Local stability and Hopf bifurcation analysis for Compound TCP

Transcription

1 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 1 Local stability and Hopf bifurcation analysis for Compound TCP Debayani Ghosh, Krishna Jagannathan and Gaurav Raina Abstract We conduct a local stability and Hopf bifurcation analysis for Compound TCP, with small Drop-tail buffers, in three topologies The first topology consists of two sets of TCP flows having different round trip times, and feeding into a core router The second topology consists of two distinct sets of TCP flows, regulated by a single edge router and feeding into a core router The third topology comprises of two distinct sets of TCP flows, regulated by two separate edge routers, and feeding into a core router In each case, we conduct a local stability analysis and obtain conditions on the network and protocol parameters to ensure stability If these conditions get marginally violated, we show that the underlying systems lose local stability via a Hopf bifurcation After exhibiting a Hopf, a key concern is to determine the asymptotic orbital stability of the bifurcating limit cycles We then present a detailed analytical framework to address the stability of the limit cycles, and the type of the Hopf bifurcation by invoking Poincaré normal forms and the center manifold theory We finally conduct packet-level simulations to corroborate our analytical insights Index Terms Compound TCP, Drop-Tail, Stability, Hopf bifurcation I INTRODUCTION Network performance and end-to-end latency are affected by a combination of the choice of TCP, the size of router buffers, and the choice of queue management implemented in Internet routers [], [5], [13] A major portion of Internet traffic is controlled by the Transmission Control Protocol TCP [8], [14] There have been proposals for different flavours of TCP and queue management strategies However, Compound TCP [] is the default protocol in Windows, and a simple Drop-Tail queue management is commonly implemented in Internet routers It has been argued that the default large buffer dimensioning rule for router buffers, combined with Drop-Tail, leads to excessive delays in the Internet [5] In our recent work [6], we conducted a performance evaluation of Compound TCP, in a small buffer regime, with particular emphasis on buffer thresholds One of the key insights obtained therein was the two-fold advantage of having small router buffers In particular, our analysis showed that small buffers are favourable for ensuring the stability of the system, in addition to reducing queueing delays Furthermore, our analysis identified that the underlying dynamical systems undergo a Hopf bifurcation, and transit from a locally stable into the locally unstable regime as the buffer size increases A dynamical system is said to undergo a Hopf bifurcation if one pair of complex conjugate eigenvalues of the linearised D Ghosh, K Jagannathan and G Raina are with the Department of Electrical Engineering, IIT Madras, Chennai 636, India {ee1s5, krishnaj, gaurav}@eeiitmacin system crosses over the imaginary axis in the complex plane, when a particular parameter is varied beyond its critical value This, in turn, results in the emergence of limit cycles isolated periodic orbits in the state space as the said parameter is varied just beyond its critical value [9] We repeatedly observed limit cycles in the queue size dynamics, in numerous packet-level simulations Fig 1 portrays one such instance; indeed, it captures the emergence of limit cycles in the queue size of the core router in a single bottleneck topology, as buffer threshold is increased This motivates us to develop an analytical framework under which the emergence of these non-linear oscillations can be better understood To that end, in this paper, we provide an analytical framework to determine the type of the Hopf bifurcation, and the orbital stability of the emergent limit cycles We consider three different topologies, and focus on analysing the dynamical properties of a fluid model of Compound TCP in conjunction with small Drop-Tail buffers Our fluid model takes the form of a non-linear, time-delayed dynamical system For our analysis, we assume that the loss probability at the routers can be reasonably approximated by the blocking probability of an M/M/1/B queue in a high bandwidth-delay product regime, with a large number of long-lived flows This assumption has been substantiated empirically in [7] The first topology which we consider is a generalisation of the single bottleneck topology studied in [6], and consists of two sets of TCP flows having different round trip times, and feeding into a core router see Fig a The second topology consists of two distinct sets of TCP flows, regulated by a single edge router and feeding into a core router see Fig b The third topology comprises of two distinct sets of TCP flows, regulated by two separate edge routers, and feeding into a common core router see Fig c We wish to emphasise that the dynamical properties of Compound TCP in a single bottleneck scenario has been addressed in the literature [6], [18] However, to the best of our knowledge, this work is the first to analyse the dynamical properties of Compound TCP in multiple bottleneck scenarios For each of these cases, we conduct a local stability analysis and outline necessary and sufficient conditions for local stability, with two simplifying assumptions In the first scenario, we assume that the network parameters are the same, and that both sets of Compound TCP flows have equal round trip times In the second scenario, we assume the network parameters to be heterogeneous, and the round trip time of one set of TCP flows to be much larger as compared to the other If the local stability conditions get marginally violated, our analysis shows that the underlying systems would lose local stability via a Hopf c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

2 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE bifurcation Motivated by this insight, we then analyse only the third topology in greater detail, to better understand the impact of heterogeneous system parameters on local stability We numerically show through DDE-BIFTOOL [4] that, even in the presence of heterogeneous network parameters and different round trip times, the dynamical system undergoes a Hopf bifurcation which leads to the emergence of limit cycles As argued in [6], the emergence of limit cycles in the system dynamics could have a number of detrimental consequences for example, it could lead to the synchronisation of TCP windows, result in a loss in link utilisation, and cause the downstream traffic to be bursty Hence, it becomes imperative to study these limit cycles in further detail To that end, an important contribution of this paper lies in providing an analytical framework to characterise the type of the Hopf bifurcation, and determine the asymptotic orbital stability of the emerging limit cycles, using Poincaré normal forms and the center manifold theory To corroborate our analytical insights, we conduct some packet-level simulations in NS [] and highlight the existence and stability of the limit cycles in the average window size, as well as queue size dynamics In particular, a key insight obtained from our theoretical stability analysis and packet-level simulations is that router buffer thresholds and round trip times of TCP flows, all impact stability Since this insight remains consistent across the three topologies considered in this paper, our proposed models are fairly effective in predicting the qualitative dynamical behaviour of Compound TCP Note that, with the style of analysis presented in this work, it is not easy to analyse the dynamical properties of Compound TCP in a general topology A comment is in order Instead of treating any particular system parameter as the bifurcation parameter, we choose a suitably motivated exogenous, non-dimensional parameter as the bifurcation parameter to aid our analysis The two main advantages of this are: first, it enables us to capture the effects of different system parameters on the system stability in a unified manner and secondly, we need not be concerned about the dimension of the bifurcation parameter There is a substantial body of literature which investigates the dynamical properties of congestion control algorithms of TCP, for instance see [11], [16] and [1] Specifically, these papers deal with TCP Reno in a single bottleneck scenario, with intermediate and large Drop-Tail buffers [17] On the contrary, we analyse the dynamical properties of Compound TCP which is implemented in Windows operating system, in a small buffer regime We provide conditions for local stability for Compound TCP in more complex topologies, and show that the underlying systems could lose stability via a Hopf bifurcation if router buffer thresholds are increased We further provide an analytical framework that can enable us to characterise the type of the Hopf bifurcation and also establish the orbital stability of the emergent limit cycles A key insight derived from our work is that, router buffer thresholds in a network should be sized even smaller than those considered in the aforementioned papers, to ensure stability In particular, our analysis shows that router buffer thresholds should be of O1; specifically, they should not scale with the number of flows feeding into the router For example, a Queue size pkts 15 1 Buffer size = 15 pkts Buffer size = 1 pkts Round trip time = 1 ms Round trip time = ms Time seconds Fig 1: Emergence of limit cycles 6 long-lived Compound flows over a Mbps link, and feeding into a core router with link capacity 1 Mbps Observe the limit cycles in the queue at the core router, for larger buffer thresholds, and larger round trip times router with link capacity 1 Mbps, and fed by a large number of long-lived flows with a round trip time of ms, should not be sized more than 15 packets Furthermore, smaller buffer thresholds would reduce queueing delay, which is highly desirable There exists another line of literature on TCP analysis, by Low et al, for instance, see [1] and [1] In [1], the authors propose a new Active Queue Management AQM strategy called Random Early Marking REM In [1], the authors analyse the stability properties of TCP Reno with Random Early Detection RED queue management strategy in a single bottleneck topology, and show that the average window and queue size exhibit deterministic oscillations, for larger round trip times However, these papers do not capture the effect of buffer thresholds in Internet routers on the network performance Since the presence of persistently full large buffers leads to excess queueing delays [5], the problem of buffer sizing is highly relevant in the context of today s Internet Hence, in this paper, we mainly focus on the dynamical properties of Compound TCP with small Drop-Tail buffers, in three topologies The primary contribution of our paper is to provide explicit bounds on buffer thresholds to ensure stability, and characterise the stability of the emerging limit cycles in the system dynamics if the stability conditions get marginally violated Furthermore, these stability conditions imply that router buffer thresholds should be significantly reduced to ensure stability, as well as to reduce end-to-end latency in today s Internet The rest of the paper is organised as follows In Section II, we outline the governing fluid models for the three cases we consider Section III deals with local stability analysis of the fluid models In Section IV, we provide an analytical framework to determine the asymptotic orbital stability of the bifurcating limit cycles, and to characterise the type of the Hopf bifurcation Packet-level simulations are presented in Section V to corroborate some of the analytical insights Finally, in Section VI, we summarise our contributions c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

