  We study the online bounded-delay packet scheduling problem (BDPS), where
packets of unit size arrive at a router over time and need to be transmitted
over a network link. Each packet has two attributes: a non-negative weight and
a deadline for its transmission. The objective is to maximize the total weight
of the transmitted packets. This problem has been well studied in the
literature, yet its optimal competitive ratio remains unknown: the best upper
bound is $1.828$, still quite far from the best lower bound of $\phi \approx
1.618$.
  In the variant of BDPS with $s$-bounded instances, each packet can be
scheduled in at most $s$ consecutive slots, starting at its release time. The
lower bound of $\phi$ applies even to the special case of $2$-bounded
instances, and a $\phi$-competitive algorithm for $3$-bounded instances was
given in Chin et al. Improving that result, and addressing a question posed by
Goldwasser, we present a $\phi$-competitive algorithm for $4$-bounded
instances.
  We also study a variant of BDPS where an online algorithm has the additional
power of $1$-lookahead, knowing at time $t$ which packets will arrive at time
$t+1$. For BDPS with $1$-lookahead restricted to $2$-bounded instances, we
present an online algorithm with competitive ratio $(\sqrt{13} - 1)/2 \approx
1.303$ and we prove a nearly tight lower bound of $(1 + \sqrt{17})/4 \approx
1.281$.
