Asymptotically Optimal Controls for Time-Inhomogeneous Networks

Abstract

A framework is introduced for the identification of controls for single-class timevarying queueing networks that are asymptotically optimal in the so-called uniform acceleration regime. A related, but simpler, first-order (or fluid) control problem is first formulated. For a class of performance measures that satisfy a certain continuity property, it is then shown that any sequence of policies whose performances approach the infimum in the fluid control problem is asymptotically optimal for the original network problem. Examples of performance measures with this property are described, and simulations implementing asymptotically optimal policies are presented. The use of directional derivatives of the reflection map for solving fluid control problems is also illustrated. This work serves to complement a large body of literature on asymptotically optimal controls for time-homogeneous networks.

Optimal Control of a Time-Varying Double-Ended Production Queueing Model

“…For time-inhomogeneous queueing systems, the asymptotic control is usually considered under fluid scaling-cf. Bassamboo et al (2005), Cudina and Ramanan (2011), Ozkan and Ward (2020), and Khademi and Liu (2019), in all of which high-volume systems were considered, and the FCP was shown to be the best performance bound for the original QCP asymptotically. In Armoney et al (2019), the authors considered the problem of scheduling appointments for a service facility with customer no-shows.…”

Optimal Control of a Time-Varying Double-Ended Production Queueing Model

“…Such t (m) exists either when the initial queue length is nonzero, or there exists a time interval where λ p (t) = λ d (t). If such t (m) does not exist, then we must have zero initial queue length and λ p (t) = λ d (t) for t ∈ [0, T ], which indicates that the expected queue length E(Q(t)) in (7) and the fluid queue length q(t) in (9) are both zero in [0, T ]. Theorem 3.2 is trivial under this case.…”

“…For time-inhomogeneous queueing systems, the asymptotic control is usually considered under fluid scaling, cf. [5,9,28], in all of which, high-volume systems were considered, and the FCP was shown to be the best performance bound for the original QCP asymptotically. Without assuming any asymptotics, in many situations, deterministic models can also be shown to be the best performance bound for the expected stochastic performance.…”

Two-parameter process limits for an infinite-server queue with arrival dependent service times

“…Nonstationary arrival processes have been extensively studied in the literature of many-server queues, and it is standard to assume that the arrival processes satisfy a functional central limit theorem (FCLT) where the limit process has a deterministic time-change with a time-varying arrival rate function (see Assumption 1). For service times, in the exponential case, it is standard to assume that service rates are time-varying; see, e.g., [27,24,25,17,6,37,20,32]. However, little has been studied for general time-varying service times.…”