Improved Utility-based Congestion Control for Delay-Constrained Communication

Transcription

1 Impoved Utility-based Congestion Contol fo Delay-Constained Communication Stefano D Aonco, Laua Toni, Segio Mena, Xiaoqing Zhu, and Pascal Fossad axiv:5.2799v3 [cs.ni] 2 Jan 27 Abstact Due to the pesence of buffes in the inne netwok nodes, each congestion event leads to buffe queueing and thus to an inceasing end-to-end delay. In the case of delay sensitive applications, a lage delay might not be acceptable and a solution to popely manage congestion events while maintaining a low end-to-end delay is equied. Delay-based congestion algoithms ae a viable solution as they taget to limit the expeienced end-toend delay. Unfotunately, they do not pefom well when shaing the bandwidth with congestion contol algoithms not egulated by delay constaints (e.g., loss-based algoithms). Ou taget is to fill this gap, poposing a novel congestion contol algoithm fo delay-constained communication ove best effot packet switched netwoks. The poposed algoithm is able to maintain a bounded queueing delay when competing with othe delay-based flows, and avoid stavation when competing with loss-based flows. We adopt the well known pice-based distibuted mechanism as congestion contol, but i) we intoduce a novel non-linea mapping between the expeienced delay and the pice function, and ii) we combine both delay and loss infomation into a single pice tem based on packet inteaival measuements. We then povide a stability analysis fo ou novel algoithm and we show its pefomance in the simulation esults caied out in the NS3 famewok. Simulation esults demonstate that the poposed algoithm is able to: achieve good inta-potocol fainess popeties, contol efficiently the end-to-end delay, and finally, potect the flow fom stavation when othe flows cause the queuing delay to gow excessively. Keywods Delay-sensitive communication, congestion contol, netwok utility maximization. I. INTRODUCTION NOWADAYS, many Intenet applications aim to wok not only at maximizing thei thoughput, but also at meeting cucial delay constaints in the tansmission of data flows. Video confeencing applications ae a good example of such delay sensitive sevices, whee an excessive playback delay with the audio/video steam can dastically affect the quality of an intenet call. On-line video gaming and desktop emote contol ae othe examples of applications that equie low latency communications. When netwok links ae congested these applications need to adjust thei sending ate such that the expeienced one-way delay is kept low and bounded, while peseving fainess with othe flows. This educes to a constained esouce allocation poblem, which has to be solved in a fully distibuted manne due to scalability issues. Congestion contol algoithms can be seen as distibuted methods to solve optimal netwok esouce allocation poblems. These algoithms ae usually categoized in pimal o dual algoithms, based on the solving method adopted [] [3]. Fom a moe pactical point of view, the pimal and dual congestion contol algoithms oughly coespond, though not exactly, to loss-based and delay-based algoithms. Lossbased contolles ae widely deployed ove the intenet (e.g., TCP) and use congestion events tiggeed by packet losses to pefom ate adaptation. Howeve this class of contolles does not take into account any type of delay measuement, such as the One-Way Delay (OWD) o the Round Tip Time (RTT). Hence, thee is no contol on the latency that the packets might expeience on thei oute and lage delays can be expeienced in the case of long buffes in the inne netwok nodes. On the othe hand, delay-based congestion contol algoithms can ovecome the inceasing delay issue by detecting congestion events fom OWD measuements. Delaybased congestion contol algoithms ae theefoe suitable fo low delay applications since they ae able to keep a low communication delay by adapting the sending ate to the evolution of the delay. Howeve, they usually suffe fom poo pefomance when shaing the netwok with loss-based contolles. Loss-based congestion contol algoithms always fill the buffes of the inne netwok nodes befoe tiggeing congestion events. Theefoe, any delay-based flow shaing the same bottleneck might expeience a too lage queuing delay and quickly each stavation (i.e., a sending ate close to zeo). Thee is the need fo a congestion contol that could enable low delay communication wheneve possible (essential fo low delay communication) and that is obust against the pesence of loss-based flows (mostly deployed in the intenet) Beyond this coexistence challenge, new congestion contol algoithms to be deployed in the Intenet need: i) to povide good inte-potocol pefomance when competing against existing contolles, such as TCP; ii) to mainly act at the endpoints athe than at the inne netwok nodes (modifications of the inne netwok nodes ae paticulaly difficult); iii) to be obust to noisy measuement of netwok paametes (i.e., popagation delay). Many congestion contol algoithms have actually been poposed in the past, but to the best of ou knowledge thee is no delay based congestion contol algoithm able to well pefom in the pesence of loss-based flows and satisfying at the same time the thee main implementation challenges listed above. In this wok, we taget to fill this gap by poposing a new distibuted Delay-Constained Congestion Contol (DCCC) algoithm that is able to adapt the sending ate to both loss and delay-based congestion events and to ovecome the afoementioned issues. The ultimate objective is to peseve the low end-to-end delay constaint that is imposed by the application, when competing with othe delay-based contolled flows, and at the same time, avoid stavation when competing with lossbased flows. In moe details, we conside a scenaio whee

