Martin I. Reiman

Strong approximations for Markovian service networks

Inspired by service systems such as telephone call centers, we develop limit theorems for a large class of stochastic service network models. They are a special family of nonstationary Markov processes where parameters like arrival and service rates, routing topologies for the network, and the number of servers at a given node are all functions of time as well as the current state of the system. Included in our modeling framework are networks of Mt/Mt/nt queues with abandonment and retrials. The asymptotic limiting regime that we explore for these networks has a natural interpretation of scaling up the number of servers in response to a similar scaling up of the arrival rate for the customers. The individual service rates, however, are not scaled. We employ the theory of strong approximations to obtain functional strong laws of large numbers and functional central limit theorems for these networks. This gives us a tractable set of network fluid and diffusion approximations. A common theme for service network models with features like many servers, priorities, or abandonment is “non-smooth” state dependence that has not been covered systematically by previous work. We prove our central limit theorems in the presence of this non-smoothness by using a new notion of derivative.

A critically loaded multirate link with trunk reservation

We consider a loss system model of interest in telecommunications. There is a single service facility with N servers and no waiting room. There are K types of customers, with type ί customers requiring Aί servers simultaneously. Arrival processes are Poisson and service times are exponential. An arriving type ί customer is accepted only if there are Rί(⩾Aί ) idle servers. We examine the asymptotic behavior of the above system in the regime known as critical loading where both N and the offered load are large and almost equal. We also assume that R1,..., RK-1 remain bounded, while RKN←∞ and RKN/√N ← 0 as N ← ∞. Our main result is that the K dimensional “queue length” process converges, under the appropriate normalization, to a particular K dimensional diffusion. We show that a related system with preemption has the same limit process. For the associated optimization problem where accepted customers pay, we show that our trunk reservation policy is asymptotically optimal when the parameters satisfy a certain relation.

Asymptotically Optimal Inventory Control for Assemble-to-Order Systems with Identical Lead Times

Optimizing multiproduct assemble-to-order (ATO) inventory systems is a long-standing difficult problem. We consider ATO systems with identical component lead times and a general “bill of materials.” We use a related two-stage stochastic program (SP) to set a lower bound on the average inventory cost and develop inventory control policies for the dynamic ATO system using this SP. We apply the first-stage SP optimal solution to specify a base-stock replenishment policy, and the second-stage SP recourse linear program to make allocation decisions. We prove that our policies are asymptotically optimal on the diffusion scale, so the percentage gap between the average cost from its lower bound diminishes to zero as the lead time grows.

A Brownian control problem for a simple queueing system in the Halfin–Whitt regime

We consider a formal diffusion limit for a control problem of a multi-type multi-server queueing system, in the regime proposed by Halfin and Whitt. This takes the form of a control problem where the dynamics are driven by a Brownian motion. In one dimension, a pathwise minimum is obtained and is characterized as the solution to a stochastic differential equation. The pathwise solution to a special multi-dimensional problem (corresponding to a multi-type system) follows.

Asymptotically optimal dynamic pricing for network revenue management

A dynamic pricing problem that arises in a revenue management context is considered, involving several resources and several demand classes, each of which uses a particular subset of the resources. The arrival rates of demand are determined by prices, which can be dynamically controlled. When a demand arrives, it pays the posted price for its class and consumes a quantity of each resource commensurate with its class. The time horizon is finite: at time T the demands cease, and a terminal reward (possibly negative) is received that depends on the unsold capacity of each resource. The problem is to choose a dynamic pricing policy to maximize the expected total reward. When viewed in diffusion scale, the problem gives rise to a diffusion control problem whose solution is a Brownian bridge on the time interval [0, T]. We prove diffusion-scale asymptotic optimality of a dynamic pricing policy that mimics the behavior of the Brownian bridge. The ‘target point’ of the Brownian bridge is obtained as the solution ...

On the use of independent base-stock policies in assemble-to-order inventory systems with nonidentical lead times

We consider the use of Independent Base Stock (IBS) replenishment policies in Assemble-to-Order (ATO) inventory systems. These policies are appealingly simple and widely used, but generally suboptimal for systems with non-identical lead times. We present an IBS policy and prove that its loss of optimality is limited by the ratio of the longest lead time to the shortest one. Our results suggest that IBS policies can work well for systems where differences between lead times are dominated by their lengths.