Two-parameter process limits for infinite-server queues with dependent service times via chaining bounds

Pang, Guodong; Zhou, Yuhang

doi:10.1007/s11134-017-9550-1

Cited by 4 publications

(5 citation statements)

References 31 publications

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“…The conditions in Assumption 1 are satisfied because F is continuous. The FWLLN and FCLT for the queueing process Q n are obtained as a consequence of the two-parameter process limits in Pang and Zhou (2018b), and also as a special case of the results in this paper.…”

Section: The Work-input Process In Infinite-server Queuesmentioning

confidence: 66%

“…In this setting, the shot shape function H takes the forms H(t, x) 1(t < x) and H(t, x) x1(t < x) for the queue length and work-input processes, respectively. The (two-parameter) queueing process has been well studied in Pang and Whitt (2013) and Pang and Zhou (2018b). As a consequence of Theorem 2, we obtain an FCLT for the work-input process for the infinite-server queues with ρ-mixing service times (Theorem 4).…”

Section: Introductionmentioning

confidence: 77%

“…Consider an infinite-server queue with an arrival process A n as in Assumption 3 and service times {Z i : i ∈ N} as in Assumption 2. This model has been studied in Pang and Zhou (2018b). The total queue length process Q n : {Q n (t) : t ≥ 0} can be written as…”

Section: The Work-input Process In Infinite-server Queuesmentioning

confidence: 99%

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Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises

Pang

Zhou

2020

Stochastic Systems

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We study shot noise processes when the shot noises are weakly dependent, satisfying the ρ-mixing condition. We prove a functional weak law of large numbers and a functional central limit theorem for this shot noise process in an asymptotic regime with a high intensity of shots. The deterministic fluid limit is unaffected by the presence of weak dependence. The limit in the diffusion scale is a continuous Gaussian process whose covariance function explicitly captures the dependence among the noises. The model and results can be applied in financial and insurance risks with dependent claims as well as queueing systems with dependent service times. To prove the existence of the limit process, we employ the existence criterion that uses a maximal inequality requiring a set function with a superadditivity property. We identify such a set function for the limit process by exploiting the ρ-mixing condition. To prove the weak convergence, we establish the tightness property and the convergence of finite dimensional distributions. To prove tightness, we construct two auxiliary processes and apply an Ottaviani-type inequality for weakly dependent sequences.

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Section: The Work-input Process In Infinite-server Queuesmentioning

confidence: 66%

Section: Introductionmentioning

confidence: 77%

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Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises

Pang

Zhou

2020

Stochastic Systems

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“…This unique feature arises from the fork-join network model, in order to provide representations for the system dynamics and characterize the limit processes (see (4.24)-(4.25) in Theorem 4.1 and (5.6)). Sequential empirical processes have played an important role in studying many-server queueing models, as first observed by Krichagina and Puhalskii [29], and further developed in [51,50,38,43,45,46,47]. It is also worth noting that in these papers, the weak convergence of the associated sequential empirical processes in the space D([0, ∞), D) with the Skorohod J 1 topology is required as first observed in [29].…”

Section: Assumption 8 the Residual Waiting Times Of The Tasks In Quementioning

confidence: 96%

Heavy-traffic limits for a fork-join networkin the Halfin-Whitt regime

Lu¹,

Pang²

2016

Stoch. Syst.

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We study a fork-join network with a single class of jobs, which are forked into a fixed number of parallel tasks upon arrival to be processed at the corresponding multi-server stations. After service completion, each task will join a buffer associated with the service station waiting for synchronization, called "unsynchronized queue". The synchronization rule requires that all tasks from the same job must be completed, referred to as "non-exchangeable synchronization". Once synchronized, jobs will leave the system immediately. Service times of the parallel tasks of each job can be correlated and form a sequence of i.i.d. random vectors with a general continuous joint distribution function. We study the joint dynamics of the queueing and service processes at all stations and the associated unsynchronized queueing processes.The main mathematical challenge lies in the "resequencing" of arrival orders after service completion at each station. As in Lu and Pang (2015) for the infinite-server fork-join network model, the dynamics of all the aforementioned processes can be represented via a multiparameter sequential empirical process driven by the service vectors for the parallel tasks of each job. We consider the system in the Halfin-Whitt regime, and prove a functional law of large number and a functional central limit theorem for queueing and synchronization processes. In this regime, although the delay for service at each station is asymptotically negligible, the delay for synchronization is of the same order as the service times.

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“…We also study an infinite-server queueing model with cluster (batch) arrivals, where customers in each batch may experience some random delay before receiving service. Using the same notation as above ( representing service times), we may express the number of customers in service at time t as In [25], heavy-traffic limits (FLLN and FCLT) were established for infinite-server queues with batch arrivals where service times within each batch may be correlated, as a consequence of infinite-server queues with weakly dependent service times in [26] (see also [28]). That approach cannot include (dependent) random delays for customers in each batch.…”

Section: Introductionmentioning

confidence: 99%

Shot noise processes with randomly delayed cluster arrivals and dependent noises in the large-intensity regime

Li¹,

Pang²

2021

Adv. Appl. Probab.

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We study shot noise processes with cluster arrivals, in which entities in each cluster may experience random delays (possibly correlated), and noises within each cluster may be correlated. We prove functional limit theorems for the process in the large-intensity asymptotic regime, where the arrival rate gets large while the shot shape function, cluster sizes, delays, and noises are unscaled. In the functional central limit theorem, the limit process is a continuous Gaussian process (assuming the arrival process satisfies a functional central limit theorem with a Brownian motion limit). We discuss the impact of the dependence among the random delays and among the noises within each cluster using several examples of dependent structures. We also study infinite-server queues with cluster/batch arrivals where customers in each batch may experience random delays before receiving service, with similar dependence structures.

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Two-parameter process limits for infinite-server queues with dependent service times via chaining bounds

Cited by 4 publications

References 31 publications

Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises

Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises

Heavy-traffic limits for a fork-join networkin the Halfin-Whitt regime

Shot noise processes with randomly delayed cluster arrivals and dependent noises in the large-intensity regime

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