2 2 Buffes Flows Sendes Fig.. Netwok system model. Receives II. NETWORK UTILITY MAXIMIZATION A. System Model We conside a system with a set of R uses (o flows) indexed by R shaing a set of netwok esouces. Each use tansmits data at a sending ate x, in a single-path unicast communication between two endpoints. Let l L be the links of the netwok, and R be the L R outing matix, whee R l = if link l is used by use and zeo othewise. The total ate passing though link l, denoted by y l, can then be defined as y l = R R l x. () Evey link has a fixed maximum channel capacity c l, which is constant ove time. Each link is peceded by a buffe that accumulates the data in suplus when y l exceeds the link capacity c l. The queueing delay at the buffe of link l, q l (t), evolves ove time as follows: q l = c l (y l c l ) + q l, (2) uses send delay-sensitive data ove a packet switched netwok, as shown in Fig.. The netwok is composed of a set of links and nodes, with the links being shaed among diffeent uses who set up unicast communications between two endpoints of the netwok. The poposed contolle measues the expeienced OWD and the inteaival time of the eceived packets at the eceive node, and adjusts the ate accodingly in ode to maximize the oveall utility of the netwok flows. The key intuition is that the inteaival time of the packets is coelated to both losses and queueing delay vaiations. Hence, by using this metic, the contolle is able to wok in both delay-based and loss-based envionments. The ability to avoid stavation when competing against loss-based flows, and still guaantee a bounded expeienced delay is made possible by the use of a non-linea mapping between the expeienced OWD and the penalty congestion signal used by the ate update equation. The DCCC algoithm has been implemented in the NS3 netwok simulato and has been tested unde diffeent topologies and woking conditions. Simulation esults show the ability of the poposed algoithm to keep bounded the value of the expeienced OWD, to achieve a good inta-potocol fainess and to avoid stavation when competing with lossbased flows, such as TCP. Note that the poposed algoithm elies on the OWD measue, which is meaningful only in the case of synchonized endpoints. Howeve, the contolle can still opeate popely in the case of unsynchonized endpoints if the desynchonization is elatively small with espect to the OWD expeienced. The emainde of this pape is oganized as follows. In Section II, we povide a desciption of the system model and the Netwok Utility Maximization (NUM) famewok that is used in this pape. In Section III, we descibe ou new congestion contol algoithm and we analyze its stability in Section IV. In Section V, we povide some last details about the coect implementation of the contolle and pesent the simulation esults. Section VI descibes the elated wok. Finally conclusions ae povided in Section VII. Whee (z) + x is a pojection that keeps the queuing delay positive. Hee, as in the emainde of the wok, we identify by ż the deivative of the vaiable z with espect to time. A data unit, o packet, that aives at link l at time t takes a time equal to q l (t) befoe being actually sent ove the link. We futhe denote by p l the popagation delay of link l, which is defined as the time equied fo the data to popagate though the physical link; we assume p l to be constant ove time. Theefoe the total time fo a data unit to tavese the link l is given by the sum of the popagation and queueing delays, i.e., d l = p l + q l. The total one-way delay expeienced by use, e, is given by the sum of all the delays encounteed on the path that connects the souce node and the eceive node, which can be mathematically defined as e = l L R l d l. (3) Finally, the netwok buffes have a maximum size ql MAX and a doptail policy is implemented. This means that when the queue length eaches ql MAX, the next incoming packets ae discaded. The loss atio fo link l when its buffe is full coesponds to: ( ) + yl c l π l =, (4) y l y l c l and the total loss atio fo use is: π = l L R l ( π l ), (5) which, if the loss atios of the links ae small, can be appoximated by π l L R l π l. () We futhe intoduce anothe notion of communication delays between the netwok nodes. We denote by τ f l the delay

3 3 expeienced by data packets fom the souce node of use to the inne node located at the end of the link l. Note that τ f l is defined only if link l is a link of the oute of flow, and τ f l = e if l = L, with L being the final link of oute. Intoducing the timing aspects in Eq. () leads to have that the total ate of link l at time t is a function of the sending ates at time t τ f l, this means : y l (t) = R l x (t τ f l ). (7) R Similaly we define τl b as the time needed fo the data to tavel fom link l to the sink node and then to be sent back to the souce node. Note that any kind of netwok event is fist detected by the eceive node and then is sent back to the souce node, theefoe the following condition holds: RTT = τ f l + τ b l, l. (8) The RTT of use can also be expessed as a function of the expeienced use delay: RTT = e + e b, (9) whee e b is the total expeienced delay of the backwad path. In the cases whee the backwad path is not congested, it is easonable to assume that e b is equal to the popagation delay of the fowad path. In any case, we assume that the backwad delay e b does not depend on the ate x. We intoduce now the concept of utility function. We denote by U (x ) the benefit fo use of sending data at ate x. The utility function is usually consideed to be a concave inceasing function of the ate. In the following we conside a logaithmic utility function given by: U (x ) = h log(x ), () whee h is a simple weight paamete that can take diffeent values depending on the use, and thus modify the impotance of the diffeent flows. Finally, we point out, that ou famewok can be extended to any diffeentiable inceasing concave utility function, leading possibly to diffeent esults. B. Optimization Poblem We now define the NUM poblem based on the famewok of []. The goal is to optimally allocate the ates of the uses in such a way that the oveall utility is maximized and that the capacity constaints of the netwok links ae espected. The NUM poblem is typically defined as follows: NUM: maximize x U (x ) R subject to l L R l x c l, l =,..., L. () The NUM poblem can ideally be solved exactly in a centalized way. In this case the exact solution would be fist Note that in case of losses Eq. (7) and Eq. () do not hold exactly due to the actual ate decease caused by packet dops. The equations can howeve still be consideed valid in low-loss egime. computed and then communicated to the uses. As a esult the capacity constaints would neve be violated, leading to no losses and no queueing delay at the bottlenecks links. The equiement of a pio full knowledge of the netwok state pevents this solving method to be actually usable. Thus, the NUM poblem is typically decoupled and then solved in a distibuted manne. C. Solution by Decomposition We discuss now the most commonly used appoaches to solve the NUM poblem, which include solutions based on penalty decomposition and on dual decomposition, and we point out thei main limitations. We efe to [4] fo a detailed tutoial on decomposition methods fo NUM. A viable method is by using penalty functions, o pices, which map the violation level of the capacity constaints into a negative utility. The poblem in () can be efomulated as follows: maximize x R U (x ) l L yl g l (z)dz (2) whee g l (z) is seen as the pice to pay fo using link l when the link ate is equal to z. It is impotant fo g l ( ) to be a positive and inceasing function of