On the Value of Job Migration in Online Makespan Minimization

Makespan minimization on identical parallel machines is a classical scheduling problem. We consider the online scenario where a sequence of $n$ jobs has to be scheduled non-preemptively on $m$ machines so as to minimize the maximum completion time of any job. The best competitive ratio that can be achieved by deterministic online algorithms is in the range $[1.88,1.9201]$. Currently no randomized online algorithm with a smaller competitiveness is known, for general $m$. In this paper we explore the power of job migration, i.e.\ an online scheduler is allowed to perform a limited number of job reassignments. Migration is a common technique used in theory and practice to balance load in parallel processing environments. As our main result we settle the performance that can be achieved by deterministic online algorithms. We develop an algorithm that is $\alpha_m$-competitive, for any $m\geq 2$, where $\alpha_m$ is the solution of a certain equation. For $m=2$, $\alpha_2 = 4/3$ and $\lim_{m\rightarrow \infty} \alpha_m = W_{-1}(-1/e^2)/(1+ W_{-1}(-1/e^2)) \approx 1.4659$. Here $W_{-1}$ is the lower branch of the Lambert $W$ function. For $m\geq 11$, the algorithm uses at most $7m$ migration operations. For smaller $m$, $8m$ to $10m$ operations may be performed. We complement this result by a matching lower bound: No online algorithm that uses $o(n)$ job migrations can achieve a competitive ratio smaller than $\alpha_m$. We finally trade performance for migrations. We give a family of algorithms that is $c$-competitive, for any $5/3\leq c \leq 2$. For $c= 5/3$, the strategy uses at most $4m$ job migrations. For $c=1.75$, at most $2.5m$ migrations are used.Comment: Revised versio

"Rotterdam econometrics": publications of the econometric institute 1956-2005

This paper contains a list of all publications over the period 1956-2005, as reported in the Rotterdam Econometric Institute Reprint series during 1957-2005.

"Rotterdam econometrics": publications of the econometric institute 1956-2005

This paper contains a list of all publications over the period 1956-2005, as reported in the Rotterdam Econometric Institute Reprint series during 1957-2005

A lower bound for randomized on-line scheduling algorithms

We prove that any randomized algorithm for on-line scheduling jobs on m identical parallel machines must have a worst case ratio at least mm/(mm-(m-1)m) for every m, which tends to e/(e-1) ˜ 1.58 as m¿8. This answers a question posed in a recent paper by Bartal, Fiat, Karloff and Vohra