Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

This tutorial paper overviews recent developments in optimization-based approaches for resource allocation problems in wireless systems. We begin by overviewing important results in the area of opportunistic (channel-aware) scheduling for cellular (single-hop) networks, where easily implementable myopic policies are shown to optimize system performance. We then describe key lessons learned and the main obstacles in extending the work to general resource allocation problems for multihop wireless networks. Towards this end, we show that a clean-slate optimization-based approach to the multihop resource allocation problem naturally results in a "loosely coupled" cross-layer solution. That is, the algorithms obtained map to different layers [transport, network, and medium access control/physical (MAC/PHY)] of the protocol stack, and are coupled through a limited amount of information being passed back and forth. It turns out that the optimal scheduling component at the MAC layer is very complex, and thus needs simpler (potentially imperfect) distributed solutions. We demonstrate how to use imperfect scheduling in the cross-layer framework and describe recently developed distributed algorithms along these lines. We conclude by describing a set of open research problems

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A tutorial on cross-layer optimization in wireless networks

In this work we consider a problem related to the equilibrium Statistical Mechanics of Spin Glasses, namely the study of the Gibbs measure of the Random Energy Model. For solving this problem new results of independent interest on sums of spacings for i.i.d. Gaussian random variables are presented. Then we give a precise description of the support of the Gibbs measure below the critical temperature.

/pdf/maxweight-scheduling-in-a-generalized-switch-state-space-7i43ko8so6.pdf

MaxWeight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic

We consider the problem of allocating resources (time slots, frequency, power, etc.) at a base station to many competing flows, where each flow is intended for a different receiver. The channel conditions may be time-varying and different for different receivers. It is well-known that appropriately chosen queue-length based policies are throughput-optimal while other policies based on the estimation of channel statistics can be used to allocate resources fairly (such as proportional fairness) among competing users. In this paper, we show that a combination of queue-length-based scheduling at the base station and congestion control implemented either at the base station or at the end users can lead to fair resource allocation and queue-length stability.

/pdf/fair-resource-allocation-in-wireless-networks-using-queue-41wueu44tj.pdf

Fair resource allocation in wireless networks using queue-length-based scheduling and congestion control

Consider the forward link of a system with K remote units and a single base transmitter with time varying connecting channels. Data to be transmitted to the remote units arrives according to some random process, and is queued according to its destination. Power is to be allocated to the K channels in a queue and channel state depend on way to minimize some cost criterion. The channel fading rate is fast and the bandwidth and data arrival rates are high. Owing to the high speed of the fading, arrival, and service rates, an asymptotic or averaging of the heavy traffic type method is promising. By heavy traffic, we mean that on the average there is little server idle time and spare power over the "average" requirements. Heavy traffic analysis eliminates inessential details and focuses on the fundamental issues of scaling and parametric dependencies. To illustrate the scope of the method, several models are considered, and more complicated systems are also treated.

Control of mobile communications with time-varying channels in heavy traffic

Consider the forward link of a mobile communications system with a single transmitter and connecting to K destinations via randomly varying channels. Data arrives in some random way and is queued according to the K destinations until transmitted. Time is divided into small scheduling intervals. Current systems can estimate the channel (e.g, via pilot signals) and use this information for scheduling. The issues are the allocation of transmitter power and/or time and bandwidth to the various queues in a queue and channel-state dependent way to assure stability and good operation. The decisions are made at the beginning of the scheduling intervals. Stochastic stability methods are used both to assure that the system is stable and to get appropriate allocations, under very weak conditions. The choice of Lyapunov function allows a choice of the effective performance criteria. The resulting controls are readily implementable and allow a range of tradeoffs between current rates and queue lengths. The various extensions allow a large variety of schemes of current interest to be covered. All essential factors are incorporated into a "mean rate" function, so that the results cover many different systems. Because of the non-Markovian nature of the problem, we use the perturbed Stochastic Lyapunov function method, which is well adapted to such problems. The method is simple and effective.

Control of mobile communication systems with time-varying channels via stability methods