the link ate. The poblem in Eq. (2) can be solved by a gadient-based algoithm with the following the ate update equation: ( ẋ = α U (x ) ) R l g l (y l ), (3) l L whee U (x ) is the deivative of the utility function with espect to x. In the eminde of the pape the notation f (z) denotes the deivative of the function f(z) with espect to a geneal vaiable z. Imposing the loss atio of link l as pice: g l (y l ) = π l (y l ), and assuming R l π l π, the poblem is decoupled since uses need to know only thei own utility function and thei loss atio π to compute the ate update. Uses can easily estimate thei loss atios by obseving the numbe of dopped packets at the eceiving node. The main dawback of this solution is that at the equilibium losses ae always expeienced, which means that bottleneck buffes ae always full and lage queuing delays might be expeienced. A lage family of congestion contol algoithms, such as the classical loss-based vesion of TCP, can be modeled as systems govened by Eq. (3). Anothe possible way to solve the NUM poblem is by dual decomposition. The optimization poblem can be solved in a distibuted way by using a pimal-dual algoithm based on the following update equations: ( ẋ = α U (x ) ) R l λ l (4a) l L λ l = ν l (y l c l ) + λ l, (4b) whee λ l is the dual vaiable associated to the capacity constaint of link l, and ν l is a simple positive paamete that contols the ate of update of the dual vaiable λ l. Note that

4 4 if we set ν l = /c l, the dual vaiables have exactly the same fom of the queueing delays defined in Eq. (2). Theefoe, uses can compute Eq. (4a) by estimating the total queueing delay of thei own oute, and adapt thei ates accodingly. Fo a given utility function the delay expeienced at the equilibium of a use depends on the bottleneck capacity. Geneally, the lage the bottleneck capacities c l, the lage the use ates x at equilibium, and the lowe the delays at equilibium. It is woth noting that a flow that conveges at the equilibium to a ate equal to x, needs to expeience (ceate) a cetain amount of queueing delay along its oute, which is usually called selfinflicted delay. The basic dual decomposition method of Eq. (4) has two main limitations: i) it is not obust against loss-based flows, as aleady explained in the intoduction; ii) it assumes a pefect knowledge of the total queueing delay (when λ l = q l ). In pactice, uses can measue only the sum of the popagation and queueing delay. and extapolate a pecise measue of the queueing delay fom the total delay is highly challenging. In summay loss-based congestion contol algoithms neglect the delay aspect in thei algoithms, while delay-based congestion contol algoithms always adapt to the delay measue, loosing in tems of pefomance when competing with loss-based algoithms. In the next section, we descibe ou congestion contol algoithm which is able to ovecome the afoementioned limitations of classical congestion contol schemes. III. DELAY-CONSTRAINED CONGESTION CONTROL ALGORITHM A. Contol Algoithm In this section, we pesent ou DCCC algoithm, showing how it ovecomes the main limitations of the existing contolles. The ate update equation that we conside fo ou algoithm is the following: ẋ = k x (U (x ) v (e ) ė π ) + x. (5) We now descibe the diffeent tems of Eq. (5) in detail. The paamete k tunes the update speed of ate x. The fist tem in the backets is the deivative of the utility function U ( ). The tem v ( ) is the delay penalty function that maps the OWD into a penalty. Similaly to the loss pice definition in Eq. (4), we wite the delay penalty as: ( ) + ) + e T v (e ) = β RTT (e T ) ( e T = β e + e b, (e T ) () whee RTT is the ound tip time of use, e and e b ae the fowad and backwad expeienced delays of use espectively, β is a scaling facto and T is the delay theshold of use. The delay theshold is elated to delay that the system expeiences at the equilibium. The value of v (e ) is equal to zeo if e T < and equal to β(e T )/RTT othewise. The nomalization of the pice by RTT is motivated by ate fainess impovements and stability conditions. Note that this function is not a linea function of the expeienced delay, e, but athe a monotonically inceasing function of it. The deivative of the expeienced delay, ė, does not modify the equilibium of the system, since the time deivative at the equilibium point will be zeo by definition. Howeve, it impoves the contolle pefomance duing the tansients, since it povides infomation about the vaiation ate of the feedback vaiable. The last tem in Eq. (5), π, takes into account the expeienced losses, following Eq. (5), so that the algoithm is able to opeate in both delay-based and loss-based scenaios. In the case of no losses (π = ), ou contolle behaves as a delay-based contolle. The expeienced delay at the equilibium is evaluated by setting the time deivatives to zeo in Eq. (5): ê = RTT β U (ˆx ) + T = RTT h βˆx + T. (7) Substituting Eq. (9) into Eq. (7) we obtain: ê = êb h/(βˆx ) + T. (8) h/(βˆx ) Fom Eq. (8), we obseve that: i) ê conveges at the equilibium to lage values than T, and appoaches to T fo inceasing values of ˆx ; ii) the ate at equilibium has to veify the following inequality: ˆx > h/β. (9) This means that the delay pice v (e ) in ou DCCC algoithm neve foces the sending ate to be lowe than h/β. The main benefits of ou penalty function can be summaized as follows: The non-lineaity of the penalty function potects the flows fom stavation when competing against loss-based algoithms. Fig. 2 depicts the shape of the penalty function of Eq. () fo diffeent values of the popagation delay, when the backwads delay, e b is consideed to be equal to the one-way popagation delay in the fowad diection. The value of the penalty satuates to β fo lage values of the expeienced delay, which is the typical scenaio that we encounte when competing with loss-based flows. As a consequence the expeienced delay can neve foce the sending ate to decease to a value lowe than h/β thus peventing stavation. The non-lineaity of the penalty function alleviates unfainess poblems caused by heteogenous popagation delays among the uses. Since ou contol algoithm uses the total expeienced one-way delay instead of the queueing delay, it may lead to unfainess when a bottleneck link is shaed among uses with diffeent popagation delays. Howeve the non-linea mapping of the delay helps to alleviate this poblem when the available capacity is low. This can easily be undestood by looking at the shape of the penalty function in Fig. 2. Since the penalty value tends to satuate fo lage delays, i.e., low available capacity, it means that uses with diffeent popagation delays will have simila penalty values in this scenaio, and as a consequence simila sending ates.

