4 research outputs found
Approximating fluid schedules in packet-switched networks
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 145-151).We consider a problem motivated by the desire to provide exible, rate-based, quality of service guarantees for packets sent over switches and switch networks. Our focus is solving a type of on-line, traffic scheduling problem, whose input at each time step is a set of desired traffic rates through the switch network. These traffic rates in general cannot be exactly achieved since they treat the incoming data as fluid, that is, they assume arbitrarily small fractions of packets can be transmitted at each time step. The goal of the traffic scheduling problem is to closely approximate the given sequence of traffic rates by a sequence of switch uses throughout the network in which only whole packets are sent. We prove worst-case bounds on the additional delay and buffer use that result from using such an approximation. These bounds depend on the network topology, the resources available to the scheduler, and the types of fluid policy allowed.by Michael Aaron Rosenblum.Ph.D
The Variable-Processor Cup Game
The problem of scheduling tasks on processors so that no task ever gets
too far behind is often described as a game with cups and water. In the
-processor cup game on cups, there are two players, a filler and an
emptier, that take turns adding and removing water from a set of cups. In
each turn, the filler adds units of water to the cups, placing at most
unit of water in each cup, and then the emptier selects cups to remove up
to unit of water from. The emptier's goal is to minimize the backlog, which
is the height of the fullest cup.
The -processor cup game has been studied in many different settings,
dating back to the late 1960's. All of the past work shares one common
assumption: that is fixed. This paper initiates the study of what happens
when the number of available processors varies over time, resulting in what
we call the \emph{variable-processor cup game}.
Remarkably, the optimal bounds for the variable-processor cup game differ
dramatically from its classical counterpart. Whereas the -processor cup has
optimal backlog , the variable-processor game has optimal
backlog . Moreover, there is an efficient filling strategy that
yields backlog in quasi-polynomial time against any
deterministic emptying strategy.
We additionally show that straightforward uses of randomization cannot be
used to help the emptier. In particular, for any positive constant ,
and any -greedy-like randomized emptying algorithm , there
is a filling strategy that achieves backlog against
in quasi-polynomial time
Optimal Time-Backlog Tradeoffs for the Variable-Processor Cup Game
The \emph{-processor cup game} is a classic and widely studied scheduling
problem that captures the setting in which a -processor machine must assign
tasks to processors over time in order to ensure that no individual task ever
falls too far behind. The problem is formalized as a multi-round game in which
two players, a filler (who assigns work to tasks) and an emptier (who schedules
tasks) compete. The emptier's goal is to minimize backlog, which is the maximum
amount of outstanding work for any task.
Recently, Kuszmaul and Westover (ITCS, 2021) proposed the
\emph{variable-processor cup game}, which considers the same problem, except
that the amount of resources available to the players (i.e., the number of
processors) fluctuates between rounds of the game. They showed that this
seemingly small modification fundamentally changes the dynamics of the game:
whereas the optimal backlog in the fixed -processor game is , independent of , the optimal backlog in the variable-processor game is
. The latter result was only known to apply to games with
\emph{exponentially many} rounds, however, and it has remained an open question
what the optimal tradeoff between time and backlog is for shorter games.
This paper establishes a tight trade-off curve between time and backlog in
the variable-processor cup game. Importantly, we prove that for a game
consisting of rounds, the optimal backlog is if and only if . Our techniques also allow for us to resolve several other
open questions concerning how the variable-processor cup game behaves in
beyond-worst-case-analysis settings.Comment: 40 pages, published in International Conference on Automata,
Languages, and Programming (ICALP), 2022. Abstract abridged for arXiv
submission: see paper for full abstract. Updated to acknowledge additional
fundin