A Distributed Utility Max-Min Flow Control Algorithm

Transcription

1 A Distributed tiity Max-Min Fow Contro Agorithm Hyang-Won Lee and Song Chong Department of Eectrica Engineering and Computer Science Korea Advanced Institute of Science and Technoogy (KAIST) Abstract A fair aocation of utiity(appication-ayer performance) is essentia in providing QoS(Quaity of Service) guarantee. However, there have been few researches in the iterature of utiity-fair network resource aocation scheme. In this paper, we propose a distributed utiity max-min fow contro agorithm which accommodates appication diversity in that it does not require the concavity of utiity functions, and is scaabe in that it does not require any per-fow operation in the network. The agorithm is proved to be convergent under the assumption that there exists a singe congested node and the communication deay between any two nodes in the network is bounded. Athough the convergence of the agorithm is anayzed for the case of a singe congested node, we show through simuations that the proposed agorithm works as designed for the case of mutipe congested nodes. I. INTRODCTION One of the key goas in the next generation network is to guarantee quaity of service (QoS) which represents the eve of appication-ayer performance (utiity). To reach this goa, it is necessary to identify appication specific demands and to provide network resource aocation schemes taking into account the demands. The appication specific demand is characterized by bandwidth, deay and deay jitter etc. which are caed QoS parameters. For exampe, rea-time mutimedia appications require strict bandwidth, deay and deay jitter guarantee since to provide acceptabe performance in paying back the encoded mutimedia, at east the encoding rate shoud be guaranteed and the timey deivery of a packet is essentia. On the other hand, data appications do not require strict deay jitter guarantee because their performances strongy depend on the time spent to compete data deiveries. A utiity function is used to describe the QoS characteristics of an appication. In this work, we dea with ony a bandwidth utiity function, which describes the appication-ayer performance with respect to the bandwidth aocated to an appication. By using the concept of utiity functions, there have been numerous researches to provide utiity-aware network bandwidth aocation schemes based on optimization theory [] []. The objective that they have in common is the maximization of aggregate utiity. nder the objective, by decomposing the This research was supported by the MIC(Ministry of Information and Communication), Korea, under the ITRC(Information Technoogy Research Center) support program supervised by the IITA(Institute of Information Technoogy Assessment). probem into network probem and user probem [], [] or by appeaing to the duaity theory in optimization theory [], it is easy to construct a distributed agorithm, which is provabe to be convergent. However, this framework has two serious drawbacks. One is that the utiity functions are restricted to be stricty concave. Because the utiity functions of rea-time or rate-adaptive appications are not concave [], those appications cannot be handed in the framework. The other is that the framework cannot achieve a utiity-fair bandwidth aocation. Achieved in the framework is ony the maximization of aggregate utiity and bandwidth proportiona fairness for ogarithmic utiity functions. Lee et a. [] showed that if such agorithms deveoped for concave utiity functions are appied to non-concave utiity functions, the system can be unstabe and can cause excessive congestion in the network. They proposed a distributed rate contro agorithm which can hande both concave and sigmoida-ike utiity functions, but the utiity fairness was not considered either. On the other hand, in [], Cao and Zegura defined a utiity max-min fairness and proposed a bandwidth aocation scheme that achieves the utiity max-min fairness. Athough their work is the first step to the utiity max-min fow contro and does not need the concavity of utiity functions which is indispensabe in the framework mentioned above, the proposed scheme has many probems. Firsty, it is not