5 5 delay penalty value [/β] T T = 8 ms, OWD min = 5 ms T = 8 ms, OWD min = 25 ms T = 8 ms, OWD min = 7 ms OWD [s] Fig. 2. Delay penalty as a function of the expeienced delay fo diffeent values of popagation delays, measued in units of β. The loss-based pat of the algoithm, π, woks as an additional penalty that lowes the value of the ate at the equilibium in case the delay penalty is not able to educe it significantly. Fo instance, when congestion is expeienced fo low ates, x h/β, the loss atio π foces the ate to futhe decease. Losses can also happen due to the pesence of lossbased flows in the netwok o to the pesence of shot buffes inside the netwok. In the exteme case whee the maximum end-to-end maximum delay of a oute is smalle than the delay theshold, i.e., e MAX < T, the delay penalty is always zeo and the ate is fully diven to the equilibium exclusively by the loss tem π. B. Contolle Implementation We now biefly discuss the pactical implementation of the DCCC descibed above. Ou theoetical analysis elates to a continuous system, while in pactice we wok in a discete context, whee entities that tavel though the links ae data packets athe than continuous flows. We theefoe need to adjust the contolle in Eq. (5) to wok in a discete domain. In a packet based model a measue that can pactically be evaluated is the inteaival time of the packets at the eceive node. We now explain how, by using this quantity, we ae able to build a tem that incopoates both aspects of ou contolle, namely the delay-based pat and the loss-based pat. Moe specifically, the two tems that will be unified ae the queueing deivative tem, ė, and the tem π that takes into account the losses of the flow. We stat the deivation of the unified tem by finding the connection between the inteaival time and the eceiving ate, then we give the expession of the meged tem and show that it coesponds to appoximate the sum of π and ė We fist conside the case of long buffes whee no losses can be expeienced. Ideally the inteaival time is invesely popotional to the eceived ate. If we measue the eceiving ate in packets pe second, we can wite: x ecv (n) = t a (n) t a (n ) = t a (n), (2) whee t a (n) is the instant of aival of the n-th packet. Noting that t a (n) = t s (n) + e(n), whee t s (n) is the sending time of packet n, and that the time between two consecutive depatues is equal to /x, we can wite x ecv (n) = /x + e (n + ) e (n) = /x + e (n) (2) whee e (n) is the vaiation of the OWD between two consecutive packets. As the popagation delay is fixed, e (n) coesponds to the vaiation of the queueing delay. Since the vaiation is evaluated between the sending time of two consecutive packets, we can wite ė (n) e (n)x. Recall that we measue the sending ate x in packets ove seconds, thus e (n)x is dimensionless, as ė. If we substitute this appoximation in Eq. (2) and solve fo ė we obtain: ė x x ecv x ecv. (22) The value of x ecv can be calculated at the eceive using Eq. (2) while the value of the tue sending ate can be epoted in the heade of the tansmitted packets. We now analyze the meaning of the ight hand side of Eq. (22) in the case of losses. We conside the case of doptail buffes, thus losses ae expeienced only when the buffes ae completely full. The loss atio fo use is defined as π = xecv x, o equivalently: x x ecv x ecv = π π. (23) Note that the left hand side of Eq. (23) is equal to the ight hand side of Eq. (22). Fo small loss atios, π, Eq. (23) becomes x x ecv x ecv π. (24) ( ) x x ecv x can ecv Accoding to Eq. (24) and Eq. (22) the tem be used in both delay-based scenaios (to appoximate ė ) and loss-based scenaios (to appoximate π ). The advantage of the meged tem is that we do not need to distinguish between the two opeation modes of ou algoithm, and we simply need to evaluate the value of this tem at evey ate update step. We can finally wite the ate update ule that govens the DCCC algoithm in the discete domain as ( x new = x + T s k x U (x ) β e T RTT x (t RTT ) x ecv x ecv ), (25) whee T s is the time inteval at which the ate is updated, and x (t RTT ) denotes the delayed sending ate used to calculate the loss and queueing deivative tem (the estimation of the eceived ate and the sending ate need to coespond to the same set of packets fo a coect calculation of this final tem, thus communication delays cannot be neglected). In Section V, we explain in detail how to effectively set the diffeent paametes of the ate update equation (25).

6 IV. STABILITY ANALYSIS In this section, we discuss the stability of the DCCC algoithm based on Eq. (5), focusing on the case of delaybased egime. In ode to model a netwok whee the only congestion signal expeienced is the OWD, we set the length of netwok buffes to be infinite. In this case, the flows expeience no losses, (i.e., π l = l) and each the equilibium point using only the delay measuements. We obviously assume that the equilibium point exists in this case. Note that even if the stability of simila delay-based contolles has aleady been studied in othe woks [3], [5], due to the non-linea mapping of the expeienced delay in Eq. (5), those poofs do not apply diectly to ou algoithm. The stability of the algoithm in lossbased egime is commented at the end of the section. We fist show the global stability of the non-linea DCCC fo an ideal scenaio when the communication delays ae negligible, i.e., τ f l τ l b, fo a geneal case of R uses and L links. The communication delays ae negligible when they ae emakably smalle than the timing chaacteistics of the contol system. In the second pat of the section, we focus instead on local the stability of the contolle when communication delays ae non-negligible. A. Negligible Delay Case In ode to study the global stability of the undelayed system we extend the analysis caied out in [] to the case of nonlinea pice function. The global stability poof is based on the passivity analysis of dynamic systems [] [7]. A dynamic system chaacteized by an input u, state x, and output y is said to be passive if thee exists a positive definite stoage function V such that its deivative can be witten as: V W (x) + u T y, (2) whee W (x) is a positive semi-definite function. If two passive systems ae inteconnected in a negative feedback loop, then the sum of thei espective