J. G. Dai

The throughput of data switches with and without speedup

In this paper we use fluid model techniques to establish two results concerning the throughput of data switches. For an input-queued switch (with no speedup) we show that a maximum weight algorithm for connecting inputs and outputs delivers a throughput of 100%, and for combined input- and output-queued switches that run at a speedup of 2 we show that any maximal matching algorithm delivers a throughput of 100%. The only assumptions on the input traffic are that it satisfies the strong law of large numbers and that it does not oversubscribe any input or any output.

Stability and convergence of moments for multiclass queueing networks via fluid limit models

The subject of this paper is open multiclass queueing networks, which are common models of communication networks, and complex manufacturing systems such as wafer fabrication facilities. We provide sufficient conditions for the existence of bounds on long-run average moments of the queue lengths at the various stations, and we bound the rate of convergence of the mean queue length to its steady-state value. Our work provides a solid foundation for performance analysis either by analytical methods or by simulation. These results are applied to several examples including re-entrant lines, generalized Jackson networks, and a general polling model as found in computer networks applications. >

Stability and instability of fluid models for reentrant lines

Reentrant lines can be used to model complex manufacturing systems such as wafer fabrication facilities. As the first step to the optimal or near-optimal scheduling of such lines, we investigate their stability. In light of a recent theorem of Dai Dai, J. G. 1995. On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid models. Ann. Appl. Probab.5 49--77. which states that a scheduling policy is stable if the corresponding fluid model is stable, we study the stability and instability of fluid models. To do this we utilize piecewise linear Lyapunov functions. We establish stability of First-Buffer-First-Served FBFS and Last-Buffer-First-Served LBFS disciplines in all reentrant lines, and of all work-conserving disciplines in any three buffer reentrant lines. For the four buffer network of Lu and Kumar we characterize the stability region of the Lu and Kumar policy, and show that it is also the global stability region for this network. We also study stability and instability of Kelly-type networks. In particular, we show that not all work-conserving policies are stable for such networks; however, all work-conserving policies are stable in a ring network.

Maximum Pressure Policies in Stochastic Processing Networks

Complex systems like semiconductor wafer fabrication facilities (fabs), networks of data switches, and large-scale call centers all demand efficient resource allocation. Deterministic models like linear programs (LP) have been used for capacity planning at both the design and expansion stages of such systems. LP-based planning is critical in setting a medium range or long-term goal for many systems, but it does not translate into a day-to-day operational policy that must deal with discreteness of jobs and the randomness of the processing environment.A stochastic processing network, advanced by J. Michael Harrison (2000, 2002, 2003), is a system that takes inputs of materials of various kinds and uses various processing resources to produce outputs of materials of various kinds. Such a network provides a powerful abstraction of a wide range of real-world systems. It provides high-fidelity stochastic models in diverse economic sectors including manufacturing, service, and information technology.We propose a family of maximum pressure service policies for dynamically allocating service capacities in a stochastic processing network. Under a mild assumption on network structure, we prove that a network operating under a maximum pressure policy achieves maximum throughput predicted by LPs. These policies are semilocal in the sense that each server makes its decision based on the buffer content in its serviceable buffers and their immediately downstream buffers. In particular, their implementation does not use arrival rate information, which is difficult to collect in many applications. We also identify a class of networks for which the nonpreemptive, non-processor-splitting version of a maximum pressure policy is still throughput optimal. Applications to queueing networks with alternate routes and networks of data switches are presented.

Existence and Uniqueness of Semimartingale Reflecting Brownian Motions in Convex Polyhedrons

We consider the problem of existence and uniqueness of semimartingale reflecting Brownian motions (SRBM’s) in convex polyhedrons. Loosely speaking, such a process has a semimartingale decomposition such that in the interior of the polyhedron the process behaves like a Brownian motion with a constant drift and covariance matrix, and at each of the

On Measuring Supplier Performance Under Vendor-Managed-Inventory Programs in Capacitated Supply Chains

(d - 1)

Asymptotic optimality of maximum pressure policies in stochastic processing networks.

-dimensional faces that form the boundary of the polyhedron, the bounded variation part of the process increases in a given direction (constant for any particular face), so as to confine the process to the polyhedron. For historical reasons, this “pushing” at the boundary is called instantaneous reflection. For simple convex polyhedrons, we give a necessary and sufficient condition on the geometric data for the existence and uniqueness of an SRBM. For nonsimple convex polyhedrons, our condition is shown to be sufficient. It is an open question as to whether our condition is also necessary in the nonsimple case. From the uniqueness, it follows that an SRB...

