Abstract
We study cost-sharing games in real-time scheduling systems where the server’s activation cost in every time slot is a function of its load. We focus on monomial cost functions and consider both the case when the degree is less than one (inducing positive congestion effect for the jobs) and when it is greater than one (inducing negative congestion effect for the jobs). For the former case, we provide tight bounds on the price of anarchy, and show that the price of anarchy grows to infinity as a polynomial of the number of jobs in the game. For the latter, we observe that existing results provide constant and tight (asymptotically in the degree of the monomial) bounds on the price of anarchy. We then turn to analyze payment mechanism with arbitrary cost-sharing, that is, when the strategy of a player includes also its payment. We show that our mechanism reduces the price of anarchy of games with n jobs and unit server costs from Θ(n) to 2. We also show that, for a restricted class of instances, a similar improvement is achieved for monomial server costs. This is not the case, however, for unrestricted instances of monomial costs, for which we prove that the price of anarchy remains super-constant for our mechanism. For systems with load-independent activation costs, we show that our mechanism can produce an optimal solution which is stable against coordinated deviations.
Highlights
The study of real-time scheduling is motivated by the emergence and popularity of cloud computing
For monomial cost functions of degree less than 1, we prove that our mechanism has super-constant PoA for general instances; we prove that it does achieve an improvement from super-constant to constant for a special family of instances
The work most closely related to our paper is the one in [6] where the model we study is introduced and various results are obtained for the case of constant server costs with respect to the price of anarchy and the related concepts of the strong price of anarchy and the price of stability
Summary
The study of real-time scheduling is motivated by the emergence and popularity of cloud computing. In the standard cost-sharing setting studied in [6], each of the jobs processed at time t assumes an equal share of the server cost c(lt ) (or a proportional share for a more general setting where each job places a different load on the server) Given this rule, we would expect each job j to optimize for its individual cost share and declare the slot that minimizes the ratio of the server cost to the number of jobs (which is precisely the individual cost share) among all slots in its interval [r j , d j ). For monomial cost functions of degree less than 1, we prove that our mechanism has super-constant PoA for general instances; we prove that it does achieve an improvement from super-constant to constant for a special family of instances For instances such that the optimal solution uses a single slot (which includes, for example, instances with a common release time or a common deadline), our mechanism reduces the PoA from Θ(n(1−d)/2 ) to Θ(1). (Recall that for monomials of degree larger than 1, it is known that the PoA of congestion games is already constant [8,10] as a function of n without applying any coordination mechanism.) we show that our mechanism can be applied to yield an optimal solution that is stable against coordinated deviations of coalitions, in an environment in which different slots may have different cost
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