3 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 3 II MODELS In this section, we consider two distinct sets of TCP flows having different round trip times τ 1 and τ in three topologies For our analysis of these models, we primarily focus on long-lived flows We assume that both sets of TCP flows can be of different flavours and hence, can have different increase and decrease rules to govern the evolution of the corresponding window sizes Let the average window sizes of the two sets of flows be w 1 t and w t respectively For each acknowledgement received, the average window sizes increase by i 1 w 1 t and i w t, and for each packet loss detected, the average window sizes decrease by d 1 w 1 t and d w t respectively Note that, the increase and decrease functions for a particular TCP flavour depend on the protocol parameters Further, the loss probability at the routers is governed by the corresponding AQM strategy A Fluid models for TCP Now, we briefly outline the fluid models for the evolution of the average window sizes of the two sets of TCP flows in the congestion avoidance phase for three topologies Case I This model consists of a single bottleneck link with two distinct sets of TCP flows feeding into a common core router, as shown in Fig a The core router has a buffer size of B, with the service rate per flow as C Thus, for generalised TCP flows, the non-linear, time-delayed fluid model of the system is given by the following equations: ẇ j t = w jt τ j i j w j t 1 qt,τ 1,τ τ j d j w j tqt,τ 1,τ, j = 1,, 1 where qt,τ 1,τ represents the packet loss probability at the core router, and depends on the sending rates of both sets of TCP flows Case II This model consists of two distinct sets of TCP flows, regulated by a single edge router and feeding into a common core router, as shown in Fig b The buffer sizes of the core router and the edge router are B 1 and B, with service rates per flow as C 1 and C respectively Thus, for generalised TCP flows, the non-linear, time-delayed fluid model of the system is given by the following differential equations: ẇ jt = wjt τj i j w jt 1 q 1t,τ 1,τ q t,τ 1,τ τ j d j w jt q 1t,τ 1,τ +q t,τ 1,τ, j = 1,, where q 1 t,τ 1,τ and q t,τ 1,τ denote the packet loss probabilities at the edge router and the core router respectively Case III This model consists of two distinct sets of TCP flows, regulated by two edge routers and feeding into a common core router, as shown in Fig c The buffer size at the core router is B, with service rate per flow as C The buffer sizes for the edge routers are B 1 and B, with service rates per flow as C 1 and C respectively Thus, for generalised TCP flows, the non-linear, time-delayed fluid model of the system is given by the following equations: ẇ j t = w jt τ j i j w j t 1 p j t τ j qt,τ 1,τ τ j d j w j t p j t τ j +qt,τ 1,τ,j = 1, The loss probabilities at the two edge routers are p 1 t and p t The loss probability at the core router is denoted by qt,τ 1,τ Recall that the increase and decrease functions are specific to the choice of a particular flavour of TCP Specifically, [18] has summarised the increase and decrease functions for different TCP flavours including Compound Compound TCP is a synergy of both loss-based and delaybased congestion avoidance algorithms If the bottleneck router buffer is sized small, then the queueing delay can be assumed to be negligible Hence, in this scenario, Compound TCP behaves essentially like a loss-based congestion control algorithm Since we primarily focus on small Drop-Tail buffers, Compound TCP would predominantly react to packet loss as the feedback signal In this regime, we can state the increase and decrease functions for Compound as follows: 3 iwt = αwt k 1, and dwt = βwt 4 Here, α, k are the increase parameters and β is the decrease parameter The default values of these parameters are α = 15, k = 75 and β = 5 [] B Packet loss probability In this paper, we focus on small buffers with Drop-Tail queue policy for the local stability analysis of the non-linear fluid models of TCP given by 1, and 3 We first consider the scenario where a large number of long-lived TCP flows having a common round trip time of τ feed into a router having a buffer size of B The bottleneck link has a capacity per flow C In this scenario, we can approximate the packet loss probability of the router by the blocking probability of an M/M/1/B queue [18] This gives rise to the following fluid model: B wt qt =, 5 Cτ where wt represents the average window size of the TCP flows Using 5, we can then obtain the functional forms of packet loss probabilities for the three scenarios, which we briefly outline as follows: c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

