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Real-Time Competitive Environments: Truthful Mechanisms for Allocating a Single Processor to Sporadic Tasks Anwar Mohammadi, Nathan Fisher, and Daniel Grosu Department of Computer Science Wayne State University Detroit, MI, USA This research has been supported in part by the US National Science Foundation.

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Non-Competitive vs. Competitive...... Allocation Algorithm Single Processor Optimal allocation 2 Execution Period Value Real-Time Competitive Environments

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Reported Parameters Agent Payments Non-Competitive vs. Competitive...... Allocation Algorithm Single Processor Optimal allocation Agent 1 Agent 2 Mechanism 3 Objective: Ensure Truthful Behavior of Agents Tool: Algorithmic Mechanism Design Real-Time Competitive Environments

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Mechanism Design What is Mechanism Design? The art of designing rules in a competitive environment to achieve – Truthfulness – Efficiency 4 Design rules such that reporting true task parameters is in each agents best interest. Why Mechanism Design for Real-Time Systems? Real-time systems are becoming more open. Many applications in computer science: – network routing – human-computer interaction – parallel & distributed systems (e.g., grid/cluster computing) – internet advertisements Spectacular commercial success: – Google: $6 billion in 2005! – Yahoo!: $2.6 billion in 2005! Optimizing some system-wide objective funtion. Real-Time Competitive Environments

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Related Work Value/Utility Allocation in Non-Competitive Real-Time Systems: – Baruah et al. [1991]: On-line scheduling in the presence of overload. – Rajkumar et al. [1997]: QoS-based Resource Allocation Model (QRAM). – Aydin et al. [2001]: Optimal reward-based scheduling. –…–… Non-Real-Time Mechanism Design: – Initiated by Nisan and Ronen [2001] – Aggarwal [2006] studied knapsack auctions. Game theory in real-time systems: – Sheikh et al. [2011]: Multiprocessor periodic scheduling using game- theoretic concepts – Porter [2004]: Mechanism design for online real-time scheduling – None of these prior works on scheduling considers traditional recurring tasks in competitive environments. 5 Real-Time Competitive Environments

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Model (Non-Competitive) Implicit-deadline Sporadic Task System (with value) 6 Real-Time Competitive Environments Relative Deadline

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Problem Statement 7 Economics: Social Welfare Real-Time Competitive Environments

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Model (Competitive) Competitive Environment 8 Real-Time Competitive Environments

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Motivational Example Allocation Algorithm Winners: Agent 1 Agent 2 Agent 5 Social welfare: 20 Winners: Agent 3 Agent 4 Social welfare: 17 Winners: Agent 2 Agent 5 Social welfare: 18 (0.1, 2)(0.2, 7)(0.4, 8)(0.6, 9)(0.7, 11) True Parameters Declared Parameters 9 Truthful Social Welfare: 20 EDF-MAXVAL-DP Pseudo-Polynomial Knapsack Algorithm [Kim & Ibarra, 1975] Real-Time Competitive Environments Implication: Lying by agents can affect overall system objective!

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Mechanism Design Concepts Mechanism Agents strategy 10 Real-Time Competitive Environments

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Truthful Mechanisms A mechanism is called truthful if for each agent, truthful revelation is a dominant strategy, that is agents maximize their utilities by reporting their true types. Vickrey-Clarke-Groves (VCG) Mechanism: a truthful mechanism given an optimal allocation algorithm. 11 Real-Time Competitive Environments

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Truthful Mechanisms EDF-MAXVAL-VCG Mechanism 12 Set of winning agents if Agent j is excluded Set of winning agents if Agent j is included VCG Payment Total marginal loss of value of including Agent j (w.r.t. other agents). Computable in Pseudo-Polynomial Time (dependent upon maximum task value). Real-Time Competitive Environments

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Approximate Mechanisms Applying VCG payments to standard knapsack approximation algorithm [Kim & Ibarra, 1975] is not truthful! Reason: VCG requires a monotonic allocation algorithm. We apply technique by Briest et al. [2005] to obtain truthful approximation called EDF-MAXVAL-APROX. 13 Real-Time Competitive Environments

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Approximate Mechanisms 14 Real-Time Competitive Environments EDF-MAXVAL-APROX Mechanism (FPTAS) Reserve Prices Reserve Price per Unit

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Frugality Metric Real-Time Competitive Environments 15 The frugality ratio of a mechanism is the total payments divided by the second disjoint optimum value. [Talwar, 2003] Frugality Ratio Measurement of over/under-payment by agents. Upper Bound (EDF-MAXVAL-VCG) The bound is tight. Upper Bound (EDF-MAXVAL-APROX)

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Evaluation Type of Comparisons: Non-Truthful Type Declaration (Case Analysis) Frugality Ratios (Simulation) Execution Time (Simulation) 16 Real-Time Competitive Environments

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Non-Truthful Type Declaration (Case Analysis) 17 We consider an environment of 10 agents. All agents are truthful except Agent 5. We consider 6 cases: Case I: Agent 5 is truthful. Cases II, III, IV: Agent 5 is declaring non-true values Cases V and VI: Agent 5 is declaring non-true utilization Real-Time Competitive Environments

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Utilities and payments of Agent 5 18 Agent 5 can not obtain better utility by lying. Real-Time Competitive Environments Non-Truthful Type Declaration (Case Analysis)

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Simulation Settings 19 Real-Time Competitive Environments U > 1

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Payments vs. Declared Values Average Payments are always less than declared values. As the number of agents increase: – More competition – Less difference between payments and declared values – Better revenue from the perspective of the mechanism 20 Real-Time Competitive Environments

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Frugality Ratios More competition, higher frugality ratio. Frugality ratios for exact and approx mechanism are close. 21 Real-Time Competitive Environments

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Execution Time of Mechanisms 22 Real-Time Competitive Environments

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Conclusion Goal: Introduce notion of competition to real-time scheduling/allocation. – Reason: systems are becoming more open. – Challenge: game theory often well-behaved utility functions. We extended existing algorithms to obtain truthful exact and approximate mechanisms with – bounded frugality ratios – reserve prices Future Plans: – Different resource owner objectives – Compositional systems – Multi-processor settings – Group-strategy proof mechanisms 23 Real-Time Competitive Environments

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Questions? Real-Time Competitive Environments 24

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Frugality The frugality ratio of a mechanism is the total payments divided by the second disjoint optimum value. Definition: Frugality Ratio Property: If a set is feasible, all its subset are feasible also. Implicit-deadline sporadic task system is inclusive. Definition: Inclusive Set System Theorem: Upper Bound for Frugality Ratio 25 Real-Time Competitive Environments

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