stll possble to argue locally n tme and space for optmzng J. C. Control Structure The error correcton control mechansms we consder can all be framed n the control structure llustrated n Fgure 3. The control sgnal s computed n real-tme based on feedforward of network state nformaton and feedback nformaton on the block recovery. The network state nformaton corresponds to the packet loss probablty p b. Ths feedforward approach s further descrbed n next secton. The feedback nformaton n Fgure 3 s defned by y b = (N )z b, the number of recovered packets per block. Note that y b can be seen as an observaton of the objectve functon J. The feedback approach s based on 9

3 Controller u p Network Fg. 3. Control structure for error correcton wth ndcaton of feedforward and feedback nformaton. ths measurement and s further descrbed n Secton IV. Obvously, the feedforward approach should respond faster but be more senstve to modellng errors, whle the feedback approach s expected to respond slower but be able to adapt to unmodelled changes. From an mplementaton pont of vew, t should be noted that both the feedforward and the feedback sgnal are avalable at the recever sde: The packet loss probablty p b can be estmated and the feedback nformaton on the block recovery y b s known. Snce the error correcton s appled on the sender sde, ths nformaton needs to be communcated to the sender over some possbly unrelable network. How the unrelable transfer of p b and y b affects control performance s not further dscussed n the paper. Note, however, that the amount of data transferred n ths drecton s substantally smaller than the applcaton data sent n the forward drecton. III. FEEDFORWARD CONTROL We develop two control strateges n ths secton, based on feedforward nformaton on the loss probablty p. If p s known, the objectve functon (1) can be maxmzed. Frst we note that, f s assumed to be fxed and thereby not stochastc, E {(N )z b } =(N )E {z b } and then usng () for the probablty of block recovery. Each term n the sum n (1) s gven by (N ) p b(1 p b ) N Hence the optmal redundancy, f losses are ndependent, s gven by u b =argmax (N ) y p b(1 p b ) N (3) Ths optmzaton problem can be solved off-lne, and then the control s a smple table look-up based on the current packet loss probablty p. Hence we wll call ths control strategy the off-lne feedforward algorthm. To effcently derve the optmum redundancy (3), we consder the dfference L u = L u L u 1, where L u =(N u) u p (1 p) N It s easy to show that ( N L u =(N u) u u 1 ) p u (1 p) p (1 p) N whch gves a frst-order condton for optmalty: f L u > 0 and L u+1 < 0, then u s the optmal redundancy. Alternatvely, we can attempt to fnd the optmum redundancy (3) usng the followng recurrence scheme { ub + α +1 = 1 (J(u b ) J()), f u b () α, otherwse where α 1 > 0 and 0 < α < 1 are tunng parameters. We wll call ths scheme the on-lne feedforward algorthm. If the redundancy s too low, t s ncreased by an amount proportonal to the dfference between the current objectve and the optmum, whereas an exponental backoff s appled when the redundancy s too hgh. Ths control scheme s smlar n sprt to the AFEC scheme proposed by Park and Wang [5]. However, snce they consder another objectve functon and do not seek the maxmum of that functon, ther soluton s slghtly dfferent from (). The two control strateges descrbed n ths secton adapt fast to changes n p, under the assumpton that a good estmate of p s avalable. If the underlyng model or the estmate of p s not accurate, the control strateges mght gve unsatsfactory performance. Next, we present a feedback strategy that tres to remedy these drawbacks. IV. FEEDBACK CONTROL The feedback control algorthm attempts to move the system towards the peak of J wthout relyng on a detaled analytcal model. Although t s hard to derve a closed-form expresson for J under a general packet loss process, t s stll possble to qualtatvely study the shape of J. We may note that J wll necessarly tend to zero as u approaches N. Moreover, f p s not zero J wll be small for =0 and wll ntally ncrease as the redundancy ncreases, at least for ndependent losses. The detaled behavor of J for ntermedate values s hard to specfy, but we wll make the assumptons that J s unmodal and monotone on each sde of the peak. As dscussed n Secton II-C, the feedback sgnal y b gves nformaton about J for the current redundancy level. Snce the locaton of the peak and even J tself are uncertan, we smply estmate on whch sde of the peak the state s, and then adjust u so that the state moves closer to the peak. The estmaton s based on the dervatves of y b and. However, snce the underlyng sgnals are very nosy, we need to flter them carefully to obtan a relable estmate. To ths end, we compute yb d and ud b by frst averagng the sgnals y b and over a sequence of blocks, and then flterng the sgnals through a low-pass flter wth a dervatve term. By comparng the sgns of yb d and ud b we can decde on whch sde of the optmum the current control value correspond to, cf., Fgure. A control algorthm based on ths argument s +1 = + β sgn(y d b u d b) (5) where β>0 s a tunng parameter. 90