4 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 4 τ 1 τ 1 τ 1 B 1 C 1 B C B 1 C 1 B C B C B C τ τ τ a b c Fig : Schematic diagrams of three topologies a Case I, a single bottleneck topology b Case II, two routers in tandem and c Case III, two routers feeding into one core router Case I: The fluid model for the loss probability at the core router is given by B w 1 t/τ 1 +w t/τ qt = 6 C Case II: The fluid models for the loss probabilities are: q 1t = w1 t τ 1 + w B1 t w1 t τ τ,and q t = 1 C 1 + w B t τ C 7 Case III: Using 5, we can approximate the loss probabilities at various routers as below: B1 B w1 t w t p 1 t =, p t =, and C 1 τ 1 C τ B w1 t/τ 1 +w t/τ qt = 8 C where C = C, C1 = C 1 and C = C Using these functional forms, we now proceed to perform a local stability and bifurcation analysis for the systems given by 1, and 3 This would enable us to understand the dynamical properties of the coupled system of Compound TCP with Drop-Tail queue policy to a greater detail III LOCAL STABILITY ANALYSIS In order to perform a local stability and bifurcation analysis for the non-linear models 1, and 3, we need to choose an appropriate bifurcation parameter The loss of stability could be induced by changes in any of the protocol or the network parameters However, instead of treating any of the system parameters as a bifurcation parameter, we introduce an exogenous non-dimensional parameter κ > as the bifurcation parameter, to push the system just into the unstable regime With the introduction of κ, a general system of delay differential equations ẋt = fxt,xt τ 1,,xt τ n results in ẋt = κfxt,xt τ 1,,xt τ n Note that the non-dimensional parameter does not affect the equilibrium of the system From a theoretical point of view, the bifurcation parameter κ is introduced to aid our stability analysis and capture the interdependence of various system parameters to ensure local stability We would like to emphasise that the exogenous parameter κ does not depend on any of the system parameters and is not physically related to the models given by 1, and 3 It acts like an artificial gain parameter for the aforementioned models, which is used to aid local stability and bifurcation analysis Recall that to conduct the local stability analysis, we primarily focus on Compound TCP with Drop-Tail queues in the small buffer regime For mathematical tractability, we assume that both sets of TCP flows in all three topologies are regulated by Compound with identical protocol parameters Further, we consider two simplifying assumptions as briefly outlined below: Scenario 1: All network parameters are the same, ie, B 1 = B = B, and C 1 = C = C Further, the round trip times of both TCP flow sets are identical, ie, τ 1 = τ = τ Scenario : In this scenario, we assume that all network parameters are distinct and the round trip time of one set of TCP flows is negligible and much smaller as compared to the round trip time of the other set, ie, τ 1 >> τ and τ Under this assumption, the dynamics of the second set of TCP flows appear almost instantaneous Hence, the expressions for loss probabilities now depend on the instantaneous window size averaged over the second set of flows We now proceed to conduct a detailed local stability analysis to obtain bounds on network, and protocol parameters to ensure stability, for the systems given by 1, and 3 Case I Fig a shows the topology considered in this case With the introduction of the non-dimensional parameter κ, system 1 becomes ẇ j t =κ w jt τ j i j w j t 1 qt,τ 1,τ τ j d j w j tqt,τ 1,τ, j = 1, 9 Scenario 1: In this scenario, the first model reduces to a single bottleneck link with only one set of TCP flows having round trip time τ 1 = τ = τ Hence, system 9 becomes iwt 1 qwt τ dwtqwt τ, 1 ẇt =κ wt τ τ c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

5 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 5 where wt is the average window size of the TCP flows The non-trivial equilibrium w of system 1 satisfies iw 1 qw = dw qw 11 Note that, under the first assumption, the fluid model for the loss probability at the core router, given by 6 reduces to w qw B =, 1 Cτ at equilibrium The critical value of κ at which system 1, with Compound TCP in the small buffer regime, loses local stability, satisfies the following equation [18]: κ c αw k 1 B k 1 qw = cos 1 k 1 qw B 13 Scenario : In this scenario, system 9 becomes ẇ 1 t = κ w 1t τ 1 iw 1 t 1 qt,τ 1,τ τ 1 dw 1 tqt,τ 1,τ, ẇ t = κ w t iw t 1 qt,τ 1,τ τ dw tqt,τ 1,τ 14 Suppose w1,w is a non-trivial equilibrium of 14 and let u 1 t = w 1 t w1 and u t = w t w be small perturbations about w1 and w respectively Linearising 14 about this equilibrium, we obtain u 1 t = κm 1 u 1 t+n 1 u 1 t τ 1 +P 1 u t, u t = κ M +N u t+p u 1 t τ 1, 15 where, the linearised coefficients M j, N j and P j, j = 1, are provided in the supplementary material Looking for exponential solutions, we obtain the characteristic equation for the linearised system 15 as where, λ +κaλ+κbλe λτ1 +κ ce λτ1 +κ d =, 16 a = M 1 +M +N, b = N 1, c = N 1 M +N P 1 P, d = M 1 M +N 17 For system 14 to be locally stable about the equilibrium w 1,w, all roots of the characteristic equation 16 should lie in the left half of the complex plane It can be shown that, for negligibly small values of the non-dimensional parameter κ, the system is stable, ie, all roots would have negative real parts However, as κ is increased beyond a critical value, one pair of complex conjugate roots would cross over the imaginary axis, and hence have positive real parts At this critical value, the system would have a pair of purely imaginary roots To deduce this point, we substitute λ = iω in 16 and separate real and imaginary parts to get ω = κ d a +b ± κ d a +b 4d c Condition 1: There exists only one positive value of ω if either of the following conditions holds i d a +b >, and d a +b = 4d c ii d c < Condition : There exist two positive values of ω if d a +b >,d a +b > 4d c and d c > hold When Condition 1 is satisfied, the system transits from the locally stable regime to instability asκincreases beyond a critical value, and never regains stability as κ is further increased On the contrary, when Condition is satisfied, the system may undergo stability switches as κ is increased [3] A stability switch is said to occur when changes in a system parameter cause the system to alternate between stable and unstable regions Mathematically, the derivative of the eigenfunction with respect to the bifurcation parameter alternates its sign In the context of TCP, considering any of the network or protocol parameters as the bifurcation parameter, a stability switch would cause the average window size across all flows and queue size, to alternate between deterministic oscillations and stable behaviour, with an increase in the bifurcation parameter However, extensive NS simulations suggest the non-existence of such alternating behaviour in the average window size of a large number of Compound TCP flows, as well as in the queue size of the bottleneck router Hence, we focus only on the case when Condition 1 is satisfied, and only one positive root of ω exists This implies that there exists a cross over frequency at which one pair of complex conjugate roots crosses over the imaginary axis, and is given by ω = κa, where d a A = +b d a +b + 4d c The critical value of κ denoted by κ c, at which this transition occurs, is given by κ c = 1 cos 1 A c ab cd Aτ 1 b A +c 18 Case II Fig b shows the topology considered in this case With the introduction of κ, system becomes ẇ jt =κ wjt τj i j w jt 1 q 1t,τ 1,τ q t,τ 1,τ τ j d j w jt q 1t,τ 1,τ +q t,τ 1,τ, j = 1, 19 Scenario 1: With this assumption, the second model reduces to a single set of TCP flows, regulated by an edge router, and feeding into a core router Observe that, the loss probabilities at both routers are the same In this scenario, system 19 reduces to the following non-linear, first-order, time-delayed differential equation ẇt = κ wt τ iwt 1 pwt τ τ dwtpwt τ, c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

