QoS Constrained Adaptation for CDMA in the Wideband Limit

1. Good morning, thank you all for coming. Today I�m going to talk about some of our research in Cross layer design for wireless networks. In traditional network models different network functions are segregated into independent layers. The reason for this segregation is mainly simplicity. It is much easier to design and optimize layer by layer rather than trying to jointly optimize all of the network functions. The price you pay for this simple network design is lost efficiency. What motivates our research into cross-layer design is the fact that the efficiency penalty you incur for a segregated network model will be far too large in future wireless networks.Good morning, thank you all for coming. Today I�m going to talk about some of our research in Cross layer design for wireless networks. In traditional network models different network functions are segregated into independent layers. The reason for this segregation is mainly simplicity. It is much easier to design and optimize layer by layer rather than trying to jointly optimize all of the network functions. The price you pay for this simple network design is lost efficiency. What motivates our research into cross-layer design is the fact that the efficiency penalty you incur for a segregated network model will be far too large in future wireless networks.

2. Motivation Future wireless application will require tight constraints on QoS (delay, min. rate, etc.) Average metrics are not sufficient A dynamic programming approach to adaptation is required Interactive interference in multi-user systems Standard dynamic programming is intractable A new set of tools for optimizing systems with interacting controllers Exploits wideband limit approximations Our contribution is a general framework for optimizing cross-layer adaptation in wireless networks. When developing this framework we had a number of important goals. The first of which was to develop a way to handle a wide variety of QoS constraints, and I�ll give you some specific examples of this later. We also wanted to develop a framework that accurately described the cross-layer interaction in wireless networks. The reason behind these first two goals is that we ultimately want to generate realistic adaptive controls. So although there is a substantial amount of theory in our work, our eventual goal is to see some of our efforts working in hardware. Another thing that differentiates our work is a focus on rigorous mathematical analysis. There are no notions of heuristics in this work. Finally, in addition to results on wireless networks with QoS constraints we also have some results on the Shannon capacity of wireless networks with complex dynamics.Our contribution is a general framework for optimizing cross-layer adaptation in wireless networks. When developing this framework we had a number of important goals. The first of which was to develop a way to handle a wide variety of QoS constraints, and I�ll give you some specific examples of this later. We also wanted to develop a framework that accurately described the cross-layer interaction in wireless networks. The reason behind these first two goals is that we ultimately want to generate realistic adaptive controls. So although there is a substantial amount of theory in our work, our eventual goal is to see some of our efforts working in hardware. Another thing that differentiates our work is a focus on rigorous mathematical analysis. There are no notions of heuristics in this work. Finally, in addition to results on wireless networks with QoS constraints we also have some results on the Shannon capacity of wireless networks with complex dynamics.

3. WCDMA System Model Our multi-user model is based on the structure of a wideband CDMA network. We will assume the system consists of multiple mobiles and multiple base stations. Each mobile user can communicate with one or more base stations. We assume that if a user can communicate with multiple base stations that the base stations cooperate on the signal processing. You can think of this as a resource pooling assumption. Finally we will assume that each base station employs a linear multi-user receiver. This is a pretty generic description. What we really want to know is what happens to our original single-user modelOur multi-user model is based on the structure of a wideband CDMA network. We will assume the system consists of multiple mobiles and multiple base stations. Each mobile user can communicate with one or more base stations. We assume that if a user can communicate with multiple base stations that the base stations cooperate on the signal processing. You can think of this as a resource pooling assumption. Finally we will assume that each base station employs a linear multi-user receiver. This is a pretty generic description. What we really want to know is what happens to our original single-user model

4. WCDMA Cross-Layer Model This is the picture I showed earlier describing a single-user system. Recall the two key points I made. First, that all of the randomness in the system comes from the traffic generator and the channel, and second, that the things we control are marked by this box. So the key difference between the single-user and multi-user models is that the channel now falls into this box. Specifically, the interference in the channel now responds to the cross-layer adaptation chosen by each user. Intuitively this should make sense. Suppose I have a two user system and user 1 raises its power. Then the interference observed by user 2 will increase and user 2 will increase power also. So the interference observed by a single user is responsive to that user�s choice of adaptive control. Just looking at the picture this may not seem like a very big deal, I�m just adding in one more box. But the channel is really what drives every problem in wireless communications, and now I�m permitting the interference in the channel to be a controlled stochastic process, which is very different from the single user formulation. As you can imagine, this extension adds in all sorts of difficulties.This is the picture I showed earlier describing a single-user system. Recall the two key points I made. First, that all of the randomness in the system comes from the traffic generator and the channel, and second, that the things we control are marked by this box. So the key difference between the single-user and multi-user models is that the channel now falls into this box. Specifically, the interference in the channel now responds to the cross-layer adaptation chosen by each user. Intuitively this should make sense. Suppose I have a two user system and user 1 raises its power. Then the interference observed by user 2 will increase and user 2 will increase power also. So the interference observed by a single user is responsive to that user�s choice of adaptive control. Just looking at the picture this may not seem like a very big deal, I�m just adding in one more box. But the channel is really what drives every problem in wireless communications, and now I�m permitting the interference in the channel to be a controlled stochastic process, which is very different from the single user formulation. As you can imagine, this extension adds in all sorts of difficulties.