Models and Insights for Hospital Inpatient Operations: Time-Dependent ED Boarding Time

As widely accepted performance measures in supply chain management practice, frequency-based service levels such as fill rate and stockout rate are often considered in supply contracts under vendor-managed-inventory (VMI) programs. Using a decentralized two-party capacitated supply chain model consisting of one manufacturer and one supplier in a VMI environment, we demonstrate that suppliers service level is in general insufficient for the manufacturer to warrant the desired service level at the customer end. The method by which the supplier achieves her service level to the manufacturer also affects customer service level.By developing bounds on the customer service level, we show that the expected backorders at the supplier should also be taken into account. We suggest a supply contract that offers a menu of different combinations of suppliers service level and expected backorders according to a linear function. Under this contract, the manufacturer can control the end customer service regardless of how the supplier manages her inventory. The supplier has complete flexibility on which combination of the two quantities on the menu to choose according to her own cost functions. Because it does not require any detailed information on suppliers operational characteristics nor her costs, this kind of contract is expected to be easily implementable. In addition, we derive an estimate of the customer service level in terms of the new measures.Our findings have direct implications to supply chain metrics in general: The local service levels are insufficient measures to guarantee the system wide performance. Alternative local measures and/or coordination mechanisms should be employed to achieve desired system performance. Our analysis illustrates a possible way to explore such alternative measures.

Dynamic Control of N-Systems with Many Servers: Asymptotic Optimality of a Static Priority Policy in Heavy Traffic

.We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each quadratic holding cost structure, there is a maximum pressure policy that asymptotically minimizes the holding cost. A key to the optimality proofs is to prove a state space collapse result and a heavy traffic limit theorem for the network processes under a maximum pressure policy. We extend a framework of Bramson [Queueing Systems Theory Appl. 30 (1998) 89–148] and Williams [Queueing Systems Theory Appl. 30 (1998b) 5–25] from the multiclass queueing network setting to the stochastic processing network setting to prove the state space collapse result and the heavy traffic limit theorem. The extension can be adapted to other studies of stochastic processing networks. 1. Introduction. This paper is a continuation of Dai and Lin (2005), in which maximum pressure policies are shown to be throughput optimal for a class of stochastic processing networks. Throughput optimality is an important, first-order objective for many networks, but it ignores some key secondary performance measures like queueing delays experienced by jobs in these networks. In this paper we show that maximum pressure policies enjoy additional optimality properties; they are asymptotically optimal in minimizing a certain workload or holding cost of a stochastic processing network. Stochastic processing networks have been introduced in a series of three papers by Harrison (2000, 2002, 2003). In Dai and Lin (2005) and this paper we consider a special class of Harrison’s model. This class of stochastic processing networks is much more general than multiclass queueing networks that have been a subject of intensive study in the last 20 years; see, for example, Harrison (1988), Williams

Many-server diffusion limits for G/Ph/n+GI queues

One key factor contributing to emergency department (ED) overcrowding is prolonged waiting time for admission to inpatient wards, also known as ED boarding time. To gain insights into reducing this waiting time, we study operations in the inpatient wards and their interface with the ED. We focus on understanding the effect of inpatient discharge policies and other operational policies on the time-of-day waiting time performance, such as the fraction of patients waiting longer than six hours in the ED before being admitted. Based on an empirical study at a Singaporean hospital, we propose a novel stochastic processing network with the following characteristics to model inpatient operations: (1) A patient’s service time in the inpatient wards depends on that patient’s admission and discharge times and length of stay. The service times capture a two-time-scale phenomenon and are not independent and identically distributed. (2) Pre- and post-allocation delays model the extra amount of waiting caused by secondary bottlenecks other than bed unavailability, such as nurse shortage. (3) Patients waiting for a bed can overflow to a nonprimary ward when the waiting time reaches a threshold, where the threshold is time dependent. We show, via simulation studies, that our model is able to capture the inpatient flow dynamics at hourly resolution and can evaluate the impact of operational policies on both the daily and time-of-day waiting time performance. In particular, our model predicts that implementing a hypothetical policy can eliminate excessive waiting for those patients who request beds in mornings. This policy incorporates the following components: a discharge distribution with the first discharge peak between 8 a.m. and 9 a.m. and 26% of patients discharging before noon, and constant-mean allocation delays throughout the day. The insights gained from our model can help hospital managers to choose among different policies to implement depending on the choice of objective, such as to reduce the peak waiting in the morning or to reduce daily waiting time statistics. This paper was accepted by Assaf Zeevi, stochastic models and simulation .