Tài liệu Pricing communication networks P10 docx

10 Charging Flexible Contracts Service contracts define guarantees that are to be met by both the service provider and the buyer of the contract. As we saw in Chapter 2, service contracts may have flexible guarantees in terms of performance that allow both the network operator and the user to vary important parameters, such as the rate at which the users may send data to the network. Both parties can benefit. If dynamic pricing is available, the user can dynamically vary the amount of resources he consumes by changing his willingness to pay. The network can control congestion by dynamically adjusting the share of resources that each user is allowed making use of the loose obligations in the contracts. Flexibility is particularly useful when the load on the network dynamically rises and falls as connections start and finish, or the available capacity fluctuates. Flexibility can also benefit a user whose resource requirements vary with time. He need not predict and reserve at the start what he thinks his maximum resource requirements might be. The network also benefits because it need not reserve resources that might not be used, and so more service requests can be accommodated. Of course, by not reserving his maximum requirements, a user runs the risk that at a future time the resources he needs may be unavailable or too expensive to obtain. Since the time-varying availability of resources means that flexible contracts do not provide strict performance guarantees, users with applications that require strict guarantees may prefer contracts whose parameters are defined at the time the contract is set up and remain static throughout their lives. Such contracts can be priced using the ideas for guaranteed services described in Chapter 8. Contracts with flexible guarantees are appropriate for elastic applications. An elastic application is one that can adapt itself to a varying availability of network resources. Examples are video and voice encoders and decoders that can change their data compression rate without significant loss of perceived quality. Of course, even elastic applications require some minimum resources and this may restrict the number of such applications that can be accommodated simultaneously. A network that shares its resources amongst elastic applications with flexible contracts must decide how to divide the network resources amongst them. It does this by setting the dynamic parameters of the contracts to make best use of the resources. If each user is allowed too large a peak rate then there may be unnecessary congestion. However, if each user is allowed only a small peak rate then resources might be wasted. In general, any parameter of a contract can be flexible, but in practice it is the peak rate that is most often flexible. Let us focus on contracts like this. As there is only a single Pricing Communication Networks: Economics, Technology and Modelling. Costas Courcoubetis and Richard Weber Copyright  2003 John Wiley & Sons, Ltd. ISBN: 0-470-85130-9 236 CHARGING FLEXIBLE CONTRACTS traffic parameter that can dynamically change, such contracts are simple to define. Suppose that each user can always fully exploit the peak rate of his contract. So if he is allowed a peak rate x he will send at rate x. Since the peak rate is not fixed in the contract, but varies during the life of the connection, the user who buys such a contract must be an elastic user, that is, a user of an elastic application, who can adapt his sending rate. The notion of an elastic user can be related to the user’s utility function. Suppose u.x/ denotes a user’s utility for a flow of rate x.Let p be the price per unit flow, so the user’s net benefit is u.x /  px. The user is said to be elastic if u.x / is an increasing function of x and has no convex parts over the range x > 0. Under these conditions the maximizing value of x increases with p. By contrast, an inelastic user might require a definite rate, say Nx. He buys that rate if u.Nx/> pNx, but otherwise does not participate. There are two main approaches for implementing flexible contracts for the peak rate. These influence the way such contracts are priced. The first is by creating a market for the peak rate, in which prices fluctuate to reflect demand and resource availability. In this dynamic pricing approach, users observe prices and react by adjusting the amount of resources they purchase, i.e. their peak rates. Such an approach requires dynamic price creation and propagation mechanisms, together with an accounting mechanism to monitor each user’s response. It can be viewed as a generalization of flow control, in which flow adaptation decisions are based on price signals instead of traditional congestion signals. The second approach is based on the differentiated services concept similar to that introduced in Section 3.3.7. The network operator simply creates several service classes, differentiated by the peak rates they are allowed (which in turn differentiates them by quality aspects such as packet loss probability, delay or blocking). He posts a price for each service class, and