Heavy-traffic limits for an infinite-server fork–join queueing system with dependent and disruptive services

Abstract

We study an infinite-server fork---join queueing system with dependent services, which experiences alternating renewal service disruptions. Jobs are forked into a fixed number of parallel tasks upon arrival and processed at the corresponding parallel service stations with multiple servers. Synchronization of a job occurs when its parallel tasks are completed, i.e., non-exchangeable. Service times of the parallel tasks of each job can be correlated, having a general continuous joint distribution function, and moreover, the service vectors of consecutive jobs form a stationary dependent sequence satisfying the strong mixing ($$\alpha $$ź-mixing) condition. The system experiences renewal alternating service disruptions with up and down periods. In each up period, the system operates normally, but in each down period, jobs continue to enter the system, while all the servers will stop working, and services received will be conserved and resume at the beginning of the next up period. We study the impact of both the dependence among service times and these down times upon the service dynamics, the unsynchronized queueing dynamics, and the synchronized process, assuming that the down times are asymptotically negligible. We prove FWLLN and FCLT for these processes, where the limit processes in the FCLT possess a stochastic decomposition property and the convergence requires the Skorohod $$M_1$$M1 topology.