reaistic because each ink in the network shoud know the utiity functions of a fows passing through the ink and maintain the set of fows saturated at the ink. Secondy, they prove that the agorithm finds a utiity max-min fair rate aocation vector in finite time, but the deay, which is essentia in the communication network, is not considered in the proof. To overcome the imitations mentioned so far, based on contro theory, Cho et a. [] proposed a utiity max-min fow contro agorithm which resoves the probems of [], and showed that the agorithm converges to a utiity max-min fair rate vector by using Dewey and Jury s stabiity criterion []. In this paper, we propose a utiity max-min fow contro agorithm based on optimization theory. We first propose a utiity max-min fow contro agorithm provided that there exists a singe congested node in the network, and then extend the agorithm to the case of mutipe congested nodes. The proposed agorithm requires neither the concavity of utiity functions nor any per-fow operation in the network. Moreover, ---//$. (C) IEEE

2 we mathematicay prove, for the case of a singe congested node, that the agorithm converges to a utiity max-min fair point with bounded communication deays. Here, we do not anayze the convergence of the agorithm accommodating mutipe congested nodes due to its compexity, but we show through simuations that the agorithm works as designed. To the best of our knowedge, our work is the first paper deaing with utiity max-min fow contro based on optimization theory. II. PROBLEM FORMLATION Consider a network in which one or more sessions (sourcedestination pairs) exist, and et L and S be the set of a inks and the set of a sessions in the network, respectivey. C, S and B respectivey denote the capacity of ink, theset of sessions sharing ink and the set of sessions bottenecked at ink. Each session i is associated with a utiity function i :[m i,m i ] R + where m i and M i respectivey denote the minimum required transmission rate and the maximum possibe transmission rate of session i. Assumption : (a) Every i ( ) is stricty increasing. (b) Every i ( ) is twice continuousy differentiabe. (c) For each i S, there exist constants a i,a i,a i > such that i ( ) a i, i ( ) a i, and i ( ) a i on [m i,m i ]. Definition : A rate vector x =[x i,i S]T is said to be utiity max-min fair if it satisfies the foowing property for any other rate vector x: if j (x j ) < j(x j ) for some j, then there exists i such that i (x i ) < i (x i ) j(x j ). Definition is equivaent to the definition of bandwidth maxmin fairness except that in utiity max-min fairness, utiities are compared instead of bandwidths. Assume that every session is bottenecked at a same node or equivaenty there is ony a singe congested node in the network. Suppose that is a congested ink at the node, then B = S = S. Letx i be the transmission rate of session i. We formuate a utiity max-min fow contro probem as an optimization probem by (P E ) : min. subject to ( av (x) i (x i ) ) i B i B x i = λc m i x i M i, i S, where λ is a target utiization and x =[x i,i S] T, av(x) = N i B i (x i ) and N = B. The optima soution of (P E ) obviousy achieves a utiity max-min fairness among competing fows and aso a target utiization of ink. However, it is not easy to obtain the optima soution due to the capacity constraint. So, instead of soving (P E ) as it is, we reax the constraint by adding a penaty function as foows: (P P ) : min. ( av (x) i (x i ) ) i B ) +µ (λc i B x i subject to m i x i M i, i S µ> where µ is a penaty factor. This probem is much easier to sove than (P E ). Moreover, the foowing theorem shows that (P E ) and (P P ) have the same and unique optima soution. For the proof, see the extended version of this paper []. Theorem : Suppose that i B m i C. Then, under part (a) of Assumption, (P E ) and (P P ) have the same and unique optima soution. This theorem says that the optimaity condition of (P E ) and (P E ) is equivaenty given as av( x )= i ( x i ), i B λc = i B x i m i x i M i, i S Hence, we may sove (P P ) to find the optima soution of (P E ). Let X = i B x i and f (x) be the objective function of (P P ), i.e., f (x) = i B ( av (x) i(x i) ) + µ (λc X ). Then, i f (x) = f (x)/ x i is written by if (x) = { i(x ( i) av(x) ) i(x i) + µ (λc X ) } () and f (x) = [ i f (x),i S] T. If f (x) is convex, the optima soution is easy to find because any stationary point of a convex function f(x), i.e., x such that f(x) =, is a goba minimizer of f(x). However, we can easiy show that f (x) is not aways convex. Nevertheess, we have the foowing theorem which shows that a stationary point of f (x) is the unique optima soution of (P P ). See [] for the proof. Theorem : nder part (a) of Assumption, x is the unique optima soution of (P P ) if and ony if f (x )=. Accordingy, our objective is now to find the unique stationary point of f (x). III. ASYNCHRONOS DISTRIBTED ALGORITHM AND ITS CONVERGENCE In this section, we discuss the distributed impementation of our utiity max-min fow contro agorithm and its convergence. We find the stationary point by gradient projection method which is most commony used to sove constrained optimization probems. A. Distributed Impementation Suppose that ink is a singe congested ink, and that each session i can be expressed as (source i, receiver i). Then, the transmission rate of source i is computed by () x i [x i γ i f (x)] + i ()

3 where γ > is a step size and [ ] + i denotes the projection on the interva [m i,m i ] and i f (x) is given by (). Link maintains two main variabes, AggRate and Avg, which respectivey contain the averaged aggregate arriva rate and the averaged utiity of sessions bottenecked at the ink. We assume that in the header of a packet, there exist two fieds, and C, used to carry the information for the computation () and that a receiver generates and sends an acknowedgement(ack) packet as soon as it receives a data(dat) packet. The foowing are the agorithms carried out by inks, sources and receivers. Here, the averaging factors α and β fa in (, ), and CurRate denotes the instantaneous aggregate arriva rate. Link pon every arriva of N a packets: Estimate CurRate. AggRate ( α) AggRate + α CurRate pon arriva of a data packet: Avg ( β) Avg + β DAT. DAT. Avg DAT. DAT.C λc AggRate Source i According to its transmission rate: Send a data packet with DAT. = i(x i). pon arriva of an ACK packet: x i [x i +γ { i(x i) ACK. + µ ACK.C}] + i Receiver i pon receipt of a data packet: Generate and transmit an ACK packet with ACK. = DAT. and ACK.C = DAT.C. As seen in the above, different from the previous work [], the proposed agorithm does not require any per-fow operation in the network, thereby being scaabe. Furthermore, the agorithm in [] estimates the number of ocay bottenecked sessions at each ink which is not simpe, but our agorithm does not need that. As we wi mention in Section IV, these features aso hod for the case of mutipe congested nodes. B. Convergence Anaysis In the rea network, the communication deays between any two nodes are inevitabe and the goba cock synchronization is impossibe. Accordingy, to anayze the convergence, we need to express the distributed impementation as an asynchronous distributed agorithm, which invoves the communication deay and the asynchronism. For the anaytica tractabiity, we assume that ony the most recent vaues of x i s are used to compute AggRate and Avg. Note, however, that AggRate and Avg are the vaues averaged over time in the impementation. Let T = {,,...} be a set of indices which correspond to the sequence of physica times at which one or more components of vector x are updated. For each i S, define T i T to be a set of times at which x i is updated. For each i, j S and each t T i, a variabe τj i (t) T denotes the time at which x j used in the update of x i is generated. Note that t τj i(t) and t τ i i (t) can be viewed as a communication deay from source i to source j via ink and the round trip deay of source i, respectivey. Let x i (t) be a oca rate vector at each source i S, that is, x i (t) = ( x j (τj i(t)),j S). Then, if t T i, the update of x i is given by x i (t +)= [ x i (t) γ i f (x i (t)) ] + () i Otherwise, x i (t +)=x i (t). Assumption : (Partia Asynchronism) There exists an integer B> such that: (a) For every i S and for every t, at east one of the eements of the set {t, t +,...,t+ B } beongs to T i. (b) There hods max{,t