Consider a system to which particles with random lifetimes flow stochastically. The system is monitored at discrete time epochs following a renewal process. When the system is detected non-empty (upon monitoring), a service procedure is initiated—clearing the system off particles. Once the service procedure is concluded, the system's evolution regenerates.

Particles may represent customers or jobs in a queueing system, contaminants in a physical system, hazardous chemical or biological agents in an environmental system, standing buy/sell orders at a brokerage center, etc.

This class of stochastic systems is modeled and analyzed. We (i) derive the joint transform of the time-to-first-detection and the number of particles present in the system at that epoch, and compute their statistics; (ii) define and calculate various path-functionals and performance measures of the system; and, (iii) study the issue of optimal monitoring schemes.