Random walk analysis of parallel queueing stations

We analyze a complex stochastic system with a single server who operates in two parallel stations. The server works in station I until the input to station I slows down. Then the server switches to station II processing a queue of jobs there and then returns to station I upon finishing the total work in station II. If during his service in station II the input to station I becomes intense so that the queue of jobs there get alarmingly long, the server begins to process two queues simultaneously. Then, dependent on the situation, the server either continues with the rest of the jobs in station II or returns to station I and rests there until its buffer fills up to a certain level prompting him to resume his service. During various switchovers, the server changes his modes, each time engaging in different distributions. The switchovers are managed at first passage times of a multivariate random walk exiting relevant sets.We use and embellish methods of fluctuation analysis for multivariate random walks to investigate the queueing processes. This enables us to arrive at closed-form functionals for the processes and related random components. Applications are found in computer networking and operating systems analysis.