Jiheng Zhang

Steady state approximations of limited processor sharing queues in heavy traffic

We investigate steady state properties of limited processor sharing queues in heavy traffic. Our analysis builds on previously obtained process limit theorems, and requires the interchange of steady state and heavy traffic limits, which are established by a coupling argument. The limit theorems yield explicit approximations of the steady state queue length and response time distribution in heavy traffic, of which the quality is supported by simulation experiments.

Law of Large Number Limits of Limited Processor-Sharing Queues

Motivated by applications in computer and communication systems, we consider a processor-sharing queue where the number of jobs served is not larger than K. We propose a measure-valued fluid model for this limited processor-sharing queue and show that there exists a unique associated fluid model solution. In addition, we show that this fluid model arises as the limit of a sequence of appropriately scaled processor-sharing queues.

Fluid models of many-server queues with abandonment

We study many-server queues with abandonment in which customers have general service and patience time distributions. The dynamics of the system are modeled using measure-valued processes, to keep track of the residual service and patience times of each customer. Deterministic fluid models are established to provide a first-order approximation for this model. The fluid model solution, which is proved to uniquely exist, serves as the fluid limit of the many-server queue, as the number of servers becomes large. Based on the fluid model solution, first-order approximations for various performance quantities are proposed.

The impact of slow ocean steaming on delivery reliability and fuel consumption

With a dominant volume of global transportation being conducted by sea, ocean container transport greatly impacts the global economy. Since sea vessels are drastically more fuel efficient when traveling at lower speeds, slow steaming has become a widely adopted practice to reduce bunker costs. However, this leads to a longer transportation time, which together with the unpredictability of the delay has been a big challenge. We propose a model to quantify the relationship among shipping time, bunker cost and delivery reliability. Our findings lead to a simple and implementable policy with a controlled cost and guaranteed delivery reliability.

Diffusion limits of limited processor sharing queues

We consider a processor sharing queue where the number of jobs served at any time is limited to K, with the excess jobs waiting in a buffer. We use random counting measures on the positive axis to model this system. The limit of this measure-valued process is obtained under diffusion scaling and heavy traffic conditions. As a consequence, the limit of the system size process is proved to be a piece-wise reflected Brownian motion.

Refined Models for Efficiency-Driven Queues with Applications to Delay Announcements and Staffing.

Data has revealed a noticeable impact of delay-time-related information on phone-customers; for example and somewhat surprisingly, delay announcements can abruptly increase the likelihood to abandon (hang up). Our starting point is that the latter phenomena can be used to support the control of queue lengths and delays. We do so by timing the announcements appropriately and determining the staffing levels accordingly. To this end, we model a service system as an overloaded GI/M/s + GI queue, in which we seek to minimize the number of servers, s, subject to quality-of-service constraints (e.g., fraction abandoning), while accounting for the instantaneous (hence discontinuous) impact of an announcement on the distribution (hazard rate) of customer patience. For tractability, our analysis is asymptotic as s increases indefinitely, and it is naturally efficiency-driven (namely the servers are highly busy, and hence essentially all customers are delayed in queue prior to service). This requires one to go beyon...

Routing and Staffing in Customer Service Chat Systems with Impatient Customers

We consider customer service chat CSC systems where customers can receive real time service from agents using an instant messaging IM application over the Internet. A unique feature of these systems is that agents can serve multiple customers simultaneously. The number of customers that an agent is serving determines the rate at which each customer assigned to that agent receives service. We consider the staffing problem in CSC systems with impatient customers where the objective is to minimize the number of agents while providing a certain service level. The service level is measured in terms of the proportion of customers who abandon the system in the long run. First we propose effective routing policies based on a static planning LP, both for the cases when the arrival rate is observable and for when the rate is unobservable. We show that these routing policies minimize the proportion of abandoning customers in the long run asymptotically for large systems. We also prove that the staffing solution obtained from a staffing LP, when used with the proposed routing policies, is asymptotically optimal. We illustrate the effectiveness of our solution procedure in systems with small to large sizes via numerical and simulation experiments.

Scaling and modeling of call center arrivals

The Poisson process has been an integral part of many models for the arrival process to a telephone call centers. However, several publications in recent years suggest the presence of a significant “overdisperson” relative to the Poisson process in real-life call center arrival data. In this paper, we study the overdispersion in the context of “heavy traffic” and identify a critical factor that characterizes the stochastic variability of the arrivals to their averages. We refer to such a factor as the scaling parameter and it potentially has a profound impact on the design of staffing rules. We propose an new model to capture the scaling parameter in this paper.

Convergence to equilibrium states for fluid models of many-server queues with abandonment

Fluid models, in particular their equilibrium states, have become an important tool for the study of many-server queues with general service and patience time distributions. However, it remains an open question whether the solution to a fluid model converges to the equilibrium state and under what condition. We show in this paper that the convergence holds under some conditions. Our method builds on the framework of measure-valued processes, which keeps track of the remaining patience and service times.

A Unified Approach to Diffusion Analysis of Queues with General Patience-Time Distributions

We propose a unified approach to establishing diffusion approximations for queues with impatient customers within a general framework of scaling customer patience time. The approach consists of two steps. The first step is to show that the diffusion-scaled abandonment process is asymptotically close to a function of the diffusion-scaled queue length process under appropriate conditions. The second step is to construct a continuous mapping not only to characterize the system dynamics using the system primitives, but also to help verify the conditions needed in the first step. The diffusion approximations can then be obtained by applying the continuous mapping theorem. The approach has two advantages: (i) it provides a unified procedure to establish the diffusion approximations regardless of the structure of the queueing model or the type of patience-time scaling; and (ii) it makes the diffusion analysis of queues with customer abandonment essentially the same as the diffusion analysis of queues without customer abandonment. We demonstrate the application of this approach via the single-server system with Markov-modulated service speeds in the traditional heavy-traffic regime and the many-server system in the Halfin-Whitt regime and the nondegenerate slowdown regime.