arXiv subjects

Martin I. Reiman

Publications and source records attributed to Martin I. Reiman.

5 recordsLinked to original sources

arXiv2022

Parallel server systems under an extended heavy traffic condition: A lower bound

The standard setting for studying parallel server systems (PSS) at the diffusion scale is based on the heavy traffic condition (HTC), which assumes that the underlying static allocation linear program (LP) is critical and has a unique solution. This solution determines the graph of basic activities, which identifies the set of activities (i.e., class-server pairs) that are operational. In this paper we explore the extended HTC, where the LP is merely assumed to be critical. Because multiple solutions are allowed, multiple sets of operational activities, referred to as modes, are available. Formally, the scaling limit for the control problem associated with the model is given by a so called workload control problem (WCP) in which a cost associated with a diffusion process is to be minimized by dynamically switching between these modes. Our main result is that the WCP's value constitutes an asymptotic lower bound on the cost associated with the PSS model.

math.OC↗

arXiv2021

Asymptotically Optimal Inventory Control for Assemble-to-Order Systems

We consider Assemble-to-Order (ATO) inventory systems with a general Bill of Materials and general deterministic lead times. Unsatisfied demands are always backlogged. We apply a four-step asymptotic framework to develop inventory policies for minimizing the long-run average expected total inventory cost. Our approach features a multi-stage Stochastic Program (SP) to establish a lower bound on the inventory cost and determine parameter values for inventory control. Our replenishment policy deviates from the conventional constant base stock policies to accommodate non-identical lead times. Our component allocation policy differentiates demands based on backlog costs, Bill of Materials, and component availabilities. We prove that our policy is asymptotically optimal on the diffusion scale, that is, as the longest lead time grows, the percentage difference between the average cost under our policy and its lower bound converges to zero. In developing these results, we formulate a broad Stochastic Tracking Model and prove general convergence results from which the asymptotic optimality of our policy follows as specialized corollaries.

math.OC↗

arXiv2021

Asymptotically Optimal Control of a Centralized Dynamic Matching Market with General Utilities

We consider a matching market where buyers and sellers arrive according to independent Poisson processes at the same rate and independently abandon the market if not matched after an exponential amount of time with the same mean. In this centralized market, the utility for the system manager from matching any buyer and any seller is a general random variable. We consider a sequence of systems indexed by $n$ where the arrivals in the $n^{\mathrm{th}}$ system are sped up by a factor of $n$. We analyze two families of one-parameter policies: the population threshold policy immediately matches an arriving agent to its best available mate only if the number of mates in the system is above a threshold, and the utility threshold policy matches an arriving agent to its best available mate only if the corresponding utility is above a threshold. Using a fluid analysis of the two-dimensional Markov process of buyers and sellers, we show that when the matching utility distribution is light-tailed, the population threshold policy with threshold $\frac{n}{\ln n}$ is asymptotically optimal among all policies that make matches only at agent arrival epochs. In the heavy-tailed case, we characterize the optimal threshold level for both policies. We also study the utility threshold policy in an unbalanced matching market with heavy-tailed matching utilities and find that the buyers and sellers have the same asymptotically optimal utility threshold. We derive optimal thresholds when the matching utility distribution is exponential, uniform, Pareto, and correlated Pareto. We find that as the right tail of the matching utility distribution gets heavier, the threshold level of each policy (and hence market thickness) increases, as does the magnitude by which the utility threshold policy outperforms the population threshold policy.

math.PR↗

arXiv2008

The heavy traffic limit of an unbalanced generalized processor sharing model

This work considers a server that processes $J$ classes using the generalized processor sharing discipline with base weight vector $α=(α_1,...,α_J)$ and redistribution weight vector $β=(β_1,...,β_J)$. The invariant manifold $\mathcal{M}$ of the so-called fluid limit associated with this model is shown to have the form $\mathcal{M}=\{x\in\mathbb{R}_+^J:x_j=0 for j\in\mathcal{S}\}$, where $\mathcal{S}$ is the set of strictly subcritical classes, which is identified explicitly in terms of the vectors $α$ and $β$ and the long-run average work arrival rates $γ_j$ of each class $j$. In addition, under general assumptions, it is shown that when the heavy traffic condition $\sum_{j=1}^Jγ_j=\sum_{j=1}^Jα_j$ holds, the functional central limit of the scaled unfinished work process is a reflected diffusion process that lies in $\mathcal{M}$. The reflected diffusion limit is characterized by the so-called extended Skorokhod map and may fail to be a semimartingale. This generalizes earlier results obtained for the simpler, balanced case where $γ_j=α_j$ for $j=1,...,J$, in which case $\mathcal{M}=\mathbb{R}_+^J$ and there is no state-space collapse. Standard techniques for obtaining diffusion approximations cannot be applied in the unbalanced case due to the particular structure of the GPS model. Along the way, this work also establishes a comparison principle for solutions to the extended Skorokhod map associated with this model, which may be of independent interest.

math.PR↗

arXiv2004

Scheduling a multi class queue with many exponential servers: asymptotic optimality in heavy traffic

We consider the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers. Service times and impatience clocks are exponential while arrival processes are renewal. Our cost is an expected cumulative discounted function, linear or nonlinear, of appropriately normalized performance measures. As a special case, the cost per unit time can be a function of the number of customers waiting to be served in each class, the number actually being served, the abandonment rate, the delay experienced by customers, the number of idling servers, as well as certain combinations thereof. We study the system in an asymptotic heavy-traffic regime where the number of servers n and the offered load r are simultaneously scaled up and carefully balanced: n\approx r+β\sqrtr for some scalar β. This yields an operation that enjoys the benefits of both heavy traffic (high server utilization) and light traffic (high service levels.)

math.PR↗