2 Mathematical moelling of the Internet ing that, with an appropriate formulation of an overall optimization problem, the network's implicit objective

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1 1 Mathematical moelling of the Internet Frank Kelly Statistical Laboratory, University of Cambrige, 16 Mill Lane, Cambrige CB2 1SB, UK Abstract Moern communication networks are able to respon to ranomly uctuating emans an failures by aapting rates, by rerouting trac an by reallocating resources. They are able to o this so well that, in many respects, large-scale networks appear as coherent, almost intelligent, organisms. The esign an control of such networks present challenges of a mathematical, engineering an economic nature. This paper outlines how mathematical moels are being use to aress current issues concerning the stability an fairness of rate control algorithms for the Internet. 1 Introuction In the current Internet, the rate at which a source sens packets is controlle by TCP, the transmission control protocol of the Internet [8], implemente as software on the computers that are the source an estination of the ata. The general approach is as follows. When a resource within the network becomes overloae, one or more packets are lost loss of a packet is taken as an inication of congestion, the estination informs the source, an the source slows own. The TCP then graually increases its sening rate until it again receives an inication of congestion. This cycle of increase an ecrease serves to iscover an utilize whatever banwith is available, an to share it between ows. In the future, resources may also have the ability to inicate congestion by marking packets, using an Explicit Congestion Notication mechanism [4], an current questions concern how packets might be marke an how TCP might be aapte to react to marke packets. Just how the available banwith within a network shoul be share raises interesting issues of stability an fairness. Traitionally stability has been consiere an engineering issue, requiring an analysis of ranomness an feeback operating on fast time-scales, while fairness has been consiere an economic issue, involving static comparisons of utility. In future networks the intelligence embee in en-systems, acting rapily on behalf of human users, is likely to lessen this istinction. In Section 2 an 3 we escribe a tractable mathematical moel of a network an use it to analyse the stability an fairness of a simple rate control algorithm, following closely the evelopment in [10], [11]. Stability is establishe by show-

2 2 Mathematical moelling of the Internet ing that, with an appropriate formulation of an overall optimization problem, the network's implicit objective function provies a Lyapunov function for the ynamical system ene by the rate control algorithm. The optimum is characterize by aproportional fairness criterion. The work reviewe in Section 2 forms part of a growing boy of research on the global optimization of networks, with important relate approaches escribe by Golestani an Bhattacharyya[5] an Low an Lapsley [16]. In Section 4 we escribe a ynamical system which represents more closely the TCP algorithm, in particular the generalization MulTCP propose by Crowcroft an Oechslin [2], an iscuss its stability an fairness. Several authors have escribe network moels of TCP base on systems of ierential equations, an the results of Section 4 are close variants on the work reporte in [5], [6], [7], [13], [14], an [15]. The ynamical system representations of Sections 3 an 4 moel only gross characteristics of the ows through a network. An yet these ows evolve asa consequence of the ne etail of software operating at the packet level. In Sections 4 an 5 we briey escribe, following [6], some of the statistical approaches use to relate these macroscopic an microscopic levels of escription, an some of the insights provie by the global moel into how packets shoul be marke at resources an how TCP might react. There are clear analogies between the approach to large-scale systems escribe in this paper an that familiar from early work on statistical physics. The behaviour of a gas can be escribe at the microscopic level in terms of the position an velocity ofeach molecule. At this level of etail a molecule's velocity appears as a ranom process, with a stationary istribution as foun by Maxwell [18] 1. Consistent withthis etaile microscopic escription of the system is macroscopic behaviour best escribe by quantities such as temperature an pressure. Similarly the behaviour of electrons in an electrical network can be escribe in terms of ranom walks, an yet this simple escription at the microscopic level leas to rather sophisticate behaviour at the macroscopic level: the pattern of potentials in a network of resistors is just such that it minimizes heat issipation for a given level of current ow (Thomson an Tait [23]). Thus the local, ranom behaviour of the electrons causes the network as a whole to solve a large-scale optimization problem, an analogy that is iscusse further in [9]. Such physical analogies are immensely provocative, but of course the mathematics of large-scale communication networks, such as the global telephone network or the Internet, iers from that of physical moels in several respects. Most notably we generally know the microscopic rules (they are coe in our software) but not their macroscopic consequences an we can choose the microscopic rules, an ability that makes it all the more important to learn how to 1 It is interesting to note the substantial inuence this paper ha upon Erlang, the early pioneer of research on telephone trac [3].