6 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 6 where wt is the average window size of the TCP flows Using the functional forms given by 7, we obtain w pw = qw B = Cτ The critical value of κ, at which system loses local stability, satisfies the following equation κ c αw k 1 B k 1 pw = cos 1 k 1 pw B Scenario : In this scenario, system 19 becomes ẇ 1t = κ w1t τ1 τ 1 dw 1t ẇ t = κ wt τ dw t 1 iw 1t 1 q 1t,τ 1,τ q t,τ 1,τ q 1t,τ 1,τ +q t,τ 1,τ, iw t 1 q 1t,τ 1,τ q t,τ 1,τ q 1t,τ 1,τ +q t,τ 1,τ Linearising about w 1,w, we obtain u 1 t = κm 1 u 1 t+n 1 u 1 t τ 1 +P 1 u t, u t = κ M +N u t+p u 1 t τ 1, 3 where, the linearised coefficients M j, N j and P j, j = 1, are provided in the supplementary material Observe that, the linearised system 3 has a similar form as 15 Hence, conducting a similar kind of analysis as done for system, we obtain the critical value of the non-dimensional parameter κ, as given by 18 Case III Fig c shows the topology considered in this case With the introduction of κ, system 3 becomes i j w jt 1 p jt τ j qt,τ 1,τ ẇ jt =κ wjt τj τ j d j w jt p jt τ j+qt,τ 1,τ,j = 1, 4 Scenario 1: For Compound TCP in the small buffer regime, the critical value of κ at which system 4 transits into a locally unstable regime, satisfies the following condition [6]: κ cαw k 1 B k 1 qw = cos 1 k 1 qw B Scenario : In this scenario, system 4 reduces to ẇ 1t = κ w1t τ1 iw 1t 1 p 1t τ 1 qt,τ 1,τ τ 1 dw 1t ẇ t = κ wt τ dw t p 1t τ 1+qt,τ 1,τ, iw t 1 p t qt,τ 1,τ p t+qt,τ 1,τ 5 Linearising 5 about its equilibrium w 1,w, we obtain u 1 t = κm 1 u 1 t+n 1 u 1 t τ 1 +P 1 u t, u t = κ M +N u t+p u 1 t τ 1, 6 where, the linearised coefficients M j, N j and P j, j = 1, are provided in the supplementary material Note that, the linearised system 6 has a similar form as 15 Hence, a similar kind of local stability analysis would yield the condition on the critical value of the non-dimensional parameter κ, and the protocol parameters as given by 18 For all three scenarios, with the simplifying assumptions, the conditions derived above essentially capture the interdependence among the non-dimensional parameter κ, and the system parameters to ensure local stability Observe that, the loss of local stability can be studied with respect to any system parameter However, we prefer to choose an exogenous parameter as the bifurcation parameter, to aid our analysis It can be explicitly shown that, for all the above cases, the system loses local stability via a Hopf bifurcation [9] if the conditions derived above get violated We prove this by verifying the transversality condition of the Hopf spectrum [15] To verify this, we show that, Redλ/dκ at κ = κ c In particular, we prove that, Redλ/dκ > at κ = κ c This implies that, one pair of complex conjugate roots crosses over the imaginary axis from the left half of the complex plane to the right half Hence, κ < κ c is the necessary and sufficient condition for local stability, for all three scenarios Further, the systems undergo a Hopf bifurcation at κ = κ c Observe that, deriving a necessary and sufficient condition with heterogeneous network parameters, and different round trip times is analytically complex, for all three scenarios discussed earlier Hence, we numerically illustrate through DDE- BIFTOOL [4], that system 4 undergoes a Hopf bifurcation if the non-dimensional parameter κ is varied beyond a critical value We fix the protocol parameters as follows: α = 38, β = 5 and k = 75 Since, we mainly focus on small buffer regime, the buffer sizes of the routers are fixed as: B 1 = 1 packets, B = 15 packets, and B = 5 packets We fix the remaining network parameters as: C 1 = C = 1 packets/second, C = 18 packets/second, τ 1 = 1 second and τ = seconds Now, we vary the non-dimensional parameter κ in the range [,] and observe that the system undergoes a Hopf bifurcation at κ c = 1 At this point, the system has one pair of complex conjugate roots on the imaginary axis Hence, the system dynamics exhibit limit cycles at κ c = 1 Stability charts: To obtain insights about the system behaviour at the stability boundary, we now demonstrate some stability charts for system 4 Fig 3a represents the Hopf condition for system 4 in the two parameter space: the nondimensional parameter κ, and the protocol parameter α Observe that, if κ is increased, α would have to reduce to ensure stability Fig 3b illustrates the Hopf condition in the two parameter space: the non-dimensional parameter κ, and the buffer size at the core router B Observe that, if κ is increased, keeping other system parameters fixed, B would have to be decreased accordingly to ensure stability of system 4 Fig 4 characterises the stability boundary of system 4 with respect c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