5. WCDMA Problem Formulation Optimize adaptation in a multi-user setting The users in the system interact through interference Creates a �Chicken and Egg� control problem We seek an optimal and stable equilibrium state and adaptation for the system of users The key is the stochastic process describing the interference Generically, we would like to solve the same cross-layer adaptation problems I talked about before but now in a multi-user setting. And in this multi-user setting the user interact through the interference that they create. The key issue that pops up here is that this creates a chicken and egg control problem. For each user to figure out what adaptation they should choose, they need to know how the other mobiles in the system will respond. This is random dynamic system evolving over time and the adaptation you choose now affects how other users act in the future. The problem is, in order to figure out how the other users will respond, I need to know what control they are using. But I don�t know the control policy, that�s the point of solving the problem. So you need the interference behavior to solve the problem, and you need to solve the problem to figure out the interference behavior. What we would hope is that there is an equilibrium somewhere, where the optimal control and interference model are consistent. And it is this equilibrium that we�re trying to find. The key is the stochastic process describing the interference.Generically, we would like to solve the same cross-layer adaptation problems I talked about before but now in a multi-user setting. And in this multi-user setting the user interact through the interference that they create. The key issue that pops up here is that this creates a chicken and egg control problem. For each user to figure out what adaptation they should choose, they need to know how the other mobiles in the system will respond. This is random dynamic system evolving over time and the adaptation you choose now affects how other users act in the future. The problem is, in order to figure out how the other users will respond, I need to know what control they are using. But I don�t know the control policy, that�s the point of solving the problem. So you need the interference behavior to solve the problem, and you need to solve the problem to figure out the interference behavior. What we would hope is that there is an equilibrium somewhere, where the optimal control and interference model are consistent. And it is this equilibrium that we�re trying to find. The key is the stochastic process describing the interference.

6. Linear Multi-User Receiver Assume each of K mobiles is assigned a N-length random spreading sequence The power of a user affects the SIR of other users forcing them to change their power The component of the system that determines the interference is the structure of the receiver at the basestation. As I mentioned, we will assume each basestation employs a linear multiuse receiver, and this is the equation for the SIR of such a receiver. Assume the system contains K mobiles and that each mobile is assigned a N length random spreading sequence. The meaning of each term isn�t important. The top is user i�s received power, the bottom two terms are the thermal noise and the sum of the effective interference generated by all other users. So based on this receiver structure it�s easy to see how each user�s power affects the SIR of everyone else in the system.The component of the system that determines the interference is the structure of the receiver at the basestation. As I mentioned, we will assume each basestation employs a linear multiuse receiver, and this is the equation for the SIR of such a receiver. Assume the system contains K mobiles and that each mobile is assigned a N length random spreading sequence. The meaning of each term isn�t important. The top is user i�s received power, the bottom two terms are the thermal noise and the sum of the effective interference generated by all other users. So based on this receiver structure it�s easy to see how each user�s power affects the SIR of everyone else in the system.

