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

Abstract

This paper introduces a deterministic fluid model that approximates the many-server G t /GI/s t + GI queueing model, and determines the time-dependent performance functions. The fluid model has time-varying arrival rate and service capacity, abandonment from queue, and non-exponential service and patience distributions. Two key assumptions are that: (i) the system alternates between overloaded and underloaded intervals, and (ii) the functions specifying the fluid model are suitably smooth. An algorithm is developed to calculate all performance functions. It involves the iterative solution of a fixed-point equation for the time-varying rate that fluid enters service and the solution of an ordinary differential equation for the time-varying head-of-line waiting time, during each overloaded interval.Simulations are conducted to confirm that the algorithm and the approximation are effective.

A Markov Model for Measuring Service Levels in Nonstationary G(t)/G(t)/s(t)+G(t) Queues

“…Additional literature on the use of fluid approximations for Markovian models, can be found in Mandelbaum et al [50,54,51,52,53], Ridley et al [55] and Jiménez and Koole [48]. For systems with general service and/or abandonment time distributions, we refer to the more recent work of Whitt [45] on G(t)/GI/s + GI models (with state-dependent arrival rates), Liu and Whitt [49,56,57,58] on the G(t)/GI/s(t) + GI queue, Liu and Whitt [59] for a network of G(t)/M(t)/s(t) + GI(t) queues and references therein. A key characteristic of fluid models is that arrivals and departures are considered as continuous flows, rather than discrete processes (an assumption that becomes more acceptable as the number of servers increases).…”

Stabilizing performance in a service system with time-varying arrivals and customer feedback

“…The delay probability starts at 1 because the stabilizing staffing algorithm does not start staffing until time w > 0. That feature ensures that all arrivals wait exactly w in the limiting fluid model (see §10 of Liu & Whitt (2012a)), but it would probably not be used in applications.…”

“…Since the external arrival rate has been constructed by simple scaling, the associated DIS staffing can be constructed by simple scaling as well; see §4 of Liu & Whitt (2012a).…”

Performance Analysis of Time-Dependent Queueing Systems: Survey and Classification

“…Hampshire et al [95] combine the fluid approximation with the MOL approach (see Section 3.3.1) to analyze the abandonments and blocking probability in an MðtÞ=M=cðtÞ=KðtÞþM system. Liu and Whitt [144] introduce the fluid approximation for a GðtÞ=G=cðtÞþG system. They separately track the fluid in the queue and on the servers.…”