On the Accuracy of Fluid Models for Capacity Sizing in Queueing Systems with Impatient Customers

Abstract

We consider queueing systems in which customers arrive according to a Poisson process and have exponentially distributed service requirements. The customers are impatient and may abandon the system while waiting for service after a generally distributed amount of time. The system incurs customer-related costs that consist of waiting and abandonment penalty costs. We study capacity sizing in such systems to minimize the sum of the long-term average customer-related costs and capacity costs. We use fluid models to derive prescriptions that are asymptotically optimal for large customer arrival rates. Although these prescriptions are easy to characterize, they depend intricately upon the distribution of the customers' time to abandon and may prescribe operating in a regime with offered load (the ratio of the arrival rate to the capacity) greater than 1. In such cases, we demonstrate that the fluid prescription is optimal up to O 1. That is, as the customer arrival rate increases, the optimality gap of the prescription remains bounded.

The G t /GI/s t +GI many-server fluid queue

“…The first equation in (5) says that the fluid in service that is not served remains in service (which requires that the staffing function be feasible, as in Assumption 4). The second equation in (5) says that the fluid waiting in queue that does not abandon and does not move into service, remains in queue.…”

“…The present paper directly extends [37], which developed a deterministic fluid model to approximate the steady-state performance of a stationary G/GI/s + GI queueing model. The accuracy of fluid models for capacity planning has been strongly supported by [5]. A novel feature here and in [37], compared to most fluid models, is that we consider a non-Markovian many-server fluid model, which involves two-parameter functions; e.g., the queue content at time t that has been in queue for a duration at most y, denoted by Q(t, y), as a function of both t and y; see (2).…”

“…Here we provide for the first time a full description of the transient behavior. The fundamental evolution equations, in (5) here, are the same as in (2.14) and (2.15) of [37], but the time-dependent performance when the system is overloaded actually depends on three features introduced for the first time here. First, for non-exponential service, the time-varying rate that fluid enters service is characterized as the unique solution to a fixed-point equation; see (18) and Theorem 2.…”

Algorithms to Find Two-Hop Routing Policies in Multiclass Delay Tolerant Networks

“…The authors resort to fluid approximation to characterize the routing policies under the assumption that the number of copies of the packet is monotonically increasing in time. However, no theoretical guarantee over the quality of the solutions is provided and, in principle, fluid approximation may provide arbitrarily inefficient solutions, see, e.g., [15]. Furthermore, it is not clear whether fluid-approximation approaches can be extended to more than two hops.…”

“…In particular, we evaluated the algorithms' scalability when the number of classes grows. To obtain these results we fixed the values of some parameters ( = 1/3, τ = 100, N c = 10, L = 500) and we generated random mobility profiles and transmission technologies by uniformly sampling from the following intervals: R c ∈ [15,50], v c ∈ [1,15],…”