7. Interference Models Jointly model the state space of every mobile in the system Problem: The system state space grows exponentially Assume unresponsive interference Avoids the �Chicken and Egg� control issue Problem: Unresponsive interference models provide misleading results Diffusion approximation [Stolyar, 2001] Diffusions do not accurately capture delay performance We propose a new approach based on wideband limit approximations Several authors have proposed solutions to this problem. The most obvious solution is to avoid trying to characterize the interference altogether and just explicitly model the state space of every user in the system. The problem there is that the state space grows exponentially with the number of users, so in a system with just 50 or 60 users the problem is totally intractable. Another idea, and this one is actually VERY common in the literature, is to assume the interference is unresponsive. This gets rid of the chicken and egg problem. However, our work has shown that solving problems using an unresponsive interference model can create misleading results. Finally, a recent paper has proposed using diffusion approximations for this problem. The problem is that diffusion approximations do not do a good job of capturing delay performance. So while they might work well for capacity problems, they just don�t work in the delay constrained case. We propose a new approach based on wideband limit approximations for CDMA networks. Several authors have proposed solutions to this problem. The most obvious solution is to avoid trying to characterize the interference altogether and just explicitly model the state space of every user in the system. The problem there is that the state space grows exponentially with the number of users, so in a system with just 50 or 60 users the problem is totally intractable. Another idea, and this one is actually VERY common in the literature, is to assume the interference is unresponsive. This gets rid of the chicken and egg problem. However, our work has shown that solving problems using an unresponsive interference model can create misleading results. Finally, a recent paper has proposed using diffusion approximations for this problem. The problem is that diffusion approximations do not do a good job of capturing delay performance. So while they might work well for capacity problems, they just don�t work in the delay constrained case. We propose a new approach based on wideband limit approximations for CDMA networks.

8. Wideband Limit Approximations

9. Optimization in the Wideband Limit We want to find the optimal multi-user cross-layer adaptation for a given performance metric, subject to QoS constraints Approximate the CDMA network dynamics through the wideband limit Optimize the control in the wideband limit We check convergence and uniqueness to ensure the solution is a good approximation to a finite bandwidth system These are the steps we want to follow to solve for the optimal cross-layer control in a multi-user system. First we approximate the Wideband cdma network using the wideband limit. Then we solve for the optimal control for the wideband system. Once we have the wideband optimal control we have to go back and check some convergence and uniqueness results to ensure that our wideband optimal control is in fact asymptotically optimal. These are the steps we want to follow to solve for the optimal cross-layer control in a multi-user system. First we approximate the Wideband cdma network using the wideband limit. Then we solve for the optimal control for the wideband system. Once we have the wideband optimal control we have to go back and check some convergence and uniqueness results to ensure that our wideband optimal control is in fact asymptotically optimal.

10. Equilibrium in the Wideband Limit At time t, the ith user in the system has a state Xi(t) ? S For any K, N, the system state vector is the fraction of users in each state Define as the single user transition matrix In the wideband limit we have deterministic non-linear dynamics for the system state Here�s the notation we�ll use. Define Pi sub K as the state of the entire CDMA system. It is just the empirical distribution of all of the users in the system. The reason we use the empirical distribution as the system state is because you can show that it is a sufficient statistic to describe the interference in the system. Define P as the single user transition matrix. Then in the wideband limit we can show that the random process describing the system Pi sub K converges to a deterministic process. And that the dynamics for this wideband approximation follow this non-linear dynamic system. Furthermore, we can show that this non-linear dynamic system has a unique fixed point.Here�s the notation we�ll use. Define Pi sub K as the state of the entire CDMA system. It is just the empirical distribution of all of the users in the system. The reason we use the empirical distribution as the system state is because you can show that it is a sufficient statistic to describe the interference in the system. Define P as the single user transition matrix. Then in the wideband limit we can show that the random process describing the system Pi sub K converges to a deterministic process. And that the dynamics for this wideband approximation follow this non-linear dynamic system. Furthermore, we can show that this non-linear dynamic system has a unique fixed point.

11. Value Function in the Wideband Limit This unique fixed point allows us to write down a familiar expression for the limiting average value function. We have the exact same vector product form that we had before. But now this vector Pi has to satisfy a non-linear fixed point equation, rather than the linear equation we used in the single user problem.This unique fixed point allows us to write down a familiar expression for the limiting average value function. We have the exact same vector product form that we had before. But now this vector Pi has to satisfy a non-linear fixed point equation, rather than the linear equation we used in the single user problem.

12. Wideband Optimal Control Problem Same �size� as a single user optimization problem The non-linear constraint can introduce significant theoretical and computational complications The non-linear program is not convex We show that it can be solved by a sequence of linear programs This also leads to a similar optimization problem. This is almost the exact same form as the single user problem, except we now have a non-linear constraint. So let�s think about this for a minute. Normally this multi-user control problem is intractable due to the curse of dimensionality. Through this wideband approximation we have collapsed the problem size down to that of a single user. The price we pay is that we now have to solve a non-linear optimization problem, and this does introduce some additional complexity. One common question is whether or not this problem is convex. As it turns out is generally is NOT convex. However, you can show that it can be solved through a sequence of linear programs.This also leads to a similar optimization problem. This is almost the exact same form as the single user problem, except we now have a non-linear constraint. So let�s think about this for a minute. Normally this multi-user control problem is intractable due to the curse of dimensionality. Through this wideband approximation we have collapsed the problem size down to that of a single user. The price we pay is that we now have to solve a non-linear optimization problem, and this does introduce some additional complexity. One common question is whether or not this problem is convex. As it turns out is generally is NOT convex. However, you can show that it can be solved through a sequence of linear programs.