lets users self-select the classes to which they want to subscribe. Once a user chooses his contract, all parameters remain fixed for the duration of the contract and he is served in a ‘best-effort’ sense according to class of service he has chosen. The task for the network is to adjust the resources in each class and the corresponding prices so that each user eventually ends up using the type of service that is most appropriate for him. In this case, prices also fluctuate, but on much slower timescales, of hours and days, rather than on the timescale of a round-trip packet propagation time in the Internet, as they do under dynamic prices. This approach is very similar to that of versioning in price discrimination. It is simpler to implement than dynamic pricing, since prices remain fixed for longer periods of time and depend upon only the peak rate and the class of the contract (although one may also include a usage charge as an incentive to reduce waste). However, it has the disadvantage that users have no explicit quality of service guarantees and no means of dynamically changing the amount of resources they are allocated, unless they switch service classes by changing their contracts. In this regard, it is an example of flexible contracts with fixed price but unknown variable quality (since the number of customers that will subscribe and hence share the resources dedicated to each class is not known apriori). An important remark is that dynamic pricing mechanisms offer a broader possibility of services than do simple differentiated services. By varying the amount of peak rate he buys, the user can obtain a fixed charge per unit time and variable quality as prices fluctuate. Or, by always buying the same amount of peak rate, he can obtain fixed quality and variable charge per unit time. Finally, he may be able to obtain both a fixed price and fixed quality if he can buy some insurance against fluctuating prices from a third party who charges the average price plus some markup. Dynamic pricing signals can also be used in call admission control, as we shall see later. NOTIONS OF FAIRNESS 237 The next few sections concern the problem of allocating rates of flows to elastic users. In Section 10.1 we discuss possible meanings of ‘fair allocation’. Section 10.2 describes a decentralized method of using congestion signals to both efficiently and fairly allocating flows. It has similarities to the TCP algorithm that is used in the present Internet. Section 10.3 explains how this might be implemented in the Internet. Section 10.4 discusses a mathematical model of the present Internet in further detail. In Section 10.5 we extend the previous concepts to sharing effective flows, i.e. flows measured in terms of their effective bandwidth. Section 10.6 is about user agents (software that runs at the edges of the network and controls price reaction on behalf of the customers), and Section 10.7 is about pricing uncertainty. Section 10.8 discusses the differentiated services approach where the issue is the allocation of quality amongst users, where quality is measured in terms of congestion. We give an example to illustrate Paris Metro Pricing and show how it can increase social welfare. In Section 10.9 we discuss certain important issues that need to be considered when price mechanisms are to be used for managing network resources. 10.1 Notions of fairness In this and the following two sections, we adopt the following model of a network of elastic users. We let L be a set of links and R be a set of routes, a route being a set of links. User r uses route r and is allowed a peak rate of x r . We assume that he can use all this peak rate, and so his flow is x r . The sum of the flows on the routes that use link j must not exceed the capacity of link j , denoted by C j . Thus, we must allocate the flows fx r g subject to the set of constraints X r: j2r x r Ä C j ; for all j 2 L (10.1) where fr : j 2 rg is the set of routes that use link j. Note that, in (10.1), we can give interpretations to x r other than the peak rate; for instance, x r may be the effective rate or the average rate. We return to this in a later section. Consider Figure 10.1. Here route 0 uses links 1;:::;n, and route i uses just the single link i, i D 1;:::;n. Suppose that C 1 DÐÐÐD C n D 1. If our goal is to maximize total throughput then we should take flows of x 0 D 0andx 1 DÐÐÐD x n D 1. However, this allocation of flows might be regarded as unfair, since route 0 is denied any flow. This suggests that throughput may not be the only consideration, but that fairness can also be important. We have already discussed problems of fairly allocating cost in Chapter 7. One could proceed similarly for the problem of allocating flows. For example, just as we defined the characteristic function c.