  We consider a variant of the online buffer management problem in network
switches, called the $k$-frame throughput maximization problem ($k$-FTM). This
problem models the situation where a large frame is fragmented into $k$ packets
and transmitted through the Internet, and the receiver can reconstruct the
frame only if he/she accepts all the $k$ packets. Kesselman et al.\ introduced
this problem and showed that its competitive ratio is unbounded even when
$k=2$. They also introduced an "order-respecting" variant of $k$-FTM, called
$k$-OFTM, where inputs are restricted in some natural way. They proposed an
online algorithm and showed that its competitive ratio is at most
$\frac{2kB}{\lfloor B/k \rfloor} + k$ for any $B \ge k$, where $B$ is the size
of the buffer. They also gave a lower bound of $\frac{B}{\lfloor 2B/k \rfloor}$
for deterministic online algorithms when $2B \geq k$ and $k$ is a power of 2.
In this paper, we improve upper and lower bounds on the competitive ratio of
$k$-OFTM. Our main result is to improve an upper bound of $O(k^{2})$ by
Kesselman et al.\ to $\frac{5B + \lfloor B/k \rfloor - 4}{\lfloor B/2k \rfloor}
= O(k)$ for $B\geq 2k$. Note that this upper bound is tight up to a
multiplicative constant factor since the lower bound given by Kesselman et al.\
is $\Omega(k)$. We also give two lower bounds. First we give a lower bound of
$\frac{2B}{\lfloor {B/(k-1)} \rfloor} + 1$ on the competitive ratio of
deterministic online algorithms for any $k \geq 2$ and any $B \geq k-1$, which
improves the previous lower bound of $\frac{B}{\lfloor 2B/k \rfloor}$ by a
factor of almost four. Next, we present the first nontrivial lower bound on the
competitive ratio of randomized algorithms. Specifically, we give a lower bound
of $k-1$ against an oblivious adversary for any $k \geq 3$ and any $B$.
