m-Balanced words: A generalization of balanced words

Finding optimally balanced words for production planning and maintenance scheduling

“…For any given set of densities, the minimal possible count balance is less than or equal to 3 (Kubiak, 2009). Sano et al (2004) gives the following definition for a different measure: given a finite alphabet {1, …, n}, a word U over this alphabet, and a nonnegative integer m, a letter a is m-balanced in U if, , the gap balance of the letters 2 and 3 equals 0, and the gap balance of the letter 1 equals 2.…”

“…Altman et al (2000) show that, for very general stochastic systems, the optimal routing of jobs to servers is a balanced sequence. Sano et al (2004) introduced a generalization of balanced words and showed that using these policies to route jobs minimizes the maximum waiting time.…”

Design and analysis of a class-aware recursive loop scheduler for class-based scheduling

“…This notion is applied in [16], where the authors defined the equivalent notion of balanced schedules for K>2. Recently, the authors in [17] introduced a new notion of m-balancedness, where the value of m (which is a non-negative integer) gives a measure of smoothness of schedules for any K, where a smaller value indicates a more balanced schedule. The authors also proposed an m-balanced scheduler that constructs a schedule with m ≤ K-1 for any given x, but highlighted the difficulty in finding a schedule with the smallest possible m.…”

“…We propose a variance-based metric and then describe an existing metric that uses the notion of m-balancedness [17].…”

“…According to [17], we have the following definition: We note that m j π = 0 if a π is perfectly-smooth with respect to user j; otherwise, the pseudocode for computing m j π is given below:…”