Optimum Monitoring and Job Assignment with Multiple Markov Machines

TL;DR

The binary freshness metric is used to quantify the quality of the ability to track the states of the MMs, and two new metrics termed false acceptance ratio (FAR) and false rejection ratio (FRR) are introduced to evaluate the effectiveness of the job assignment strategy.

Abstract

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 $\textit{binary freshness metric}$ to quantify the quality of our ability to track the states of the MMs, and introduce two new metrics termed $\textit{false acceptance ratio}$ (FAR) and $\textit{false rejection ratio}$ (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.

Figures (6)

• Figure 1: System model of a single MM: The MM oscillates between "free" (0) and "busy" (1) with rates $\alpha$ and $\beta$ due to internal jobs. When an external job is assigned, the state goes to "process external job" (2) and recovers to "free" with rate $\gamma$. MM is sampled with rate $\mu$. External jobs arrive with rate $\lambda$.
• Figure 2: State transition diagram of $Y(t)$.
• Figure 3: Jump chain of $Y(t)$.
• Figure 4: State transition diagram of a feedback free system.
• Figure 5: Variation of ${\mathbb{E}}[\Delta]$, $\text{FAR}$ and $\text{FRR}$ with $\mu$ for $\lambda=\beta=\gamma=0.5$.

Theorems & Definitions (12)

• Theorem 1
• Lemma 1
• Corollary 1
• Theorem 2
• Theorem 3
• Remark 1
• Remark 2
• Lemma 2
• Remark 3
• Remark 4