6,201 research outputs found
Asymptotically Optimal Load Balancing Topologies
We consider a system of servers inter-connected by some underlying graph
topology . Tasks arrive at the various servers as independent Poisson
processes of rate . Each incoming task is irrevocably assigned to
whichever server has the smallest number of tasks among the one where it
appears and its neighbors in . Tasks have unit-mean exponential service
times and leave the system upon service completion.
The above model has been extensively investigated in the case is a
clique. Since the servers are exchangeable in that case, the queue length
process is quite tractable, and it has been proved that for any ,
the fraction of servers with two or more tasks vanishes in the limit as . For an arbitrary graph , the lack of exchangeability severely
complicates the analysis, and the queue length process tends to be worse than
for a clique. Accordingly, a graph is said to be -optimal or
-optimal when the occupancy process on is equivalent to that on
a clique on an -scale or -scale, respectively.
We prove that if is an Erd\H{o}s-R\'enyi random graph with average
degree , then it is with high probability -optimal and
-optimal if and as , respectively. This demonstrates that optimality can
be maintained at -scale and -scale while reducing the number of
connections by nearly a factor and compared to a
clique, provided the topology is suitably random. It is further shown that if
contains bounded-degree nodes, then it cannot be -optimal.
In addition, we establish that an arbitrary graph is -optimal when its
minimum degree is , and may not be -optimal even when its minimum
degree is for any .Comment: A few relevant results from arXiv:1612.00723 are included for
convenienc
Overshoots in stress strain curves: Colloid experiments and schematic mode coupling theory
The stress versus strain curves in dense colloidal dispersions under start-up
shear flow are investigated combining experiments on model core-shell
microgels, computer simulations of hard disk mixtures, and mode coupling
theory. In dense fluid and glassy states, the transient stresses exhibit first
a linear increase with the accumulated strain, then a maximum ('stress
overshoot') for strain values around 5%, before finally approaching the
stationary value, which makes up the flow curve. These phenomena arise in
well-equilibrated systems and for homogeneous flows, indicating that they are
generic phenomena of the shear-driven transient structural relaxation.
Microscopic mode coupling theory (generalized to flowing states by integration
through the transients) derives them from the transient stress correlations,
which first exhibit a plateau (corresponding to the solid-like elastic shear
modulus) at intermediate times, and then negative stress correlations during
the final decay. We introduce and validate a schematic model within mode
coupling theory which captures all of these phenomena and handily can be used
to jointly analyse linear and large-amplitude moduli, flow curves, and
stress-strain curves. This is done by introducing a new strain- and
time-dependent vertex into the relation between the the generalized shear
modulus and the transient density correlator.Comment: 21 pages, 13 figure
Routing and Staffing when Servers are Strategic
Traditionally, research focusing on the design of routing and staffing
policies for service systems has modeled servers as having fixed (possibly
heterogeneous) service rates. However, service systems are generally staffed by
people. Furthermore, people respond to workload incentives; that is, how hard a
person works can depend both on how much work there is, and how the work is
divided between the people responsible for it. In a service system, the routing
and staffing policies control such workload incentives; and so the rate servers
work will be impacted by the system's routing and staffing policies. This
observation has consequences when modeling service system performance, and our
objective is to investigate those consequences.
We do this in the context of the M/M/N queue, which is the canonical model
for large service systems. First, we present a model for "strategic" servers
that choose their service rate in order to maximize a trade-off between an
"effort cost", which captures the idea that servers exert more effort when
working at a faster rate, and a "value of idleness", which assumes that servers
value having idle time. Next, we characterize the symmetric Nash equilibrium
service rate under any routing policy that routes based on the server idle
time. We find that the system must operate in a quality-driven regime, in which
servers have idle time, in order for an equilibrium to exist, which implies
that the staffing must have a first-order term that strictly exceeds that of
the common square-root staffing policy. Then, within the class of policies that
admit an equilibrium, we (asymptotically) solve the problem of minimizing the
total cost, when there are linear staffing costs and linear waiting costs.
Finally, we end by exploring the question of whether routing policies that are
based on the service rate, instead of the server idle time, can improve system
performance.Comment: First submitted for journal publication in 2014; accepted for
publication in Operations Research in 2016. Presented in select conferences
throughout 201
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