7 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 7 to two sets of system parameters: a round trip time of the first set of TCP flows τ 1 and the buffer size at the core router B, b increase protocol parametersα and k It is evident from Fig 4a that for large round trip times, larger buffer thresholds at the core router could de-stabilise the system Similarly, it can be easily observed from Fig 4b that there exists a trade-off between the increase parameters α and k to ensure stability Hence, we conclude that, both protocol parameters, and network parameters, need to be co-designed carefully to maintain stability of system 4 If these Hopf conditions get violated, the system would lose stability leading to the emergence of limit cycles in the system dynamics This in turn results in the emergence of deterministic oscillations in the queue size dynamics Since these limits cycles are detrimental to the overall network performance, it becomes imperative to study the type of the Hopf bifurcation and the stability of the emergent limit cycles in greater detail IV HOPF BIFURCATION ANALYSIS We now provide a detailed analytical framework to characterise the type of Hopf bifurcation and the asymptotic orbital stability of the emergent limit cycles, for system 4 Note that we have motivated the exogenous, nondimensional parameter κ as the bifurcation parameter This enables us to capture the effect of the different system parameters on the system stability in a unified manner Note that, κ = 1 yields the original system We can set the values of the system parameters of 4 such that κ c = 1, to obtain the stability boundary of the original system 4 If we now marginally increase κ, then the system would lose local stability via a Hopf bifucation As a result, the system dynamics would exhibit limit cycles The Hopf bifurcation analysis enables us to analyse the system dynamics in its locally unstable regime, in the neighbourhood of the Hopf condition Using Poincaré normal forms and the center manifold theory, we present an analytical framework to determine the type of the Hopf bifurcation and the orbital stability of the emergent limit cycles Our analysis closely follows the analysis presented in [9], [1], [15] Let κ = κ c + µ, where µ R Observe that, the system undergoes a Hopf bifurcation at µ =, where κ = κ c We can now consider µ as the bifurcation parameter An incremental change in κ from κ c to κ c +µ where µ >, pushes the system to its locally unstable regime Step 1: Using Taylor series expansion, we segregate the right hand side of 4 into linear and non-linear parts We then cast this into the standard form of an operator differential equation Step : At the critical value of the bifurcation parameter, ie, atµ =, the system has one pair of purely imaginary eigenvalues with non-zero velocity The linear eigenspace spanned by the eigenvectors corresponding to these eigenvalues is called the critical eigenspace The center manifold theorem [1, Chapter 5, Theorem 51] guarantees the existence of a locally invariant dimensional manifold which is tangent to the critical eigenspace at the equilibrium of the system Step 3: Next, we project the system onto its critical eigenspace and its complement at the critical value of the bifurcation parameter This enables us to capture the dynamics of the system on the center manifold, with the help of an ordinary differential equation in a single complex variable Step 4: Finally, using Poincaré normal forms, we evaluate the lyapunov coefficient and the floquet exponent, which characterise the type of the Hopf bifurcation and the asymptotic orbital stability of the emergent limit cycles respectively Suppose w1,w is an equilibrium for 4 Let u 1t = w 1 t w1 and u t = w t w be small perturbations about the equilibrium Thus, a Taylor series expansion of 4 about its equilibrium w1,w is as follows u 1 t =κ ξ a u 1 t+ξ b u 1 t τ 1 +ξ d u t τ +κ ξ aa u 1 t+ξ bbu 1 t τ 1+ξ dd u t τ +ξ ab u 1 tu 1 t τ 1 +ξ ad u 1 tu t τ +ξ bd u 1 t τ 1 u t τ +κ ξ aaa u 3 1 t+ξ bbbu 3 1 t τ 1+ξ ddd u 3 t τ +ξ aab u 1 tu 1t τ 1 +ξ aad u 1 tu t τ +ξ abb u 1 tu 1t τ 1 +ξ bbd u 1t τ 1 u t τ +ξ add u 1 tu t τ +ξ bdd u 1 t τ 1 u t τ +ξ acd u 1 tu 1 t τ 1 u t τ, u t =κ χ c u t+χ d u t τ +χ b u 1 t τ 1 +κ χ cc u t+χ ddu t τ +χ bb u 1 t τ 1 +χ cd u tu t τ +χ bc u 1 t τ 1 u t +χ bd u 1 t τ 1 u t τ +κ χ ccc u 3 t+χ dddu 3 t τ +χ bbb u 3 1 t τ 1 +χ ccd u tu t τ +χ bcc u 1 t τ 1 u t +χ cdd u tu t τ +χ bdd u 1 t τ 1 u t τ +χ bbc u 1t τ 1 u t+χ bbd u 1t τ 1 u t τ +χ bcd u 1 t τ 1 u tu t τ 7 The Taylor series coefficients are presented in the supplementary material Using the notation u = [u 1 u ] T, we reduce equation 7 to the following form ut = L µ u t +Fu t,µ, 8 where t >,µ R For τ >, we define u t θ = ut+θ, u t : [ τ,] R, θ [ τ,] For system 4, without loss of generality, we assume that τ 1 > τ L : C[ τ 1,] R denotes a family of continuous and bounded functionals parametrised by µ Here, C[a, b] denotes the set of all continuous functions on the interval[a, b] The operator F : C[ τ 1,] R consists of the non-linear terms Further, we assume that F is analytic and bothland F c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

8 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 8 Protocol parameter, α Hopf condition Non-dimensional parameter, κ Core router buffer, B 16 8 Hopf condition Non-dimensional parameter, κ a Fig 3: Stability chart Hopf condition for 4 with Compound TCP in the small buffer regime with respect to two sets of parameters: a the non-dimensional parameter κ, and the protocol parameter α, b the non-dimensional parameter κ, and the buffer size at the core router B The shaded region below the Hopf condition curve represents the stable region b Core router buffer, B 4 3 Hopf Condition 15 Round trip time, τ 1 Protocol parameter, k Hopf Condition Protocol parameter, α 1 a Fig 4: Stability chart Hopf condition for 4 with Compound TCP in the small buffer regime with respect to two sets of model parameters: a round trip time of the first set τ 1, and the buffer size of the core router B, b protocol parameters α and k The shaded region below the Hopf condition curve represents the stable region b depend analytically on the bifurcation parameter µ for small µ The linear operator is L µ u t = [ ] ξa u κ 1 t+ξ b u 1 t τ 1 ξ d u t τ 9 χ b u 1 t τ 1 χ c u t+χ d u t τ We now cast equation 8 into the following standard form of an operator differential equation, u = Aµu t +Ru t 3 Note that, 3 has onlyu t rather than bothu t andu Now, using the Riesz representation theorem [19, Chapter 6, Theorem 619], we transform the linear problem d/dtut = L µ u t The Riesz representation theorem guarantees the existence of an matrix-valued measure η,µ : [ τ 1,] R 4, such that each component of η has bounded variation and for all φ C[ τ 1,], L µ φ = dηθ,µφθ θ= τ 1 In particular, we have L µ u t = θ= τ 1 dηθ,µu t θ Observe that, for system 4, the matrix dη is dηθ,µ = [ ] ξa δθ+ξ κ b δθ +τ 1 ξ d δθ +τ dθ χ b δθ +τ 1 χ c δθ+χ d δθ +τ 31 Here, δ is the Dirac-delta measure Let C 1 [ τ 1,] denote the space of all functions defined on [ τ 1,], with continuous first derivatives For φ C 1 [ τ 1,], we then define the following linear and non-linear operators { dutθ Aµu t θ = dθ, θ [ τ 1,, L µ u t, θ = { Ru t θ =, θ [ τ 1,, Fu t,µ, θ = 3 Note that, du t /dθ du t /dt Hence, equation 8 can be transformed into 3 Further, recall that κ = κ c + µ, and the system undergoes a Hopf bifurcation at the critical point µ = Hence, we fix µ = to perform the necessary analysis at the point of bifurcation At µ =, the system has a pair of complex eigenvalues on the imaginary axis: λ = ±iω, where ω > Let qθ denote the eigenvector for A corresponding to the eigenvalue λ = iω We assume that c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

