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Two-parameter process limits for an infinite-server queue with arrival dependent service times

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  • Pang, Guodong
  • Zhou, Yuhang

Abstract

We study an infinite-server queue with a general arrival process and a large class of general time-varying service time distributions. Specifically, customers’ service times are conditionally independent given their arrival times, and each customer’s service time, conditional on her arrival time, has a general distribution function. We prove functional limit theorems for the two-parameter processes Xe(t,y) and Xr(t,y) that represent the numbers of customers in the system at time t that have received an amount of service less than or equal to y, and that have a residual amount of service strictly greater than y, respectively. When the arrival process and the initial content process both have continuous Gaussian limits, we show that the two-parameter limit processes are continuous Gaussian random fields. In the proofs, we introduce a new class of sequential empirical processes with conditionally independent variables of non-stationary distributions, and employ the moment bounds resulting from the method of chaining for the two-parameter stochastic processes.

Suggested Citation

  • Pang, Guodong & Zhou, Yuhang, 2017. "Two-parameter process limits for an infinite-server queue with arrival dependent service times," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1375-1416.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:5:p:1375-1416
    DOI: 10.1016/j.spa.2016.08.003
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    References listed on IDEAS

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    1. Avishai Mandelbaum & Petar Momčilović, 2012. "Queues with Many Servers and Impatient Customers," Mathematics of Operations Research, INFORMS, vol. 37(1), pages 41-65, February.
    2. Hongyuan Lu & Guodong Pang & Yuhang Zhou, 2016. "$$G/{ GI}/N(+{ GI})$$ G / G I / N ( + G I ) queues with service interruptions in the Halfin–Whitt regime," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 127-160, February.
    3. Anatolii Puhalskii, 2013. "On the $$M_t/M_t/K_t+M_t$$ queue in heavy traffic," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(1), pages 119-148, August.
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    5. William A. Massey, 1985. "Asymptotic Analysis of the Time Dependent M/M/1 Queue," Mathematics of Operations Research, INFORMS, vol. 10(2), pages 305-327, May.
    6. Hongyuan Lu & Guodong Pang & Yuhang Zhou, 2016. "$$G/{ GI}/N(+{ GI})$$ G / G I / N ( + G I ) queues with service interruptions in the Halfin–Whitt regime," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 127-160, February.
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    Cited by:

    1. Pang, Guodong & Zhou, Yuhang, 2018. "Functional limit theorems for a new class of non-stationary shot noise processes," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 505-544.
    2. Guodong Pang & Yuhang Zhou, 2018. "Two-parameter process limits for infinite-server queues with dependent service times via chaining bounds," Queueing Systems: Theory and Applications, Springer, vol. 88(1), pages 1-25, February.

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