Kevin Ross

Scheduling bursts in time-domain wavelength interleaved networks

A time-domain wavelength interleaved network (TWIN) (Widjaja, I. et al., IEEE Commun. Mag., vol.41, 2003) is an optical network with an ultrafast tunable laser and a fixed receiver at each node. We consider the problem of scheduling bursts of data in a TWIN. Due to the high data rates employed on the optical links, the burst transmissions typically last for very short times compared with the round trip propagation times between source-destination pairs. A good schedule should ensure that: 1) there are no transmit/receive conflicts; 2) propagation delays are observed; 3) throughput is maximized (schedule length is minimized). We formulate the scheduling problem with periodic demand as a generalization of the well-known crossbar switch scheduling. We prove that even in the presence of propagation delays, there exist a class of computationally viable scheduling algorithms which asymptotically achieve the maximum throughput obtainable without propagation delays. We also show that any schedule can be rearranged to achieve a factor-two approximation of the maximum throughput even without asymptotic limits. However, the delay/throughput performance of these schedules is limited in practice. We consequently propose a scheduling algorithm that exhibits near optimal (on average within /spl sim/7% of optimum) delay/throughput performance in realistic network examples.

Convergent Numerical Scheme for Singular Stochastic Control with State Constraints in a Portfolio Selection Problem

We consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs. Numerical approximations for the value function using the Markov chain approximation method of Kushner and Dupuis are studied. The main result of the paper shows that the value function of the Markov decision problem (MDP) corresponding to the approximating controlled Markov chain converges to that of the original stochastic control problem as various parameters in the approximation approach suitable limits. All our convergence arguments are probabilistic; the main assumption that we make is that the value function be finite and continuous. In particular, uniqueness of the solutions of the associated HJB equations is neither needed nor available (in the generality under which the problem is considered). Specific features of the problem that make the convergence analysis nontrivial include unboundedness of the state and control space and the cost function; degeneracies in the dynamics; mixed boundary (Dirichlet-Neumann) conditions; and presence of both singular and absolutely continuous controls in the dynamics. Finally, schemes for computing the value function and optimal control policies for the MDP are presented and illustrated with a numerical study.

Dynamic scheduling of optical data bursts in time-domain wavelength interleaved networks

We consider the problem of scheduling bursts of data in an optical network with an ultra-fast tunable laser and a fixed receiver at each node. In (K. Ross et al, Technical Report SU NETLAB-2002-12/1, Eng. Lib., Stanford Uni., Stanford, CA (2002)) we considered the static scheduling problem of meeting demand in the minimal time. Here we substantially extend these results to the case of online, dynamic scheduling. Due to the high data rates employed on the optical links, the burst transmissions typically last for very short times compared to the round trip propagation times between source-destination pairs. A good schedule ensures that (i) there are no transmit/receive conflicts, (ii) throughput is maximized, and (iii) propagation delays are observed. We formulate the scheduling problem as a generalization of the well-known crossbar switch scheduling problem. We show that the algorithms presented in the previous work can be implemented in dynamic form to give 100% throughput. Further, we show that one of the more intuitive solutions does not lead to maximal throughput. In particular, we show advantages of adaptive batch sizes rather than fixed batch sizes for both throughput and performance.

Dynamic quality of service control in packet switch scheduling

Recent research in packet switch scheduling algorithms has moved beyond throughput maximization to quality of service (QoS) control. Several classes of algorithms have been shown to achieve maximal throughput under certain system conditions. Between classes and within each class, QoS performance varies based on arrival traffic and properties of the scheduling algorithm being utilized. Here we compare two classes of throughput-maximizing algorithms and their performance with respect to buffer sizes. These classes are randomized algorithms, which can be characterized as offline algorithms, and projective cone scheduling algorithms, which are online since they respond to the current workload in the system. In each class, parameters can be fine-tuned to reflect the priorities of individual switch ports. We show how the online algorithms lead to significantly better quality of service performance.

