eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-09-23
62:1
62:18
10.4230/LIPIcs.ESA.2024.62
article
Optimizing Throughput and Makespan of Queuing Systems by Information Design
Griesbach, Svenja M.
1
https://orcid.org/0000-0001-8018-3289
Klimm, Max
1
https://orcid.org/0000-0002-9061-2267
Warode, Philipp
2
https://orcid.org/0000-0002-2878-6872
Ziemke, Theresa
1
https://orcid.org/0000-0001-8812-9041
Institute of Mathematics, Technische Universität Berlin, Germany
School of Business and Economics, Humboldt-Universität zu Berlin, Germany
We study the optimal provision of information for two natural performance measures of queuing systems: throughput and makespan. A set of parallel links (queues) is equipped with deterministic capacities and stochastic offsets where the latter depend on a realized state, and the number of states is assumed to be constant. A continuum of flow particles (agents) arrives at the system at a constant rate. A system operator knows the realization of the state and may (partially) reveal this information via a public signaling scheme to the flow particles. Upon arrival, the flow particles observe the signal issued by the system operator, form an updated belief about the realized state, and decide on which link they use. Inflow into a link exceeding the link’s capacity builds up in a queue that increases the cost (total travel time) on the link. Dynamic inflow rates are in a Bayesian dynamic equilibrium when the expected cost along all links with positive inflow is equal at every point in time and not larger than the expected cost of any unused link. For a given time horizon T, the throughput induced by a signaling scheme is the total volume of flow that leaves the links in the interval [0,T]. The public signaling scheme maximizing the throughput may involve irrational numbers. We provide an additive polynomial time approximation scheme (PTAS) that approximates the optimal throughput by an arbitrary additive constant ε > 0. The algorithm solves a Lagrangian dual of the signaling problem with the Ellipsoid method whose separation oracle is implemented by a cell decomposition technique. We also provide a multiplicative fully polynomial time approximation scheme (FPTAS) that does not rely on strong duality and, thus, allows to compute the optimal signals. It uses a different cell decomposition technique together with a piecewise convex under-estimator of the optimal value function. Finally, we consider the makespan of a Bayesian dynamic equilibrium which is defined as the last point in time when a total given value of flow leaves the system. Using a variational inequality argument, we show that full information revelation is a public signaling scheme that minimizes the makespan.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol308-esa2024/LIPIcs.ESA.2024.62/LIPIcs.ESA.2024.62.pdf
Information Design
Dynamic Flows
Public Signals
Convex Envelope