4 u u βl 1 1 L t Fg.. Prncpal sketch of the soluton to (). The number of redundant packets wll vary around u wth perod L and ampltude βl. 9 β = β = 0.01 Optmal redundancy β = β = 0.00 β = 0.01 Optmal redundancy x 7 Fg.. Transents of the control sgnal u wth three dfferent values of β x 0. Effectve throughput 0. Fg. 5. Smulaton of control algorthm (5) usng ndependent packet losses To determne good values of β and to nvestgate closedloop system characterstcs, we consder a flud model of (5): u = β sgn(y d u d ) () The control sgnal u s hence lnearly ncreasng or decreasng wth slope β. Ths controller fts nto the class of swtchng extremum controllers, whch have been tradtonally used n other control applcatons []. The control sgnal wll ncrease untl the fltered sgnals ndcate that the peak has been reached. Gven that there s a delay L after the peak s passed untl t s actually dscovered, the soluton of () wll be a sawtooth sgnal wth perod L and ampltude βl, see Fgure. Smulatons wth ndependent packet losses confrm ths behavor. Fgure 5 show smulatons for two dfferent values of β. The sgnals are sawtooth-shaped, but there s dscrepancy due to nose caused by the loss process. For β =0.01 we have a perod of about 00 blocks, so our analyss predcts the ampltude β, whch corresponds well wth the smulaton result. For β =0.05, we have perod 1300 and thus an ampltude just above 3, whch also agrees well wth the smulatons. Note that u wll vary around u wth an ampltude dependng lnearly on β. The smaller the ampltude, the hgher the effectve throughput. Wth u varyng n an nterval, the effectve throughput wll be the mean value of the functon n that nterval. Ths analyss suggests that control performance mproves as β decreases. However, the value of β nfluences the gan n the control loop, and a hgher gan gves faster response durng transents. Furthermore, f the value of β s too small, the controller wll not excte the system enough to detect changes n network condtons and mght fal to fnd the optmal redundancy. To llustrate ths, we nvestgate how the effectve throughput vares wth β durng a step n packet β Fg. 7. The effectve throughput as a functon of β durng a step n packet loss probablty. loss probablty. Responses for three dfferent values of β are shown n Fgure. Here t s clearly llustrated that a small β gves small oscllatons but slow response to changes n p, whereas a large β gves faster response but large oscllatons. To generalze ths observaton, one can fnd the effectve throughput as a functon of β for ths partcular step response. Ths s presented n Fgure 7, where both the threshold for enough exctaton (small β) and the performance degradaton due to large oscllatons (large β) s shown. V. SIMULATIONS The control algorthms have been evaluated va extensve smulatons n ns-. We report results for ndependent packet losses as well as correlated losses generated by the Glbert model descrbed n Secton II. In both cases, we evaluate the performances for a scenaro wth two sudden changes n the average packet loss probablty. Ths behavor of the packet loss process s motvated from observatons of real Internet traffc [13]. To avod effects of naccurate flterng, the two feedforward controllers are assumed to know the correct value for the packet loss probablty n all smulatons. Fgure shows a smulaton for the case of ndependent losses and Fgure 9 shows the assocated redundancy for the three control algorthms. From the detaled vew of the transent shown n Fgure we see that the feedforward algorthms react quckly to changes n packet loss probablty. 91

5 1 1 Lost packet/block Optmal redundancy 1 13 Lost packets/block x x Fg.. Lost packets per block and optmal redundancy for the smulaton wth ndependent packet losses. Fg.. Transents of the control sgnal durng a sudden ncrease n packet loss probablty. Smulaton wth ndependent losses y b x x Fg. 9. Control sgnals durng sudden change n packet loss. Smulaton wth ndependent losses. Note the overlap between the two feedforward algorthms. Fg. 11. Output sgnals durng sudden change n packet loss. The sgnals are fltered. Smulaton wth ndependent losses. Ths s natural snce they have drect knowledge of the network state. The feedback algorthm s a bt slower, but t stll manages to fnd the optmal redundancy, thereby achevng good steady state performance. The effects of ths slower adapton s also clearly shown n Fgure 11. The reason for the large dp durng the ncrease n packet loss probablty s that t s much worse to use too lttle redundancy than too much, f the dscrepancy s more than a few packets, cf., Fgure. From the output sgnals, we can also conclude that the oscllatons n the control sgnal have a mnor effect on the effectve throughput durng steady state. Numercal values for the effectve throughput and block recovery rates acheved by the dfferent strateges are below. J Recovery rate Off-lne feedforward % On-lne feedforward % % Even though the on-lne feedforward algorthm has slghtly slower transents than the off-lne soluton usng optmal redundancy all the tme, t performs nearly as well. The feedback algorthm, wth ts slower transents and sustaned oscllatons around the optmal redundancy, performs