17 Intelligent Flow Control under a Game Theoretic Framework Huimin Chen and Yanda Li 17.1 Introduction In recent years, flow control and network management in high speed networks have drawn much attention by researchers in the telecommunications field. In large-scale network environments with high complexity, decentralized control and decision making are required. Many research papers concerned with multidisciplinary approaches to modeling, controlling and managing networks from a computational intelligence point of view appear in the key telecommunication journals every year. However, only a few of them concentrate on the similarity between resource allocation in a network environment and the market mechanism in economic theory. By taking an economic (market based and game theoretic) approach in the flow control of a communication network, we seek solutions where the intelligence and decision making is distributed, and is thus scalable, and the objective of a more efficient and fair utilization of shared resources results from the induced market dynamics. Unlike other decentralized approaches, our main focus is on using computational intelligence to model the distributed decision agents and to study the dynamic behavior under game-theoretic framework in allocating network resources. By viewing a network as a collection of resources which users are selfishly competing for, our research aims at finding efficient, decentralized algorithms, leading to network architectures which provide explicit Quality of Service (QoS) guarantees, the crucial issue in high speed multimedia networks. There are several parallel research projects from different institutions dealing with this complicated issue. COMET at Columbia Unversity (http://comet.columbia.edu/research/) Telecommunications Optimization: Heuristic and Adaptive Techniques, edited by D.W. Corne, M.J. Oates and G.D. Smith © 2000 John Wiley & Sons, Ltd Telecommunications Optimization: Heuristic and Adaptive Techniques. Edited by David W. Corne, Martin J. Oates, George D. Smith Copyright © 2000 John Wiley & Sons Ltd ISBNs: 0-471-98855-3 (Hardback); 0-470-84163X (Electronic) Telecommunications Optimization: Heuristic and Adaptive Techniques 308 calls the decentralized decision making process Networking Games. Using network game approaches, they intend to solve the crucial issue of pricing from the ground-up of the network from the engineering point of view, rather than having, as in existing communication networks, a necessarily arbitrary price structure imposed post facto. By taking a market-based approach to distributed systems, the Michigan MARX Project (http://ai.eecs.umich.edu/MARX/) seeks solutions to enable adaptive allocation of resources in large-scale information systems. A very useful collection of links and various resources on network economics and related matters is maintained at Berkeley by H.R. Varian (http://www.sims.berkeley.edu/resources/infoecon/Networks.html). However, there are few published works which attempt to solve call admission control as well as traffic management in telecommunication networks from a game-theoretic point of view by using computational intelligence. In this chapter, we organize our work into two parts. The first part is dedicated to the Connection Admission Control (CAC) process in ATM networks. By taking dynamic CAC as a resource sharing problem, we use a cooperative game model with the product form of a certain combination of user preferences to optimize the call block probability while maintaining the negotiated QoS at the call setup stage. After deriving the product form of the user preference function, we use a genetic algorithm to optimize this objective function to maintain fair shares during the traffic transmission phase. The second part of our work models non-cooperative user behavior in the resource allocation process under a generalized auction framework. We propose an optimal auction rule that leads to equilibrium in a repeated dynamic game. We also use neural networks to model the user strategy in bidding, and briefly discuss the formation of common knowledge in repeated games and the allocation efficiency. The common trait for these two parts is that we use some computational intelligence techniques to solve the complicated modeling and optimization problems in resource allocation. 