stoage functions is a Lyapunov function fo the inteconnected system. Indeed, the output of one system is the opposite of the input of the othe one, so the two tems u T y vanish: V W (x ) W 2 (x 2 ) + u T y + u T 2 y 2 (27a) = W (x ) W 2 (x 2 ) y2 T y + y T y 2 (27b) = W (x ) W 2 (x 2 ). (27c) Ou dynamic system can actually be modeled as an inteconnection of two systems (namely, the uses and links dynamics in Fig. 3), which dive the behavio of ou congestion contol famewok. By poving that these two sepaate systems ae both passive, then we also pove the global stability in the sense of Lyapunov fo the entie system. Let fist intoduce some notations that ae used in ou stability poof. The opeato ˆ denotes the value of the vaiables at the equilibium point, and δ denotes the deviation of a vaiable fom its equilibium point, e.g., δx = x ˆx. We denote with V use and V link the stoage function of the uses and links subsystems espectively. The inputs of the uses system ae the Fig. 3. R e, ė d, d ẋ U (x) pice(e, ė) q /c(y c) Block scheme of the congestion contol Uses Dynamic x y Links Dynamic expeienced delays e and thei deivatives ė, while its output coesponds to the sending ates x. Similaly, the links system has the link ates, y, as input vaiables and the links delays, d and d, as output vaiables. The outing matix R maps the use ates to the link ates and the link delays to the uses expeienced delays. Finally, the equilibium point of the system is eached when the dynamic equations (2) and (5) ae set to zeo. This is equivalent to having v (ê ) = U (ˆx ) and ŷ l = c l if ˆq l >, o ŷ l c l if ˆq l =. Since we ae at equilibium, q l = ; we futhe have ė = l L R ld l =, whee d l is the sum of the queueing delay q l and the popagation delay p l (which is constant ove time). We conside the following candidate stoage functions fo the use and link subpat espectively: V use = kα ( R x ˆx z ˆx dz z ) R (28) V link = δd 2 l c l + (c l ŷ l )d l, (29) 2 α l L l L fo some α >. Note that since d l the two stoage functions ae positive definite functions fo δx and δq espectively. It can be veified, by taking the time deivative of the two stoage functions, that inequality of Eq. (27c) holds fo two valid functions W use and W link, poving the stability of the system. The full poof can be found in Appendix A. B. Non-negligible Delay Case We now study the local stability of the system when the communication between netwok nodes is affected by non negligible delays, as it is typically the case in pactice. In this case we fist lineaize the ate update equations of the nonlinea delayed system at the equilibium point, then we study the local stability of the delayed system similaly to [3], [8].

7 7 When communication delays ae non-negligible, Eq. (5) can be witten as: ( ẋ (t) = k x (t) U (x (t)) (3) v ( l L R l d l (t τ b l)) l L R l d l (t τ b l) while the queueing delay given by Eq. (2) becomes: ( q l (t) = R l x (t τ f l c ) c l l R ) + ) + x q l. (3) The new equations (3) and (3) epesent the uses and links dynamics taking into account the communication delays. Recall that τ f l and τ l b captue the time needed fo the packets to go fom souce to link l, and fo the feedback infomation to be eceived by the souce espectively. Recall also that the congestion infomation fom link l needs to each the end node of use befoe being sent back to the souce node. As a consequence, the sum τ f l + τ l b does not depend on l and is equal to the RTT of the oute, i.e., τ f l + τ l b = RTT. When we lineaize Eq. (3) and Eq. (3) at the equilibium point, we obtain the following equations: δx(t) = k ˆx (Û (ˆx )δx(t) ˆv (ê ) l L R l δq l (t τ b l)) ) R l δ q l (t τl) b, l L ( ) δq l (t) = R l δx(t τ f l c ), (32) l R whee U coesponds to the second deivative of the utility function calculated at the equilibium point, and v is the deivative of the penalty function calculated at the equilibium point. By taking the Laplace tansfom of the above equation we obtain: sδx(s) = K ˆX (Û δx(s) ˆV R T b (s)δq(s) sr T b (s)δq(s)) sδq(s) = C (R f (s)δx(s)), (33) whee δx indicates the deviation of the vaiable x fom the equilibium point, i.e., δx = x ˆx, and δx(s) efes to the Laplace tansfom of δx. The same notation applies to δq. The system of equations (33) is witten in a matix fom, whee C is an L L squae diagonal matix whose enties ae equal to c l. The outing matices R f (s) and R b (s) embed the delay infomation and ae defined as: R f l (s) = e sτ f l if use employs link l and othewise, and analogously R b l (s) = e sτ b l if use employs link l and othewise. Then, K and ˆX ae R R squaed diagonal matices which contain the values of the gains k and the values of the ates at equilibium x espectively. ˆV and Û ae R R squaed diagonal matices whose enties coespond to the values at the equilibium point of the fist deivative of the penalty delay functions and to the second ode deivative of the utility functions espectively. Next, solving (33) fo δx(s) we obtain: δq(s) = C R f (s)s (Is K ˆX Û ) K ˆX ( ˆV ) s R T b (s)δq(s). (34) Fom Eq. (34) we can easily detemine the etun loop atio F (s) [9], [], of ou multivaiable feedback system: F (s) = C R f (s)s (Is K ˆX Û ) K ˆX ( ˆV + s) R T b (s). (35) Noting that R T b (s) = diag(e srtt )R T f ( s) and using the popety that diagonal matices commute, we can wite: F (s) = C R f (s)s (Is K ˆX Û ) ( ) K ˆV + s diag(e srtt ) ˆX R T f ( s). (3) Fo the sake of claity, we intoduce the matix G(s): G(s) = s (Is K ˆX ( ) Û ) K ˆV + s diag(e srtt ), (37) this is an R R diagonal matix. Similaly to [5], we futhe intoduce the matix R(s) = diag(/ c l )R f (s)diag( x ). (38) Since the eigenvalues of matix poduct does not depend on the ode of the matices we can wite: ( ) σ(f (s)) = σ R(s)G(s) RT ( s), (39) whee σ( ) denotes the set of eigenvalues of a matix. We can now veify the stability of the system using the Nyquist stability citeion. We set s = jω, whee j is the imaginay unit, and we vay its value on the Nyquist path. The tajectoies of the eigenvalues of the system as a function of jω must not encicle the point + j fo the system to be stable. We use now the main esult of []: σ(f (jω)) ρ( R T (jω) R( jω)) co( σ(g(jω))), (4) whee ρ( ) denotes to the spectal adius of a matix and co( ) denotes the convex hull. Eq. (4) states that the eigenvalues of F (jω) ae located on the convex hull made by the eigenvalues of G(jω) scaled by the spectal adius of R T (jω) R( jω). Fom [2], we futhe know that the spectal adius ρ( R T (jω) R( jω)) M, with M being the maximum numbe of links used by any use. Hence, to pove the stability of the system, we need to show that the eigenvalues of G(jω) in Eq. (39) do not encicle the point /M + j in the complex plane. Since G(jω) is a diagonal matix, as can be seen fom Eq. (37), we need to show that all the enties on its diagonal veify the Nyquist stability citeion. Defining G (jω) as the element of the diagonal of matix G(jω) we have e jωrtt v + jω G (jω) = k jω jω k ˆx U. (4) As the utility functions ae concave, U is negative, and so both poles of Eq. (4) ae non positive. With a utility function defined as U (x ) = h log(x ), the value of the second