3 Mathematical moelling of the Internet 3 preict consequences. Finally, a note on the title of this paper. The topic iscusse here is just one of many that might have been covere in a paper with this title, an no attempt is mae here to survey all the important moels that arise in the stuy of communication networks. Some inication of the huge scope for mathematical moelling may be foun in the collections [19], [12] an in the recent article by Willinger an Paxson [25]. 2 Rate control of elastic trac Consier a network with a set J of resources. Let a route r be a non-empty subset of J, an write R for the set of possible routes. Associate a route r with a user, an suppose that if aratex r > 0isallocate to user r then this has utility U r (x r ) to the user. Assume that the utility U r (x r ) is an increasing, strictly concave function of x r over the range x r > 0 (following Shenker [21], we call trac that leas to such a utility function elastic trac). To simplify the statement of results, assume further that U r (x r ) is continuously ierentiable, with Ur(x 0 r )! 1 as x r # 0 an Ur(x 0 r )! 0 as x r " 1. Suppose that as a resource becomes more heavily loae the network incurs an increasing cost, perhaps expresse in terms of elay or loss, or in terms of aitional resources the network must allocate to assure guarantees on elay an loss: let C j (y) is the rate at which cost is incurre at resource j when the loa through it is y. Suppose that C j (y) is ierentiable, with y C j(y) =p j (y) (2.1) where the function p j (y) is assume throughout to be a positive, continuous, strictly increasing function of y boune above byunity, for j 2 J. Thus C j (y) is a strictly convex function. Next consier the system of ierential equations 0 t x r(t) = X 1 wr (t) ; x r (t) j (t) A (2.2) j2r for r 2 R, where j (t) =p j X s:j2s 1 x s (t) A (2.3) for j 2 J. We interpret the relations (2.2){(2.3) as follows. Suppose that resource j generates a continuous stream of feeback signals at rate yp j (y) when the total ow through resource j is y that resource j sens a proportion x r =y of these feeback signals to a user r with a ow of rate x r through resource j an that user r views each feeback signal as a congestion inication requiring some reuction in the ow x r. Then equation (2.2) correspons to a rate control

4 4 Mathematical moelling of the Internet algorithm for user r that comprises two components: a steay increase at rate proportional to w r (t), an a steay ecrease at rate proportional to the stream of congestion inication signals receive. Initially we shall suppose that the weights w are xe. The following theorem is taken from [11]. Theorem 2.1 If w r (t) =w r > 0 for r 2 R then the function X X U(x) = w r log x r ; r2r j2j C j X s:j2s x s 1 A (2.4) is a Lyapunov function for the system of ierential equations (2.1){(2.3). The unique value x maximizing U(x) is a stable point of the system, to which all trajectories converge. Proof The assumptions on w r r 2 R, anp j j 2 J, ensure that U(x) is strictly concave on the positive orthantwithaninterior maximum the maximizing value of x is thus unique. Observe U(x) = w X X r ; p j x r x r j2r s:j2s setting these erivatives to zero ienties the maximum. Further t U; x(t) = X r2r = r t x r(t) r x r (t) X X 2 w r ; x r (t) p j x s (t) j2r s:j2s establishing that U(x(t)) is strictly increasing with t, unless x(t) = x, the unique x maximizing U(x). The function U(x) is thus a Lyapunov function for the system (2.1){(2.3), an the theorem follows. 2 At the stable point x r = P w r j2r j : (2.5) This equation has a simple interpretation in terms of a charge per unit ow: the variable j is the shaow price perunitofow through resource j. Next suppose that each userr is able to vary its own weight w r = w r (t), but is require to pay the network at the rate w r (t) per unit time. The parameter w r (t) can be viewe as the willingness to pay of user r Songhurst [22] provies a iscussion of relate charging an accounting mechanisms, an of their implementation. In particular, we suppose that user r is able to monitor its rate x r (t) continuously, anchooses to vary smoothly the parameter w r (t) so as to satisfy w r (t) =x r (t)u 0 r(x r (t)) : (2.6)

