generating function

The congestion problems with processor vacations have confronted with increasing intricacy, and their explicit transient solutions are exceptionally hard to compute. The transient solution is more significant for studying the dynamical behavior of computing systems over a finite period and predominantly utilizes within the state-of-the-art design architect for a real-time I/O system. Motivated from this, we adopt the mathematical concepts, namely continued fractions and generating function, to derive explicit expressions for transient-state probabilities. Transient-state probabilities of the processing delay problem with a single processor which adopts the multiple vacations policy to save power consumption and thermal trip error with discouragement and feedback are obtained in terms of modified Bessel functions using the properties of the confluent hypergeometric function. Due to the inaccessibility of processor, discouragement behaviors balking and reneging of the job requests are prone to exhibit. Routing back for the service feedback for the processed job request is also critical to maintaining the quality of service $(QoS)$. For the glance of the I/O system performance, the expected value of the state of the computing system using stationary queue-size distribution is also derived.