A Heavy-Traffic Limit for the Cycle Counting Process in G/G/1, Optional Interruptions and Elastic Screen Brownian Motion

We consider a queue in which the server is available for primary customers for a random number of busy cycles, after which he leaves for a random amount of time as soon as the system becomes empty. Under appropriate normalization we establish a heavy traffic limit which turns out to be a regenerative generalized elastic screen process RGESP with random jumps or linear parts of the trajectories. These jumps or linear parts occur when the local time at zero accumulates to a certain random level. As a main tool we first establish a heavy traffic limit theorem for the renewal counting process associated with the busy cycles in the underlying queueing system. In particular we show that this limit is proportional to the local time at zero of a reflected Brownian motion. We compare the stationary distribution for the RGESP with the corresponding one for the M/G/1 queue, where for both explicit expressions are obtained, and show that the results are consistent.