slghtly worse. The second set of smulatons nvestgates the performance of the dfferent schemes under correlated losses generated by the Glbert model. The control sgnals of the three control algorthms are shown n Fgures and 13. In ths case, the feedforward algorthms are computed under the assumpton of ndependent losses and thus underestmate the optmal redundancy. The feedback algorthm, on the other hand, stablzes on a hgher level of redundancy than the other schemes and acheves the best performance of the three: J Recovery rate Off-lne feedforward % On-lne feedforward % % The output sgnals are shown n Fgure 1, where t s seen that the feedback controller performs worse durng the transents, but fnds better steady state values. The underestmaton of the redundancy by the feedforward algorthms s natural snce, f packets are lkely to be lost n bursts, the probablty for losng a block ncreases even though the average packet loss probablty s the same. Note also that the correlaton of losses causes a general performance degradaton. Ths s a well-known phenomenon n forward error correcton [1], [15], and s due to the 9

6 y b x x Fg.. Control sgnals durng sudden change n packet loss. Smulaton wth correlated losses. Note the overlap between the two feedforward algorthms. Fg. 1. Output sgnals durng sudden changes n packet loss. The sgnals are fltered. Smulaton wth correlated losses x Fg. 13. Transents durng a sudden ncrease n packet loss probablty. Smulaton wth correlated losses. ncreased block loss probablty: Keepng the same level of redundancy as f the losses were ndependent wll decrease the effectve throughput (snce more blocks are lost); ncreasng the redundancy gves hgher block recovery probablty but less applcaton data s sent and the effectve throughput s decreased. VI. CONCLUSIONS Ths paper has studed several control structures for error correcton. The frst two were feedforward controllers, one on-lne and one off-lne, based on packet loss probablty for the connecton. A drawback wth ths approach s the dependency on accurate network state nformaton and the relance on uncertan models. To cope wth these ssues, we proposed a new feedback soluton that tracks the optmum usng gradent estmaton. The controllers are evaluated wth smulatons n ns-, demonstratng that the feedforward controller performs well when there s no model error, whereas the feedback controller performs better n the presence of model errors. There are several ssues that warrants further nvestgatons. One s how to best estmate the gradents despte the hgh nose level n sgnals, another one s to develop systematc ways for tunng the feedback algorthm accountng for both stablty and performance. It would also be nterestng to apply other extremum control approaches to the forward error correcton problem, as well as control structures combnng feedforward and feedback components. Moreover, the senstvty wth respect to lost packets n the feedback sgnal needs further studes. VII. ACKNOWLEDGMENTS The authors wsh to thank the revewers for ther useful comments. We are also ndebted to Inés Cabrera Molero for her help wth the ns- smulatons. REFERENCES [1] D. Estrn, Ed., Embedded, Everywhere: A Research Agenda for Networked Systems of Embedded Computers. U.S. Natonal Research Councl, 001. [] P. Antsakls and J. Balleul, Specal ssue on networked control systems, IEEE Trans. on Automatc Control, vol. 9, no. 9, 00. [3] S. Graham, G. Balga, and P. R. Kumar, Issues n the convergence of control wth communcaton and computng: Prolferaton, archtecture, desgn, servces, and mddleware, n IEEE CDC, 00. [] J. Nlsson, B. Bernhardsson, and B. Wttenmark, Stochastc analyss and control of real-tme systems wth random tme delays, Automatca, vol. 3, Jan [5] K. Park and W. Wang, AFEC: an adaptve forward error-correcton protocol and ts analyss, Techncal Report CSD-TR97-03, Department of Computer Scences, Purdue Unversty, Tech. Rep., [] J.-C. Bolot, S. Fosse-Parss, and D. F. Towsley, Adaptve FEC-based error control for nternet telephony, n INFOCOM (3), 1999, pp [7] K. Park and W. Wang, QoS-senstve transport of real-tme MPEG vdeo usng adaptve forward error correcton, n ICMCS, Vol., 1999, pp. 3. [] K. B. Aryur and M. Krstć, Real-Tme Optmzaton by Extremum Seekng Control. John Wley & Sons Inc, 003. [9] The network smulator, [] W. Jang and H. Schulzrnne, Modelng of packet loss and delay and ther effect on real-tme multmeda servce qualty, n Proc. NOSSDAV, 000. [11] R. E. Blahut, Theory and Practce of Error Control Codes. Addson- Wesley, 193. [] J. Sternby, A revew of extremum control, Department of Automatc Control, Lund Insttute of Technology, Tech. Rep., Aprl [13] K. Jacobsson, N. Möller, K. H. Johansson, and H. Hjalmarsson, Some modelng and estmaton ssues n traffc control of heterogeneous networks, n Internatonal Symposum on Mathematcal Theory of Networks and Systems, Leuven, Belgum, 00. [1] I. Cdon, A. Khamsy, and M. Sd, Analyss of packet loss processes n hgh-speed networks, IEEE Transactons on Informaton Theory, vol. 39, no. 1, [15] N. Shacham and P. McKenny, Packet recovery n hgh-speed networks usng codng and buffer management, n IEEE INFOCOM,