5 Mathematical moelling of the Internet 5 this correspons to a user who euces from observation of the network that its current charge per unit ow is r = w r (t)=x r (t), an then chooses w r = w r (t) to solve the optimization problem maximize over w r 0: U r wr r ; w r In [11] a similar proof to that of Theorem 2.1 is use to establish the following result. Theorem 2.2 The strictly concave function X X U(x) = U r (x r ) ; r2r j2j C j X s:j2s x s 1 A : (2.7) is a Lyapunov function for the system of ierential equations (2.1){(2.3), (2.6), an hence the unique value x maximizing U(x) is a stable point of the system, to which all trajectories converge. Thus if each userr is able to choose its own weight, or willingness to pay, w r, an oes this so as to optimize its own utility less payment, then the overall system will converge to the rate allocation x maximizing the aggregate utility (2.7). At the optimum the relation (2.5) will again hol, but where now theweights w have been chosen by users themselves. 3 Fairness The expression for an aggregate utility (2.7) requires that utilities an costs are aitive. In this Section we consier what can be one without this assumption. The vector Ur (x r ) r 2 R ; X j2j X 1 C j x s A (3.1) s:j2s lists the utilities of the varioususers an the cost of the network. A rate allocation x is Pareto ecient if any alteration that strictly increases at least one of the components of the vector (3.1) must simultaneously strictly ecrease at least one other component [24]. By the strict monotonicity of the components of the vector (3.1), every allocation x is Pareto ecient. Say that an allocation x is achievable by weights w if x maximizes the function (2.4), an hence is a stable point of the system (2.1){(2.3). From the strict concavity of the components of the vector (3.1), together with the supporting hyperplane theorem, it follows that every allocation x is achievable by some choice of weights w. Pareto eciency becomes more interesting when routing choices are allowe. If a user's utility is a function of the sum of its ows over several routes, then it is quite possible to construct ow patterns which are not Pareto ecient. The

6 6 Mathematical moelling of the Internet ow patterns which emerge as stable points of the routing generalization [11] of the system (2.1){(2.3) are, however, just those that are Pareto ecient. Thus many, or even all, rate allocations x are Pareto ecient. Is there a rationale for choosing any particular one? This question is often couche in terms of fairness, an the following construction is familiar in contexts as iverse as political philosophy [20] an ata networks [1]. Say that the allocation x is max{min fair if for any other vector x that leaves the nal component of (3.1) constant an for which there exists r such thatx r >x r, there also exists s such that x s <x s x r. The max{min fairness criterion gives an absolute priority to the smaller ows, in the sense that if x s x r then no increase in x r, no matter how large, can compensate for any ecrease in x s, no matter how small. A max{min fair allocation is necessarily Pareto ecient, an thus achievable by some choice of the weights w. It is possible to escribe the stable point of the system (2.1){(2.3) in the language of fairness. Say that the allocation x is proportionally fair for the weightvector w if for any other vector x that leaves the nal component of (3.1) unaltere, the aggregate of weighte proportional changes is zero or negative: X r2r w r x r ; x r x r 0: This criterion favours smaller ows, but less emphatically than max{min fairness. The allocation (2.5) is proportionally fair for the weight vector w. 4 Network moels of TCP The ierential equations (2.1){(2.3) represent a system that shares several characteristics with Jacobson's TCP algorithm [8] operating in the current Internet, but also has several ierences, which wenow iscuss. Aow through the Internet will receive congestion inication signals, whether from roppe or marke packets, at a rate roughly proportional to the size of the ow an the response of Jacobson's congestion avoiance algorithm to a congestion inication signal is to halve the size of the ow. Thus there are two multiplicative eects: both the number of congestion inication signals receive an the response to each signal scale with the size of the ow. A further important feature of Jacobson's algorithm [8] is that it is self-clocking: the sener uses an acknowlegement from the receiver to prompt a step forwar an this prouces an important epenence on the roun-trip time T of the connection. In more etail, TCP maintains a winow of transmitte but not yet acknowlege packets the rate x an the winow size cwn satisfy the approximate relation cwn = xt. Each positive acknowlegement increases the winow size cwn by 1=cwn each congestion inication halves the winow size. Crowcroft an Oechslin [2] have propose that users be allowe to set a parameter m, which woul inter alia multiply by m the rate of aitive increase an make 1; 1=2m the multiplicative ecrease factor in Jacobson's algorithm. The resulting algorithm, MulTCP, woul behave inmany respects as a collection of m single TCP