Ð/ in Section 7.1.1, we could define the characteristic function, v.S/, as the maximal total flow that can be accommodated by the network if it is used only by the users in the set S, S Â N ,whereN Df1;:::;ng. Notions such as Shapley value payoff, or nucleolus, that are concerned with ways to share v.N/, could be used C 1 C 2 C 3 C n x 0 x 1 x 2 x 3 x n Figure 10.1 A problem of fairly sharing bandwidth. Here route 0 uses links 1;:::;n, route i uses just the single link i, i D 1;:::;n, and the link capacities C 1 ;:::;C n are all 1. Throughput is maximized by taking flows of x 0 D 0andx 1 DÐÐÐDx n D 1. However, this allocation of flows might be regarded as unfair, since route 0 is denied any flow. 238 CHARGING FLEXIBLE CONTRACTS to define a fair allocation of flows. Of course, we prefer flow allocation methods that are easily understood, and which can be implemented by a simple algorithm or by pricing. We say that an allocation satisfies the max-min fairness criterion if it is not possible to increase some flow without simultaneously decreasing another flow that is already smaller. This gives absolute priority to small flows. No increase in a larger flow can compensate for a decrease in a smaller flow. Equivalently, a flow allocation is max-min fair if for every route r there exists some link on that route, say l, such that l is filled to capacity and x r is maximal amongst the flows that use link l. This condition clearly identifies a max-min fair flow, because if one were to try to increase the flow on route r it would necessarily require a reduction in the flow on some other route through link l, and that could only decrease the value of a smaller or equal flow. In Figure 10.1 the max-min fair allocation is x 0 D x 1 DÐÐÐD x n D 1=2. The following algorithm can be used to find the max-min fair allocation of flows. Start with all flows set at zero and then gradually increase them at the same rate, until some link (or set of links) becomes full. Fix the flows on the routes that use that link (or set of links). Note that these flows are equal. The flows on all other routes are allowed to increase equally, until some other link (or links) becomes full. The flows on the routes that use these links are now fixed. The algorithm continues in this manner until the values of the flows on all routes are fixed. At the point that the flow on a route becomes fixed, it is maximal amongst the flows that use that link. Thus, the condition of the previous paragraph is satisfied. We can make other definitions of fairness. We say an allocation of flows exhibits proportional fairness when there is no change in the allocation of flows that can increase the sum of the proportional rate changes. That is, the allocation fx r g is proportionally fair, if for every other feasible allocation, fx 0 r g, X r2R x 0 r  x r x r Ä 0 Note that this is equivalent to fx r g maximizing P r log x r , subject to fx r g feasible. In the example of Figure 10.1, the proportionally fair allocation is x 0 D 1=.n C 1/ and x 1 DÐÐÐD x n D n=.n C 1/. Note that flow 0, which uses many links, is penalized. Similarly, we say an allocation of flows exhibits weighted proportional fairness when there is no change in the allocation of flows that can increase a weighted sum of the proportional rate changes. That is, for every other feasible allocation,fx 0 r g, X r2R w r x 0 r  x r x r Ä 0 Now fx r g must maximize P r w r log x r subject to fx r g feasible. Proportional fairness arises naturally in the solution of a number of interesting problems. Recall the solution of the Nash bargaining game described in Section 7.2.1. This concerns the choice of a point u D .u 1 ;:::;u k / within a bargaining set U. The choice of u is to be made subject to four reasonable assumptions about what constitutes a fair choice. If players have equal bargaining power then the Nash bargaining game is solved by maximizing P r log u r over u 2 U .Takingu r D x r and U equal to the set of feasible flows, we see that the Nash solution is the same as a proportionally fair allocation. If players have unequal bargaining powers then u should be chosen to maximize P r w r log u r , and this corresponds to a weighted proportionally fair allocation. The next example illustrates that proportional fairness arises naturally as a result of existing flow control procedures used in the Internet. THE PROPORTIONAL FAIRNESS MODEL 239 Example 10.1 (A fair window flow control) Suppose the flow x r on route r is controlled by a window of size B r bytes (as described in Section 3.3.7). In general, the use of window flow control leads to fluctuating rates. That is, x r varies in time and the resulting traffic is bursty. However, as an approximation to the actual case, suppose that the network is equipped with additional mechanisms to smooth bursts so that there is a static regime in which x r remains constant. Let T r be the round-trip time on route r, excluding any queueing delay on the forward path or backward path due to packets waiting in buffers (i.e. the round-trip time if the system is lightly loaded). Let B r and T r be fixed. Also assume that inside the network packets are queued at the routers in order of their arrival, and buffers are large enough to imply no packet losses. In this model, sources do not react to congestion indication signals and send packets at constant continuous rates x r D T total r =B r ,where T total r is the total round-trip delay on route r . It can be shown that