9 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 9 qθ has a form: qθ = [ 1 φ 1 ] T e iω θ Since it satisfies Aqθ = iω qθ, we obtain φ 1 = κχ b e iωτ1 κχ c +κχ d e iωτ iω We now define the following adjoint operator { A dαs µαs = ds, s,τ 1], t= τ 1 dη T t,α t, s = where η T denotes the transpose of η Observe that, the domains of A and A are C 1 [ τ 1,] and C 1 [,τ 1 ] respectively Then, λ = iω is an eigenvalue of A and for some nonzero vector p, we have A pζ = iω pζ Assuming pθ to have the form: pθ = D [ φ 1 ] T e iω θ, and solving the aforementioned operator equation, we obtain φ = κχ b e iωτ1 κξ a +κξ b e iωτ1 +iω Let us define the inner-product of the functions ψ C[,τ 1 ] and φ C[ τ 1,] as ψ,φ = ψφ θ= τ 1 θ ζ= ψ T ζ θdηθ,µφζdζ 33 Using the above definition of inner product, we can easily verify that the eigenvectors p and q satisfy the conditions p,q = 1 and p, q = when D = φ 1+κξb τ 1 e iωτ1 +κξ d φ 1 τ e iωτ +φ 1 1+κχd τ e iωτ +κχ b τ 1 e iωτ The critical eigenspace corresponding to the pair of eigenvalues ±iω, denoted by T c, is now dimensional and is spanned by {Req, Imq}, where Req and Imq denote the real and imaginary parts of q respectively Further, we denote the complement of the critical eigenspace T c as T su We now project system 3 onto T c and T su For u t, a solution of 3 at µ =, define zt = p,u t, and wt,θ = u t θ Re ztqθ 35 Recall that the center manifold, C is tangent to the critical eigenspace at the equilibrium The representation of the center manifold is wt,θ = w zt, zt,θ, where wz, z,θ = w θ z +w 11θz z +w θ z + 36 Here, w ij θ = [ w ij1 θ w ij θ ] T for i,j {,1,} We observe that, z and z are the local coordinates on the manifold C in the direction of the eigenvectors p and p respectively Further, note that the existence of the center manifold C ensures that equation 3 can now be reduced to an ordinary differential equation for a single complex variable z on C At µ =, in the coordinates 35, the dynamics of z can be represented as żt = p,au t +Ru t = iω zt+gz, z 37 Now, we can expand the function gz, z in powers of z and z as z gz, z = g +g 11z z +g z +g z z We now need to determine the coefficients w 11 θ, w θ, w θ in equation 36 to solve the differential equation 37 for z Following [9] we can write ẇ = u t żq z q, and using 3 and 37 we obtain ẇ = { Aw Re p F qθ, θ [ τ 1,, Aw Re p F q +F, θ =, which, using 36, can be rewritten as ẇ = Aw+Hz, z,θ Here, Hz, z,θ can be expanded in powers of z and z as Hz, z,θ = H θ z +H 11θz z+h θ z +, 39 where, for i,j {,1,}, H ij θ = [ H ij1 θ H ij θ ] T Now, on the center manifold C, near the origin ẇ = w z ż +w z z 4 We now use equations 36 and 37 to replace w z and ż and their conjugates and equate this with 4 to get iω Aw θ = H θ, Aw 11 θ = H 11 θ, iω +Aw θ = H θ, 41 as in [9] Now, we observe that u t θ = wz, z,θ+zqθ+ z qθ = w θ z +w 11θz z +w θ z +ze iωθ + ze iωθ +, 4 from which we obtain u t, u t τ 1, and u t τ We now proceed to expand the non-linear terms present in equation 7 using equation 4 and retain only the coefficients of z,z z, z,z z Using the definition gz, z = p F z, z we then determine the coefficients ofz,z z, z andz z, which are provided in the supplementary material, but not presented here due to space constraints Note that, the expression for g 1 has w θ and w 11 θ which we need to evaluate Now, for θ [ τ, from 39, we have Hz, z,θ = Re q F qθ z = g +g 11z z +g z + ḡ z +ḡ z 11z z +ḡ + which when compared with 39 gives H θ = g qθ ḡ qθ, qθ qθ, H 11 θ = g 11 qθ ḡ 11 qθ c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

10 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 1 wt τ 13 wt τ w t w t 136 a Fig 5: Phase portraits Emergence of limit cycle in the dynamics of w t in 4, for Compound TCP in the small buffer regime, with the variation in the non-dimensional parameter κ Observe that, a Trajectories converge to stable equilibrium for κ = 95, b Trajectories converge to a stable limit cycle for κ = 15 b Using equations 3 and 41, we have ẇ θ = iω w θ+g qθ+ḡ qθ, ẇ 11 θ = g 11 qθ+ḡ 11 qθ 44 Solving the differential equations in 44, we get w θ = g iω qe iωθ ḡ 3iω qe iωθ +ee iωθ, w 11 θ = g 11 iω qe iωθ ḡ11 iω qe iωθ +f 45 The objective now is to determine e and f We define, Hz, z, = Re q F q +F, 46 wheref represents the non-linear terms that can be expanded in powers of z as z F = F +F 11z z +F z +F z z Substituting the coefficients from the expansion of F gives H = g q ḡ q+ [ F 1 F ] T, H 11 = g 11 q ḡ 11 q+ [ F 111 F 11 ] T 48 From 3 and 48, we obtain g q+ḡ q = [ ] T F 1 F + [ ] κa11 iω w 1 +κa 1 w 1 τ 1 +κa 13 w τ1, κa 3 w 1 τ+κa 1 iω w +κa w τ g 11 q+ḡ 11 q = [ ] T F 111 F 11 [ ] κa11 w κa 1 w 111 τ 1 +κa 13 w 11 τ κa 3 w 111 τ 1 +κa 1 w 11 +κa w 11 τ 49 Substituting w,w τ,w 11 and w 11 τ from 45 in 49 we get e and f of the form e = [ e 1 e ] T and f = [ f 1 f ] T 5 Note that,e 1,e,f 1 andf can be derived explicitly in terms of system parameters, which are provided in the supplementary material but are not presented here due to space constraints Using e and f we evaluate w and w 11, using which we Amplitude Non-dimensional parameter, κ Fig 6: Bifurcation diagram Variation in the amplitude of the emergent limit cycles in w t at κ = 1 for system 4, with Compound TCP flows in the small buffer regime compute g 1 We now have all the terms required for the analysis of Hopf bifurcation as follows, see [9] c 1 = i g g 11 g ω g + g 1, 51 µ = Re c 1 α, β = Re c 1, 5 where c 1 is the lyapunov coefficient and α = Redλ/dκ κ=κc The following conditions enable us to verify the type of the Hopf bifurcation, and the asymptotic orbital stability of the limit cycles [9] The Hopf bifurcation is supercritical if µ > and subcritical if µ < The limit cycles are asymptotically orbitally stable if β < and unstable if β > Substituting the expression forg 1 in 51 yields the expression for c 1, which is the lyapunov coefficient We can then compute µ and β using 5 We now present a numerical example, and compute the values of µ and β for Compound TCP in the small buffer regime Example: We first fix the system parameters as follows: α = 38, k = 75, β = 5, B 1 = 1 packets, B = 15 packets, c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

