A Truthful Mechanism for Value-Based Scheduling in Cloud Computing

Abstract

We introduce a novel pricing and resource allocation approach for batch jobs on cloud systems. In our economic model, users submit jobs with a value function that specifies willingness to pay as a function of job due dates. The cloud provider in response allocates a subset of these jobs, taking into advantage the flexibility of allocating resources to jobs in the cloud environment. Focusing on social-welfare as the system objective (especially relevant for private or in-house clouds), we construct a resource allocation algorithm which provides a small approximation factor that approaches 2 as the number of servers increases. An appealing property of our scheme is that jobs are allocated non-preemptively, i.e., jobs run in one shot without interruption. This property has practical significance, as it avoids significant network and storage resources for checkpointing. Based on this algorithm, we then design an efficient truthful-in-expectation mechanism, which significantly improves the running complexity of black-box reduction mechanisms that can be applied to the problem, thereby facilitating its implementation in real systems.

Appendix: The Configuration LP: A Different View of the LP Formulation

We present further insights on the choice of (6). An alternative formulation of BFS is its representation as a configuration LP. For every job j and every allocation $a_{j}\in\mathcal{A}_{j}$ we have a variable z _j (a _j ) representing whether job j has been fully allocated according to a _j or not. The configuration LP is defined as follows:

The first constraints allow us to choose at most one allocation per job and the second constraints are capacity constraints. Note that since allocations are defined over the reals, the number of allocations in a set $\mathcal{A}_{j}$ is uncountable. However, the following proposition shows that (LP-D), is actually an efficient representation of (CONF-LP-D).

Proposition 1

The optimal values of fractional social welfare of (LP-D) and (CONF-LP-D) are equal.

Proof

Consider a solution y of the relaxed linear program and recall that $x_{j}=\frac{1}{D_{j}}\sum_{t}y_{j} (t )$. For every job j we construct an allocation a _j matching the values {y _j (t)} by setting: $a_{j} (t )=\frac{y_{j} (t )}{x_{j}}$. This gives us a feasible allocation, since ∑ _t a _j (t)=D _j and a _j (t)≤k _j for every $t\in\mathcal{T}_{j}$ by (6). Now, set z _j (a _j )=x _j and z _j (a)=0 for every $a\in\mathcal{A}_{j}\setminus \{ a_{j} \}$. It is easy to see that z is a feasible solution of the configuration LP.

In the opposite direction, consider a solution z of the configuration LP. Define for every $t\in\mathcal{T}_{j}$:

where $x_{j}=\sum_{a\in\mathcal{A}_{j}}z_{j} (a )$. Notice that a _j is a feasible allocation, since a _j (t)≤k _j for every $t\in\mathcal{T}_{j}$ and since:

By the definition of a _j , the total capacity taken by a _j is identical to the capacity taken by allocations according to z. Moreover, the contribution of a _j to the objective function is the sum of contributions by allocations in $\mathcal{A}_{j}$. To conclude, we translate a _j to its matching vector y _j by setting y _j (t)=a _j (t)⋅x _j for every $t\in\mathcal{T}_{j}$. □