in the corresponding fluid model there is a unique static regime and in this regime the resulting set of flows fx r g is the unique solution to the optimization problem maximize x½0 X r2R .B r log x r  x r T r /; subject to X r: j2r x r Ä C j for all j This is similar to the problem of allocating flows in a weighted proportionally fair way, but with a penalty term of  P r x r T r , that has the interpretation of the total rate of delay incurred by packets. If the round trip times are negligible, then the static flows will comprise a weighted proportionally fair allocation with weights equal to the window sizes. Thus, we see that fixed size window control can achieve a fair sharing of the bandwidth according to this criterion, provided scheduling at each link is performed in an appropriate manner. In more general set-ups, it can also be shown that flow control mechanisms that use additive increase/multiplicative decrease tend to produce flow allocations that are proportionately fair. Such mechanisms control the window size so that in the absence of congestion the sending rate increases at a constant rate, while if capacity is exceeded on some link in the path and congestion occurs, then the sending rate decreases at a rate proportional to its size at the time congestion was sensed. Further details are given in Section 10.4. 10.2 The proportional fairness model Let us continue with the model of the previous section. Recall that each route is associated with a user and requires some subset of the links. User r is an elastic user, whose utility for a flow of x r is u r .x r /, which we assume to be increasing, strictly concave and continuously differentiable in x r . Let us define SYSTEM as the global optimization problem of maximizing the social welfare, subject to the capacities of the links. It is SYSTEM : maximize x½0 X r2R u r .x r /; subject to X r: j2r x r Ä C j ; for all j 2 L We can rewrite this as SYSTEM : maximize x½0 X r u r .x r /; subject to Ax Ä C (10.2) where A jr D 1or0as j 2 r or j 62 r , respectively. 240 CHARGING FLEXIBLE CONTRACTS Although it is easy enough to formulate SYSTEM, it is not practical for the network operator to solve it. The problem is that he does not know all the utility functions, u r . Similarly, the users cannot solve SYSTEM because they do not know each other’s utilities or the constraint set Ax Ä C. Fortunately, if the network and users exchange a little information they can solve SYSTEM together. Suppose user r is willing to pay an amount w r per unit time, and that if he does this he receives a flow x r D w r =½ r .Here½ r is the charge per unit flow on route r and is controlled by the network. Imagine that the network somehow chooses the ½ r s so that the resulting flows are feasible. Given that user r has been told the price ½ r his problem is USER r : maximize w r ½0 u r .w r =½ r /  w r (10.3) Note that he has all the information he needs to solve this problem. What problem should the network solve? It cannot easily solve SYSTEM because it does not know the users’ utility functions. However, suppose the users communicate their w r s and the network then allocates flows so they are weighted proportionally fair. That is, it solves the problem NETWORK : maximize x½0 X r2R w r log x r ; subject to Ax Ä C (10.4) A key result is that the solution of SYSTEM occurs when the NETWORK and USER r problems are solved simultaneously, with their solutions in equilibrium. That is, there exist f½ r g, fx r g, fw r g such that x r D w r =½ r and these are simultaneously solutions to SYSTEM, NETWORK,andUSER r for all r . This result is important because it means that if the users and network communicate their w r sand½ r s, then the solution of SYSTEM can be obtained using information that is available to each agent. Let us prove the key result. It follows from solving SYSTEM by Lagrangian methods. Given the value of ½ r the solution of USER r is straightforward: w r satisfies u 0 r .w r =½ r /  ½ r D 0 (10.5) Consider the solution of NETWORK. Imagine that the fw r g are given and fixed. The fact that P r w r log x r is concave in x and the constraints are linear, means that NETWORK can be solved by maximization of its Lagrangian, i.e. by solving maximize x;z½0 L ; where L D X r2R w r log x r C ¼ > .C  Ax  z/ (10.6) Since for this objective function x r D 0 cannot be optimal, L is maximized where it is stationary with respect to x r ,andso @ L @x r D w r x r  X j: j2r ¼ j D 0 Thus necessary and sufficient conditions for fx r g to be a solution to NETWORK are that we can find f¼ j g such that x r D w r P j: j2r ¼ j ;¼½ 0 ; x ½ 0 ; and ¼ > .C  Ax/ D 0 (10.7) THE PROPORTIONAL FAIRNESS MODEL 241 These equations have a nice interpretation as defining a set of market clearing prices f¼ j g when user r has an initial amount of money w r and purchases as much bandwidth as possible in route r . As necessary conditions these imply the existence of fx r g and f¼ j g satisfying (10.7). Similarly, consideration of the solution of SYSTEM by maximizing its Lagrangian gives that a sufficient condition for its solution is the existence of fx r g, f¼ j g such that u 