G/G/∞ queues with renewal alternating interruptions

Guodong Pang, Yuhang Zhou

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

We study G/G/8 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 f-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, 8), (D, L1)) endowed with the Skorokhod M1 topology. The limit processes in the FCLT possess a stochastic decomposition property.

Original languageEnglish (US)
Pages (from-to)812-831
Number of pages20
JournalAdvances in Applied Probability
Volume48
Issue number3
DOIs
StatePublished - Sep 1 2016

Fingerprint

Filling stations
Renewal
Distribution functions
Queue
Dynamical systems
Servers
Topology
Decomposition
Functional Central Limit Theorem
Customers
Stochastic Decomposition
Marginal Function
Mixing Conditions
Heavy Traffic
Law of large numbers
D-space
Continuous Distributions
Marginal Distribution
System Dynamics
Identically distributed

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Applied Mathematics

Cite this

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G/G/∞ queues with renewal alternating interruptions. / Pang, Guodong; Zhou, Yuhang.

In: Advances in Applied Probability, Vol. 48, No. 3, 01.09.2016, p. 812-831.

Research output: Contribution to journalArticle

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