Many-server heavy-traffic limit for queues with time-varying parameters

Liu, Yunan; Whitt, Ward

doi:10.1214/13-aap927

Cited by 46 publications

(75 citation statements)

References 24 publications

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“…Since MSHT FCLT's for IS models can be fruitfully applied to establish associated MSHT FCLT's for finite-server models [24,32,34], our results here have broader implications. In particular, we intend to apply the results here to establish a MSHT FCLT for the G t /GI/s t + GI model with timevarying arrival rate and staffing, customer abandonment (the +GI) and alternating overloaded (OL) and underloaded (UL) intervals, extending the FCLT for the G t /M/s t + GI model in [24].…”

Section: Introductionmentioning

confidence: 85%

“…In particular, we intend to apply the results here to establish a MSHT FCLT for the G t /GI/s t + GI model with timevarying arrival rate and staffing, customer abandonment (the +GI) and alternating overloaded (OL) and underloaded (UL) intervals, extending the FCLT for the G t /M/s t + GI model in [24]. The present results apply in three ways.…”

Section: Introductionmentioning

confidence: 88%

“…Nevertheless, the assumption is natural. Extensions are possible, as illustrated by §10 of [24]. Remark 2.3 (joint convergence and independence of the limits).…”

Section: The Modelmentioning

confidence: 99%

“…The integrals in Theorem 3.2 should be interpreted just as in [30], as explained in Remark 3.2 there. In particular, the deterministic integrals in (3.7) and (3.12) are all Stieltjes integrals, while the integrals in (3.6), (3.11), (3.16) and (3.20) are two-parameter stochastic integrals, just as in [23,24,30]. As in Theorem 3.2 and Remark 3.3 of [30], the continuity assumption on the cdf G ν in Assumption 2 is used to get the representation in terms of the Kiefer process.…”

Section: ) Being a Zero-mean Gaussian Process With Covariance Functionmentioning

confidence: 99%

“…We remark that this kind of separation of variability has been observed in the past. For instance, in [24], the FCLT limit of the waiting time process solves a stochastic differential equation driven by three independent BMs that are associated with the arrival process, abandonment times and service process.…”

Section: ) Being a Zero-mean Gaussian Process With Covariance Functionmentioning

confidence: 99%

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Heavy-traffic Limit for the Initial Content Process

2017

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To understand the performance of a queueing system, it can be useful to focus on the evolution of the content that is initially in service at some time. That necessarily will be the case in service systems that provide service during normal working hours each day, with the system shutting down at some time, except that all customers already in service at the termination time are allowed to complete their service. Also, for infinite-server queues, it is often fruitful to partition the content into the initial content and the new input, because these can be analyzed separately. With i.i.d service times having a nonexponential distribution, the state of the initial content can be described by specifying the elapsed service times of the remaining initial customers. That initial content process is then a Markov process. This paper establishes a many-server heavy-traffic (MSHT) functional central limit theorem (FCLT) for the initial content process in the space D D , assuming a FCLT for the initial age process, with the number of customers initially in service growing in the limit. The proof applies a symmetrization lemma from the literature on empirical processes to address a technical challenge: For each time, including time 0, the conditional remaining service times, given the ages, are mutually independent but in general not identically distributed. (FCLT's) for the standard G/G/s queueing model, with unlimited waiting space and service in order of arrival, expose the impact of the stochastic variability in the arrival and service processes on the transient and steady-state performance. This is important because the general G/G/s model is far less tractable than its Markovian M/M/s counterpart, even for the special case in which the interarrival times and service times come from independent sequences of i.i.d. random variables. From [14,41], we know that conventional heavy-traffic theory tells a simple story: With conventional heavy-traffic, where the arrival rate increases to the maximum possible service rate with a fixed number of servers, the arrival and service processes contribute via Introduction. Heavy-traffic (HT) functional central limit theorems

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Section: Introductionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 88%

“…Nevertheless, the assumption is natural. Extensions are possible, as illustrated by §10 of [24]. Remark 2.3 (joint convergence and independence of the limits).…”

Section: The Modelmentioning

confidence: 99%

Section: ) Being a Zero-mean Gaussian Process With Covariance Functionmentioning

confidence: 99%

Section: ) Being a Zero-mean Gaussian Process With Covariance Functionmentioning

confidence: 99%

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Heavy-traffic Limit for the Initial Content Process

2017

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Time‐varying Queues: a Two‐time‐scale Approach

Yin

Zhang

2021

Queueing Theory 1

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Approximations for heavily loaded G/GI/n + GI queues

Liu

Whitt

2016

Naval Research Logistics

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Abstract:Motivated by applications to service systems, we develop simple engineering approximation formulas for the steadystate performance of heavily loaded G/GI/n+GI multiserver queues, which can have non-Poisson and nonrenewal arrivals and non-exponential service-time and patience-time distributions. The formulas are based on recently established Gaussian many-server heavy-traffic limits in the efficiency-driven (ED) regime, where the traffic intensity is fixed at ρ > 1, but the approximations also apply to systems in the quality-and-ED regime, where ρ > 1 but ρ is close to 1. Good performance across a wide range of parameters is obtained by making heuristic refinements, the main one being truncation of the queue length and waiting time approximations to nonnegative values. Simulation experiments show that the proposed approximations are effective for large-scale queuing systems for a significant range of the traffic intensity ρ and the abandonment rate θ, roughly for ρ > 1.02 and θ > 2.0.

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Many-server heavy-traffic limit for queues with time-varying parameters

Cited by 46 publications

References 24 publications

Heavy-traffic Limit for the Initial Content Process

Heavy-traffic Limit for the Initial Content Process

Time‐varying Queues: a Two‐time‐scale Approach

Approximations for heavily loaded G/GI/n + GI queues

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