0 r .x r /  X j: j2r ¼ j D 0 ;¼½ 0 ; x ½ 0 ; and ¼ > .C  Ax/ D 0 (10.8) Now if USER r and NETWORK have solutions that coincide, we must have ½ r D P j: j2r ¼ j . This means that the sufficient condition (10.8) of SYSTEM can be satisfied using the nec- essary conditions for the NETWORK and USER r problems in (10.5) and (10.7). We stress again that the advantage of approaching the solution of SYSTEM by way of the NETWORK and USER r problems is that the network and users only need information to which they have access. The network does not need to know the users’ utility functions, and the users need not know the resource constraints of the links. The network communicates with the user r by announcing a new value of the rate x r , or equivalently announcing a new price ½ r . Using this data, user r solves USER r and then tells the network his new selection of w r . In practice, however, it is not guaranteed that this iterative computation between the users and the network converges and that its solution is stable. This is because the computations are asynchronous and the communication of the values of fw r g and fx r g are subject to random delays which differ from user to user. It would be better if we could avoid having to solve the NETWORK problem in some central location, but could instead solve it in some decentralized way. Ideally, the network could communicate some information to the users that would give them the incentives to make their own adjustments to flows in a way to solve NETWORK themselves. This would be similar to the way that the TCP protocol operates in the Internet, in which software running on each user’s computer adjusts the local sending rate and results in an allocation of flows inside the network by operating in a distributed fashion. The challenge is to understand how this can be done so that the network as a whole reacts intelligently to perturbations and the resulting allocation of flows solves the desired network optimization problem. We begin by presenting two algorithms that solve NETWORK in a distributed fashion. We then discuss some convergence and stability results, which suggest that such a distributed solution to NETWORK, combined with users solving their own local USER r problem, does eventually converge to the solution of the SYSTEM problem. 10.2.1 A Primal Algorithm One way to find fx r g and f¼ j g that satisfy the sufficient conditions (10.7) of NETWORK is by use of a so-called primal algorithm. In both this algorithm and the dual algorithm which follows, we assume the w r s are held fixed by the users at constant values. Consider the system d dt x r .t/ D Ä r w r  x r .t/ X j: j2r ¼ j .t/ ! (10.9) ¼ j .t/ D p j X s: j2s x s .t/ ! (10.10) 242 CHARGING FLEXIBLE CONTRACTS y p j (y) C j C j v q j (v) Figure 10.2 Functions p j .y/ and q j .¹/ are inverse functions of one another. p j .y/ is the rate of charge when the total flow through link j is y. q j .¹/ is the rate of flow in link j that produces a rate of charge in that link of ¹. Here Ä r > 0 is a route-dependent parameter which affects the convergence rate of the algorithm, and p j .y/ is a nonnegative, continuous, increasing function of y, not identically zero, which is close to 0 when link j is not full and rises rapidly to infinity as the flow in link j tends to C j from below. See the left part of Figure 10.2. There are some interesting interpretations to (10.9)–(10.10). We can think of resource j as marking a proportion p j .y/ of packets with a feedback signal when the total flow passing through is y. These feedback signals are sent back to the users at the rate they are produced, indicating congestion. In actuality, there is a delay between the time a signal is produced and the time it is received at the edge of the network. This can affect the stability of the solution, an issue we return to later. User r responds to each congestion signal by reducing his sending rate by a constant amount. Alternatively, we can think of ¼ j .t/ D p j .t/ as the price per unit flow charged by link j. These prices are communicated to the users. In fact, user r need only see the sum of the prices of the links along route r . In accordance with (10.9), user r decreases or increases x r as his rate of charge is respectively more or less than w r . Prices can also be communicated using the following mechanism. Assume that resource j marks passing packets in proportion to p j .y/, but these marks have the meaning of a unit charge. Then the second term within the parenthesis on the right hand side of (10.9) is the total rate of charge observed by user r. Observe that using marks to convey price information is much easier to implement than actually informing users about price values. A mark can be conveyed by setting a bit in a packet header. Also, this interpretation is naturally related to the definition of sample path congestion prices discussed in Section 9.4. A packet is marked if it contributes to congestion. In this case, p j .y/ refers to the probability of marking a packet. One can show that the system defined by (10.9)–(10.10) converges to a stable equilibrium point. To do this, we consider the Lyapunov function U.x/ D X r w r log x r  X j Z P s: j2s x s 0 ¼ j .y/ dy One can check that U is strictly concave in fx r g, with a single interior maximum. Using (10.9) we find d dt U.x.t// D X r @U @x r d dt x r D Ä r X r 1 x r .t/ w r  x r .t/ X j: j2r ¼ j .t/ ! 