Browsing by Author "Jennings, Otis B"

Many types of services are provided using some equipment or machines, e.g. transportation systems using vehicles. Designs of these systems require capacity decisions, e.g., the number of vehicles. In my dissertation, I use many-server and conventional heavy-traffic limit theory to derive asymptotically optimal capacity decisions, giving the desired level of delay and availability performance with minimum investment. The results provide near-optimal solutions and insights to otherwise analytically intractable problems.

The dissertation will comprise two essays. In the first essay, &ldquoAsymptotic Analysis of Delay-based Performance Metrics and Optimal Capacity Decisions for the Machine Repair Problem with Spares,&rdquo I study the M/M/R machine repair problem with spares. This system can be represented by a closed queuing network. Applications include fleet vehicles' repair and backup capacity, warranty services' staffing and spare items investment decisions. For these types of systems, customer satisfaction is essential; thus, the delays until replacements of broken units are even more important than delays until the repair initiations of the units. Moreover, the service contract may include conditions on not only the fill rate but also the probability of acceptable delay (delay being less than a specified threshold value).

I address these concerns by expressing delays in terms of the broken-machines process; scaling this process by the number of required operating machines (or the number of customers in the system); and using many-server limit theorems (limits taken as the number of customers goes to infinity) to obtain the limiting expected delay and probability of acceptable delay for both delay until replacement and repair initiation. These results lead to an approximate optimization problem to decide on the repair and backup-capacity investment giving the minimum expected cost rate, subject to a constraint on the acceptable delay probability. Using the characteristics of the scaled broken-machines process, we obtain insights about the relationship between quality of service and capacity decisions. Inspired by the call-center literature, we categorize capacity level choice as efficiency-driven, quality-driven, or quality- and efficiency-driven. Hence, our study extends the conventional call center staffing problem to joint staffing and backup provisioning. Moreover, to our knowledge, the machine-repair problem literature has focused mainly on mean and fill rate measures of performance for steady-state cost analysis. This approach provides complex, nonlinear expressions not possible to solve analytically. The contribution of this essay to the machine-repair literature is the construction of delay-distribution approximations and a near-optimal analytical solution. Among the interesting results, we find that for capacity levels leading to very high utilization of both spares and repair capacity, the limiting distribution of delay until replacement depends on one type of resource only, the repair capacity investment.

In the second essay, &ldquoDiffusion Approximations and Near-Optimal Design of a Make-to-Stock Queue with Perishable Goods and Impatient Customers,&rdquo I study a make-to-stock system with perishable inventory and impatient customers as a two-sided queue with abandonment from both sides. This model describes many consumer goods, where not only spoilage but also theft and damage can occur. We will refer to positive jobs as individual products on the shelf and negative jobs as backlogged customers. In this sense, an arriving negative job provides the service to a waiting positive job, and vice versa. Jobs that must wait in queue before potential matching are subject to abandonment. Under certain assumptions on the magnitude of the abandonment rates and the scaled difference between the two arrival rates (products and customers), we suggest approximations to the system dynamics such as average inventory, backorders, and fill rate via conventional heavy traffic limit theory.

We find that the approximate limiting queue length distribution is a normalized weighted average of two truncated normal distributions and then extend our results to analyze make-to-stock queues with/without perishability and limiting inventory space by inducing thresholds on the production (positive) side of the queue. Finally, we develop conjectures for the queue-length distribution for a non-Markovian system with general arrival streams. We take production rate as the decision variable and suggest near-optimal solutions.

The connection between elliptic stochastic diffusion processes and partial differential equations is rich and well understood. This connection is not very well understood when the stochastic differential equation takes values in an infinite dimensional space such as a function space. In this case, the diffusion is a stochastic partial differential equation (SPDE) and the notion of ellipticity is ambiguous. We establish a sufficient condition on the diffusion coefficient of a class of nonlinear SPDEs, which is analogous to the nondegeneracy condition in finite dimensions, that allows for the existence of a Markov transition density that is absolutely continuous with respect to an infinite dimensional Gaussian measure.

In the second part of this work, we consider a two-station queueing network that processes K job types. The first station in this network is a polling station, and we assume that the second station is operating under any nonidling service discipline. We consider diffusion-scaled versions of many of the processes governing this system, and we show that the scaled two-dimensional total workload process converges to Brownian motion in a wedge. We also show that the scaled immediate workload process for station 2 does not converge, but admits an averaging principle.