Total stretch minimization on single and identical parallel machines

We consider the classical problem of scheduling $n$ jobs with release dates on both single and identical parallel machines. We measure the quality of service provided to each job by its stretch, which is defined as the ratio of its response time to processing time. Our objective is to schedule these jobs non-preemptively so as to minimize sum stretch. So far, there have been very few results for sum stretch minimization especially for the non-preemptive case. For the preemptive version, the Shortest remaining processing time (SRPT) algorithm is known to give $2$-competitive for sum stretch on single machine while its is $13$-competitive on identical parallel machines. Leonardi and Kellerer provided the strong lower bound for the more general problem of \textit{sum (weighted) flow time} in single machine and identical parallel machines, respectively . Therefore, we study this problem with some additional assumptions and present two new competitive ratio for existing algorithms. We show that the Shortest processing time (SPT) algorithm is $\Delta - \frac{1}{\Delta}+1$-competitive for non-preemptive sum stretch minimization on single machine and it is $\Delta - \frac{1}{\Delta}+ \frac{3}{2} -\frac{1}{2m}$ on $m$ identical parallel machines, where $\Delta$ is the upper bound on the ratio between the maximum and the minimum processing time of the jobs.