2 ½ 0 Thus, x converges to the maximizer of U. One can show that by sharpening the form of p j this maximizer can be made an arbitrarily good approximation to the solution of THE PROPORTIONAL FAIRNESS MODEL 243 NETWORK. For example, one can take p j .y/ D maxf0; y  C j C žg=ž 2 . Then maximizing U is an arbitrarily close approximation to the SYSTEM problem as ž ! 0. 10.2.2 A Dual Algorithm An alternative approach uses a dual algorithm, based on the Lagrangian dual problem to NETWORK. This problem is: minimize ¼½0 max x;z½0 L,whereL is defined in (10.6). It reduces to maximize ¼½0 X r w r log X j: j2r ¼ j !  X j ¼ j C j and the objective function can be approximated by V.¼/ D X r w r log X j: j2r ¼ j !  X j Z ¼ j 0 q j .¹/ d¹ whereweimaginethatq j .0/ D 0andq j .¹/ increases rapidly to C j for ¹>0, as shown at the right of Figure 10.2. To find fx r g, f¼ j g, consider the system d dt ¼ j .t/ D Ä j X r: j2r x r .t/  q j .¼ j .t// ! (10.11) x r .t/ D w r X k:k2r ¼ k .t/ (10.12) Again, Ä j > 0, and q j .¼ j / is interpreted as a rate of flow in link j corresponding to a rate of charge in that link of ¼ j . This rate of charge adjusts itself proportionally to the excess demand (the term in the right-hand side of (10.11)). This is similar to the tatonnement process of Section 5.4.1. As with the primal algorithm, it can be shown that V.¼/ is strictly concave in ¼ and that it is a Lyapunov function for the system (10.11)–(10.12). Thus, ¼ converges to the maximizer of V. 10.2.3 User Adaptation So far, we have assumed that the w r s are fixed on the timescales of the operation of the pri- mal and dual algorithms. In this case NETWORK is solved by assigning each user r aratex r . Now suppose that the w r s are allowed to change. A simple iteration between users and network could take place as follows. The users could wait until the x r sinNETWORK converge. Then, after computing the resulting prices based on their previous selections of w r s and the newly defined x r s, they could solve their local USER r problems and make new choices of w r s. These could become new inputs to NETWORK, which could be resolved, and so on. A more interesting iteration is when both problems are solved at the same time. Observe that x r .t/ is defined by the differential equation (10.9), which user r can affect only through his choice of w r .t/. Suppose that he monitors x r .t/ continuously and varies w r .t/ so that w r .t/ D x r .t/u 0 r .x r .t// (10.13) 244 CHARGING FLEXIBLE CONTRACTS This choice is consistent with continuously maximizing his net benefit. To see this, suppose that at time t he observes the implied price ½ r .t/ D w r .t/=x r .t/.ForaŽ that is imagined to be infinitesimally small, he chooses w r .t C Ž/ for time t C Ž so to maximize his net benefit, i.e. so w r .t C Ž/ D arg max w r [u r .w r =½ r .t//  w r ]. This happens if and only if w r .t C Ž/ satisfies u 0 r .w r .t C Ž/=½ r .t// D ½ r .t/, which is (10.13) as Ž ! 0. Thus, we see that choosing w r .t/ by (10.13) is consistent with his continually maximizing his net benefit at the presently implied price. In summary, user r responds to the varying prices by solving USER r and using (10.13) in the place of w r in (10.9). Using revised Lyapunov functions, one can again establish the stability of both the primal and dual algorithms in this case, i.e. when users solve their local optimization problems while they perform flow control. The appropriate Lyapunov function for the primal is U.x.t// D X r u r .x r /  X j Z P s: j2s x s 0 p j .y/ dy (10.14) where again, this can be made an arbitrarily close approximation to the SYSTEM problem by appropriate choice of p j .Ð/ 10.2.4 Stochastic Effects and Time Lags Further things can be said when there are delays and stochastic effects. One can still prove convergence but the coefficient Ä r is important. When there are no delays or stochastic effects, the speed of convergence increases with both Ä r and the magnitude of the derivatives of p j .Ð/ at the point where U.x/ is maximized. One can think of p 0 j .Ð/ as the sensitivity of the resource’s load response, and of Ä r as the sensitivity of the response of end-systems to congestion marks. To model delays we would write ¼.t  d jr / and x s .t  d js / on the right-hand sides of (10.9) and (10.10), where d jk is a number expressing the time delay in communication between user k and link j. When there are stochastic effects, decreasing Ä r or increasing p 0 j .Ð/ leads to a smaller variance in the distribution of x.t/ at equilibrium. Increasing p 0 j .Ð/ and Ä r increases the speed of convergence but may make the system unstable. Thus, there is a trade-off between the speed of convergence and stability. In broad terms, the speed of convergence decreases as the magnitude of the d jr increase, and the equilibrium point of the resulting system is asymptotically stable provided Ä r is sufficiently small (for a given choice of p j .