Fluid Limit for the Machine Repairman Model with Phase-Type Distributions
We consider the Machine Repairman Model with N working units that break randomly and independently according to a phase-type distribution. Broken units go to one repairman where the repair time also follows a phase-type distribution. We are interested in the behavior of the number of working units when N is large. For this purpose, we explore the fluid limit of this stochastic process appropriately scaled by dividing it by N.
This problem presents two main difficulties: two different time scales and discontinuous transition rates. Different time scales appear because, since there is only one repairman, the phase at the repairman changes at a rate of order N, whereas the total scaled number of working units changes at a rate of order 1. Then, the repairman changes N times faster than, for example, the total number of working units in the system, so in the fluid limit the behavior at the repairman is averaged. In addition transition rates are discontinuous because of idle periods at the repairman, and hinders the limit description by an ODE.
We prove that the multidimensional Markovian process describing the system evolution converges to a deterministic process with piecewise smooth trajectories. We analyze the deterministic system by studying its fixed points, and we find three different behaviors depending only on the expected values of the phase-type distributions involved. We also find that in each case the stationary behavior of the scaled system converges to the unique fixed point that is a global attractor. Proofs rely on martingale theorems, properties of phase-type distributions and on characteristics of piecewise smooth dynamical systems. We also illustrate these results with numerical simulations.
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