Optimum Monitoring and Job Assignment with Multiple Markov Machines

We study a class of systems termed Markov Machines (MM) which process job requests with exponential service times. Assuming a Poison job arrival process, these MMs oscillate between two states, free and busy. We consider the problem of sampling the states of these MMs so as to track their states, subject to a total sampling budget, with the goal of allocating external job requests effectively to them. For this purpose, we leverage the $\mathit{\text{binary}}\mathit{\text{freshness}}\mathit{\text{metric}}$ to quantify the quality of our ability to track the states of the MMs, and introduce two new metrics termed $\mathit{\text{false}}\mathit{\text{acceptance}}\mathit{\text{ratio}}$ (FAR) and $\mathit{\text{false}}\mathit{\text{rejection}}\mathit{\text{ratio}}$ (FRR) to evaluate the effectiveness of our job assignment strategy. We provide optimal sampling rate allocation schemes for jointly monitoring a system of $N$ heterogeneous MMs.