There is a large literature on the rate of convergence problem for general unconstrained stochastic approximations. Typically, one centers the iterate $\theta_n$ about the limit point $\bar\theta$ and then normalizes by dividing by the square root of the step size $\epsilon_n$. Then some type of convergence in distribution or weak convergence of Un, the centered and normalized iterate, is proved. For example, one proves that the interpolated process formed by the Un, converges weakly to a stationary Gaussian diffusion, and the variance of the stationary measure is taken to be a measure of the rate of convergence. See the references in [A. Benveniste, M. Metivier, and P. Priouret, Adaptive Algorithms and Stochastic Approximation, Springer-Verlag, Berlin, New York, 1990; L. Gerencer, SIAM J. Control Optim., 30 (1992), pp. 1200--1227; H. J. Kushner and D. S. Clark, Stochastic Approximation for Constrained and Unconstrained Systems, Springer-Verlag, Berlin, New York, 1978; H. J. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications, Springer-Verlag, Berlin, New York, 1997; M. T. Wasan, Stochastic Approximation, Cambridge University Press, Cambridge, UK, 1969] for algorithms where the step size either goes to zero or is small and constant. Large deviations provide an alternative approach to the rate of convergence problem [P. Dupuis and H. J. Kushner, SIAM J. Control Optim., 23 (1985), pp. 675--696; P. Dupuis and H. J. Kushner, SIAM J. Control Optim., 27 (1989), pp. 1108--1135; P. Dupuis and H. J. Kushner, Probab. Theory Related Fields, 75 (1987), pp. 223--244; A. P. Korostelev, Stochastic Recurrent Processes, Nauka, Moscow, 1984; H. J. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications, Springer-Verlag, Berlin, New York, 1997]. When the iterates of the algorithm are constrained to lie in some bounded set, the limit point is frequently on the boundary. With the exception of the large deviations type [P. Dupuis and H. J. Kushner, SIAM J. Control Optim., 23 (1985), pp. 675--696; P. Dupuis and H. J. Kushner, Probab. Theory Related Fields, 75 (1987), pp. 223--244], the rate of convergence literature is essentially confined to the case where the limit point is not on a constraint boundary.
When the limit point is on the boundary of the constraint set the usual steps are hard to carry out. In particular, the stability methods which are used to prove tightness of the normalized iterates cannot be carried over in general, and there is the problem of proving tightness of the normalized process and characterizing the limit process. 
This paper develops the necessary techniques and shows that the stationary Gaussian diffusion is replaced by an appropriate stationary reflected linear diffusion, whose variance plays the same role as a measure of the rate of convergence. An application to constrained function minimization under inequality constraints $q^i(x)\le 0,i\le p$, is given, where both the objective function and the constraints are observed in the presence of noise. The iteration is on both the basic state variable and a Lagrange multiplier, which is constrained to be nonnegative. If a limit multiplier value for an active constraint is zero, then the classical method for computing the rate cannot be used, but (under appropriate conditions) it is a special case of our results. Rate of convergence results are important because, among other reasons, they immediately yield the advantages of iterate averaging methods, as noted in [H. J. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications, Springer-Verlag, Berlin, New York, 1997].

Rate of Convergence for Constrained Stochastic Approximation Algorithms

This paper is motivated by the work of Altman and Shimkin (1998). Customers arrive at a service center and must choose between two types of service: a channel that is shared by all currently in it and a dedicated line. The mean service cost (or time) for any customer entering the shared resource depends on the decisions of all future arrivals up to the time of departure of that customer, and so has a competitive aspect. The decision rule of each arriving customer is based on its own immediate self-interest, given the available data on the past performance. If the current estimate of the cost for the shared resource equals that of the dedicated line, any decision is possible. The procedure is a type of learning algorithm. The convergence problem is one in asynchronous stochastic approximation, where the ODE may be a differential inclusion. It is shown that, asymptotically, the performance of the learning system is that for the symmetric Nash strategy, despite the allowed arbitrariness and lack of coordination.

Stochastic approximation and user adaptation in a competitive resource sharing system

This paper is motivated by the work of Altman and Shimkin (1998). Customers arrive at a service center and must choose between two types of service; a channel that is shared by all currently in it, and a dedicated line. The mean service cost (or time) for any customer entering the shared resource depends on the decisions of all future arrivals up to the time of departure, and so has a competitive aspect. The system keeps a record of the (discounted) mean sample service costs (or times) for the shared resource, as a function of the number there when a new arrival joins, and the arriving customers use this to make their decisions. The decision rule of each arriving customer is based on its own immediate self interest, given the available data. They select the service with the smallest estimated service cost. But, if the current estimate of the cost for the shared resource equals that of the dedicated line, any decision is possible. The procedure is a type of learning algorithm. The long term behavior of the arrivals and its effect on the system averages is of interest. The convergence problem is one in asynchronous stochastic approximation, where the ODE has a set-valued right hand side. The set value arises due to the arbitrariness of the decision at certain values of the current estimates. It is shown that, asymptotically, the performance of the learning system is that for the symmetric Nash strategy, despite the allowed arbitrariness and lack of coordination.

http://ieeexplore.ieee.org/iel5/6103/16425/00760587.pdf

R. Buche

Papers

Control of mobile communications with time-varying channels in heavy traffic

Control of mobile communication systems with time-varying channels via stability methods

Rate of Convergence for Constrained Stochastic Approximation Algorithms

Stochastic approximation and user adaptation in a competitive resource sharing system

Stochastic approximation and user adaptation in a competitive resource sharing system