An Optimal Control Framework for Online Job Scheduling with General Cost Functions

TL;DR

The analysis offers a principled method of estimating dual variables in a general setting of online job scheduling and can achieve state-of-the-art competitive ratios for several special cases and provide new competitive ratios which are the first in their settings.

Abstract

We consider the problem of online job scheduling on a single machine or multiple unrelated machines with general job/machine-dependent cost functions. In this model, each job $j$ has a processing requirement (length) $v_{ij}$ and arrives with a nonnegative nondecreasing cost function $g_{ij}(t)$ if it has been dispatched to machine $i$, and this information is revealed to the system upon arrival of job $j$ at time $r_j$. The goal is to dispatch the jobs to the machines in an online fashion and process them preemptively on the machines so as to minimize the generalized completion time $\sum_{j}g_{i(j)j}(C_j)$. Here $i(j)$ refers to the machine to which job $j$ is dispatched, and $C_j$ is the completion time of job $j$ on that machine. It is assumed that jobs cannot migrate between machines and that each machine can work on a single job at any time instance. In particular, we are interested in finding an online scheduling policy whose objective cost is competitive with respect to a slower optimal offline benchmark, i.e., the one that knows all the job specifications a priori and is slower than the online algorithm. We first show that for the case of a single machine and special cost functions $g_j(t)=w_jg(t)$, with nonnegative nondecreasing $g(t)$, the highest-density-first rule is optimal for the generalized fractional completion time. We then extend this result by giving a speed-augmented competitive algorithm for the general nondecreasing cost functions $g_j(t)$ by utilizing a novel optimal control framework. This approach provides a principled method for identifying dual variables in different settings of online job scheduling with general cost functions. Using this method, we also provide a speed-augmented competitive algorithm for multiple unrelated machines with convex functions $g_{ij}(t)$, where the competitive ratio depends on the curvature of cost functions $g_{ij}(t)$.

Figures (4)

• Figure 1: An illustration of the $\alpha,\beta$-plots in Example \ref{['ex:alpha-beta']}. The blue line segments on the left figure correspond to steps (i.e., subjobs) in the optimal split instance. The red line segments are those associated with the optimal dual variables in the original HGFCS instance, which are obtained at the end of Algorithm \ref{['alg:update']}. The right figure illustrates the optimal $\beta$-curves for the split and original instances.
• Figure 2: An illustration of the residual network flow $\hbox{RNF}(r_n)$ with three alive jobs $\mathcal{A}(r_n)=\{1,2,3\}$ and residual job lengths $v_1(r_n)=2$, $v_2(r_n)=3$, and $v_3(r_n)=1$. The optimal flow cost is equal to the sum of the colored-edge costs.
• Figure 3: The left and right figures illustrate the optimal flow before and after the addition of the unit length job $n$, respectively. $j_1$ and $j_2$ are two alive jobs at time $r_n$ with lengths $v_{j_1}(r_n)=2$ and $v_{j_2}(r_n)=3$, which are represented by red and blue colors, respectively. Each solid edge $(j,t)$ shows whether a unit of job $j$ is scheduled at time slot $t$, with a cost given by the value on that edge. By adding job $n$, the new optimal flow in the right figure can be obtained by removing (unscheduling) the dashed edges with negative signs and adding (scheduling) the new solid edges with plus signs. The result is an alternating path $P:=n,t_0,j_1,t_1,j_{2},t_{2}$. The change in the optimal flow cost is the sum of the edge costs along $P$ with respect to plus/minus signs.
• Figure 4: Upon the arrival of a new job $n$ at time $r_n$, the old tail $\{\hat{\beta}_t\}_{t\ge r_n}$ is updated to the new one $\{\hat{\beta}'_t\}_{t\ge r_n}$. Here the $\hat{\boldsymbol{\beta}}$-curve is given by the upper envelope of the other three curves. The increase in the size of the area under the $\hat{\boldsymbol{\beta}}$-curve due to the arrival of job $n$ is precisely the increase in the integral flow cost of the algorithm $\Delta_n(\hbox{Alg})$, which is also set for the dual variable $\hat{\alpha}_n$. Summing over all jobs gives us the overall area under the $\hat{\beta}$-curve and is equal to the integral flow cost of the algorithm.

Theorems & Definitions (27)

• Definition 1
• Lemma 1
• Remark 1
• Lemma 2
• Definition 2
• Lemma 3
• Example 1
• Theorem 1
• Theorem 2: Minimum Principle
• Corollary 1