B +} τ i j(t) t, () for a i, j S and a t. By (a), the interva of any consecutive update is bounded, and by (b), the communication deay between any two sources is bounded. Let m =[m i,i S] T and M =[M i,i S] T, then we have the foowing theorem. See [] for the proof. Theorem : nder Assumption and, there exists a constant γ > such that if <γ<γ, then starting from any initia rate vector m x() M, every imit point of {x(t)} generated by the asynchronous agorithm () is the unique minimizer of f (x). From this theorem, we can notice that with a sufficienty sma step size γ>, the agorithm converges to the unique optima soution. IV. EXTENSION TO THE CASE OF MLTIPLE CONGESTED NODES So far, we have proposed and anayzed a utiity max-min fow contro agorithm for the case of a singe congested node. In this section, we discuss the extension of the utiity maxmin fow contro agorithm to the case of mutipe congested nodes. Suppose that there exist mutipe congested nodes in the network and one of the botteneck inks is, then B is not equa to S any more. Thus, if av(x) is updated as stated in the above subsection, i.e., updated whenever a data packet arrives on ink, then av(x) becomes the average vaue of the utiities of a sessions passing through. However, what av(x) is expected to be is the average utiity of sessions bottenecked at ink. It is therefore necessary to identify which sessions are bottenecked at ink. To do this, we need three more fieds, BL, NewBL and Min, in the header of a data packet. BL denotes the current botteneck ink, thus if DAT.BL is equa to ink, Avg is updated by DAT.. Min is set to a big number Max by the source and is compared with Avg of each ink on the path to find a new botteneck ink. That is, if Min > Avg, then NewBL and Min is respectivey set to ink and Avg. Beow is the pseudocode of our agorithm to accommodate mutipe congested nodes. Link

4 s s s s RT s s RT L L R R RT R R R R tiity Fig.. The network topoogy Fig.. The arriva and departure of sessions pon every arriva of N a packets: Estimate CurRate. AggRate ( α) AggRate + α CurRate pon arriva of a data packet: If DAT.BL == Avg ( β) Avg + β DAT. If DAT.Min > Avg DAT.Min Avg DAT.C λc AggRate DAT.NewBL Source i According to its transmission rate: Send a data packet with DAT. = i(x i), DAT.Min = Max and DAT.BL = BL. pon receipt of an ACK packet: x i [x i +γ { i(x i) ACK. + µ ACK.C}] + i BL = ACK.NewBL Receiver i pon receipt of a data packet: Generate and transmit an ACK packet with ACK. = DAT.Min DAT., ACK.C = DAT.C and ACK.NewBL = DAT.NewBL. The scaabiity sti hods here because the agorithm does not require any per-fow operation, but more fieds in the packet and more operations at the ink are needed compared to the case of a singe congested node. Once the botteneck ink of a session is identified, it takes more than RTT (round trip time) for the botteneck ink to be known to the source. Nevertheess, the agorithm converges as we show in the next section. V. SIMLATION RESLTS In this section, we show through simuations that the proposed utiity max-min fow contro agorithm works as designed. We used NS- network simuator []. The step size, the penaty factor and the target utiization used for simuations are γ =., µ =. and λ =.. We examine the proposed agorithm in the network topo- Bandwidth(Mbps) Fig.. tiity functions used in the simuation TABLE I THE THEORETICAL FAIR RATES(MBPS) OVER TIME session ogy shown in Fig.. S i, R i and RT i denote source i, receiver i and router i, respectivey. The pair (S i,r i ) constitutes session i and there are totay sessions. Each utiity function is given as: (x ) =. og(x +), (x ) =. og(x( +), (x ) =.x ), (x ) =.x, (x )= +e.(x ) +e and (x )= ( +e.