2,015 research outputs found
A Matrix-Analytic Solution for Randomized Load Balancing Models with Phase-Type Service Times
In this paper, we provide a matrix-analytic solution for randomized load
balancing models (also known as \emph{supermarket models}) with phase-type (PH)
service times. Generalizing the service times to the phase-type distribution
makes the analysis of the supermarket models more difficult and challenging
than that of the exponential service time case which has been extensively
discussed in the literature. We first describe the supermarket model as a
system of differential vector equations, and provide a doubly exponential
solution to the fixed point of the system of differential vector equations.
Then we analyze the exponential convergence of the current location of the
supermarket model to its fixed point. Finally, we present numerical examples to
illustrate our approach and show its effectiveness in analyzing the randomized
load balancing schemes with non-exponential service requirements.Comment: 24 page
Doubly Exponential Solution for Randomized Load Balancing Models with General Service Times
In this paper, we provide a novel and simple approach to study the
supermarket model with general service times. This approach is based on the
supplementary variable method used in analyzing stochastic models extensively.
We organize an infinite-size system of integral-differential equations by means
of the density dependent jump Markov process, and obtain a close-form solution:
doubly exponential structure, for the fixed point satisfying the system of
nonlinear equations, which is always a key in the study of supermarket models.
The fixed point is decomposited into two groups of information under a product
form: the arrival information and the service information. based on this, we
indicate two important observations: the fixed point for the supermarket model
is different from the tail of stationary queue length distribution for the
ordinary M/G/1 queue, and the doubly exponential solution to the fixed point
can extensively exist even if the service time distribution is heavy-tailed.
Furthermore, we analyze the exponential convergence of the current location of
the supermarket model to its fixed point, and study the Lipschitz condition in
the Kurtz Theorem under general service times. Based on these analysis, one can
gain a new understanding how workload probing can help in load balancing jobs
with general service times such as heavy-tailed service.Comment: 40 pages, 4 figure
Decentralized List Scheduling
Classical list scheduling is a very popular and efficient technique for
scheduling jobs in parallel and distributed platforms. It is inherently
centralized. However, with the increasing number of processors, the cost for
managing a single centralized list becomes too prohibitive. A suitable approach
to reduce the contention is to distribute the list among the computational
units: each processor has only a local view of the work to execute. Thus, the
scheduler is no longer greedy and standard performance guarantees are lost.
The objective of this work is to study the extra cost that must be paid when
the list is distributed among the computational units. We first present a
general methodology for computing the expected makespan based on the analysis
of an adequate potential function which represents the load unbalance between
the local lists. We obtain an equation on the evolution of the potential by
computing its expected decrease in one step of the schedule. Our main theorem
shows how to solve such equations to bound the makespan. Then, we apply this
method to several scheduling problems, namely, for unit independent tasks, for
weighted independent tasks and for tasks with precendence constraints. More
precisely, we prove that the time for scheduling a global workload W composed
of independent unit tasks on m processors is equal to W/m plus an additional
term proportional to log_2 W. We provide a lower bound which shows that this is
optimal up to a constant. This result is extended to the case of weighted
independent tasks. In the last setting, precedence task graphs, our analysis
leads to an improvement on the bound of Arora et al. We finally provide some
experiments using a simulator. The distribution of the makespan is shown to fit
existing probability laws. The additive term is shown by simulation to be
around 3 \log_2 W confirming the tightness of our analysis
Performance driven distributed scheduling of parallel hybrid computations
AbstractExascale computing is fast becoming a mainstream research area. In order to realize exascale performance, it is necessary to have efficient scheduling of large parallel computations with scalable performance on a large number of cores/processors. The scheduler needs to execute in a pure distributed and online fashion, should follow affinity inherent in the computation and must have low time and message complexity. Further, it should also avoid physical deadlocks due to bounded resources including space/memory per core. Simultaneous consideration of these factors makes affinity driven distributed scheduling particularly challenging. We attempt to address this challenge for hybrid parallel computations which contain tasks that have pre-specified affinity to a place and also tasks that can be mapped to any place in the system. Specifically, we address two scheduling problems of the type Pm|Mj,prec|Cmax. This paper presents online distributed scheduling algorithms for hybrid parallel computations assuming both unconstrained and bounded space per place. We also present the time and message complexity for distributed scheduling of hybrid computations. To the best of our knowledge, this is the first time that distributed scheduling algorithms for hybrid parallel computations have been presented and analyzed for time and message bounds under both unconstrained space and bounded space
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