17.2 A Connection Admission Control Scheme Based on Game Theoretic Model in ATM Networks 17.2.1 Preliminaries of Congestion Control in ATM Networks Asynchronous Transfer Mode (ATM) has gradually become the standard in Broadband Integrated Services Digital Networks (B-ISDN). It aims to integrate all of the digital communication services, such as voice, image, video and data transfer, into a single integrated network. A fixed-length packet in an ATM network is called a cell. The network takes advantage of statistical multiplexing to improve the link utilization, while ensuring the requirements of different types of Quality of Service (QoS) from various traffic sources. The demand for intelligent flow control to prevent possible network congestion has become a major issue for network management. The congestion control schemes of the ATM network can be classified into two types: preventive and reactive control. The first approach is mainly applied to Constant Bit Rate (CBR) and Variable Bit Rate (VBR) services, while the second is applied to Available Bit Rate (ABR) services. Preventive congestion control includes Connection Admission Control (CAC) and traffic enforcement. CAC decides whether to accept or reject a user request for connection establishment. If a connection is established, the bandwidth (and possibly the cell buffers at each switching node) for the Intelligent Flow Control under a Game Theoretic Framework 309 incoming service along this connection has to be explicitly allocated during the call holding period. The number of connections accepted at any time has an upper bound due to finite network resources, which mainly depends upon statistical characteristics of traffic sources provided by the end users, QoS constraints and the pricing scheme added on each connection. Traffic enforcement regulates the traffic load in each connection by shaping or smoothing the incoming traffic according to the user declared traffic descriptors. There exist a lot of CAC schemes for bandwidth allocation. Early attempts often used a static approach to model the incoming traffic: during the call setup phase, the end user provides two sets of parameters, one indicating the QoS requirements (e.g. cell loss rate, maximal tolerable delay) and the other containing all the statistical descriptors of the traffic. If the network has enough bandwidth and buffers to transmit the new traffic source, then it will accept this user request and allocate the required resources. Otherwise, the call will be blocked or rejected. However, because of its reliance on the traffic descriptors provided by the end user, the above method has the following problems (Hsu and Walrand, 1996): • How to choose statistical parameters for different traffic sources has not yet been agreed upon. • Usually, at the call setup phase, the end user is not able to determine the traffic descriptors accurately. • Once the incoming traffic is multiplexed with other cell streams, its statistical characteristics will change, and the user declared parameters may not have the same accuracy for all traversed links during transmission. To overcome these problems, one simple solution is to reduce the parameters of the traffic descriptors from the end user (e.g. peak cell rate and mean cell rate), and the network makes a conservative resource allocation by assuming the ‘worst’ traffic pattern the user will provide. Details of various traffic models for ‘worst’ case service patterns can be found in Perros and Elsayed (1996). Although quite simple, this approach often leads to inefficient bandwidth utilization. Thus, it is necessary to develop a scheme using a dynamic allocation mechanism which can estimate the required bandwidth of each source by monitoring user-specified QoS parameters during cell transmission. A review of dynamic allocation strategies for ATM networks can be found in Chong et al. (1995). However, the diversity of traffic characteristics and the variety of QoS constraints make the optimization effort of dynamic bandwidth allocation a complicated problem. Various analytical methods and queuing models have been used, but most of them are highly computationally expensive (e.g. even the simple heterogeneous on/off queuing model in Lee and Mark (1995) has no analytical solution in closed form). Recently, heuristic approaches based on equivalent bandwidth have been presented using neural networks and fuzzy set theory. For example, Hiramatsu (1991) proposed a CAC and a traffic enforcement framework using a neural network; Chang and Cheng (1994) proposed a method of traffic regulation based on a fuzzy neural network; Ndousse (1994) implemented a modified leaky bucket to smooth the traffic by using fuzzy set theory; in Zhang and Li (1995), the authors derived a CAC scheme based on a fuzzy neural network, and integrated it with a genetic algorithm to optimize the network parameters. All of these schemes can be seen as a trade-off between the accuracy of traffic modeling and the implementation complexity. A major drawback is Telecommunications Optimization: Heuristic and Adaptive Techniques 310 the lack of fairness