  \textit{Weighted shortest processing time first} (WSPT) is one of the best
known algorithms for total weighted completion time scheduling problems. For
each job $J_j$, it first combines the two independent job parameters weight
$w_j$ and processing time $p_j$ by simply forming the so called Smith ratio
$w_j/p_j$. Then it schedules the jobs in order of decreasing Smith ratio
values. The algorithm guarantees an optimal schedule for a single machine and
the approximation factor $1.2071$ for parallel identical machines.
  For the corresponding online problem in a single machine environment with
preemption, the \textit{weighted shortest remaining processing time first}
(WSRPT) algorithm replaces the processing time $p_j$ with the remaining
processing time $p_j(t)$ for every job that is only partially executed at time
$t$ when determining the Smith ratio. Since more than 10 years, we only know
that the competitive ratio of this algorithm is in the interval $[1.2157,2]$.
  In this paper, we present the tight competitive ratio $1.2259$ for WSRPT. To
this end, we iteratively reduce the instance space of the problem without
affecting the worst case performance until we are able to analyze the remaining
instances. This result makes WSRPT the best known algorithm for deterministic
online total weighted completion time scheduling in a preemptive single machine
environment improving the previous competitive ratio of $1.5651$. Additionally,
we increase the lower bound of this competitive ratio from $1.0730$ to
$1.1038$.
