Final answer:
The Pollaczek-Khintchine formulas apply to queuing systems with a single-server, FCFS discipline, exponentially distributed service times, where the traffic intensity is less than one.
Step-by-step explanation:
The Pollaczek-Khintchine formulas are applicable to queueing systems that are single-server, operate under a first-come, first-served discipline (FCFS), and have exponentially distributed service times. These formulas are used to calculate various characteristics of the queuing system, such as the average number of items in the queue, the average waiting time in the queue, and the idle time probability of the server. The traffic intensity, which is the ratio of the average arrival rate to the average service rate, must be less than one for the system to reach a steady state. These conditions are suitable for mathematical modeling where the arrival and service processes exhibit randomness, which typically occurs in telecommunication systems, computer networks, and certain types of manufacturing processes. The Pollaczek-Khintchine formulas are rooted in probabilistic and stochastic processes, and their implementation often involves complex calculations that take into account the inherent randomness of the input and service operations.