among different classes of service, especially when different pricing schemes are introduced to the individual user. The recent development of the differentiated service Internet model recalls the interest of fair shares among different services, and the CAC scheme also requires improvement to fit this need. In the following, we first briefly describe the measurement-based method to monitor the equivalent bandwidth of traffic sources; then we propose a cooperative game model to derive the measure of fair shares in resource allocation. Under the game theoretic framework, the fairness criterion of resource allocation can be derived through maximizing the product form of the user preference function (also called the utility function). Given the analytical form of the utility function based on the bandwidth-delay-product, we use a modified genetic algorithm to solve the optimization problem and derive the available bandwidth for each type of traffic. The CAC criterion is modified to maintain the fairness of accepting/rejecting the incoming calls. Simulation results show that our method can achieve a desired fairness for different traffic types and also maintain good link utilization. 17.2.2 Cooperative Game Model for Dynamic Bandwidth Allocation The equivalent bandwidth model of traffic sources We first briefly introduce the method to estimate the equivalent bandwidth of multiplexed traffic based on traffic measurement. Due to the diversity of the traffic arrival process, when heterogeneous traffic sources are multiplexed, the bandwidth allocated for each single traffic source has an equivalent representation, assuming that no multiplexing is induced. In terms of equivalent bandwidth, the estimation is based on large quantities of traffic multiplexing. Denote the traffic arrival process as {X 1 , X 2 , …} and the service rate as S. In a given traffic measurement interval, there are W such discrete arrivals, and we assume W is large enough. With regard to its asymptotic distribution, we have: ∑∑ =+= ⋅⋅⋅== W k W Wk kk XXXX 1 2 1 21 , ˆ , ˆ (17.1) Assume the arrival process is stationary; the new process in equation 17.1 asymptotically has identical independent distribution. According to large deviation theory, the equivalent bandwidth b can be calculated by the following equation (Chen, 1996): ∑ = = Wn i X W n i e n W W / 1 ˆ log 1 )( ˆ θ θλ (17.2) })( ˆ {sup ˆ 0 θθλδ θ S W n W n ≤= > (17.3) W n W n W n n b δ δλ ˆ ) ˆ ( ˆ lim ∞→ = (17.4) Intelligent Flow Control under a Game Theoretic Framework 311 Details about the equivalent bandwidth can be found in Lee and Mark (1995), and measurement-based admission control procedures in Jamin et al. (1996). It is necessary to point out that the equivalent bandwidth of the individual traffic represents the required resource capacity that the network manager has to allocate for the established connection in order to meet the QoS requirement. The accepted traffic sources can also be regulated via end-to-end traffic enforcement (Chen, 1996). The above provides a brief illustration of dynamic bandwidth allocation, which is also the preliminary in developing a CAC strategy that we will discuss next. The fair share criterion based on the cooperative game model Early research on dynamic resource allocation mainly concentrated on improving link utilization under certain QoS constraints. Less attention has been paid to the fair shares issue among different service types when many incoming calls are competing for network connections and some calls must be blocked. Recently, more work has emphasized the problem of fairness among end users, as well as different types of traffic streams using network resources. For example, in Bolla et al. (1997), the authors try to balance the call block probability among the incoming calls of different types. In Jamin et al. (1998), the authors combine the CAC strategy with a usage-based pricing scheme to achieve a certain degree of fairness. In general, these works propose different objective functions that have to be optimized when admission control and the models of the objective function can be classified into two classes. One formulates the optimization problem taking the Grades of Service (GoS) as constraints of the objective function. The other approach assigns each class of traffic a certain weight or a reward parameter, and the distribution of the GoS is adjusted in resource allocation by changing the value of the weights. The implementation of the above approaches is by no means easy due to the lack of an efficient algorithm to optimize the objective function on-the-fly. Thus, it is difficult to satisfy the needs of a real- time CAC decision using the above schemes. However, from the user-preference point of