7 Mathematical moelling of the Internet 7 connections its smoother behaviour is more plausibly moelle by a system of ierential equations. For MulTCP the expecte change in the congestion winow cwn per upate step is approximately m cwn (1 ; p) ; cwn 2m p (4.1) where p is the probability of congestion inication at the upate step. Since the time between upate steps is about T=cwn, the expecte change in the rate x perunittimeisapproximately m cwn (1 ; p) ; cwn 2m p =T T=cwn = m x2 (1 ; p) ; T 2 2m p: Motivate by this calculation, we moel MulTCP by the system of ierential equations t x r(t) = m r mr ; + x r(t) X 2 j (t) (4.2) Tr 2 Tr 2 2m r j2r where T r is the roun-trip time for the connection of user r, an where j (t) is again given by equation (2.3). We again view p j (y) as the probability apacket prouces a congestion inication signal at resource j, but we o not now insist that p j (y) satises relation (2.1). Note that if congestion inication is provie by ropping a packet, then equation (4.2) approximates the probability of a packet rop along a route by thesum of the packet rop probabilities at each of the resources along the route. To hol x(t) to the positive orthant, augment equation (4.2) with the interpretation that its left-han sie is set to zero if x r (t) is zero an its right-han sie is negative. Theorem 4.1 U(x) = X r The function p 2mr T r arctan xr T X Z P r s:j2s xs p ; p j (y)y (4.3) 2mr j2j 0 is a Lyapunov function for the system of ierential equations (4.2), (2.3). The unique value x maximizing U(x) is a stable point of the system, to which all trajectories converge. Proof Thus The function U(x) is strictly concave on the positive r U(x) = m r T 2 r t U; x(t) = mr T 2 r X r2r ;1 X X + x2 r ; p j x s : 2m r r t x r(t)

8 8 Mathematical moelling of the Internet = X r2r mr T 2 r + x r(t) 2 ;1 2 2m r t x r(t) establishing that U(x(t)) is strictly increasing with t, unless x(t) = x, the unique x maximizing U(x). The function U(x) is thus a Lyapunov function for the system, an the theorem follows. 2 Comparing the form (4.3) with the aggregate utility function (2.7), we see that MulTCP can be viewe as acting as if the utility function of user r is p 2mr T r arctan xr T r p 2mr an as if the network's cost is the nal term of the form (4.3). The system (4.2) has stable point x r = m r T r 2(1 ; pr ) p r 1=2 where X p r = p j : j2r If we were to omit the factor (1 ; p) in the expression (4.1) then the implicit utility function for user r woul be ; (2m r) 2 T 2 r x r an the stable point of the system woul be x r = m r T r r 2 p r (4.4) recovering the inverse epenence on roun trip time an the inverse square root epenence on packet loss familiar from the literature on TCP [17] 2. We have notpresume that the function p j (y) is ene by relation (2.1) inee p j (y) coul not in general satisfy this relation if congestion can only be signalle by ropping packets. But the functions p j (y) an C j (y) are both strictly monotone functions of y, an this is enough to ensure that the stable point is Pareto ecient. The same conclusion cannot be reache when users or 2 The constant of proportionality, p 2, is sensitive to the ierence between MulTCP an m single TCP connections, as well as other features of TCP implementations iscusse in [17]. With m single TCP connections the connection aecte by a congestion inication signal is more likely to be one with a larger congestion winow. This bias towars the larger of the m connections increases the nal term of the expectation (4.1) an ecreases the constant of proportionality.