13. Example: Power Adaptation With Deadline Constrained Traffic Assume deadline sensitive data (100ms) 50 km/h Microcell (COST 207) Minimize average transmission power subject to a deadline constraint What happens as system load increases? Let �number of users per Hz� vary between 0 and 1

14. Power vs. System Load vs. Deadline Constraint

15. Technical Issues We show that we can always find a global optimal solution to the non-linear program. We have proven convergence and asymptotic optimality of the finite user controls No discontinuity in the wideband limit The nature of the wideband limit Technically we have applied an �mean-field� approximation: LLN over bandwidth The CLT version induces a non-stationary Gauss-Markov process There are a number of technical issues that I did not discuss, and I�ll briefly mention them now. We have shown that we can always find a solution to that non-linear program. In addition, we have also shown that the finite user optimal controls converge to the wideband approximation for the optimal control. This is actually a very important technical point. If we did not have this, then as the number of users increased we could have the interference in the channel increasing and then have the interference drop at the limit. This would result in totally useless approximation for the optimal control. We have performed simulations using this approximation to the optimal control and so far the suggest that this is indeed a pretty good approximation. There are a number of technical issues that I did not discuss, and I�ll briefly mention them now. We have shown that we can always find a solution to that non-linear program. In addition, we have also shown that the finite user optimal controls converge to the wideband approximation for the optimal control. This is actually a very important technical point. If we did not have this, then as the number of users increased we could have the interference in the channel increasing and then have the interference drop at the limit. This would result in totally useless approximation for the optimal control. We have performed simulations using this approximation to the optimal control and so far the suggest that this is indeed a pretty good approximation.

16. Summary A general framework for adaptation in multi-user wireless systems Characterization of of �QoS constrained capacity� A rigorous methodology for approximating the interaction between users. Powerful new machinery for analyzing the optimal control of dynamic systems with loosely interacting users

17. Extensions Multi-user wireless networks Nonlinear receivers and/or multiple antennas Uplink/Downlink duality Non-stationary Gauss-Markov approximation for the interference (CLT approximation) Channel capacity with complex dynamics Multiple antenna channels Multiple access and broadcast channels

18. Future Wireless Networks The reason future wireless networks will suffer so much from a segregated network design is not because of the technology that will be employed. The problem is the types of traffic that will be carried on the network. Future networks will carry a vast array of new services, ranging from traditional voice to high-speed data, streaming audio and video, or even multiplayer games. The point is that simple voice connectivity is not longer sufficient. Customers and service providers want content and services. So what are the key challenges?The reason future wireless networks will suffer so much from a segregated network design is not because of the technology that will be employed. The problem is the types of traffic that will be carried on the network. Future networks will carry a vast array of new services, ranging from traditional voice to high-speed data, streaming audio and video, or even multiplayer games. The point is that simple voice connectivity is not longer sufficient. Customers and service providers want content and services. So what are the key challenges?

19. Challenges Provisioning for many types of traffic Voice, video, streaming audio, data Each traffic type requires different delay constraints and rate requirements Time varying network quality All traffic goes over the same network! Introduces the need for cross-layer design. We need to find a way to provision a wireless network for may types of traffic. Different traffic types will have different performance requirements. For example, real-time voice requires a round trip delay of less than 200ms but a relatively low bit-rate. Whereas streaming audio can tolerate much larger delays, but requires a much higher bit-rate since the dynamic range of music is substantially higher than that of voice. In addition to the random nature of wireless networks we also have to deal with the fact that all of this traffic must go onto the same network. We are not going to build one network for voice, one for data, one for streaming audio, and so forth. It is the combination of all of these problems that motivates our interest in cross-layer design.We need to find a way to provision a wireless network for may types of traffic. Different traffic types will have different performance requirements. For example, real-time voice requires a round trip delay of less than 200ms but a relatively low bit-rate. Whereas streaming audio can tolerate much larger delays, but requires a much higher bit-rate since the dynamic range of music is substantially higher than that of voice. In addition to the random nature of wireless networks we also have to deal with the fact that all of this traffic must go onto the same network. We are not going to build one network for voice, one for data, one for streaming audio, and so forth. It is the combination of all of these problems that motivates our interest in cross-layer design.