Existence of Optimal Controls for Singular Control Problems with State Constraints

1. Introduction. This paper is concerned with a class of singular control problems with state constraints. The presence of state constraints, a key feature of the problem, refers to the requirement that the controlled diffusion process take values in a closed convex cone at all times [see (3)]. We consider an infinite horizon discounted cost of the form (4). The main objective of the paper is to establish the existence of an optimal control. Singular control is a well-studied but rather challenging class of stochastic control problems. We refer the reader to [7], especially the sections at the end of each chapter, for a thorough survey of the literature. Classical compactness arguments that are used for establishing the existence of optimal controls for problems with absolutely continuous control terms (cf. [8]) do not naturally extend to singular control problems. For one-dimensional models, one can typically establish existence constructively, by characterizing an optimally controlled process as a reflected diffusion (cf. [2, 3, 15]). In higher dimensions, one approach is to study the regularity of solutions of variational inequalities associated with singular control problems and the smoothness of the corresponding free boundary. Such smoothness results are the starting points in the characterization of the optimally controlled process as a constrained diffusion with reflection at the free boundary. Excepting specific models (cf. [30, 31]), this approach encounters substantial difficulties, even for linear dynamics (cf. [32]); a key difficulty is that little is known about the regularity of the free boundary in higher dimensions. Alternative approaches for establishing the existence of optimal controls based on compactness arguments are developed in [12, 17, 25]. The first of these papers considers linear dynamics, while the last two consider models with nonlinear coefficients. In all

Optimizing quality of service in packet switch scheduling

Recently, extensive analytic research into packet scheduling in crossbar switches has yielded interesting throughput maximizing algorithms. Surprisingly, however, quality of service (QoS) performance associated with these algorithms has only been approximated through simulation. We present here certain randomized algorithms with analytic QoS. These are simple to implement and possess closed form expressions for various performance measures. By fine tuning particular parameters of these algorithms, one can vary the QoS associated with the individual ports as desired. This allows cost and utility optimization, a feature which was not feasible under previously studied packet scheduling algorithms.

OPTIMAL STOPPING AND FREE BOUNDARY CHARACTERIZATIONS FOR SOME BROWNIAN CONTROL PROBLEMS

A singular stochastic control problem with state constraints in two-dimensions is studied. We show that the value function is C 1 and its directional derivatives are the value functions of certain optimal stopping problems. Guided by the optimal stopping problem we then introduce the associated no-action region and the free boundary and show that, under appropriate conditions, an optimally controlled process is a Brownian motion in the no-action region with re∞ection at the free boundary. This proves a conjecture of Martins, Shreve and Soner [21] on the form of an optimal control for this class of singular control problems. An important issue in our analysis is that the running cost is Lipschitz but not C 1 . This lack of smoothness is one of the key obstacles in establishing regularity of the free boundary. We show that the free boundary is Lipschitz and if the Lipschitz constant is su‐ciently small, a certain oblique derivative problem on the no-action region admits a unique viscosity solution. This uniqueness result is key in characterizing an optimally controlled process as a re∞ected difiusion in the no-action region.

Characterization of the Differentiation and Leptin Secretion Profile of Adult Stem Cells on Patterned Polylactide Films

Several issues need to be better understood before breast tissue engineering becomes a clinically viable option. One of the most important aspects is the interaction between cells and the microtopography of the implant surface. The aim of this study was to evaluate the efficacy of D1 cells, multipotent mouse bone marrow stromal precursors, in differentiating to adipocytes and to characterize their metabolic activity (lactic acid released and glucose consumed), leptin secretion and lipid production when cultured on patterned poly(L-lactide) (PLLA) films. It was determined that, by appropriate stimulation, the D1 cells displayed morphological characteristics of adipocytes and produced lipid. The results showed that a patterned surface did affect the rate of lipid production. Polynomial models were proposed to predict the amount of leptin secreted by the cells over a period of time.