8 8 deivative of the utility and the deivative of the pice at the equilibium point ae espectively equal to U v = β h/ˆx RTT, (42) = hˆx 2 whee in the second equation we used the popety that v (ê ) = U (ˆx ) at equilibium. Note that, in ode to have a negative zeo in the tansfe function G (jω) we need the inequality ˆx > h/β to hold. This condition suppots in fact that h/β is the minimum equilibium ate fo the puely delaybased subpat as descibed in Section III. Finally, substituting Eq. (42) in Eq. (4), we obtain G (jω) = η e jωrtt jωrtt β h/ˆx RTT + jω, (43) h jω + η ˆx RTT whee we have intoduced the nomalized gain η = k RTT. Eq. (43) is a tansfe function with two poles and one zeo. The location of the second pole and the zeo ae actually linked since they depend on the same quantities. In paticula, due to the pevious consideations about the minimum value of the equilibium ate, the pole cannot be located at an angula fequency lage than η β/rtt, and the zeo has to be located between and β/rtt. By selecting appopiate values fo the contolle paametes h, β and η, we can ensue that point /M + j is not encicled. The tuning of the paametes is complex because in geneal the setting of one paamete affects the value of the othes. One possible way to poceed is to allocate the zeo-pole couple at low fequencies and the coss fequency at which G (jω) = /M to a highe value. In this case at the coss fequency the zeo and the pole compensate each othe decoupling and facilitating the tuning of the est of the paametes. C. Summay of stability analysis In this section we have studied the stability of ou system in the case whee the only congestion event eceived fom the netwok is the expeienced delay. The main esult is that the non-linea mapping of the expeienced delay leads to a stable delay-based system. The stability analysis can also be extended to lossy scenaios. In the case of doptail buffe policy, when losses ae expeienced the queue size is constant and equal to the maximum one (we do not conside the case of Random Ealy Detection (RED) policy [3] o any othe Advanced Queue Management (AQM) mechanism). Since a constant delay plays no ole in the placement of eigenvalues of the linea dynamic system at equilibium, the delay-based tems of the ate update equation disappea in the lineaization pocess of the system. Hence, fo the loss-based pat of ou algoithm, the lineaized update equation is consistent with the one used in [], [4] and the stability poof can be caied out by following the steps descibed in these woks. V. FINAL IMPLEMENTATION AND SIMULATIONS We now povide simulations esults fo the poposed DCCC algoithm. Fist, we explain how the contolle pefomance is affected by each paamete and how to efficiently set them. TABLE I. PARAMERS SETTING SUMMARY Paamete name Used value Suggested ange k /(2.5RTT) /(2.5T s)-/(t s) β T s RTT 5-5 ms T 5- ms 5-25 ms h 2 kbps 5-5 kbps Then we povide the simulation esults whee we analyze the basic behavio of ou algoithm, the inta-potocol fainess, the TCP coexistence and finally we povide a compaison with othe simila congestion contol algoithms. A. Paametes and Simulation Setup The paametes of the DCCC ae set by using the esults obtained fom the stability analysis of Subsection IV-B, and by empiical knowledge on the effect of the paametes on the contolle behavio. The gain k, the sampling peiod T s and the paamete β of Eq. (25) stongly affect the coss fequencies of the tansfe functions that compose the matix G (defined in Eq. (37)). They theefoe affect the speed and the stability of the closed loop system and ae set to impose the stability of the system. Convesely, the paametes h and T do not stongly affect the stability of the system. The value of h affects the aggessiveness of the contolle, as the pice at equilibium is popotional to h. The pice eflects both the oveshoot of the delay theshold in delay-based envionments and the fainess level with othe loss-based potocols. The value of h eflects also the minimum ate that is achieved with extemely long buffes by the delay penalty, which is equal to h/β. Theefoe the value of h has to be popely tuned to find the best tadeoff between lage guaanteed ates (i.e., lage h) and not too lage pices (i.e., small h). The delay theshold T is set by the application. Ideally, if the two endpoints ae synchonized, the application should set T such that an acceptable end-to-end delay is achieved. It is woth noting that in ode to guaantee a fully utilization of the channel, T should be lage than the minimum OWD. In Table I we list fo each contolle paamete the value adopted in the conducted simulations as well as a suggested opeation ange. Finally, we point out that the paametes setting that we use ae valid in a wide ange of scenaios, even if they might not always lead to optimal pefomance. Fo example, a T value between 5 ms and ms epesents a valid setting in a wide ange of netwok condition, povided that it is toleated by the application. Fo a moe thoough discussion on the setting of the paametes, and on how they affect the algoithm behavio, we efe the eade to Appendix B. To evaluate the pefomance of ou contolle, we caied out expeiments using the NS3 netwok simulato platfom. We have tested the contolle in diffeent netwok topologies and scenaios in ode to show that the algoithm is able to wok in loss-based as well as delay-based envionments. In paticula, we conside a single link topology, Topology (see Fig. 4) and Topology 2 (see Fig. 9) in ou simulations. The fist one is the