11 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 11 Average window size pkts 8 Buffer size = 15 pkts NS 8 Buffer size = 1 pkts Time seconds NS Fig 7: Case I: 6 Compound TCP flows with an average round trip time of ms and access speed of Mbps, and feeding into a core router with a link capacity of 1 Mbps B = 5 packets, C 1 = C = 1 packets/second, C = 18 packets/second, τ 1 = 1 second, and τ = seconds With these parameter values, the system undergoes a Hopf bifurcation at κ c = 1 We now increase the value of the nondimensional parameter to κ = 15, and push the system just beyond the edge of stability We then compute the required expressions as Rec 1 = 738<, α = 3467 >, µ = 19 >, β = 1477< Thus, the Hopf bifurcation is supercritical and the emergent limit cycles are asymptotically orbitally stable Phase portraits and bifurcation diagram: We present the phase portrait for system 4, for Compound TCP in the small buffer regime, in Fig 5 First, we fix a point α = 38, κ = 1, on the stability boundary, as shown in Fig 3a The remaining system parameter values are fixed, as in the numerical example above We now plot the phase portrait for the average window size for the second set of TCP flows, w t for κ = 95 and κ = 15 respectively Observe that, for κ = 95, w t converges to its equilibrium value, as expected For κ = 15, it exhibits orbitally stable limit cycles, having undergone a Hopf bifurcation at κ = 1 Note that, the average window size of the first set of Compound flows can be shown to exhibit qualitatively similar dynamical behaviour We now present the bifurcation diagram for system 4, in Fig 6 Observe that, the amplitude of the limit cycles increases as κ is increased beyond 1 V PACKET-LEVEL SIMULATIONS We now validate our proposed models 1, and 3 for Compound TCP in a small buffer regime To achieve this, we compare results obtained from packet-level simulations conducted in the Network Simulator NS [] and numerical computations conducted in MATLAB We show numerically and via packet-level simulations that as buffer thresholds are Average window size pkts 7 Buffer size = 15 pkts NS 7 Buffer size = 1 pkts Time seconds NS Fig 8: Case II: 6 Compound TCP flows with an average round trip time of ms and access speed of Mbps, regulated by an edge router with a link capacity of 1 Mbps, and feeding into a core router with a link capacity of 8 Mbps Average window size pkts 7 1 Buffer size = 15 pkts NS NS Edge router Edge router Buffer size = 1 pkts Time seconds Fig 9: Case III: Two sets of 6 Compound TCP flows with an average round trip time of ms and access speed ofmbps, regulated by two edge routers each with a link capacity of 1 Mbps, and feeding into a core router with a link capacity of 18 Mbps NS NS increased, the underlying dynamical systems lose stability for large round trip times This leads to the emergence of deterministic oscillations in the average window as well as queue size dynamics A Average window size dynamics The first topology consists of 6 long-lived Compound TCP flows each with an access speed of Mbps, and feeding into a core router with a link capacity of 1 Mbps The size of each packet is fixed to be 15 bytes Hence, the available bandwidth per flow is C = 14 packets/second We further consider the average round trip time of the flows to be ms To illustrate the emergence of deterministic nonlinear oscillations in the average window size of all flows, we vary the buffer size of the bottleneck router from 15 to 1 packets With these parameter values, we present the evolution c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

12 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 1 of the average window size of the TCP flows, obtained from numerical computations and packet-level simulations, see Fig 7 Our numerical computation shows that the system would be stable for a buffer threshold of 15 packets, whereas for a buffer threshold of 1 packets, it shows the onset of deterministic oscillations in the average window size It can be immediately observed that packet-level simulations corroborates this insight For a buffer size of 15 packets, packetlevel simulations show small random fluctuations, whereas for a buffer size of 1 packets, deterministic oscillations in the average window size can be observed We can similarly validate our proposed models for the second Case II and the third Case III topologies, see Fig 8 and Fig 9 respectively We next demonstrate the emergence of deterministic oscillations in the queue size dynamics of the core router, as buffer thresholds are increased, for the topology depicted by Case III B Queue size dynamics The system consists of two sets of 6 long-lived Compound TCP flows each with an access speed of Mbps, regulated by two edge routers and feeding into a core router Each edge router has a link capacity of 1 Mbps, and the core router has a link capacity of 18 Mbps Since our primary focus is on small buffers, we fix the buffer size of each edge router to be 15 packets, and vary the buffer size of the core router from 15 to 1 packets Further, we fix the average round trip time of one set of flows to be 1 ms, and vary the average round trip time of the other set from 1 ms to ms The simulations are illustrated in Fig 1 Observe that, if the buffer sizes at all routers are fixed at 15 packets, the queue at the core router is completely random and stable, since the queue does not exhibit any deterministic oscillations When the buffer size at the core router is increased to 1 packets and the round trip time of the second set of flows is ms, the queue dynamics exhibits limit cycles Hence, larger queue thresholds are prone to inducing limit cycles, for larger round trip times These limit cycles in the queue size lead to synchronisation among TCP flows and make the downstream traffic bursty We end this section with some pertinent remarks, which are as follows: Remark 1: Relevance of the exogenous bifurcation parameter: We note that the parameter κ is purely a theoretical construct which is introduced essentially to aid our mathematical analysis It is not physically related to the models 1, and 3 considered in this paper However, it does help to capture the combined effect the various model parameters have on local stability Since these model parameters affect the equilibrium of the system, it is theoretically rather convenient to use κ as the bifurcation parameter to push the system just beyond the edge of stability We established that varying κ beyond its critical value would lead the system to lose local stability via a Hopf bifurcation, which would lead to the emergence of limit cycles Our stability conditions clearly highlight that model parameters such as buffer thresholds and round trip times impact local stability So, for the packet-level simulations, we consider two Queue size pkts 15 1 Buffer size = 15 pkts Buffer size = 1 pkts τ 1 = 1 ms, τ = 1 ms τ 1 = 1 ms, τ = ms Time seconds Fig 1: Emergence of limit cycles Two sets of 6 Compound flows over a Mbps link, regulated by two edge routers, and feeding into a core router with link capacity 18 Mbps Observe the limit cycles in the queue at the core router, for larger buffer thresholds, and larger round trip times physically significant model parameters, ie, buffer thresholds and round trip times as bifurcation parameters Remark : Impact of buffer sizing on throughput: A key motivation behind employing large buffers in Internet routers is to achieve high link utilisation In our work, we have shown that with larger buffer thresholds, the underlying dynamical systems lose stability for large round trip times This leads to the emergence of deterministic oscillations in the queue size dynamics see Fig 1 and results in the synchronisation of TCP flows Consequently, all TCP senders would back off simultaneously whenever congestion is detected and the bottleneck queue would almost be empty before it becomes full again This would render the available bandwidth under-utilised which would lead to the intermittent loss of throughput This defeats the very purpose of employing large buffers in Internet routers Our theoretical stability analysis recommends buffer thresholds at the Internet routers be sized small Hence, it becomes imperative to study the impact of buffer sizing on an important performance metric for a network; namely, throughput To that end, we present a simulation conducted in NS in the single bottleneck scenario; see Fig 11 We consider 6 longlived Compound TCP flows feeding into a core router We then vary the buffer size of the core router from 1 to 3 packets and plot the time-averaged throughput We consider three cases: i link capacity = 1 Mbps, ii link capacity = 3 Mbps and iii link capacity = 5 Mbps For these cases, we assume the access speed of each flow to be Mbps, 6 Mbps and 1 Mbps respectively Further, we consider the average round trip time of all Compound TCP flows to be ms, 1 ms and 5 ms respectively For all three cases, with smaller buffer thresholds, the underlying dynamical systems are stable As buffer threshold of the core router is increased, the underlying dynamical systems would lose local stability leading to the emergence of deterministic oscillations c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