view, the problem of fair shares among different traffic types is suitable for modeling as multi-player cooperative games. Early studies such as Sairamesh et al. (1995) and Dziong and Mason (1996) discussed the call block probability in resource allocation under the game theoretic model. In Dziong and Mason (1996) the Raiffa, Nash and modified Thomson solutions were compared with regard to their characteristics and the CAC boundary to apply the solution, but only a few simple traffic models are used in simulation to achieve nearly equal call blocking probabilities among the different traffic types. In the following, we first introduce the cooperative game model, and then define the utility function based on the form of bandwidth-delay-product of the incoming traffic. Using this model, we propose a CAC scheme to achieve a certain trade-off of call block probabilities among different traffic types. Finally, we try to optimize the objective function used in the cooperative game model by applying a genetic algorithm. To simplify the notation, suppose there are two types of incoming traffic competing for the network resources. All possible bandwidth allocation strategies between the different types of traffic form a strategy set, and the outcome of the game is evaluated with two players’ utilities namely u = {u 1 , u 2 }, u i ∈ R. Denote U as the set of all possible outcomes of the game between the two players, and assume that U is convex and compact. Note that this assumption is introduced in order to derive the unique optimal solution under the Telecommunications Optimization: Heuristic and Adaptive Techniques 312 cooperative game model. However, in solving the optimization problem using a genetic algorithm, we do not need this property of the set U. The outcome of the cooperative game with two players has the following properties: the increase of one player’s utility will decrease that of the other; each player’s cooperative utility is no less than the non-cooperative outcome. Suppose that when two types of traffic sources demand network bandwidth, the network manager uses a centralized decision rule to ensure the fairness of resource sharing, such as certain coordination between the users transmitting different types of traffic. This procedure is called a cooperative game. For convenience, assuming that the elements in U have been normalized to [0, 1], we define the player’s preference as his utility function, given below: )1( 211 uuv −+= β (17.5) )1( 122 uuv −+= β (17.6) where β is a weight factor. In the cooperative game model with two players, by solving the optimization problem: }{max 21 vv Uu ⋅ ∈ (17.7) the two players’ bandwidth utilization may achieve a ‘relatively’ fair share at some operation point with respect to β . When β = 0, the solution of equation 17.7 is called the Nash point; when β = 1, the solution is called the Raiffa point; and when β = −1, the solution is called the modified Thomson point. In the multi-player cooperative game model, player j 's utility function has the form ∑ ≠ −−+= ij ijj uNuv |)1(| β (17.8) The economic meanings of each solution corresponding to equation 17.7 can be found in Shubik (1982). When using the cooperative game model, choosing an appropriate utility function is the key issue in ensuring fairness among the various services. Except for the diversity of the bandwidth requirements of the traffic sources, the statistical characteristics of the traffic and utilization of the resources along the end-to-end links should also be considered. In deriving a utility function of an appropriate form, we may want to include the above factors, and also impose a weight factor that controls the influence on each player's satisfaction at different GoS levels. Denote C to be the original data length of the traffic source; denote C′ to be the average length after compensating the data retransmission due to cell error or cell loss. We use the empirical formula C′ = C·[1+α(L)] to model the traffic amount, where L is the cell loss rate at one switching node and α(L) is a function of L called the influence factor, due to cell retransmission. Assume that B is the available bandwidth to be allocated to the user, and T is the mean end-to-end cell delay, then the average cell transmission time can be written as follows: Intelligent Flow Control under a Game Theoretic Framework 313 LCn LCn T B C k B C BLD ′ − ′       + ′ + ′ ≈ 1 2),( (17.9) In equation 17.9, k is a coefficient modeling the sliding window for traffic shaping, and can be chosen as half of the window size under a window-based traffic shaping mechanism. Notice that k may also be a variable in connection with the burst curve of the arrival traffic. The upper bound of k should not exceed the