Solving the Open Problem of Finding the Exact Pointwise Stable Fluid Flow Approximation (PSFFA) State Variable of a Non-Stationary M/M/1 Queue with Potential Real-Life PSFFA Applications to Computer Engineering

Abstract It is a real mathematical challenge to analytically find the closed-form expression of the temporal first- order differential equation expressing the steady state variable, x(t), in a non-stationary or time- changing M/M/1 queueing system is addressed. The simulation was used to solve the only known solution to this open problem, namely the closed-form expression of PSFFA model of the non-stationary M/M/1 queueing system. This work proposes a constant ratio, namely β(Ismail's ratio) of both time-dependent mean arrival and mean service rates to provide a precise analytic solution to this open topic. Numerical experiments were carried out to capture the temporal impact combined with Ismail's ratio β, and queueing parameters on the stability dynamics of the non-stationary M/M/1 queue. Potential applications of PSFFA to Computer engineering are provided. Some challenging open problems are addressed combined with closing remarks and future research directions.