Randomized Assignment of Jobs to Servers in Heterogeneous Clusters of Shared Servers for Low Delay

Abstract

We consider the job assignment problem in a multi-server system consisting of N parallel processor sharing servers, categorized into M (≪ N ) different types according to their processing capacity or speed. Jobs of random sizes arrive at the system according to a Poisson process with rate N λ. Upon each arrival, a small number of servers from each type is sampled uniformly at random. The job is then assigned to one of the sampled servers based on a selection rule. We propose two schemes, each corresponding to a specific selection rule that aims at reducing the mean sojourn time of jobs in the system.We first show that both methods achieve the maximal stability region. We then analyze the system operating under the proposed schemes as N → ∞ which corresponds to the mean field. Our results show that asymptotic independence among servers holds even when M is finite and exchangeability holds only within servers of the same type. We further establish the existence and uniqueness of stationary solution of the mean field and show that the tail distribution of server occupancy decays doubly exponentially for each server type. When the estimates of arrival rates are not available, the proposed schemes offer simpler alternatives to achieving lower mean sojourn time of jobs, as shown by our numerical studies.

On the Throughput Optimization in Large-Scale Batch-Processing Systems

“…When started below 𝐿 1 , the evolution is governed by ( 24) initially and the system moves towards (𝑧 * 11 , 𝜂𝑧 * 11 ). This eventually changes the evolution dynamics to (25) Finally Part (iii) of the theorem follows by the same line arguments as in the proof of Part (iii) of Theorem 4.1. A more general result under the assumption of equal batch sizes is given in Appendix A.3.…”

“…Case 2, we see that if the initial point lies above 𝐿 1 , the evolution is given by (25) and the system converges to (𝑧 * 12 , 𝜂𝑧 * 12 ). When started below 𝐿 1 , the evolution is governed by ( 24) initially and the system moves towards (𝑧 * 11 , 𝜂𝑧 * 11 ).…”

“…denotes the fraction of active clients in the system. The process (𝑤 (𝑛) (𝑡), 𝑡 ≥ 0) is a density dependent jump Markov process [21,24,25] with rates…”

Asymptotics of Replication and Matching in Large Caching Systems

“…We are interested in the limiting behavior of the process x (n) as n → ∞. We first notice from its transition rates, that x (n) is a density dependent jump Markov process (see [26]- [28] for definition) with a conditional drift given by the mapping (14) and h i,j,K,r (w) = 0 if r = 0. For systems in which the drift h is Lipschitz continuous and satisfies certain regularity conditions, the classical results of Kurtz (Theorem 3.1 of [26]) imply that the process x (n) converges in distribution (and hence in probability) to the unique deterministic process x = (x(t), t ≥ 0) satisfyingẋ = h(x).…”

Large Time Behaviour and the Second Eigenvalue Problem for Finite State Mean-Field Interacting Particle Systems

“…In this paper, we study large time behaviour and the second eigenvalue problem for Markovian mean-field interacting particle systems with jumps. Our motivation is to provide an understanding of metastable phenomena in engineered systems such as load balancing networks [1,2,27,26,17], wireless local area networks [6,5,9,20,30,7], and in natural systems involving grammar acquisition, sexual evolution [29,28], epidemic spread [21,13], etc. These systems are briefly described in Section 1.…”