capacity of the leaky bucket under leaky bucket flow control. n is the number of nodes and/or switches that the traffic traverses from source to destination. In equation 17.9 we potentially assume that the cell loss rate at each node is approximately the same. Using the result derived above, the utility function based on the bandwidth-delay-product is defined as follows       ′ − ′ + ′ + ′ = LCn LCn BTCkCLfBLC 1 )2()(),( (17.10) In equation 17.10, f (L) is a weight function that decreases with the increase in the cell loss rate, depending on different QoS requirements. From equation 17.10, we can see that the number of traffic sources and the bandwidth-delay-product represent the availability of network resources to a certain type of traffic during the call setup phase. f (L) is considered as the nominal QoS given that the connection is established for the incoming traffic. Hence we consider C(L,B) as the utility function of various types of traffic sources. Note that the utility function for each type of traffic is given as: NiBLCu iiii , ,2,1 ),,( == (17.11) then the outcome of the cooperative game given in equation 17.7 ensures the fair share of bandwidth among different traffic sources in a quantitatively simplified measurement. Note that utility functions of other forms may also be introduced for differentiated or best effort services following a similar derivation; the genetic algorithm does not require the objective function to have a specific analytical form. Note also that in our approach, the cooperative game is not played by the end users but the network manager, who sets certain operation points of the CAC boundary according to the fair shares criterion. The CAC strategy with fair shares among various traffic sources After introducing the cooperative game model, we propose the CAC strategy, which includes the following stages: when the ith class of traffic arrives, the network estimates its equivalent bandwidth i b ~ using equations 17.1–17.4 proposed in section 17.2.2. If the statistical parameters of the incoming traffic are available, the equivalent bandwidth can also be calculated by means of queuing analysis through various numerical approaches. At the same time, the network manager uses a genetic algorithm to search for the optimal solution of the optimization problem of equation 17.7, thus achieving the fair shares bandwidth B i available for this class of service. In case the bandwidth allocated to each traffic is per-flow guaranteed (e.g. static allocation), the network end can run genetic search off-line or at an earlier stage; while a network using dynamic bandwidth allocation, B i is adjusted online via genetic optimization to achieve the desired optimal point. Assume the Telecommunications Optimization: Heuristic and Adaptive Techniques 314 bandwidth for the ith class of traffic sources is , ~ i B as a result, the network decides to accept or reject the incoming traffic depending on whether i B ~ − B i is greater than . ~ i b In fact, when the ith and jth classes of traffic compete for the network resources, the network will accept the class at a higher priority whose allocated bandwidth has not exceeded its maximal bandwidth in fair share. In the next subsection, we will give the detailed procedure using a GA to solve the optimization problem of equation 17.7 and we will show that by properly choosing the weight factors of the utility function in equations 17.5 and 17.6, the desired fairness among various classes of service can be achieved. 17.2.3 Call Admission Control with Genetic Algorithm The basic principles of genetic algorithm Genetic Algorithms (GAs) are stochastic optimization methods that usually require the objective function to have the form given below: }|)(min{ N IBCCf ∈ (17.12) Usually, we have: ∞< ′ <=∈∀ )(0 ,}1,0{ CfIBC NN We can see that equation 17.7 satisfies the above requirement. Chapter 1 has introduced the basic operation of a genetic algorithm. In the following, we concentrate on the results of applying the technique in our application. Computer Simulation Results In our work we use the GA proposed in Zhang et al. (1994; 1995), which uses modified mutation and crossover operators. The parameters coded into strings for GA optimization are the bandwidth available for each type of traffic source. When there are n i sources accepted by the network with service type i, the cell loss rate of this type of service at the switch can be estimated by using equivalent bandwidth (Chen, 1996). Normalizing equations 17.10 and 17.11 and substituting them into equation 17.7, we can get the fitness value for GA optimization. The simulation results show that by using an improved GA approach, the global optimum (strictly speaking, it is