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.
Original language | English (US) |
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Pages (from-to) | 1375-1416 |
Number of pages | 42 |
Journal | Stochastic Processes and their Applications |
Volume | 127 |
Issue number | 5 |
DOIs | |
State | Published - May 1 2017 |
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics