Many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment

Aras, Ahmet Korhan; Liu, Yunan

doi:10.1007/s11134-018-9575-0

Cited by 17 publications

(7 citation statements)

References 37 publications

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“…Table 6.1 implies that our choice of simulation run length grant us satisfactory statistical precision for demonstration purposes. Our result here is consistent with Table 1 and (10) in [140].…”

Section: Notation and Simulation Methodologysupporting

confidence: 93%

“…We conjecture that the relevant many-server heavy-traffic limit for the stationary departure process is a Gaussian process with the covariance function of the stationary renewal processes associated with the service times, as in the CLT for renewal processes in Theorems 7.2.1 and 7.2.4 of [143]. Partial support comes from [10], Appendix F of [11] and [65].…”

Section: Extensionsmentioning

confidence: 90%

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A robust queueing network analyzer based on indices of dispersion

Whitt

You

2021

Naval Research Logistics

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This dissertation concludes five years of doctoral study and research. I am deeply indebted to many exceptional scholars and inspiring individuals, who guide me through an voyage of discovery that never meant to be a downwind sailing.First and foremost, I would like to thank my advisor, Professor Ward Whitt, for the enlightening guidance throughout my life as a Ph.D. student. He is always a walking encyclopedia to me with his accurate pointers to the literature and his prompt and constructive feedback. His generosity in sharing his knowledge from both the mathematics and engineering point of view helped me greatly in understanding the big picture of the subject matter of this work. As a great educator and scholar, he has been and will always be a role model that I look up to.

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Section: Notation and Simulation Methodologysupporting

confidence: 93%

Section: Extensionsmentioning

confidence: 90%

A robust queueing network analyzer based on indices of dispersion

Whitt

You

2021

Naval Research Logistics

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“…We conjecture that the relevant manyserver heavy-traffic limit for the stationary departure process is a Gaussian process with the covariance function of the stationary renewal processes associated with the service times, as in the CLT for renewal processes in Whitt (2002, Theorems 7.2.1 and 7.2.4). Partial support comes from Aras et al (2018), Aras et al (2017b, Apendix F), and Gamarnik and Goldberg (2013).…”

Section: Extensionsmentioning

confidence: 97%

Heavy-Traffic Limit of theGI/GI/1 Stationary Departure Process and Its Variance Function

Whitt

You

2018

Stochastic Systems

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Abstract. Heavy-traffic limits are established for the stationary departure process from a GI/GI/1 queue and its variance function. The limit process is a function of the Brownian motion limits of the arrival and service processes plus the stationary reflected Brownian motion (RBM) limit of the queue-length process. An explicit expression is given for the variance function, which depends only on the first two moments of the interarrival times and service times plus the previously determined correlation function of canonical (drift −1, diffusion coefficient 1) RBM. The limit for the variance function here is used to show that the approximation for the index of dispersion for counts of the departure process used in our new robust queueing network analyzer is asymptotically correct in the heavy-traffic limit.

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“…In addition, letting H ( t ) denote the head‐of‐line waiting time at time t , that is, the waiting time of the customer who has been waiting the longest (if there is any). Following Aras et al (2018) and Liu and Whitt (2014) we depict E ( t ) and Q ( t ) as

E false(t false) = {false\sum}_{i = 1}^{A false(t - H false(t false) false)} {normal𝟙}_{{γ_{i} > V (τ_{i})}} and

Q false(t false) = {false\sum}_{i = A false(t - H false(t false) false)}^{A false(t false)} {normal𝟙}_{{τ_{i} + γ_{i} > t}} for t \geq 0,

where the random variables 0 ≤ τ 1 ≤ τ 2 ≤ ⋯ denote arrival epochs, and γ 1 , γ 2 , … represent the abandonment times of successive customers that arrived to the system.…”

Section: Model With Customer Abandonmentmentioning

confidence: 99%

Staffing many‐server queues with autoregressive inputs

Sun

Liu

2020

Naval Research Logistics

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Recent studies reveal significant overdispersion and autocorrelation in arrival data at service systems such as call centers and hospital emergency departments. These findings stimulate the needs for more practical non‐Poisson customer arrival models, and more importantly, new staffing formulas to account for the autocorrelative features in the arrival model. For this purpose, we study a multiserver queueing system where customer arrivals follow a doubly stochastic Poisson point process whose intensities are driven by a Cox–Ingersoll–Ross (CIR) process. The nonnegativity and autoregressive feature of the CIR process makes it a good candidate for modeling temporary dips and surges in arrivals. First, we devise an effective statistical procedure to calibrate our new arrival model to data which can be seen as a specification of the celebrated expectation–maximization algorithm. Second, we establish functional limit theorems for the CIR process, which in turn facilitate the derivation of functional limit theorems for our queueing model under suitable heavy‐traffic regimes. Third, using the corresponding heavy traffic limits, we asymptotically solve an optimal staffing problem subject to delay‐based constraints on the service levels. We find that, in order to achieve the designated service level, such an autoregressive feature in the arrival model translates into notable adjustment in the staffing formula, and such an adjustment can be fully characterized by the parameters of our new arrival model. In this respect, the staffing formulas acknowledge the presence of autoregressive structure in arrivals. Finally, we extend our analysis to queues having customer abandonment and conduct simulation experiments to provide engineering confirmations of our new staffing rules.

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Many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment

Cited by 17 publications

References 37 publications

A robust queueing network analyzer based on indices of dispersion

A robust queueing network analyzer based on indices of dispersion

Heavy-Traffic Limit of theGI/GI/1 Stationary Departure Process and Its Variance Function

Staffing many‐server queues with autoregressive inputs

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