9 Mathematical moelling of the Internet 9 the network have routing choices, an choose routes generating fewer congestion inication signals. For example, suppose that p j (y) = y N j Mj (4.5) a form that woul arise if resource j were moelle as an M=M=1 queue with service rate N j packets per unit time at which apacket is marke with a congestion inication signal if it arrives at the queue to n at least M j packets alreay present. Suppose the network suers unit loss for each such arriving packet: then y C j(y) =(M j +1) y N j Mj an so the packet marking probability unerstimates the shaow price by a factor M j +1. This iscrepancy is enough to allow the construction of examples exhibiting Braess' paraox, where the aition of capacitytoanetwork, followe by routing aaptation to the extra capacity, leas to a new stable point at which every user is worse o (for a iscussion of Braess' paraox couche in the optimization framework of this paper see [9]). Such a stable point is certainly not Pareto ecient. We en this section with such an example. Consier the network illustrate in the Figure, where TCP connections sen packets from noe a to noe, with some connections route via noe b an others via noe c. Suppose congestion inication signals are generate at the various queues at rates (4.5), an returne a b M 1 N 1 M 2 M 2 N 2 N 2 c M 1 N 1 Braess' paraox for a network. Aing a link between noes b an c, followe by routing aaptation to the extra capacity,may reuce the throughput of all connections. Fig. 1.

10 10 Mathematical moelling of the Internet to the sources at noe a via other routes not explicitly moelle. Suppose that there are 2000 connections in total, that roun-trip times are 1000 time units, an that throughputs are etermine by relation (4.4). Suppose that connections are route so as to equalize the rate at which congestion inication signals are generate along ierent routes. Then approximately 1000 connections will be route along each of the two possible routes, an, with the parameter choices (N 1 M 1 N 2 M 2 ) = ( ), p 1 = 0:006 for the horizontal links, while p 2 =0:019 for the vertical links. Next suppose a high capacity link, at which no congestion inication signals are generate, is introuce between noes b an c. This allows some connections to be route along a thir route, from a to b to c to. At the new equilibrium, the loa on the horizontal links becomes about 1044 connections an the loa on the vertical links about 956 connections, with p 1 = p 2 =0:013, an the throughput of each connection rops by nearly 3 per cent. 5 Packet marking strategies We have briey inicate in Section 4 how TCP software on users' computers can perform operations, at the packet level, that implement the en-system behaviour moelle in the ierential equations (4.2) similarly a relatively minor variation, iscusse in [6], [13], [14], is enough to achieve the en-system behaviour moelle by the ierential equations (2.2). In this Section we consier how the behaviour require of resources may be implemente at the packet level. We have argue in earlier sections that the rate of packet marking at resource j shoul signal its shaow price, ene as the erivative (2.1), an we next escribe how this be one in a simple robust manner, at the packet level. For simplicity of exposition let us suppose that packets are all of equal length. A simple moel escribing a queue with a nite buer is as follows. Let Y t;1 be the number of packets that arrive at the resource in the interval (t ; 1 t], an let Q t be the queue size at time t. Then the recursion Q t = min fm Q t;1 ; IfQ t;1 > 0g + Y t;1 g escribes a queue with a buer capacityofm that is able to serve a single packet per unit time the number of packets lost at time t is [Q t;1 ; IfQ t;1 > 0g + Y t;1 ; M] + : Dene a busy perio to en at time t if Q t;1 = 1, an a busy perio to begin at time t if Q t;1 1anQ t 1. The impact of an aitional packet upon the total number of packets lost, its shaow price, is relatively easy to escribe. Consier the behaviour of the queue with the aitional packet inclue in the escription of the queue's sample path. Then the aitional packet increases the number of packets lost by one if an only if the time of arrival of the aitional packet lies within a critical congestion interval, ene as a perio between the start of a busy perio an the loss,

