G/G/∞ Queues with Renewal Alternating Interruptions

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: X(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: (i) i.i.d. with a distribution of a finite support, or (ii) a stationary and weakly dependent sequence satisfying the φ-mixing condition and having a continuous marginal distribution function. We consider the system in a heavytraffic 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 FLLN and FCLT for the process X(t, y) in this regime, where the convergence is in the space D([0,∞), (D, L1)) endowed with the Skorohod M1 topology. The limit processes in the FCLT possess a stochastic decomposition property. K eywords: G/G/∞ queue; dependent service times; service interruptions; two-parameter stochastic processes; FLLN; FCLT; Skorohod M1 topology