We show that if performance measures in a stochastic scheduling problem satisfy a set of so-called partial conservation laws (PCL), which extend previously studied generalized conservation laws (GCL), then the problem is solved optimally by a priority-index policy for an appropriate range of linear performance objectives, where the optimal indices are computed by a one-pass adaptive-greedy algorithm, based on Klimov's. We further apply this framework to investigate the indexability property of restless bandits introduced by Whittle, obtaining the following results: (1) we identify a class of restless bandits (PCL-indexable) which are indexable

We consider a single class open queueing network, also known as a generalized Jackson network (GJN). A classical result in heavytraffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic intensity approaches unity. However, barring simple instances, it is still not known whether the stationary distribution of RBM provides a valid approximation for the steady-state of the original network. In this paper we resolve this open problem by proving that the re-scaled stationary distribution of the GJN converges to the stationary distribution of the RBM, thus validating a so-called “interchange-of-limits” for this class of networks. Our method of proof involves a combination of Lyapunov function techniques, strong approximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN.

We study the distribution of steady-state queue lengths in multiclass queueing networks under a stable policy. We propose a general methodology based on Lyapunov functions, for the performance analysis of infinite state Markov chains and apply it specifically to Markovian multiclass queueing networks. We establish a deeper connection between stability and performance of such networks by showing that if there exist linear and piecewise linear Lyapunov functions that show stability, then these Lyapunov functions can be used to establish geometric type lower and upper bounds on the tail probabilities, and thus bounds on the expectation of the queue lengths. As an example of our results, for a re-entrant line queueing network with two processing stations operating under a work-conserving policy we showthat E[L] =O 1 (1; ) 2 � where L is the total number ofcustomers in the system, and is the maximal actual or virtual traffic intensity inthenetwork. In a Markovian setting, this extends a recent result by Daiand Vande Vate, which states that a re-entrant line queueing network with two stations is globally stable if < 1: We also present several results on the

This paper considers a multi-product, single-server production system where both setup times and costs are incurred whenever the server changes product. The system is make-to-order with a per unit backlogging cost. The objective is to minimize the long-run average cost per unit time. Using a fluid model, we provide a closed-form lower bound on system performance. This bound is also shown to provide a lower bound for stochastic systems when scheduling is static, but is only an approximation when scheduling is dynamic. Heavy-traffic analysis yields a refined bound that includes second-moment terms. The fluid bound suggests both dynamic and static scheduling In this paper we consider a production environment where a number of different products are produced on a single machine and setup activities are necessary when switches of product type are made. These setup activities require both time and cost that depend on the specific product type. Throughout the paper we assume that the setups do not depend on the previous product produced

Our focus is on a class of Markov chains which have a polyhedral translation invariance property for the transition probabilities. This class can be used to model several applications of interest which feature complexities not found in usual models of queueing networks, for example failure prone manufacturing systems which are operating under hedging point policies, or enhanced wafer fab models featuring batch tools and setups or affine index policies. We present a new family of performance bounds which is more powerful both in expressive capability as well as the quality of the bounds than some earlier approaches.

We introduce penalty function based admission control policies to approximately maximize the expected reward rate in a loss network. These control policies are easy to implement and perform well both in the transient period as well as in steady state. A major advantage of the penalty approach is that it avoids solving the associated dynamic program. However, a disadvantage of this approach is that it requires the capacity requested by individual requests to be su#ciently small compared to total available capacity. We first solve a related deterministic linear program (LP) and then translate an optimal solution of the LP into an admission control policy for the loss network via an exponential penalty function. We show that the penalty policy is a target-tracking policy -- it performs well because the optimal solution of the LP is a good target ! We demonstrate that the penalty approach can be extended to track arbitrarily defined target sets. Results from preliminary simulation studies are included.

Our model is a constrained homogeneous random walk in + . The convergence to stationarity for such a random walk can often be checked by constructing a Lyapunov function. The same Lyapunov function can also be used for computing approximately the stationary distribution of this random walk, using methods developed by Meyn and Tweedie in [34]. In this paper we show that, for stationary homogeneous random walks, computing the stationary probability exactly is an undecidable problem, even if a Lyapunov function is available. That is no algorithm can exist to achieve this task. We then prove that computing large deviation rates for this model is also an undecidable problem. We extend these results to a certain type of queueing systems. The implication of these results is that no useful formulas for computing stationary probabilities and large deviations rates can exist in these systems.

Motivated by one of the leading intermodal logistics suppliers in the United States, we consider an internal pricing mechanism for managing a fleet of service units (shipping containers) flowing in a closed queueing network. Nodes represent geographic locations and arcs represent travel between them. Customer requests for arcs arrive over time, and the problem is to find an accept/reject policy that maximizes the long-run time average reward rate from accepting requests. We formulate the problem as a semi-Markov decision process, and give a simple linear program that provides an upper bound on the optimal reward rate. Using Palm calculus, we derive a nonlinear program that approximately captures queueing and stockout effects on the network. We show that this nonlinear program provides a new, alternative derivation of an upper bound due to Harel (1988) for the Erlang loss probability. We also interpret the optimality conditions as optimizing the behaviour of decentralized agents assigned to units. Using optimal solution information in an auxiliary pricing problem, we construct an easily computable functional approximation to the dynamic programming value function. The resulting policy is computationally efficient and produces superior economic performance as compared with other policies and the upper bound. Furthermore, it provides a methodologically grounded solution to the firm’s internal pricing problem. 1

Performance analysis of queueing networks is one of the most challenging areas of queueing theory. Barring very specialized models such as product-form type queueing networks, there exist very few results which provide provable non-asymptotic upper and lower bounds on key performance measures. In this paper we propose a new performance analysis method, which is based on the robust optimization. The basic premise of our approach is as follows: rather than assuming that the stochastic primitives of a queueing model satisfy certain probability laws, such as, for example, i.i.d. interarrival and service times distributions, we assume that the underlying primitives are deterministic and satisfy the implications of such probability laws. These implications take the form of simple linear constraints, namely, those motivated by the Law of the Iterated Logarithm (LIL). Using this approach we are able to obtain performance bounds on some key performance measures. Furthermore, these performance bounds imply similar bounds in the underlying stochastic queueing models.

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