sub-optimum or near optimum) is achieved with around only 20 iterations in GA evolution. The simulation scenario is chosen as follows. We assume there are two types of traffic source. One (traffic I) has a peak bit rate of 64kbps, a mean bit rate of 22kbps and an average burst length of 100 cells. The source-destination route traverses four switching nodes and the mean cell delay is 4ms. The QoS of traffic I requires a cell loss rate less than 10 -4 . The other (traffic II) has a peak bit rate of 10Mbps, a mean bit rate of 1Mbps and an average burst length of 300 cells. The source-destination route traverses two switching nodes and the average delay is 0.28 ms. The traffic source has the QoS constraint of a cell loss rate less than 10 -8 . Assume that the link capacity of the network is 155.5 Mbps; the weight function in equation 17.10 takes the empirical formula as below: Intelligent Flow Control under a Game Theoretic Framework 315 LLeLf QoSL 005.1)( ,)( )(10 == −− α (17.13) The length of one string representing the normalized bandwidth in the GA is 15 bits. In computer simulation, the CAC boundary given different traffic scenarios (data length of traffic sources) as well as different buffer sizes is calculated via the measurement-based approach discussed in earlier. Concerning the fair shares between traffic I and II, the maximal number of connections for each type of traffic during a heavy load period is derived using GA optimization, and is listed in Table 17.1. Table 17.1 The CAC boundaries with different weight factors of two types of traffic sources. Weight factor β Buffer size ( in cells ) Data length of traffic I Data length of traffic II The number of traffic I The number of traffic II Nash 0 10 3 10 5 10 7 4046 3 Raiffa 1 10 3 10 5 10 7 4748 1 Thomson -1 10 3 10 5 10 7 2430 8 -0.25 10 3 10 5 10 7 3460 4 -0.75 10 3 10 5 10 7 2729 7 0.25 10 3 10 5 10 7 4328 2 0.75 10 3 10 5 10 7 4625 1 Nash 10 3 10 5 10 8 2628 7 Raiffa 10 3 10 5 10 8 3296 3 Thomson 10 3 10 5 10 8 1368 13 Nash 10 3 10 3 10 8 1966 12 Raiffa 10 3 10 3 10 8 4338 4 Thomson 10 3 10 3 10 8 76 17 Nash 10 2 10 5 10 7 2495 3 Raiffa 10 2 10 5 10 7 3868 1 Thomson 10 2 10 5 10 7 1245 5 From Table 17.1, we can see that the traffic sources with a longer data length (indicating a longer connection holding period) will get a relatively larger bandwidth. It is clear that traffic II needs a much larger bandwidth than traffic I. If fair share is not considered between the two types of traffic sources, traffic II is more likely to be blocked than traffic I during the heavy load period when both types of incoming traffic compete for network connection. The solution of the cooperative game has such properties that it guarantees certain reserved bandwidth for both of the traffic sources, despite the diversity Telecommunications Optimization: Heuristic and Adaptive Techniques 316 of their required bandwidth and the number of hops in their routes. The greater the amount of traffic, the more the reserved bandwidth for the usage of this traffic type at the network operation point. Table 17.1 also reveals that the CAC boundary will become smaller when the buffer size decreases, provided other configurations unchanged. However, the solution of the game theoretic model will not exceed the boundary without the constraint of fair shares in a CAC decision. The selection of the weight factor β of the utility function has a significant influence on the fair share among different types of traffic sources. The network manager may choose an appropriate β to make the trade-off of call block probabilities among different traffic sources. In Table 17.1, we can see that the available bandwidth for traffic I increases as β increases from −1 to 1. When β = 1, the utility function of each type of traffic considers its own gain as well as the loss of other types of traffic for the amount of bandwidth available. When β = −1, the net gains of the players’ cooperation in sharing the bandwidth are considered. When β = 0, each player only cares about individual profits through cooperation. Thus, the above three conditions have a clear economic meaning representing certain fair share criteria in a CAC decision. Through various simulation studies, we find that using a Nash solution as an additional constraint to the CAC decision may be a good candidate in making a balance of call block probabilities among different traffic types. As β = 0 and the data length of traffic varies, the upper bound of the bandwidth available for traffic I is derived from 28% to 87%, while for traffic II it is from 46% to 78%, and the link utilization remains relatively high. In short, the simulation results show that a certain trade- off between call block distribution among different traffic sources and the network efficiency can be achieved by setting an appropriate operation point at the CAC boundary, which is an optimization problem under the cooperative game framework. 