Robert D. Foley

Confidence intervals using orthonormally weighted standardized time series

We extend the standardized time series area method for constructing confidence intervals for the mean of a stationary stochastic process. The proposed intervals are based on orthonormally weighted standardized time series area variance estimators. The underlying area estimators possess two important properties: they are first-order unbiased, and they are asymptotically independent of each other. These properties are largely the result of a careful choice of weighting functions, which we explicitly describe. The asymptotic independence of the area estimators yields more degrees of freedom than various predecessors; this, in turn, produces smaller mean and variance of the length of the resulting confidence intervals. We illustrate the efficacy of the new procedure via exact and Monte Carlo examples. We also provide suggestions for efficient implementation of the method.

Large deviations of a modified Jackson network: Stability and rough asymptotics

Consider a modified, stable, two node Jackson network where server 2 helps server 1 when server 2 is idle. The probability of a large deviation of the number of customers at node one can be calculated using the flat boundary theory of Schwartz and Weiss [Large Deviations Performance Analysis (1994), Chapman and Hall, New York]. Surprisingly, however, these calculations show that the proportion of time spent on the boundary, where server 2 is idle, may be zero. This is in sharp contrast to the unmodified Jackson network which spends a nonzero proportion of time on this boundary.

Dual command travel times and miniload system throughput with turnover-based storage

We analyze travel times for automated storage/retrieval systems. In particular, we apply our travel time model to turnover-based storage systems and determine the mean and variance of dual command travel times. We present detailed numerical results for selected rack shape factors and ABC inventory profiles. We then investigate the effect of alternative rack configurations on travel time performance measures. We also show how to determine the throughput of miniload systems with turnover-based storage and exponentially distributed pick times.

Performance of miniload systems with two-class storage

To increase throughput, the higher turnover items are often stored near the input/output point of a miniload system. We analyze the performance of such a miniload with a square-in-time rack containing two storage zones: high turnover and low turnover. First, we derive the distribution of the dual command travel time. System performance measures such as the throughput depend upon the distribution of the dual command travel time and the distribution of pick times. We work out the details and obtain closed-form expressions for throughput for two important families of pick time distributions: deterministic and exponential. Finally, we investigate how the size of the high turnover region affects throughput.

Bridges and networks: Exact asymptotics

We extend the Markov additive methodology developed in (1, 3) to obtain the sharp asymptotics of the steady state probability of a queueing network when one of the nodes gets large. We focus on a new phenomenon we call a bridge. The bridge cases occur when the Markovian part of the twisted Markov additive process is one null recurrent or one transient while the jitter cases treated in (1, 3) occur when the Markovian part is (one) positive recurrent. The asymptotics of the steady state is an exponential times a polynomial term in the bridge case but is purely exponential in the jitter case. We apply this theory to a modified, stable, two node Jackson network where Server Two helps Server One when Server Two is idle. We derive the sharp asymptotics of the steady state distribution of the number of customers queued at each node as the number of customers queued at the Server One grows large. In so doing we get an intuitive understanding of the companion paper (2) which gives a large deviation analysis of this problem using the flat boundary theory in (4). Unlike the (unscaled) large deviation path of a Jackson network which jitters along the boundary, the unscaled large deviation path of the modified network tries to avoid the boundary where Server Two helps Server One (and forms a bridge). In the fluid limit this bridge does collapse to a straight line but the proportion of time spent on the flat boundary tends to zero. This bridge phenomenon is ubiquitous. We also treated the bathroom problem described in (4) and found the bridge case is present. Here we derive the sharp asymptotics of the steady state of the bridge case and we obtain the results consistent with those obtained in (? ) using complex variable methods.

QUEUES WITH DELAYED FEEDBACK

Queues with delayed feedback have been little studied in queueing theory. Presented here is a rather complete discussion of such problems including queue length processes, busy period processes and several customer flow processes (e.g., departure processes). The case in which the delay mechanism is an M-server queue is studied in detail but it is shown later that many of the results carry over to a more general delay mechanism.

Exact asymptotics for the stationary distribution of a Markov chain : a production model

We derive rough and exact asymptotic expressions for the stationary distribution π of a Markov chain arising in a queueing/production context. The approach we develop can also handle “cascades,” which are situations where the fluid limit of the large deviation path from the origin to the increasingly rare event is nonlinear. Our approach considers a process that starts at the rare event. In our production example, we can have two sequences of states that asymptotically lie on the same line, yet π has different asymptotics on the two sequences.

BIAS OPTIMALITY IN A QUEUE WITH ADMISSION CONTROL

We consider a finite capacity queueing system in which each arriving customer offers a reward. A gatekeeper decides based on the reward offered and the space remaining whether each arriving customer should be accepted or rejected. The gatekeeper only receives the offered reward if the customer is accepted. A traditional objective function is to maximize the gain, that is, the long-run average reward. It is quite possible, however, to have several different gain optimal policies that behave quite differently. Bias and Blackwell optimality are more refined objective functions that can distinguish among multiple stationary, deterministic gain optimal policies. This paper focuses on describing the structure of stationary, deterministic, optimal policies and extending this optimality to distinguish between multiple gain optimal policies. We show that these policies are of trunk reservation form and must occur consecutively. We then prove that we can distinguish among these gain optimal policies using the bias or transient reward and extend to Blackwell optimality.

Optimal prices for finite capacity queueing systems

We prove a lower bound on the optimal price for a fairly large class of blocking systems with general arrival and service processes, determine optimal price expressions for M/M/1/m and M/GI/s/s systems, and investigate how optimal prices change with changes in the size of the waiting room and service capacity.

Throughput bounds for miniload automated storage/retrieval systems

To compute miniload system throughput, the distribution of the pick times is needed. Unfortunately, during the design phase, only partial information may be available such as the mean pick time. In this paper, we determine tight upper and lower bounds on throughput for several different types of partial information. We also give numerical examples to show how to apply the bounds.