20. Cross-Layer Design Application layer Adaptive source coding and flexible QoS Network Layer Adaptive routing Medium Access (MAC) layer Multi-class queueing and prioritization Link layer Adaptive techniques (rate, power, coding) When we talk about cross-layer design it�s useful to review what functions are performed by different layers in the network. These are just some examples, the application layer can perform source coding, the network layer contains routing functions, the medium access layer can perform queueing and prioritization. The link layer layer contains many of the traditional notions of adaptation in wireless communications. Here you can have adaptive power control, coding, modulation, multi-user detection and interference cancellation. The idea behind cross-layer design is that we can uncover substantial gains in performance and efficiency by joint optimizing the behavior of these layers. For example, source compression at the application layer can improve with knowledge of the transmission rate being used at the link layer. Or the network layer can gain by looking both up and down the stack. Where the routing algorithm might add redundant links if the link layer provides an unreliable channel or if the QoS constraints from the application layer are particularly tight.When we talk about cross-layer design it�s useful to review what functions are performed by different layers in the network. These are just some examples, the application layer can perform source coding, the network layer contains routing functions, the medium access layer can perform queueing and prioritization. The link layer layer contains many of the traditional notions of adaptation in wireless communications. Here you can have adaptive power control, coding, modulation, multi-user detection and interference cancellation. The idea behind cross-layer design is that we can uncover substantial gains in performance and efficiency by joint optimizing the behavior of these layers. For example, source compression at the application layer can improve with knowledge of the transmission rate being used at the link layer. Or the network layer can gain by looking both up and down the stack. Where the routing algorithm might add redundant links if the link layer provides an unreliable channel or if the QoS constraints from the application layer are particularly tight.

21. Future Research Cross-layer adaptation Multimedia Ad-Hoc networks Admissions control and routing New methodologies for solving these problems Intersection of dynamic systems, communications, and information theory

22. Conclusions Cross-layer design is critical for delay constrained wireless networks Simple average constraints can be misleading! Developed a general framework for optimal cross-layer adaptation in multi-user networks Presented a new method for computing Shannon capacity of complex systems The techniques we develop can be applied across a wide range of problems in communications, control, and information theory.

23. Capacity, Separation and Cross-Layer Design One of Shannon�s most famous results is a �separation theorem� for a class of channels Perhaps this might tell us something about the fundamental optimality of cross-layer design We consider the fundamental Shannon capacity of channels with complex dynamics Time-varying channels with memory Time-varying input processes with memory To obtain these capacity limits we have developed a new method of computing capacity using Lyapunov exponents

24. Outline Introduction Cross-layer adaptation (single-user) Performance results Multi-user networks Capacity of time-varying systems This is the structure of the rest of the talk. First I�ll go over our framework for cross-layer adaptation in the context of a single user system. This makes the framework a bit easier explain and it also facilitates some specific performance results that clearly show the benefit of cross-layer adaptation. Then I�ll talk about how we extend the single-user framework to a multi-user wireless network. Finally I�ll briefly discuss some recent results on the Shannon capacity of time-varying systems.This is the structure of the rest of the talk. First I�ll go over our framework for cross-layer adaptation in the context of a single user system. This makes the framework a bit easier explain and it also facilitates some specific performance results that clearly show the benefit of cross-layer adaptation. Then I�ll talk about how we extend the single-user framework to a multi-user wireless network. Finally I�ll briefly discuss some recent results on the Shannon capacity of time-varying systems.

25. Single User System Model This is the model we will use for a single-user system. Each of these blocks represents a different component of the system model. Traffic is generated here and these arrows represent the signal path of the data as it flows from the transmitter to the receiver. There are a couple of important things I want you to remember from this slide. The first is that everything inside this box is something we would like to control in order to meet whatever our performance constraints are. The second thing to note is that all of the randomness in this model is contained in the traffic generator and the channel, both of which are uncontrolled. Finally, in order to enforce stringent QoS constraints we need to have a detailed model for the dynamics of this system as it evolves from one time slot to the next. We construct these detailed dynamics through a state space model for the single user system.This is the model we will use for a single-user system. Each of these blocks represents a different component of the system model. Traffic is generated here and these arrows represent the signal path of the data as it flows from the transmitter to the receiver. There are a couple of important things I want you to remember from this slide. The first is that everything inside this box is something we would like to control in order to meet whatever our performance constraints are. The second thing to note is that all of the randomness in this model is contained in the traffic generator and the channel, both of which are uncontrolled. Finally, in order to enforce stringent QoS constraints we need to have a detailed model for the dynamics of this system as it evolves from one time slot to the next. We construct these detailed dynamics through a state space model for the single user system.