11 Mathematical moelling of the Internet 11 within the same busy perio, of a packet otherwise the aitional packet oes not aect the number of packets lost. Packets arriving uring critical congestion intervals shoul, ieally, be marke. It will often be icult to etermine the shaow price of a packet while the packet is passing through the queue it will, in general, be unclear whether or not the current busy perio will en before a packet is lost. Nevertheless the above ieal behaviour gives consierable insight into the form of sensible marking strategies, an suggests simple, robust strategies which result in a rate of packet marking at a resource which is equal to its shaow price [6]. The above iscussion shows that the network shaow prices, the variables j (t) appearing in equation (2.3), may beientie, at least statistically, onthe sample path escribing loa at resource j. This ientication provies the require linkage between microscopic, packet-level, marking strategies, often of an apparently crue an statistical nature, an sophisticate macroscopic behaviour interpretable as the global optimization of a large-scale network. Bibliography 1. D. Bertsekas an R. Gallager (1987) Data Networks. Prentice-Hall. 2. J. Crowcroft an P. Oechslin (1998) Dierentiate en-to-en Internet services using a weighte proportionally fair sharing TCP. ACM Computer Communications Review 28, 53{ A.K. Erlang (1925) A proof of Maxwell's law, the principal proposition in the kinetic theory of gases. In E. Brockmeyer, H.L Halstrom, H.L. an A. Jensen, The life an works of A.K. Erlang, Copenhagen, AcaemyofTechnical Sciences, { S. Floy (1994) TCP an Explicit Congestion Notication, ACM Computer Communication Review 24, S.J. Golestani an S. Bhattacharyya (1998) A class of en-to-en congestion control algorithms for the Internet. In Proc. Sixth International Conference on Network Protocols R.J. Gibbens an F.P. Kelly (1999) Resource pricing an congestion control, Automatica P. Hurley, J. Y. Le Bouec an P. Thiran (1998) A note on the fairness of aitive increase an multiplicative ecrease. In Proc. 16th International Teletrac Congress, Einburgh, P. Key an D. Smith (es). Elsevier, Amsteram V. Jacobson (1998) Congestion avoiance an control. In Proc. ACM SIG- COMM '88, A revise version, joint with M.J. Karels, is available via ftp://ftp.ee.lbl.gov/papers/congavoi.ps.z. 9. F.P. Kelly (1991). Network routing. Phil. Trans. R. Soc. Lon. A,337, F.P. Kelly (1997). Charging an rate control for elastic traf-

12 12 Mathematical moelling of the Internet c. European Transactions on Telecommunications 8, F. P. Kelly, A. K. Maulloo, an D. K. H. Tan (1998) Rate control in communication networks: shaow prices, proportional fairness an stability. Journal of the Operational Research Society, 49, F. P. Kelly, S. Zachary an I. Zieins (es) (1996) Stochastic Networks: Theory an Applications. Royal Statistical Society Lecture Notes Series, Vol 4. Oxfor University Press, Oxfor. 13. P. Key an D. McAuley (1999) Dierential QoS an pricing in networks: where ow control meets game theory. IEE Proc Software P. Key, D. McAuley, P. Barham, an K. Laevens. Congestion pricing for congestion avoiance. Microsoft Research report MSR-TR S. Kunniyar an R. Srikant. En-to-en congestion control schemes: utility functions, ranom losses an ECN marks. University of Illinois. 16. S. H. Low an D. E. Lapsley. Optimization ow control, I: basic algorithm an convergence. To appear, IEEE/ACM Transactions on Networking M. Mathis, J. Semke, J. Mahavi, an T. Ott (1997) The macroscopic behaviour of the TCP congestion avoiance algorithm. Computer Communication Review J.C. Maxwell (1860) Illustrations of the ynamical theory of gases. Philosophical Magazine 20, D. Mitra (e) (1995) Avances in the funamentals of networking. IEEE J. Selecte Areas in Commun. 13, J. Rawls (1971) A Theory of Justice, Harvar University Press. 21. S. Shenker (1995) Funamental esign issues for the future Internet. IEEE J. Selecte Areas in Commun. 13, D.J. Songhurst (e) (1999) Charging Communication Networks: from Theory to Practice, Elsevier, Amsteram. 23. W. Thomson an P.G. Tait (1879) Treatise on Natural Philosophy, Cambrige. 24. H.R. Varian (1992). Microeconomic Analysis, thir eition, Norton, New York. 25. W. Willinger an V. Paxson (1998) Where Mathematics meets the Internet. Notices of the American Mathematical Society, 45,