Approximations and Bounds for (n, k) Fork-Join Queues: A Linear Transformation Approach

Compared to basic fork-join queues, a job in (n, k) fork-join queues only needs its k out of all n sub-tasks to be finished. Since (n, k) fork-join queues are prevalent in popular distributed systems, erasure coding based cloud storages, and modern network protocols like multipath routing, estimating the sojourn time of such queues is thus critical for the performance measurement and resource plan of computer clusters. However, the estimating keeps to be a well-known open challenge for years, and only rough bounds for a limited range of load factors have been given. In this paper, we developed a closed-form linear transformation technique for jointly-identical random variables: An order statistic can be represented by a linear combination of maxima. This brand-new technique is then used to transform the sojourn time of non-purging (n, k) fork-join queues into a linear combination of the sojourn times of basic (k, k), (k+1, k+1), ..., (n, n) fork-join queues. Consequently, existing approximations for basic fork-join queues can be bridged to the approximations for non-purging (n, k) fork-join queues. The uncovered approximations are then used to improve the upper bounds for purging (n, k) fork-join queues. Simulation experiments show that this linear transformation approach is practiced well for moderate n and relatively large k.Comment: 10 page

Cohomology of generalized restricted Lie algebras

AbstractIn this note, the generalized restricted Lie algebra, which was introduced by Shu Bin in [J. Algebra 194 (1997) 157–177], is studied. By generalizing the concept of restricted subalgebras and the concept of restricted homomorphism, we show that the second generalized restricted cohomology HϕL2(L,M) is isomorphic to the equivalence classes of those generalized restricted extension of M by L. For any generalized restricted Lie algebra (L,BL,ϕL) and any generalized restricted L-module M, we show that the sequence 0→HϕL1(L,M)→H1(L,M)→homFL,ML→HϕL2(L,M)→H2(L,M)→homFL,H1(L,M) is exact

Statistically Steady State Large‐Eddy Simulations Forced by an Idealized GCM: 1. Forcing Framework and Simulation Characteristics

Using large‐eddy simulations (LES) systematically has the potential to inform parameterizations of subgrid‐scale processes in general circulation models (GCMs), such as turbulence, convection, and clouds. Here we show how LES can be run to simulate grid columns of GCMs to generate LES across a cross section of dynamical regimes. The LES setup approximately replicates the thermodynamic and water budgets in GCM grid columns. Resolved horizontal and vertical transports of heat and water and large‐scale pressure gradients from the GCM are prescribed as forcing in the LES. The LES are forced with prescribed surface temperatures, but atmospheric temperature and moisture are free to adjust, reducing the imprinting of GCM fields on the LES. In both the GCM and LES, radiative transfer is treated in a unified but idealized manner (semigray atmosphere without water vapor feedback or cloud radiative effects). We show that the LES in this setup reaches statistically steady states without nudging to thermodynamic GCM profiles. The steady states provide training data for developing GCM parameterizations. The same LES setup also provides a good basis for studying the cloud response to global warming