26. State-Space Model The interaction between the components of the cross-layer system is modeled as a single FSMC Judicious modeling choices are required to prevent intractable models The traffic generator and channel are the �sources of randomness� and are uncontrolled The joint adaptation policy defines the transition probabilities for the cross-layer system In order to accurately characterize the cross-layer interactions between all of those components we construct one large finite-state Markov chain for the entire system. Though I won�t go into details today, a fair amount of effort goes into this step to ensure that we construct a model that is both realistic and tractable. As I mentioned on the previous slide, I want reiterate that the traffic generator and channel are uncontrolled policies and are also the sources of randomness in this model. So you can think of our model for everything on the previous slide as one giant Markov chain, the joint adaptation that we define for the system defines a transition matrix for this Markov chain.In order to accurately characterize the cross-layer interactions between all of those components we construct one large finite-state Markov chain for the entire system. Though I won�t go into details today, a fair amount of effort goes into this step to ensure that we construct a model that is both realistic and tractable. As I mentioned on the previous slide, I want reiterate that the traffic generator and channel are uncontrolled policies and are also the sources of randomness in this model. So you can think of our model for everything on the previous slide as one giant Markov chain, the joint adaptation that we define for the system defines a transition matrix for this Markov chain.

27. Problem Formulation Find the optimal adaptation for a given performance metric For example, minimum average transmit power Subject to appropriate QoS constraints: Maximum delay Probability of packet loss or consecutive packet losses Dynamic programming is a natural approach to solve this problem The complexity of the cross-layer system model complicates the DP solution In general, we want to find the optimal cross-layer adaptation for a given performance metric. In this talk I�ll use some typical metrics such as minimize average power. Subject to appropriate QoS constraints. Now when I say appropriate constraints, I mean we need to use constraints that accurately characterize performance. This means that we need to move away from traditional notions like average delay and look at tougher constraints like maximum delay or probability of consecutive packet loss. I�ll give you a specific example of why this is so important in a few slides. As it turns out, the state space model we�ve constructed and the general nature of our performance constraints make Dynamic programming a good candidate for solving this problem. As it turns out, the complexity of our model still makes this a on-trivial problem.In general, we want to find the optimal cross-layer adaptation for a given performance metric. In this talk I�ll use some typical metrics such as minimize average power. Subject to appropriate QoS constraints. Now when I say appropriate constraints, I mean we need to use constraints that accurately characterize performance. This means that we need to move away from traditional notions like average delay and look at tougher constraints like maximum delay or probability of consecutive packet loss. I�ll give you a specific example of why this is so important in a few slides. As it turns out, the state space model we�ve constructed and the general nature of our performance constraints make Dynamic programming a good candidate for solving this problem. As it turns out, the complexity of our model still makes this a on-trivial problem.

28. Definitions State of the cross-layer system: The adaptive policy at time t is chosen by a control g: The control determines the transition matrix P(g) of the cross-layer system. Now I�m going to introduce a few definitions that we need to set up a dynamic program so that we can solve our cross-layer adaptation problem. Let x(t) be the state of the single user system at time t. We�ll define an adaptive policy a(t) as a function of the state. So the control g maps states into adaptive policies. Finally, the control g determines a transition matrix for the Markov chain describing the cross-layer system.Now I�m going to introduce a few definitions that we need to set up a dynamic program so that we can solve our cross-layer adaptation problem. Let x(t) be the state of the single user system at time t. We�ll define an adaptive policy a(t) as a function of the state. So the control g maps states into adaptive policies. Finally, the control g determines a transition matrix for the Markov chain describing the cross-layer system.

