In this paper, we focus on simple exponential approximations for steady-state tail probabilities in G/GI/1 queues based on large-time asymptotics. We relate the large-time asymptotics for the steady-state waiting time, sojourn time and workload. We evaluate the exponential approximations based on the exact asymptotic parameters and their approximations by making comparisons with exact numerical results for BMAP/GI/1 queues. Numerical examples show that the exponential approximations are remarkably accurate at the 90 th percentile and beyond. Key words: queues; approximations; asymptotics; tail probabilities; sojourn time and workload.

Motivated by queues with service interruptions, we consider an infinite-capacity storage model with a two-state random environment. The environment alternates between "up" and "down" states. In the down state, the content increases according to one stochastic process; in the up state, the content decreases according to another stochastic process. We describe the steady-state behavior of this system under assumptions on the component stochastic elements. For the special case of deterministic linear flow during the up and down states, we show that the steady-state content is directly related to the steady-state workload or virtual waiting time in an associated G/G/1 queue, thus supplementing results of Gaver and Miller (1962), Miller (1963) and Chen and Yao (1990). Subject classifications: Probability, regenerative processes: a storage model in a two-state random environment. Queues, approximations: a fluid model with random disruptions. Inventory/production, operating characteristics: a...

We establish a heavy-traffic asymptotic expansion (in powers of one minus the traffic intensity) for the asymptotic decay rates of queue-length and workload tail probabilities in stable infinite-capacity multi-channel queues. The specific model has multiple independent heterogeneous servers, each with i.i.d. service times, that are independent of the arrival process, which is the superposition of independent non-identical renewal processes. Customers are assigned to the first available server in the order of arrival. The heavy-traffic expansion yields relatively simple approximations for the tails of steady-state distributions and higher percentiles, yielding insight into the impact of the first three moments of the defining distributions.

We propose an enhancement to the parametric-decomposition method for calculating approximate steady-state performance measures of open queueing networks with non-Poisson arrival processes and non-exponential service-time distributions. Instead of using a variability parameter c a 2 for each arrival process, we suggest using a variability function c a 2 (ρ) , 0 < ρ < 1, for each arrival process; i.e., the variability parameter should be regarded as a function of the traffic intensity ρ of a queue to which the arrival process might go. Variability functions provide a convenient representation of different levels of variability in different time scales for arrival processes that are not nearly renewal processes. Variability functions enable the approximations to account for long-range effects in queueing networks that cannot be addressed by variability parameters. For example, the variability functions provide a way to address the heavy-traffic bottleneck phenomenon, in which exceptional variability (either high or low) in the input has little impact in a series of queues with low-to-moderate traffic intensities, and then has a big impact when it reaches a later queue with a relatively high traffic intensity. The variability functions also enable the approximations to characterize irregular periodic deterministic external arrival processes in a reasonable way; i.e., if there are no batches, then c a 2 (ρ) should be 0 for ρ near 0 or 1, but c a 2 (ρ) can assume arbitrarily large values for appropriate intermediate ρ. We present a full network algorithm with variability functions, showing that the idea is implementable. We also show how simulations of single queues can be effectively exploited to determine variability functions for difficult external arrival processes. Key words: queueing networks, tandem queues, approximations, parametric-decomposition approximations, two-moment approximations, heavy traffic, squared coefficient of variation.

The complexity of parallel applications and parallel processor scheduling policies makes both exact analysis and simulation difficult, if not intractable, for large systems. In this paper we propose a new approach to performance modeling of multiprogrammed processor scheduling policies, that of interpolation approximations. We first define a workload model that contains parameters for the essential properties of parallel applications with respect to scheduling discipline performance, yet lends itself to mathematical analysis. Key features of the workload model include general distribution of total job processing time, general distribution of available job parallelism, and a simple characterization of parallelism overheads. We then show that one can find specific values of the system parameters for which the parallel system under a given scheduling policy reduces to a queueing system with a known (closed-form) solution. Finally, interpolation between the points with known solutions is u...

To appear in Operations Research. Motivated by queues with service interruptions, we consider an infinite-capacity storage model with a two-state random environment. The environment alternates between ‘‘up’ ’ and ‘‘down’ ’ states. In the down state, the content increases according to one stochastic process; in the up state, the content decreases according to another stochastic process. We describe the steady-state behavior of this system under assumptions on the component stochastic elements. For the special case of deterministic linear flow during the up and down states, we show that the steady-state content is directly related to the steady-state workload or virtual waiting time in an associated G/G/1 queue, thus supplementing results of Gaver and Miller (1962), Miller (1963) and Chen and Yao (1990). Subject classifications: Probability, regenerative processes: a storage model in a two-state random environment. Queues, approximations: a fluid model with random disruptions. Inventory/production, operating characteristics: a fluid model with alternating up and down times.

The performance evaluation of many complex manufacturing, communication and computer systems has been made possible by modeling them as queueing systems. Many approximations used in queueing theory have been drawn from the behavior of queues in light and heavy tra c conditions. In this paper, we propose a new approximation technique, which combines the light and heavy tra c characteristics. This interpolation approximation is based on the theory of multipoint Pade approximation which is applied at two points: light and heavy tra c. We show how this can be applied for estimating the waiting time moments of the GI/G/1 queue. The light tra c derivatives of any order can be evaluated using the MacLaurin series analysis procedure. The heavy tra c limits of the GI/G/1 queue are well known in the literature. Our technique generalizes the previously developed interpolation approximations and can be used to approximate any order of the waiting time moments. Through numerical examples, we show that the moments of the steady state waiting time can be estimated with extremely high accuracy under all ranges of tra c intensities using low orders of the approximant. We also present a framework for the development of simple analytical approximation formulas. Keywords: GI/G/1 queue, heavy tra c limits, MacLaurin series, multipoint Pade approximation. 1