Otis B. Jennings

Nurse Staffing in Medical Units: A Queueing Perspective

In this paper, we present a closed queueing model to determine efficient nurse staffing policies. We explicitly model the workload experienced by s nurses within a single medical unit with n homogeneous patients as a closed M/M/s//n queueing system, where each patient alternates between requiring assistance and not. The performance of the medical unit is based on the probability of excessive delay, the relative frequency with which the delay between the onset of patient neediness and the provision of care from a nurse exceeds a given time threshold. Using new many-server asymptotic results, we find that effective staffing policies should deviate from threshold-specific nurse-to-patient ratios by factors that take into account the total number of patients present in the unit. In particular, our staffing rule significantly differs from California Bill AB 394, legislation that mandates fixed nurse-to-patient staffing ratios. Simulations show that our results are robust to delay-dependent service times, generally distributed service times, and nonhomogeneous patients, i.e., those with different acuity levels.

A time-varying call center design via lagrangian mechanics

We consider a multiserver delay queue with finite additional waiting spaces and time-varying arrival rates, where the customers waiting in the buffer may abandon. These are features that arise naturally from the study of service systems such as call centers. Moreover, we assume rewards for successful service completions and cost rates for service resources. Finally, we consider service-level agreements that constrain both the fractions of callers who abandon and the ones who are blocked. Applying the theory of Lagrangian mechanics to the fluid limit of a related Markovian service network model, we obtain near-profit-optimal staffing and provisioning schedules. The nature of this solution consists of three modes of operation. A key step in deriving this solution is combining the modified offered load approximation for loss systems with our fluid model. We use them to estimate effectively both our service-level agreement metrics and the profit for the original queuing model. Second-order profit improvements are achieved through a modified offered load version of the conventional square root safety rule.

A modified offered load approximation for nonstationary circuit switched networks

In this paper, we develop approximation methods to analyze blocking in circuit switched networks with nonstationary call arrival traffic. We formulate generalizations of the pointwise stationary and modified offered load approximations used for the nonstationary Erlang loss model or M(t)/M/c/c queue. These approximations reduce the analysis of nonstationary circuit switched networks to solving a small set of simple differential equations and using the methods for computing the steady state distributions for the stationary versions of such loss networks. We also discuss how the use of time varying arrival rates literally adds a new dimension to the class of telecommunication networks we can model. For example, we can model the behavior of alternate routing due to link‐failure, which is a feature that the classical stationary version of the model cannot capture. Our nonstationary model can also describe aspects of the dynamic calling traffic behavior arising in cellular mobile traffic. For the special case of a two‐link, three node network, we present numerical results to compare the various approximation methods to calculations of the exact blocking probabilities. We also adapt these calculations to approximate the behavior of rerouting calling traffic due to link‐failure. The results are achieved by formulating some new recursions for evaluating the steady state blocking probabilities of such networks. We also generalize these techniques to develop analogous formulas for a linear N‐node circuit switched network.

An Overloaded Multiclass FIFO Queue with Abandonments

In this paper we consider a single-server queue fed by K independent renewal arrival streams, each representing a different job class. Jobs are processed in a FIFO fashion, regardless of class. The total amount of work arriving to the system exceeds the servers capacity. That is, the nominal traffic intensity of the system is assumed to be greater than one. Jobs arriving to the system grow impatient and abandon the queue after a random amount of time if service has not yet begun. Interarrival, service, and abandonment times are assumed to be generally distributed and class specific. We approximate this system using both fluid and diffusion limits. To this end, we consider a sequence of systems indexed by n in which the arrival and service rates are proportional to n; the abandonment distribution remains fixed across the sequence. In our first main result, we show that in the limit as n tends to infinity, the virtual waiting time process converges to a limiting deterministic process. This limit may be characterized as the solution to a first-order ordinary differential equation ODE. Specific examples are then presented for which the ODE may be explicitly solved. In our second main result, we refine the deterministic fluid approximation by showing that the fluid-centered and diffusion-scaled virtual waiting time process weakly converges to an Ornstein-Uhlenbeck process whose drift and infinitesimal variance both vary over time. This process may also be solved for explicitly, thus yielding approximations to the transient as well as steady-state behavior of the virtual waiting time process.

Dimensioning Large-Scale Membership Services

Motivated by workforce planning problems in health care, professional, warranty, and repair services, we propose modeling service centers that are exclusively dedicated to fixed client constituencies as closed multiserver queueing systems, a framework we refer to as membership services. We provide fluid and diffusion approximations of the number of users within the membership who are requesting service. The approximations are obtained via many-server limit theorems, where the limiting regime assumptions of each theorem correspond to a particular staffing strategy a manager might employ. Accordingly, we propose staffing rules designed to meet a certain desired performance criterion. In particular, when the objective is to minimize the staffing size subject to a constraint on the probability of delay for a service-requesting customer, we suggest staffing rules inspired by the so-called quality-and efficiency-driven (QED), or Halfin-Whitt, limiting regime. Numerical evaluations of our proposed QED scheme indicate that, although justified for large systems, the staffing rule performs well for memberships of all sizes.

Stability of General Processing Networks

Multiclass queueing networks are effective tools for capturing the dynamics of complex manufacturing systems. For instance, one can model multiple product lines as well as processes with highly reentrant flows. In addition to their industrial relevance, multiclass queueing networks present theoretical challenges absent from their single class precursors. Not surprisingly, the research community has devoted considerable effort recently to the study of such network models, both as an academic exercise as well as for practical purposes.

Averaging Principles for a Diffusion-Scaled, Heavy-Traffic Polling Station with K Job Classes

This paper provides heavy traffic limit theorems for a polling station serving jobs from K exogenous renewal arrival streams. It is a standard result that the limiting diffusion-scaled total workload process is semimartingale reflected Brownian motion. For polling stations, however, no such limit exists in general for the diffusion-scaled, K-dimensional queue length or workload vector processes. Instead, we prove that these processes admit averaging principles, the natures of which depend on the service discipline employed at the polling station. Parameterized families of exhaustive and gated service disciplines are investigated. Each policy under consideration has K stochastic matrices associated with it—one for each job class—that describe the transition of the workload vector while a given job class is being polled. These matrices give rise to K probability vectors that are vertices of a K - 1-simplex. Loosely speaking, each point of this simplex acts as an instantaneous lifting operator, converting th...

Heavy-Traffic Limits of Queueing Networks with Polling Stations: Brownian Motion in a Wedge

We consider two serial single-server stations in heavy traffic. There are two job types: All jobs visit station 1 and then station 2. Station 1 processes jobs in an exhaustive service or gated service fashion; station 2 uses an arbitrary nonidling service discipline. Neither station incurs switchover delays. We prove two heavy-traffic limit theorems (HTLT) for the diffusion-scaled, two-dimensional total workload process: one for when the first station implements exhaustive service and the other for when gated service is employed. Our limiting processes are two-dimensional Brownian motions in a wedge, a type of reflected Brownian motion (RBM). The limiting process under exhaustive service is equal in distribution to the limiting process that one obtains when the first station performs one of two buffer priority policies.