Decentralized Throughput Scheduling

Motivated by the organization of online service systems, we study models for throughput scheduling in a decentralized setting. In throughput scheduling, we have a set of jobs j with values wj , processing times pj , and release dates rj and deadlines and dj , to be processed nonpreemptively on a set of unrelated machines. The goal is to maximize the total value of jobs scheduled within their time window [rj , dj ]. While several approximation algorithms with different performance guarantees exist for this and related models, we are interested in the situation where subsets of servers are governed by selfish players. We give a universal result that bounds the price of decentralization, in the sense that any local α-approximation algorithms, α ≥ 1, yield equilibria that are at most a factor (α + 1) away from the global optimum, and this bound is tight. For models with identical machines, we improve this bound to α √ e/( α √ e− 1) ≈ (α + 1/2), which is shown to be tight, too. We also address some variations of the problem. 1 Model and Notation We consider a non-preemptive scheduling problem with unrelated machines, to which we refer as decentralized throughput scheduling problem throughout the paper. The input of an instance I ∈ I consists of a set of jobs J , a set of machines M, and a set of players N . Each job j ∈ J comes with a release time rj , deadline dj , nonnegative value wj and processing time pij if scheduled on machine i ∈ M. Machines can process only one job at a time. Job j is feasibly scheduled (on any of the machines) if its processing starts no earlier than rj and finishes no later than dj . For any set of jobs S ⊆ J , we let w(S) = ∑ j∈S wj be the total value. Each player n ∈ N controls a subset of machines Mn ⊆M and aims to maximize the total value of jobs that can be feasibly scheduled on its set of machines Mn. Here Mn, n ∈ N , is a partition of the set of machines M. In this paper we are primarily interested in equilibrium allocations, which we define as a situation in which none of the players n can improve the total value of jobs that can be feasibly scheduled on its set of machines Mn by removing some of its jobs and adding some of the yet unscheduled jobs. Note that a player cannot make a claim on jobs that are scheduled on machines of other players. Let us denote by w(S) the total weight of jobs in S, for S ⊆ J . An equilibrium ? Research supported by CTIT, Centre for Telematics and Information Technology, University of Twente, project “Mechanisms for Decentralized Service Systems”. 2 Jasper de Jong, Marc Uetz, and Andreas Wombacher allocation is a (pure) Nash equilibrium in a strategic form game where player n’s strategies are the subsets of jobs Sn ⊆ J that can be feasibly scheduled on machines Mn, and valuations are w(Sn), but with the condition that a strategy profile (Sn)n∈N is feasible if and only if the sets Sn, n ∈ N , are pairwise disjoint. We will refer to these allocations as Nash equilibrium (NE) allocations. The question if a given player can improve, generally describes an NP-hard optimization problem. Therefore, we will also consider a relaxed equilibrium condition: We say an allocation is an α-approximate Nash equilibrium (α-NE) if none of the players n can improve the total value of jobs that can be feasibly scheduled on its set of machinesMn by a factor larger than α by removing some of its jobs and adding some of the yet unscheduled jobs. By the existence of constant factor approximation algorithms for (centralized) throughput scheduling, e.g. [2, 4], the players are thus equipped with polynomial time algorithms to indeed reach an α-NE in polynomial time, for certain, constant values of α. Our main focus will be the analysis of the price of anarchy (PoA) [14], lower bounding the quality of any Nash equilibrium relative to the quality of a globally optimal solution. More specifically, we are interested in the ratio PoA = max I∈I max NE∈NE(I) w(OPT ) w(NE) , (1) where OPT denotes a globally optimal allocation for I, and NE(I) denotes the set of all Nash equilibria of instance I. Note that OPT is a Nash equilibrium, too, hence the price of stability, as proposed in [1], trivially equals 1. 2 Motivation, Related Work and Contribution Our motivation to study this problem is to analyze the performance of decentralized service systems, where jobs are posted, e.g. on an online portal, and service providers can select these on an take-it-or-leave-it basis. The problem we address can be seen as a stylized version of coordination problems that appear in several application domains. We give three examples: (1) When operating micro grids for decentralized energy production and consumption, the goal is to consume locally produced energy as much as possible. Here, jobs can be defined as the operation of appliances, e.g. operating a washing machine), typically bounded by a time window and attached with a certain $-value. Machines, on the other side, are local energy producers like PV-panels, micro CHPs or local micro windmills [3, 12]. (2) In cloud computing, service providers such as Amazon and Google provide an infrastructure service, that is, provide a virtual machine with a specific service level for a certain period of time. The aim of a federated cloud computing environment, e.g. [18], is to “coordinate load distribution among different cloudbased data centers in order to determine optimal location for hosting application services ”. (3) In private car sharing portals like Tamyca [17] or Autonetzer [16], 1 This is achieved by introducing externalities, letting the valuation of player n be −∞ whenever Sn is not disjoint with the sets S−n chosen by all other players −n. Decentralized Throughput Scheduling 3 clients post their requests to rent a car for a certain time period, and the price they are willing to pay. Car owners (private or not) in the neighborhood can select these requests and rent their car(s). The underlying non-strategic optimization problem is known in the optimization literature and sometimes referred to as throughput scheduling. See for example [2], and follow-up papers, e.g. [4]. In the 3-field notation of [8], the problem reads R|rj | ∑ wjUj , where R denotes the unrelated machine model, rj specifies that there are nontrivial release dates, and the objective is to minimize the total weight of late jobs (which is equivalent to the maximization objective considered here). Approximation algorithms for several versions of the problem have been discussed in the literature (e.g., with or without weights, identical or unrelated machines), most notably [2, 4]. Special cases that are of particular interest are the single machine case with unit weights and zero release dates, solved in polynomial time by the Moore-Hodgson algorithm [13], and the case with identical machines and unit processing times, which can be cast and solved as an assignment problem [5]. To the best of our knowledge, the decentralized version that we propose here has not been addressed before. Our main contribution lies in the the informal claim that, in general, the price of decentralization is very moderate: If local decisions of all players are approximately optimal with relative performance guarantee α, then any equilibrium allocation is not worse than an (α + 1)-fraction of the global optimum. We improve this to ≈ (α + 1/2) when all machines are identical, and the equilibria are obtained in a special, sequential way which will be explained later. In fact, except for the identical machine case, the proofs for the upper bounds are fairly simple, yet surprisingly universal. Most work lies in the corresponding tight lower bound constructions. 3 A First Encounter Example 1. There are two playersN = {1, 2}, each controlling exactly one of two related machinesM = {1, 2}, with machine speeds s1 = 1, s2 = 2 3 , respectively. There are two jobs J = {1, 2} with processing times p1 = p2 = 1, deadlines d1 = 1, d2 = 3 2 and values w1 = w2 = 1. Release times are 0. When job 1 is allocated to machine 1 and job 2 to machine 2, both jobs can meet their respective deadlines. This is obviously an optimal allocation. However when job 2 is allocated to machine 1, only one job can be scheduled before its deadline. See also Figure 1. Note that both allocations are a Nash equilibrium. Now w(OPT )/w(NE) = 2/1 = 2 for the second allocation, and we see from this simple example that PoA ≥ 2 in (1), even for the case of related machines, unit weights, unit processing times and zero release dates. u t 2 The related machine model is a special case of the unrelated machine model by letting pij = pj si . 4 Jasper de Jong, Marc Uetz, and Andreas Wombacher s1 = 1 p2 = 1, d2 = 1 1 2