9 9 Flow bottleneck link Flow 2 Flow 3 UDP Flow Netwok Topology. Fig x Rate [bps].5 flow flow 2 flow OWD LOSSES 5 loss atio OWD [ms] Fig. 5. Sending ate and OWD fo the thee DCCC flows shaing a common bottleneck link when the equilibium is diven by the expeienced delay. As second scenaio, we still conside the netwok topology of the pevious simulations, but in this case the flows have diffeent popagation delays, theefoe diffeent pices. We denote by [p p2 p3 p4 ] the popagation delays of the fou consideed flows. The capacity of the bottleneck link vaies fom Mbps to Mbps. The maximum queueing delay is set to 5 ms (the buffe size anges fom about packets to 3 packets depending on the capacity value), and T is set to ms. We measue the aveage sending ate at equilibium and we compute the Jain s fainess index [9], which fo the Rate [bps] 2 x.5 flow flow 2 flow OWD [ms] B. Fainess Analysis In this set of simulations, we evaluate the pefomance of the DCCC contolle when it shaes the bottleneck links with othe flows contolled by the same algoithm. Fainess is an impotant metic fo congestion contol algoithms; since it eflects the ability of flows to faily shae the available bandwidth without penalizing ealy o late stating flows. We conside a case with thee DCCC flows that shae a link with an unesponsive UDP flow with a constant ate of 5 kbps. The netwok topology is shown in Fig. 4. Two of the flows ( and 2) stat at 2 and 4 s espectively and last until the end of the simulation. The flow 3 stats at s and stops tansmitting at 2 s. The unesponsive UDP flow is always active duing the simulation. The bottleneck link has a one way popagation delay of 25 ms, and a total capacity of 3.5 Mbps. The theshold delay of the thee DCCC flows is set to ms. In a fist simulation we set the maximum buffe length to 3 packets (appoximately 3 ms of maximum queueing delay). Since T 3 ms, no losses ae expeienced by the flows at equilibium. Results ae povided in Fig. 5. The DCCC flows faily shae the bottleneck link without penalizing ealy o late stating flows. When all thee flows ae active the delay at equilibium is slightly highe, as expected since the sending ate at the equilibium goes fom.5 Mbps pe use to Mbps pe use. This behavio is indeed expected: accoding to Eq. (25) the algoithm needs a highe pice, i.e., a lage delay, in ode to each a lowe ate at equilibium. We then conduct a simulation with the same settings, but with a maximum buffe length of 25 packets (appoximately 45 ms of maximum queueing delay). In this scenaio T is geate than the maximum expeienced delay, theefoe the DCCC flows ae diven to equilibium by the expeienced losses athe than by the expeienced delay. The sending ates and OWD fo this simulation ae depicted in Fig.. We obseve that, also when the system is in a loss-based egime, the flows faily shae the available bandwidth OWD.2 Losses 4 2 loss atio classic dumbbell topology, whee seveal uses shae the same unique bottleneck link. The second topology is the so-called paking lot topology, with two bottleneck links, whee two uses employ only one of these links while a thid use, with a longe data path, employs both congested links. All the links connecting the endpoints to the bottleneck ae set to a high connection speed, e.g. Mbps. We focus on low/medium values of the bottleneck capacity since this is the typical, and most citical, scenaio fo eal-time applications, see [5]. We evaluate the pefomance of the DCCC algoithm unde diffeent metics such as thoughput, self-inflicted delay and fainess. We also compae the algoithm with othe delay-based congestion contols, namely: the Netwok-Assisted Dynamic Adaptation (NADA) congestion contol algoithm [], the Google congestion contol (GCC) algoithm [7] and the Low Exta Delay Backgound Taffic (LEDBAT) algoithm [8]. Fo a moe detailed desciption of the congestion contol algoithms implementation and fo futhe simulation test cases, we efe the eade to Appendix C and Appendix D Fig.. Thee flows shaing a common bottleneck, equilibium is diven by losses. Dop tail bottleneck buffe.

10 Jain's fainess index Fig. 7. delays one-way popagation delays [ ] ms one-way popagation delays [4 5 7 ] ms one-way popagation delays [ ] ms nomilized capacity [bps] 5 Jain s fainess index of fou DCCC flows with diffeent popagation case of N flows is defined as: ( N ) 2 n= x n J(x) = N. (44) N n= x2 n The Jain s Fainess index is a well known index to measue the fainess of ate allocation and it anges between (full fainess) and /N (poo fainess). The simulation esults ae shown in Fig. 7, whee the fainess index is povided as a function of the nomalized capacity, which coesponds to the total capacity nomalized by the numbe of flows. The most heteogenous scenaio (e.g. OWDs equal to [ ]) is the least fai especially fo lage capacity values. Howeve, when the available bandwidth is limited, which is the most constained and the most citical scenaio, the fainess is impoved. This behavio is motivated by the fact that, even if heteogenous popagation delays lead to heteogenous pices, the mismatch is educed fo lage pices (small capacities) leading to a moe fai ate allocation. Fig. 8 depicts the uses lowest and highest sending ate at equilibium as a function of the nomalized capacity. In the ideal case of full fainess, this plot should featue a staight line with all the ates equal to the nomalized capacity. Even if cuves in Fig. 8 deviates fom this ideal behavio fo high values of capacity, still no flow is expeiencing stavation and a elatively good sending ate is achieved. Fo example, when the maximum popagation delay of one flow is thee times the minimum one, the most penalized flow is still able to achieve a sending ate equal to appoximately half nomalized capacity. We conclude these esults by pointing out that, unlike loss-based algoithms, fo delay-based algoithms achieve full inta-potocol fainess in all possible scenaios is paticulaly challenging. The challenge is caused by the fact that flows might not commonly agee on the value of the pice. Note that this issue is not a peculiaity of ou algoithm, fo example, in othe potocols [], [2], latecomes flows 2 tend to undeestimate the queuing delay and convege to lage sending ates. In ou algoithm we ae not