29. Value Function The desired performance metric determines a cost function r(g) The expected cost (value) function is minimized over a finite or infinite horizon We consider infinite horizon to avoid inappropriate time-scale issues Our goal is to choose the optimal control. In order to do that we need to define a Value function for the control. Let�s define the vector r(g) as the cost incurred by g. Every element in the vector r(g) corresponds to the cost of g for each possible state of the system. The value function we will use is an infinite horizon value function that assigns equal weight to the average cost incurred in each time slot. We choose this particular value function as opposed to a finite horizon or discounted cost formulation to avoid some sticky time scale problems. For example, a discounted cost value function places more weight on the cost that�s incurred in the current time slot than the cost 10 time slots from now. This isn�t realistic in our problem setting. Another nice thing about this value function is that we can simplify it to a simple vector product, where the vector pi is the stationary distribution associated with the transition matrix P(g).Our goal is to choose the optimal control. In order to do that we need to define a Value function for the control. Let�s define the vector r(g) as the cost incurred by g. Every element in the vector r(g) corresponds to the cost of g for each possible state of the system. The value function we will use is an infinite horizon value function that assigns equal weight to the average cost incurred in each time slot. We choose this particular value function as opposed to a finite horizon or discounted cost formulation to avoid some sticky time scale problems. For example, a discounted cost value function places more weight on the cost that�s incurred in the current time slot than the cost 10 time slots from now. This isn�t realistic in our problem setting. Another nice thing about this value function is that we can simplify it to a simple vector product, where the vector pi is the stationary distribution associated with the transition matrix P(g).

30. Optimization Problem This simplified form of the value function leads to a linear program we can solve for the optimal control. Here we have the value function we want to optimize. The first two constraints require that pi be a stationary distribution for the transition matrix P(g). This last constraint is a set of performance constraints, so we could jointly constrain the probability that delay exceeds some maximum value as well as average delay. One thing to note is that this can be a challenging optimization problem in terms of sheer size. This first constraint is actually a vector constraint. So the actual LP could have tens of thousands of states and hundreds of thousands of variables. It�s not so large that it is computationally intractable, but it can be a challenge. This simplified form of the value function leads to a linear program we can solve for the optimal control. Here we have the value function we want to optimize. The first two constraints require that pi be a stationary distribution for the transition matrix P(g). This last constraint is a set of performance constraints, so we could jointly constrain the probability that delay exceeds some maximum value as well as average delay. One thing to note is that this can be a challenging optimization problem in terms of sheer size. This first constraint is actually a vector constraint. So the actual LP could have tens of thousands of states and hundreds of thousands of variables. It�s not so large that it is computationally intractable, but it can be a challenge.

31. Applications We have proposed a general problem formulation We illustrate the techniques with a specific example Power control and source/channel coding This example illustrates two key features The �flaw of averages� The benefit of cross-layer adaptation Admittedly, this is a bit of a generic formulation that doesn�t really convey much intuition nor does it motivate the need for cross-layer design. For that we�ll turn to a specific example of power control and joint-source channel coding for delay sensitive traffic. This example demonstrates two critical issues. The first is what I call the flaw of averages. What this means is that typical constraints on averages, average delay for example, lead to misleading results. The second important point is the concrete gains in efficiency resulting from cross-layer design.Admittedly, this is a bit of a generic formulation that doesn�t really convey much intuition nor does it motivate the need for cross-layer design. For that we�ll turn to a specific example of power control and joint-source channel coding for delay sensitive traffic. This example demonstrates two critical issues. The first is what I call the flaw of averages. What this means is that typical constraints on averages, average delay for example, lead to misleading results. The second important point is the concrete gains in efficiency resulting from cross-layer design.

32. Power and Joint Source-Channel Coding for EDGE Traffic arrives according to an On/Off DTMC Source can be coded into 56 byte or 112 byte packets with a deadline of 100 milliseconds Channel code options are MCS-5 and MCS-7 (Rate 0.37 and .74 8PSK) Power: 20mW to 800mW in 2 dB increments TU-50 channel model within a microcell shadowing environment Here are the details of our model. We constructed it to model EDGE, the next generation of GSM. Traffic arrives according to a standard on/off Markov chain model. Here you can see the source and channel coding options. If the packet size and channel coding rates seem a bit strange, that�s what is in the specification for EDGE so that�s what we used. These are the power choices available and the channel model is a typical urban multi-path model for a 50km/h mobile. The first result I want to show is a typical problem that you will see in the literature. Minimize average delay subject to an average power constraint. But rather than look at average delay, one thing you do not see very often is how delay varies with channel gain.Here are the details of our model. We constructed it to model EDGE, the next generation of GSM. Traffic arrives according to a standard on/off Markov chain model. Here you can see the source and channel coding options. If the packet size and channel coding rates seem a bit strange, that�s what is in the specification for EDGE so that�s what we used. These are the power choices available and the channel model is a typical urban multi-path model for a 50km/h mobile. The first result I want to show is a typical problem that you will see in the literature. Minimize average delay subject to an average power constraint. But rather than look at average delay, one thing you do not see very often is how delay varies with channel gain.

