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G/G/∞ queues with renewal alternating interruptions

Part of: Markov processes Limit theorems Special processes Stochastic processes Operations research and management science Generalities

Published online by Cambridge University Press:  19 September 2016

Guodong Pang  and
Yuhang Zhou
Guodong Pang*
Affiliation:
Pennsylvania State University
Yuhang Zhou*
Affiliation:
Pennsylvania State University
*
* Postal address: The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, Pennsylvania State University, University Park, PA 16802, USA.
* Postal address: The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, Pennsylvania State University, University Park, PA 16802, USA.
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Abstract

We study G/G/∞ queues with renewal alternating service interruptions, where the service station experiences `up' and `down' periods. The system operates normally in the up periods, and all servers stop functioning while customers continue entering the system during the down periods. The amount of service a customer has received when an interruption occurs will be conserved and the service will resume when the down period ends. We use a two-parameter process to describe the system dynamics: Xr(t,y) tracking the number of customers in the system at time t that have residual service times strictly greater than y. The service times are assumed to satisfy either of the two conditions: they are independent and identically distributed with a distribution of a finite support, or are a stationary and weakly dependent sequence satisfying the ϕ-mixing condition and having a continuous marginal distribution function. We consider the system in a heavy-traffic asymptotic regime where the arrival rate gets large and service time distribution is fixed, and the interruption down times are asymptotically negligible while the up times are of the same order as the service times. We show the functional law of large numbers and functional central limit theorem (FCLT) for the process Xr(t,y) in this regime, where the convergence is in the space 𝔻([0,∞), (𝔻, L1)) endowed with the Skorokhod M1 topology. The limit processes in the FCLT possess a stochastic decomposition property.

Keywords

G/G/∞queuedependent service timeservice interruptiontwo-parameter stochastic processFLLNFCLTSkorokhod M1 topology

MSC classification

Primary: 60K25: Queueing theory 60F17: Functional limit theorems; invariance principles 90B22: Queues and service 60J75: Jump processes
Secondary: 60G44: Martingales with continuous parameter 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 3 , September 2016 , pp. 812 - 831
Copyright
Copyright © Applied Probability Trust 2016 

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