able to guaantee optimal inta potocol fainess in all possible scenaios, but we mitigate unfainess issues in the 2 A latecome flow is a flow that stats sending data ove an aleady congested (i.e., standing queueing delay gate than zeo) netwok link. ate at equilibium [bps] 4 min ate, popagation delays [ ] ms max ate, popagation delays [ ] ms min ate, popagation delays [4 5 7] ms 3 max ate, popagation delays [4 5 7] ms min ate, popagation delays [ ] ms max ate, popagation delays [ ] ms nomalized capacity [bps] 5 Fig. 8. Minimum and maximum ates at equilibium vesus nomalized capacity fo fou DCCC flows shaing a bottleneck with diffeent popagation delays. most constained scenaios, sacificing the pefomance in less constained scenaios, i.e., when capacities ae lage. We now analyze the fainess of ou contolle fo the paking lot netwok topology depicted in Fig. 9. Both bottleneck links have a channel capacity of 2.5 Mbps and a popagation delay of 25 ms. An unesponsive UDP flow steams though both links at a constant ate of 5 Kbps. Fo the equilibium to be diven by the expeienced delay, we set the theshold T to ms fo all the uses, and the maximum length of the buffes inside the netwok to packets (equivalent to a maximum queueing delay of appoximately 33 ms), which ensues that no losses ae expeienced duing this simulation. Flow 3 stats at s and stops at 2 s while the othe flows ae active duing the entie simulation. The simulation esults ae shown in Fig.. Flow always gets a lowe ate compaed to the othe flows. This is expected and due to the fact that its oute is longe, hence it is penalized compaed to the othe flows. Finally, we conduct a last simulation fo a mixed egime: the maximum queuing delay is set to 5 packets (equivalent to a maximum queueing delay of appoximately 5 ms), while the theshold delay T is set to ms. Also in this case we obseve that flow achieves a lowe ate than the othe two competing flows, as shown in Fig.. In this expeiment flow is expeiencing losses and having a positive value of the delay penalty, hence we call this opeation mode mixed egime. Note that the maximum queuing delay is set on a pe buffe basis, thus flow 3, which passes though two congested links, expeiences a total queuing delay of ms. In this test case, flows and 2 convege to the same ate when use 3 is not active because both pass though the same unique bottleneck, and the OWD is lowe then the theshold delay fo both of them. On the othe hand, when all thee flows ae active, flow taveses two bottlenecks links, expeiencing a double value of losses and conveging to a lowe equilibium ate. C. TCP Coexistence We now study the pefomance of the DCCC when it competes against TCP flows. We again use a single link topology, with a capacity of 2.5 Mbps and a popagation delay of 5 ms. Thee flows shae simultaneously the link: an

11 aveage ate at equilibium [bps] 8 Flow bottleneck link bottleneck link 2 UDP Flow Flow 2 poposed algoithm (T = ms) HSTCP max one way delay [s] Flow 3 Fig. 2. Aveage ate at equilibium of ou algoithm and HSTCP when competing fo a bottleneck fo diffeent dop tail buffe size. Fig. 9. Netwok Topology 2. The flow DCCC passes though two bottlenecks links and competes with anothe DCCC flow on each of these links OWD [ms] OWD 5. Losses 5 loss atio Rate [bps] 2 x flow flow 2.5 flow Fig.. Time evolution of the sending ates and delays fo the paking lot topology when equilibium is diven by the expeienced delay. unesponsive UDP flow with a constant sending ate of 5 kbps, a flow unning the DCCC algoithm and a HSTCP [2] flow (we povide test cases with TCP New Reno and TCP Westwood in Appendix D). The delay theshold of the DCCC algoithm has been set to ms. We un simulations fo diffeent doptail buffe size anging fom 3 to 8 packets (coesponding oughly to ms and ms of maximum queueing espectively). The simula-.5 flow flow 2 flow OWD [ms] OWD LOSSES 5 5 loss atio Rate [bps] 2 x Fig.. Time evolution of the sending ates and delays fo the paking lot topology when equilibium is diven by losses tion esults in Fig. 2 show the aveage ate at equilibium fo the DCCC and TCP algoithms. We can see that the level of fainess against TCP depends on the buffe size. This dependency is caused by the delay-based pat of the congestion algoithm, since the buffe size has an impact on the expeienced delay and theefoe on the ate at the equilibium. In the pesence of small buffes, ou algoithm eaches a highe ate at equilibium than TCP one. This is due to the fact that the loss-based pat of DCCC is moe aggessive than the TCP congestion contol. On the othe hand, in the case of lage buffe size, the DCCC algoithm suffes against TCP. Howeve, due to the non-lineaity of the DCCC penalty function, the DCCC flow is potected, and it neve staves, eaching the expected lowe bound ate of h/β (h/β = 2 kbps in the simulations). By modifying the value of h, we can tune the guaanteed ate that is eached in lage delay envionments and theefoe we can limit pefomance degadation when competing with TCP. In conclusion, when shaing the bottleneck link with TCP, the DCCC algoithm cannot guaantee TCP fainess, but it still compaes favoably with espect to othe delay-based algoithms poposed in the liteatue [22] [24], which ae note able to guaantee a lowe bound on the delay-based sending ate. D. Compaison with Othe Congestion Contol Algoithms We now conduct some expeiments to compae ou algoithm with othe delay-based congestion contolles. We concentate on the behavio of the algoithms when they opeate in delay-based mode and not on how they pefom in lossy envionments. We conside two candidate algoithms of the IETF RMCAT (RTP Media Congestion Avoidance Techniques) Woking Goup [25]: NADA and the GCC. NADA is implemented accoding to the desciption of the IETF daft [2] (vesion 3) and the scientific publicatio []; the GCC instead is implemented following the desciption of the IETF daft [7] (vesion 2) and the detailed analysis of the scientific publications [23], [27], [28]. Howeve, both algoithms ae moving tagets, and the implementation follows the desciption found in the cited documents. To have a moe exhaustive evaluation we also include the LEDBAT algoithm in the compaison esults. Fo a moe detailed discussion of these congestion contol algoithms we efe the eade to Section VI, while fo