33. Delay Vs. Channel Gain For Different Power Constraints That�s what we show here. READ THE AXES. Each of these lines represents an average power constraint. As you can see, as channel gain decreases, the conditional expectation of delay increases. The problem is that if you just look at average delay you might think that the performance here is fine. This demonstrates the flaw of averages. When trying to characterize QoS, average constraints are not just inadequate, they are often misleading. Now, suppose instead that we would like to constrain delay across all channel gains. In our formulation we can impose a constraint like this�.. So that the conditional expectation of delay is 50ms everywhere. Now let�s minimize average power subject to this new delay constraint.That�s what we show here. READ THE AXES. Each of these lines represents an average power constraint. As you can see, as channel gain decreases, the conditional expectation of delay increases. The problem is that if you just look at average delay you might think that the performance here is fine. This demonstrates the flaw of averages. When trying to characterize QoS, average constraints are not just inadequate, they are often misleading. Now, suppose instead that we would like to constrain delay across all channel gains. In our formulation we can impose a constraint like this�.. So that the conditional expectation of delay is 50ms everywhere. Now let�s minimize average power subject to this new delay constraint.

34. Here you can the power cost for imposing this tighter constraint on delay. EXPLAIN AXES. This first line shows the average power necessary to meet the delay constraint when we only use power control. However, if we allow cross-layer design through source-channel coding we get this second line which shows a substantial gain in power consumption. On thing that�s nice about this formulation is that the question �What is cross-layer design worth?� has an answer, about 200mw.Here you can the power cost for imposing this tighter constraint on delay. EXPLAIN AXES. This first line shows the average power necessary to meet the delay constraint when we only use power control. However, if we allow cross-layer design through source-channel coding we get this second line which shows a substantial gain in power consumption. On thing that�s nice about this formulation is that the question �What is cross-layer design worth?� has an answer, about 200mw.

35. Single User Summary We develop a general framework for cross-layer adaptation. We have applied this frame work to a number of cross-layer optimization problems Permits a wide variety of cross-layer adaptation to meet QoS constraints. �Flaw of Averages� Cross-layer design provides substantial gains in both performance and efficiency Up to this point we have talked about the single-user version of our cross-layer design framework. I should mention that we have applied this to a number of other interesting problems, including how to perform cross-layer adaptation when faced with inaccurate and delayed channel estimates. The key points to take away from this section is that our formulation permits very general QoS constraints, which is necessary since constraints on simple averages can be misleading. Moreover, cross-layer design in wireless networks is critical due to the substantial gains in performance that can be achieved. Now, let�s see how we extend the single-user formulation to a multi-user wireless networkUp to this point we have talked about the single-user version of our cross-layer design framework. I should mention that we have applied this to a number of other interesting problems, including how to perform cross-layer adaptation when faced with inaccurate and delayed channel estimates. The key points to take away from this section is that our formulation permits very general QoS constraints, which is necessary since constraints on simple averages can be misleading. Moreover, cross-layer design in wireless networks is critical due to the substantial gains in performance that can be achieved. Now, let�s see how we extend the single-user formulation to a multi-user wireless network

36. Channels with Memory We consider the case of no channel state information (the transition dynamics are known) Time-varying channels with finite memory induce infinite memory in the channel output. Capacity for time-varying infinite memory channels is only known in terms of a limit Closed-form capacity solutions are only known in a few cases Gilbert Elliot Channel and Finite State Markov Channels

37. A New Characterization of Channel Capacity Capacity using Lyapunov exponents Similar definitions hold for l(Y) and l(X;Y)

38. Intuition and Connections In some cases the Lyapunov exponent is entropy: The vector pn is the �direction� associated with l(X) and the conditional channel state probability This vector has a number of interesting properties It is the standard prediction filter in hidden Markov models Under certain conditions we can use its stationary distribution to directly compute l(X)

39. Computing Lyapunov Exponents Define p as the stationary distribution of the �direction vector� pn We prove that we can compute these Lyapunov exponents in closed form as This result is a significant advance in the theory of Lyapunov exponent computation

40. Computing Capacity Closed-form formula for mutual information We prove continuity of the Lyapunov exponents with respect to input distribution and channel Can thus maximize I(X;Y) relative to p(x), which yields the channel capacity We also develop a new CLT for sample entropy Rigorous confidence interval methodology for simulation-based estimates of entropy