A Numerical Approach to Stability of Multiclass Queueing Networks

The Multi-class Queueing Network (McQN) arises as a natural multi-class extension of the traditional (single-class) Jackson network. In a single-class network subcriticality (i.e. subunitary nominal workload at every station) entails stability, but this is no longer sufficient when jobs/customers of different classes (i.e. with different service requirements and/or routing scheme) visit the same server; therefore, analytical conditions for stability of McQNs are lacking, in general. In this note we design a numerical (simulation-based) method for determining the stability region of a McQN, in terms of arrival rate(s). Our method exploits certain (stochastic) monotonicity properties enjoyed by the associated Markovian queue-configuration process. Stochastic monotonicity is a quite common feature of queueing models and can be easily established in the single-class framework (Jackson networks); recently, also for a wide class of McQNs, including first-come-first-serve (FCFS) networks, monotonicity properties have been established. Here, we provide a minimal set of conditions under which the method performs correctly. Eventually, we illustrate the use of our numerical method by presenting a set of numerical experiments, covering both single and multi-class networks

Restless bandit marginal productivity indices II: multiproject case and scheduling a multiclass make-to-order/-stock M/G/1 queue

This paper develops a framework based on convex optimization and economic ideas to formulate and solve approximately a rich class of dynamic and stochastic resource allocation problems, fitting in a generic discrete-state multi-project restless bandit problem (RBP). It draws on the single-project framework in the author's companion paper "Restless bandit marginal productivity indices I: Single-project case and optimal control of a make-to-stock M/G/1 queue", based on characterization of a project's marginal productivity index (MPI). Our framework significantly expands the scope of Whittle (1988)'s seminal approach to the RBP. Contributions include: (i) Formulation of a generic multi-project RBP, and algorithmic solution via single-project MPIs of a relaxed problem, giving a lower bound on optimal cost performance; (ii) a heuristic MPI-based hedging point and index policy; (iii) application of the MPI policy and bound to the problem of dynamic scheduling for a multiclass combined MTO/MTS M/G/1 queue with convex backorder and stock holding cost rates, under the LRA criterion; and (iv) results of a computational study on the MPI bound and policy, showing the latter's near-optimality across the cases investigated

RESTLESS BANDIT MARGINAL PRODUCTIVITY INDICES II: MULTIPROJECT CASE AND SCHEDULING A MULTICLASS MAKE-TO-ORDER/-STOCK M/G/1 QUEUE

This paper develops a framework based on convex optimization and economic ideas to formulate and solve approximately a rich class of dynamic and stochastic resource allocation problems, fitting in a generic discrete-state multi-project restless bandit problem (RBP). It draws on the single-project framework in the author´s companion paper “Restless bandit marginal productivity indices I: Single-project case and optimal control of a make-to-stock M/G/1 queue”, based on characterization of a project´s marginal productivity index (MPI). Our framework significantly expands the scope of Whittle (1988)´s seminal approach to the RBP. Contributions include: (i) Formulation of a generic multi-project RBP, and algorithmic solution via single-project MPIs of a relaxed problem, giving a lower bound on optimal cost performance; (ii) a heuristic MPI-based hedging point and index policy; (iii) application of the MPI policy and bound to the problem of dynamic scheduling for a multiclass combined MTO/MTS M/G/1 queue with convex backorder and stock holding cost rates, under the LRA criterion; and (iv) results of a computational study on the MPI bound and policy, showing the latter´s near-optimality across the cases investigated.

A Hierarchical Approach to Robust Stability of Multiclass Queueing Networks

We re-visit the global - relative to control policies - stability of multiclass queueing networks. In these, as is known, it is generally insufficient that the nominal utilization at each server is below 100%. Certain policies, although work conserving, may destabilize a network that satisfies the nominal load conditions; additional conditions on the primitives are needed for global stability. The global-stability region was fully characterized for two-station networks in [13], but a general framework for networks with more than two stations remains elusive. In this paper, we offer progress on this front by considering a subset of non-idling control policies, namely queue-ratio (QR) policies. These include as special cases also all static-priority policies. With this restriction, we are able to introduce a complete framework that applies to networks of any size. Our framework breaks the analysis of QR-global stability into (i) global state-space collapse and (ii) global stability of the Skorohod problem (SP) representing the fluid workload. Sufficient conditions for both are specified in terms of simple optimization problems. We use these optimization problems to prove that the family of QR policies satisfies a weak form of convexity relative to policies. A direct implication of this convexity is that: if the SP is stable for all static-priority policies (the "extreme" QR policies), then it is also stable under any QR policy. While QR-global stability is weaker than global stability, our framework recovers necessary and sufficient conditions for global stability in specific networks

Mathematical programming algorithms for equilibrium road traffic assignment

The equilibrium approach to representing interactions between the supply and demand sides of traffic assignment has been used widely in the estimation of traffic flows on road networks. Although this approach is quite reasonable, there is a considerable gap between the observed and modelled values of cost and flow. This gap can be reduced by relaxing some of the restrictive assumptions behind the models used in order to enhance their realism. This study investigates the solutions of various advanced road traffic assignment models. Priority and signal controlled junctions are modelled in traffic assignment in order to enhance the realism of junction analysis. A multiclass assignment is modelled to represent different groups of users. These problems are known to be non-separable because traffic cannot be segmented in such a way that the costs incurred by any one segment vary only with the flow within that segment. Existence, uniqueness and stability properties of solutions to these problems are investigated. These analyses are important to know the reliability and repeatability of any solutions that are calculated. Analyses of these properties lead to some guidelines for using these detailed models. A number of new solution algorithms are developed to solve the resulting traffic assignment problems. These algorithms belong to the general category of simplicial decomposition which solves the problem by dividing it into two subproblems: a linear and a master subproblem which are solved alternately. One of the advantages of these algorithms is that they operate in a lower dimensional space than that of original feasible region and hence allow large-scale problems to be solved with improved accuracy and speed of convergence. These improved algorithms give many choices to the traffic management studies. Two substantial networks have been used to compare the performance of new algorithms on the various models developed. They have performed favourably by comparison with existing algorithms. A small example network has been used to investigate existence, uniqueness and stability properties using the models. In a priority controlled model, a unique stable solution has been obtained using the model whilst in a signal controlled model, multiple and unstable solutions have been obtained. In a multiclass model, a unique solution has been obtained in terms of the total class flow whilst multiple solutions have been obtained in terms of each class flow. These results correspond well to the theoretical analyses of these models, which has shown to have indeterminate behaviour and by the nature of these models assumed, the degree of non-separability is ordered according to priority controlled, multiclass and signal controlled models