@article{buche_ghosh_pipiras_2012, title={HEAVY TRAFFIC APPROXIMATIONS OF A QUEUE WITH VARYING SERVICE RATES AND GENERAL ARRIVALS}, volume={28}, ISSN={["1532-4214"]}, DOI={10.1080/15326349.2012.646526}, abstractNote={Heavy traffic limit theorems are established for a class of single server queueing models including those with heavy-tailed or long-range dependent arrivals and time-varying service rates. The models are motivated by wireless queueing systems for which there is an increasing evidence of the presence of heavy-tailed or long-range dependent arrivals, and where the service rates vary with the changes in the wireless medium. The main focus of the paper is to obtain the different possible limit processes that can arise depending on the relationship between scalings for both the arrival and departure processes. The limit processes obtained here are driven by either Brownian motion (when the contribution from the departure process dominates the limit) or the limits of properly scaled arrivals (when the contribution from the arrival process dominates the limit), typical examples being stable Lévy motion or fractional Brownian motion. In particular, for the case where arrival process is given by the infinite source Poisson process, this relationship, which determines the type of the limiting queue-length process, is a simple condition involving the heavy tail exponent, arrival rate and channel variation parameter in the wireless medium model. To establish these limit results, two approaches are studied. In one approach, when the limit is driven by Brownian motion, the perturbed test function method is extended to incorporate reflection. In contrast, the second approach allows for non-Markovian and/or non-Gaussian driving processes in the limit. Both approaches involve averaging in the drift term arising from random service rates at the departures. In the second approach, this averaging is carried out directly and pathwise, thus sidestepping the assumption of driving Brownian motion used in the perturbed test function method.}, number={1}, journal={STOCHASTIC MODELS}, author={Buche, Robert and Ghosh, Arka P. and Pipiras, Vladas}, year={2012}, pages={63–108} }
 @article{buche_kushner_2005, title={Adaptive optimization of least-squares tracking algorithms: With applications to adaptive antenna arrays for randomly time-varying mobile communications systems}, volume={50}, ISSN={["1558-2523"]}, DOI={10.1109/TAC.2005.858682}, abstractNote={Adaptive antenna arrays are used for reducing the effects of interference and increasing capacity in mobile communications systems. Typical algorithms recursively compute the antenna weights that minimize the weighted error function (at discrete times kh, k=1,2,..., for a sampling interval h) /spl sigma//sub l=1//sup k//spl alpha//sup k-l/[e/sub l/(W)]/sup 2/, where e/sub l/(W) is a measure of the reception error at time lh with antenna weight vector W, and /spl alpha/<1. The forgetting factor /spl alpha/<1 allows tracking as conditions change and the minimization is used only to get the weights. The average detection error rate depends heavily on the chosen value of /spl alpha/, whose optimal value can change rapidly in time, perhaps significantly in seconds. We add another adaptive loop that tracks the optimal value of /spl alpha/ and greatly improves the operation when the environment is randomly time-varying. The additional adaptive loop is based on an approximation to a natural "gradient descent" method. The algorithm is practical and can improve the performance considerably. In terms of average detection error rates and for all of the scenarios tested, the new system tracks the optimal value of /spl alpha/ well, and always performs better (sometimes much better) than the original algorithm that uses any fixed value of /spl alpha/. Although the initial motivation arises in adaptive antennas, the method can be used to improve algorithms for tracking parameters of time-varying nonlinear systems, where similar issues are involved.}, number={11}, journal={IEEE TRANSACTIONS ON AUTOMATIC CONTROL}, author={Buche, R and Kushner, HJ}, year={2005}, month={Nov}, pages={1749–1760} }
 @article{buche_kushner_2004, title={Control of mobile communication systems with time-varying channels via stability methods}, volume={49}, ISSN={["0018-9286"]}, DOI={10.1109/TAC.2004.837590}, abstractNote={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.}, number={11}, journal={IEEE TRANSACTIONS ON AUTOMATIC CONTROL}, author={Buche, R and Kushner, HJ}, year={2004}, month={Nov}, pages={1954–1962} }