17.2.4 Summary of Connection Admission Control using a Cooperative Game In this section, the CAC mechanisms based on dynamic bandwidth allocation are briefly discussed. The major issue of fair shares of the bandwidth among different traffic sources is analyzed under a cooperative game model. The utility function concerning the bandwidth- delay-product of each type of traffic is proposed, and the final optimization problem is solved using an improved genetic algorithm. To speed up our CAC scheme, the optimization problem to achieve fair shares can be calculated offline. In online tuning, we can set the initial population to include the previous solution for GA optimization. The proposed CAC scheme is still a centralized mechanism to regulate the traffic flows from end users, and we potentially assume that the network decision-maker has enough information (traffic statistics and QoS requirements) on the incoming traffic to force the link utilization to the desired operation point. The cooperative game model also has a clear economic meaning to interpret certain types of solution as the negotiation result among the users to improve all of their profits in using the available network bandwidth. When such information to derive the utility function of each type of traffic is unavailable, a distributed mechanism for resource allocation is required. We will model the decentralized resource allocation process as a generalized auction game in the next section. [...]... discussion of auction design in section 3 can be generalized to other scheduling problems in distributed computing systems, while the allocation rule remains the same 318 Telecommunications Optimization: Heuristic and Adaptive Techniques In the following discussion, we assume that the resource to be allocated to the potential bidders is a quantity of link capacity, and does not assume any specific mapping... sold or consumed Thus, if bidder i knows that his value estimate of unit capacity is ti , then his expected utility from the auction mechanism described by (p, x) is: 320 Telecommunications Optimization: Heuristic and Adaptive Techniques U i ( p , x, t i ) = bi ti ∫ [ti pi (t ) − xi (t )] f −i (t −i )dt −i (17.14) T− i where dt −i = dt 1L dt i −1dt i +1 L dt n Similarly, the expected utility for the network... ti ti − t0 −  ∈N   T  1 − Fi (t i )    pi (t ) f (t )dt f i (t i )    subject to the constraints of equations 17.16 and 17.20 Assume also that (17.23) 322 Telecommunications Optimization: Heuristic and Adaptive Techniques  xi (t ) = ti  pi (t ) −  ti ∫a i pi (t −i , si )  dsi  si  (17.24) Then (p, x) represents an optimal auction With a simple regularity assumption, we can compute... auction game using software agents Unlike the scenarios of interconnected networks studied in Lazar and Semret (1998a), our simulations are limited to a single route 324 Telecommunications Optimization: Heuristic and Adaptive Techniques with a divisible resource capacity, and add learning strategies to the bidders as well as the auctioneer to formulate certain system dynamics (e.g convergence of probability... Theoretic Framework Figure 17.1 The PDFs estimation from bidder 5 using the Parzen window Figure 17.2 The PDFs estimation from bidder 5 using the neural network approach 325 Telecommunications Optimization: Heuristic and Adaptive Techniques 326 17.4 Concluding Remarks In this chapter, we propose a game theoretic framework to analyze flow control and resource allocation in high speed networks We organize our... (t )t j ] f (t )dt j∈N T − = dj tj aj aj = − ∑ ∫ [U j ( p, x, a j ) + b j ∫ j∈N dj ∑ ∫a U j ( p, x, t j ) f j (t j )dt j j∈N j Q j ( p, s j ) ds j ] f j (t j )dt j sj Telecommunications Optimization: Heuristic and Adaptive Techniques 328 = − ∑ U j ( p , x, a j ) − b j ∫ dj dj ∫ aj sj j∈N f j (t j ) Q j ( p, s j ) dt j ds j sj dj = − ∑ U j ( p, x, a j ) − b j ∫ (1 − F j ( s j )) aj j∈N = − ∑ U j ( p, . (http://comet.columbia.edu/research/) Telecommunications Optimization: Heuristic and Adaptive Techniques, edited by D.W. Corne, M.J. Oates and G.D. Smith © 2000 John Wiley & Sons, Ltd Telecommunications Optimization: Heuristic and Adaptive. Ltd ISBNs: 0-471-98855-3 (Hardback); 0-470-84163X (Electronic) Telecommunications Optimization: Heuristic and Adaptive Techniques 308 calls the decentralized decision making process Networking. on/off queuing model in Lee and Mark (1995) has no analytical solution in closed form). Recently, heuristic approaches based on equivalent bandwidth have been presented using neural networks and