13 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 13 Throughput Mbps Buffer size pkts Capacity = 5 Mbps Capacity = 3 Mbps Capacity = 1 Mbps Fig 11: Impact of buffer sizing on throughput 6 long-lived Compound TCP flows feeding into a core router in the queue size dynamics Further, it can be easily observed from Fig 11 that the average throughput is fairly good for smaller buffer thresholds Hence, smaller buffer thresholds are favourable for stability and do not degrade throughput significantly VI CONCLUDING REMARKS In today s Internet, router buffers are dimensioned according to the bandwidth-delay product rule yielding rather large buffers This leads to large queueing delays [5], [13], which are undesirable and detrimental for quality of service Thus, we study the impact of router buffer thresholds on the dynamical properties of Compound TCP, a practically implemented TCP flavour We showed through analysis and packet-level simulations that larger buffer thresholds would induce instabilities in the system dynamics in the form of limit cycles This could have a number of detrimental effects on the network performance; namely, synchronisation of TCP flows and loss of link utilisation Thus, smaller router buffers are favourable for stability with an added advantage of reduced latency Our work could help set the stage for design guidelines for lowlatency stable networks In this paper, we considered three different topologies, and conducted a detailed local stability analysis with two simplifying assumptions, to obtain necessary and sufficient conditions for stability To aid our analysis, we motivated a suitable non-dimensional bifurcation parameter, and illustrated that, the underlying dynamical systems lose stability if the bifurcation parameter is varied Further, in the multiple bottleneck scenario, even without any simplifying assumptions on the system parameters, we numerically identified that the system loses stability via a Hopf bifurcation A key insight obtained was the trade-off between different system parameters to ensure stability, as illustrated through some stability charts Further, since a Hopf bifurcation results in the emergence of limit cycles, we conducted a detailed Hopf bifurcation analysis To that end, we provide a framework to determine the asymptotic orbital stability of the limit cycles using Poincaré normal forms and the center manifold theory We then conducted some packet-level simulations that highlighted the existence and stability of limit cycles in the average window size, as well as queue size dynamics as system parameters vary The insights obtained in this paper could have important consequences for the modelling and the performance evaluation of communication networks From a theoretical perspective, this opens many challenging questions centred around the development of accurate fluid models for TCP and queue management policies From a practical perspective, the emergence of stable limit cycles could have an impact on the quality of service these issues merit further investigation REFERENCES [1] S Athuraliya, SH Low, VH Li, and Q Yin, REM: active queue management, IEEE network, vol15, pp 48-53, 1 [] VG Cerf, Bufferbloat and other Internet challenges, IEEE Internet Computing, vol 5, pp 79 8, 14 [3] KL Cooke, and Z Grossman, Discrete delay, distributed delay and stability switches, Journal of Mathematical Analysis and Applications, vol 86, pp 59 67, 198 [4] K Engelborghs, T Luzyanina, and D Roose, bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Transactions on Mathematical Software, vol 8, pp 1 1, [5] J Gettys and K Nichols, Bufferbloat: dark buffers in the Internet, Communications of the ACM, vol 55, pp 57 65, 1 [6] D Ghosh, K Jagannathan, and G Raina, Right buffer sizing matters: stability, queuing delay and traffic burstiness in compound TCP, in Proceedings of 5nd Annual Allerton Conference on Communication, Control, and Computing, 14 [7] D Ghosh, K Jagannathan, and G Raina, Right buffer sizing matters: stability and queuing dynamics in TCP, arxiv preprint arxiv: , 16 [8] S Ha, I Rhee and L Xu, CUBIC: a new TCP-friendly high-speed TCP variant, ACM SIGOPS Operating Systems Review, vol 4, pp 64 74, 8 [9] BD Hassard, ND Kazarinoff and Y-H Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, 1981 [1] YA Kuznetsov, Elements of Applied Bifurcation Theory, Springer Science & Business Media, 13 [11] F Liu, ZH Guan, and HO Wang, Stability and Hopf bifurcation analysis in a TCP fluid model, Nonlinear Analysis: Real World Applications, vol 1, pp , 11 [1] SH Low, F Paganini, J Wang, and JC Doyle, Linear stability of TCP/RED and a scalable control, Computer Networks, vol 43, pp , 3 [13] K Nichols and V Jacobson, Controlling queue delay, Communications of the ACM, vol 55, pp 4 5, 1 [14] J Padhye, V Firoiu, D Towsley and JF Kurose, Modeling TCP Reno performance: a simple model and its empirical validation, IEEE/ACM Transactions on Networking, vol 8, pp , [15] G Raina, Local bifurcation analysis of some dual congestion control algorithms, IEEE Transactions on Automatic Control, vol 5, pp , 5 [16] GRaina and O Heckmann, 7 TCP: local stability and Hopf bifurcation, Performance Evaluation, vol 64, pp 66-75, 7 [17] G Raina and D Wischik, Buffer sizes for large multiplexers: TCP queueing theory and instability analysis, in Proceedings of Next Generation Internet Networks, 5 [18] P Raja and G Raina, Delay and loss-based transport protocols: buffersizing and stability, in Proceedings of International Conference on Communication Systems and Networks, 1 [19] W Rudin, Real and Complex Analysis, Tata McGraw-Hill, 1987 [] K Tan, J Song, Q Zhang and M Sridharan, A Compound TCP approach for high-speed and long distance networks, in Proceedings of IEEE INFOCOM, 6 [1] YG Zheng and ZH Wang, Stability and Hopf bifurcation of a class of TCP/AQM networks, Nonlinear Analysis: Real World Applications, vol 11, pp , 1 [] The Network Simulator NS [Online] Available: Information c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

14 This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS , IEEE 14 Debayani Ghosh obtained her B Tech in Electronics and Communication Engineering from Cochin University of Science and Technology, Cochin, India in 1 She is a PhD Scholar with the Department of Electrical Engineering at the Indian Institute of Technology Madras, Chennai, India Her research interests lie in the performance evaluation of communication networks and control theory Gaurav Raina obtained his PhD in Mathematics from the University of Cambridge in 6 He currently holds a faculty position in the Department of Electrical Engineering at the Indian Institute of Technology Madras, and is also a visiting research fellow in Mathematics at the University of Cambridge, UK His research interests lie in the design and performance evaluation of communication networks, intelligent transportation systems, control theory, and non-linear systems Krishna Jagannathan obtained his B Tech in Electrical Engineering from IIT Madras in 4, and the SM and PhD degrees in Electrical Engineering and Computer Science from Massachusetts Institute of Technology MIT in 6 and 1 respectively During 1-11, he was a visiting post-doctoral scholar in Computing and Mathematical Sciences at Caltech, and an off-campus post-doctoral fellow at MIT Since November 11, he has been an assistant professor in the Department of Electrical Engineering, IIT Madras He worked as a consultant at the Mathematical Sciences Research Center, Murray Hills, NJ in 5, an engineering intern at Qualcomm, Campbell, CA in 7 His research interests lie in the stochastic modeling and analysis of communication networks, transportation networks, network control, and queueing theory Dr Jagannathan serves on the editorial boards of the journals IEEE/ACM Transactions on Networking and Performance Evaluation He is the recipient of a best paper award at WiOpt 13, and the Young Faculty Recognition Award for excellence in Teaching and Research at IIT Madras c 17 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information