Conservation Laws and Related Applications

Abstract

Work conservation is a fundamental invariance principle governing the behavior of a wide variety of queueing systems. In certain queueing models, the work conservation principle yields work conservation laws, which are linear constraints on performance measures that give an exact or relaxed linear programming formulation of the region of achievable performance spanned under admissible scheduling policies. Such laws are useful for exact or approximate performance analyses of such models, as well as for design and computation of optimal or nearly optimal scheduling policies relative to given performance objectives.

Age-optimal updates of multiple information flows

“…By Lemma 3, the state process {∆ P1 (t), t ≥ 0} of policy P 1 has the same distribution with the state process {∆ P (t), t ≥ 0} of policy P ; the state process {∆ π1 (t), t ≥ 0} of policy π 1 has the same distribution with the state process {∆ π (t), t ≥ 0} of policy π. By (35) and Theorem 6.B.30 in [21], ( 16) holds for all policy π ∈ Π np . Finally, the equivalence between ( 16) and ( 17) follows from (2).…”

A Fast-Pivoting Algorithm for Whittle’s Restless Bandit Index

“…For that purpose, the author introduced the PCL-indexability conditions in [3,4,24] because they are grounded on the satisfaction of partial conservation laws (PCLs). See [38] for a survey on conservation laws in stochastic scheduling and multi-class queueing systems.…”

“…This has its early roots in Klimov's algorithm in [36] for calculating the optimal priority indices for scheduling a multiclass queue with Bernoulli feedback, which was extended in [37] to a framework of stochastic scheduling systems satisfying so-called generalized conservation laws, such as the classic (non-restless) multi-armed bandit problem and branching bandits. See also [38,39]. Yet, the aforementioned work has not addressed the efficient computational implementation of such an algorithm, which is necessary for its actual deployment and widespread application.…”

A Verification Theorem for Threshold-Indexability of Real-State Discounted Restless Bandits

“…PCLs are an extension of classic conservation laws satisfied by certain stochastic scheduling models, which have been used to establish optimality of index policies through analysis of linear programming (LP) formulations of their achievable performance regions. See, e.g., Coffman and Mitrani (1980), Shanthikumar and Yao (1992), Bertsimas and Niño-Mora (1996), and the survey Niño-Mora (2011).…”