Ð/s). In our discussion so far, we have assumed that the choice of the function p j .Ð/ that is to be implemented at link j is simply an engineering decision, whose aim is to realize a robust and efficient flow control procedure that will lead to a good approximation to the optimal solution of SYSTEM. We now discuss how such a choice may be related to actual congestion costs. 10.2.5 Proportional Fairness with a Congestion Cost In Chapter 9 we considered models in which social welfare is reduced by a congestion cost. A model with a congestion cost that is similar to (10.2) is SYSTEM c : maximize x½0 X r u r .x r /  X j c j X s: j2s x s ! (10.15) [...]... interconnected networks produce congestion marks, there should be mechanisms that allow the payments of the users to be shared fairly among these networks There are more difficult issues when intermediate networks deploy different technologies for providing quality of service, which may create problems in propagating congestion marks generated by other networks There are also interesting incentive issues Networks. .. (2002b) The TCP algorithm in its present version is due to Jacobson (1998) Odlyzko (1997) is the proposer of Paris Metro Pricing described in Section 10.8 An interesting paper on the evolution of pricing in telecommunication networks is Odlyzko (2001) A proposal for a billing architecture for pricing the Internet is in Edell, Mckeown and Varaiya (1995) For an example of a screen-based global commodities exchange... separate as possible from the transmission system Pricing should merely be applied to events already occurring in the network, rather than introducing elements into network protocols for charging purposes ž Use split-edge pricing Split-edge pricing is a way to avoid dealing with your customers’ customers or your suppliers’ suppliers In this type of pricing, the price that a provider offers his customer... which networks are chosen by the other users Suppose that the users were to partition themselves between networks 1 and 2 as f1; 3g, f2g, respectively User 2 increases his net benefit by moving to network 1 if 1 3 > v 0:5/ p C C That is, if p > 1=C So suppose p > 1=C, so that f1; 3g, f2g cannot be an equilibrium of the system Similarly, suppose that the users were to partition themselves between networks. .. differentiation can take place through users’ choices alone, with no need for the network to implement any kind of complex priority mechanism in the routers 10.8.1 Paris Metro Pricing A novel method proposed for Internet pricing is so-called Paris Metro Pricing (PMP) The name derives from a time when the Paris Metro had two types of metro car The cars were identical, except that it was more expensive to buy a ticket... of onward charges from onward networks in providing the service However, onward charges are not passed on unaltered directly to the customer Instead, the provider consolidates his pricing schedules, and so avoids things becoming increasingly complex as network paths becomes longer This requires that for each invocation of a service every contractrelated relationship (pricing, responsibility for service... riding in a less crowded car Similarly, we might partition one physical network into two or more logically separate networks, such that the only difference in the networks is the price charged per packet Each user chooses for himself the network he will use Levels of congestion in the networks depend upon the numbers of users they attract A network that charges more than another network will attract... can trade congestion amongst themselves Example 10.3 (A model of Paris Metro Pricing) Let us make a simple model of PMP and illustrate how it can increase social welfare Suppose there are just three users We have available a resource of size 2C which we can use to build either (a) a single network with capacity 2C, or (b) two networks, each with capacity C Suppose user i has a utility of the form ui... heuristic can be applied a few times, and then the Intelligent Agent can run the Pool-Adjacent-Violators algorithm from the beginning, with all points available 10.7 Pricing uncertainty In this chapter, we have been considering the problem of pricing flexible contracts The price of bandwidth fluctuates with the level of usage and a customer can balance the bandwidth he obtains against the amount he is willing... market ž Handle rapid price fluctuations Dynamic pricing during congestion results in the most efficient market, but customers do not like dealing with rapidly fluctuating prices The network should provide its customers with software that ‘absorbs’ the rapid price fluctuations and implements their buying policies It may also offer insurance services by selling communications services at constant prices to . flexible. Let us focus on contracts like this. As there is only a single Pricing Communication Networks: Economics, Technology and Modelling. Costas Courcoubetis. interconnected networks produce congestion marks, there should be mechanisms that allow the payments of the users to be shared fairly among these networks.