Advances in Applied Probability

G/G/∞ queues with renewal alternating interruptions

Guodong Pang and Yuhang Zhou

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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.

Article information

Source
Adv. in Appl. Probab., Volume 48, Number 3 (2016), 812-831.

Dates
First available in Project Euclid: 19 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aap/1474296316

Mathematical Reviews number (MathSciNet)
MR3568893

Zentralblatt MATH identifier
1351.60123

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60F17: Functional limit theorems; invariance principles 90B22: Queues and service [See also 60K25, 68M20] 60J75: Jump processes
Secondary: 60G44: Martingales with continuous parameter 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)

Keywords
G/G/∞ queue dependent service time service interruption two-parameter stochastic process FLLN FCLT Skorokhod 𝑀₁ topology

Citation

Pang, Guodong; Zhou, Yuhang. G/G/∞ queues with renewal alternating interruptions. Adv. in Appl. Probab. 48 (2016), no. 3, 812--831. https://projecteuclid.org/euclid.aap/1474296316


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