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 X e (t, y) and X r (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.
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 timetthat have residual service times strictly greater thany. 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 processXr(t,y) in this regime, where the convergence is in the space 𝔻([0,∞), (𝔻,L1)) endowed with the SkorokhodM1topology. The limit processes in the FCLT possess a stochastic decomposition property.
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