  Partitioned multiprocessor scheduling has been widely accepted in academia
and industry to statically assign and partition real-time tasks onto identical
multiprocessor systems. This paper studies fixed-priority partitioned
multiprocessor scheduling for sporadic real-time systems, in which
deadline-monotonic scheduling is applied on each processor. Prior to this
paper, the best known results are by Fisher, Baruah, and Baker with speedup
factors $4-\frac{2}{M}$ and $3-\frac{1}{M}$ for arbitrary-deadline and
constrained-deadline sporadic real-time task systems, respectively, where $M$
is the number of processors. We show that a greedy mapping strategy has a
speedup factor $3-\frac{1}{M}$ when considering task systems with arbitrary
deadlines. Such a factor holds for polynomial-time schedulability tests and
exponential-time (exact) schedulability tests. Moreover, we also improve the
speedup factor to $2.84306$ when considering constrained-deadline task systems.
We also provide tight examples when the fitting strategy in the mapping stage
is arbitrary and $M$ is sufficiently large. For both constrained- and
arbitrary-deadline task systems, the analytical result surprisingly shows that
using exact tests does not gain theoretical benefits (with respect to speedup
factors) for an arbitrary fitting strategy.
