## Abstract

In order to obtain Markov heavy-traffic approximations for infinite-server queues with general non-exponential service-time distributions and general arrival processes, possibly with time-varying arrival rates, we establish heavy-traffic limits for two-parameter stochastic processes. We consider the random variables Q^{e}(t,y) and Q^{r}(t,y) representing the number of customers in the system at time t that have elapsed service times less than or equal to time y, or residual service times strictly greater than y. We also consider W^{r}(t,y) representing the total amount of work in service time remaining to be done at time t+y for customers in the system at time t. The two-parameter stochastic-process limits in the space D([0,∞),D) of D-valued functions in D draw on, and extend, previous heavy-traffic limits by Glynn and Whitt (Adv. Appl. Probab. 23, 188-209, 1991), where the case of discrete service-time distributions was treated, and Krichagina and Puhalskii (Queueing Syst. 25, 235-280, 1997), where it was shown that the variability of service times is captured by the Kiefer process with second argument set equal to the service-time c. d. f.

Original language | English (US) |
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Pages (from-to) | 325-364 |

Number of pages | 40 |

Journal | Queueing Systems |

Volume | 65 |

Issue number | 4 |

DOIs | |

State | Published - 2010 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Computer Science Applications
- Management Science and Operations Research
- Computational Theory and Mathematics