(x ) +e ). As mentioned in many papers, the ogarithmic utiity function represents an eastic appication such as FTP whereas the sigmoida function approximates the utiity of a rea-time appication. The inear utiity function represents an appication whose satisfaction increases ineary to the bandwidth aocated to it. For better presentation, we show the utiity function in Fig.. The ink capacity between any two routers is set to Mbps. On the other hand, the ink capacities of the access inks are a set to Mbps so that no sessions are throtted there. The propagation deay of each ink is dispayed in the figure. Fig. shows the arriva-departure scenario of sessions over time, and the theoretica fair rates under the scenario are summarized in Tabe I. The simuation resuts are shown in Fig.. Observe from Fig. (a) that the utiizations of L and L are aways a target vaue.. In the period [,), session and share ink L, and session and share ink L, so the simuation scenario can be considered as two independent singe-botteneck scenarios. As seen in Fig. (c), in that period, the transmission rates of session and are equa to the theoretica ones and, we can see from Fig. (d) that the utiities of session and are equa to the average utiity at L shown in Fig. (b). For session and, the same resut is achieved at ink L. This resut shows that the utiity max-min fairness is achieved for the case of singe congested node whie the ink utiization is kept at a target vaue. From the time when session

5 aggregate arriva rate(mbps) (a) Aggregate arriva rates at L at L average utiity (b) Average utiities at L at L transmission rate(mbps) (c) Transmission rates session session session session session session utiity (d) tiities session session session session session session Fig.. Resuts with mutipe botteneck ink configuration joins the network, the simuation scenario becomes to refect the case of mutipe congested nodes. In the period [,), the resut agrees with what we expected theoreticay. Notice that the utiity of session in that period is equa to the average utiity at L, which impies that session is bottenecked at L. In the period [,), session joins the network, and we can see from Fig. (d) that the botteneck ink of session and is L because the utiities of them are equa to the average utiity at L. The expected resut is achieved for the period [,). In brief, the proposed utiity max-min fow contro agorithm works as designed when there exist mutipe congested nodes in the network. VI. CONCLDING REMARKS In this paper, we deveoped a utiity max-min fow contro agorithm which is based on the optimization theory. The agorithm operates in asynchronous and distributed manner and does not require any per-fow operation in the network. Different from most of the researches deaing with utiity functions, we did not assume that every utiity function is stricty concave. Nevertheess, for the case of a singe congested node, we mathematicay proved that the agorithm converges to a unique goba optima point under some conditions. Athough the anaysis was performed for the case of a singe congested node, we showed through simuations that the agorithm works as designed for the case of mutipe congested nodes as we. [] F. Key, A. K. Mauoo, and D. K. H. Tan, Rate contro for communication networks: shadow prices, proportiona fairness and stabiity, J. Oper. Res. Soc., vo., no., pp., Mar.. [] S. H. Low and D. E. Lapsey, Optimization fow contro-ii: Impementation, Interna Report, Mebourne niversity, May. [] S. Athuraiya and S. H. Low, Optimization fow contro-i: Basic agorithm and convergence, IEEE/ACM Trans. Networking, vo., no., pp., Dec.. [] R. J. La and V. Anantharam, tiity-based rate contro in the internet for eastic traffic, IEEE/ACM Trans. Networking, vo., no., pp., Apr.. [] S. Shenker, Fundamenta design issues for the future internet, IEEE J. Seected Areas Comm., vo., no., pp., Sep.. [] J.-W. Lee, R. R. Mazumdar, and N. B. Shroff, Non-convexity issues for internet rate contro with muti-cass services: Stabiity and optimaity, in Proc. IEEE INFOCOM, Hong Kong, Mar.. [] Z. Cao and E. W. Zegura, tiity max-min: An appication-oriented bandwidth aocation scheme, in Proc. IEEE INFOCOM, New York, Mar., pp.. [] J. w. Cho and S. Chong, tiity max-min fow contro using soperestricted utiity functions, Submitted for Pubication. [] A. G. Dewey and E. I. Jury, A stabiity inequaity for a cass of noninear feedback systems, IEEE Trans. Automatic Contro, vo., no., pp., Jan.. [] H.-W. Lee and S. Chong, A distributed utiity max-min fow contro agorithm, Extended version of this paper,. [Onine]. Avaiabe: mshw/iccextd.pdf [] Ns-. [Onine]. Avaiabe: REFERENCES [] F. Key, Charging and rate contro for eastic traffic, Euro. Trans. Teecommun., vo., pp., Dec..