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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, todoi:10.24200/sci.2020.52933.2955 fatcat:wpke4ehnc5bjffa6skd6pffaze