Asymptotic optimality of a greedy randomized algorithm in a large-scale service system with general packing constraints

Abstract

We consider a service system model primarily motivated by the problem of efficient assignment of virtual machines to physical host machines in a network cloud, so that the number of occupied hosts is minimized.There are multiple types of arriving customers, where a customer's mean service time depends on its type. There is an infinite number of servers. Multiple customers can be placed for service into one server, subject to general "packing" constraints. Service times of different customers are independent, even if served simultaneously by the same server. Each new arriving customer is placed for service immediately, either into a server already serving other customers (as long as packing constraints are not violated) or into an idle server. After a service completion, each customer leaves its server and the system.We propose an extremely simple and easily implementable customer placement algorithm, called Greedy-Random (GRAND). It places each arriving customer uniformly at random into either one of the already occupied servers (subject to packing constraints) or one of the so-called zero-servers, which are empty servers designated to be available to new arrivals. One instance of GRAND, called GRAND(aZ), where a ≥ 0 is a parameter, is such that the number of zero-servers at any given time t is aZ(t), where Z(t) is the current total number of customers in the system. We prove that GRAND(aZ) with a > 0 is asymptotically optimal, as the customer arrival rates grow to infinity and a → 0, in the sense of minimizing the total number of occupied servers in steady state. In addition, we study by simulations various versions of GRAND and observe the dependence of convergence speed and steadystate performance on the number of zero-servers.

Overcommitment in Cloud Services Bin packing with Chance Constraints

“…We therefore leave this question open. Note that several works consider bin-packing with item departures (see, e.g., Stolyar and Zhong (2015) and the references therein). In this work, the authors design a simple greedy algorithm for general packing constraints and show that it can be asymptotically optimal.…”

On Non-Preemptive VM Scheduling in the Cloud

“…[3,24,31,36,42]). The VM placement in an infinite server system has been studied in [13,[33][34][35]. Four closely related papers are [23], [21], [22], [11] where a finite model of the cloud is studied and preemptive [23] and non-preemptive [11,21,22] scheduling algorithms to stabilize the system are proposed.…”

Large-scale heterogeneous service systems with general packing constraints

“…The infinite-server variant of our model is a generalization of the homogeneous (one server type) model studied in [13][14][15], which focused on the problem of minimizing the number of occupied servers in steadystate. In particular, GRAND algorithm was proposed and shown to be asymptotically optimal in [15].…”

“…Note that the setting of the algorithm parameters a s , that achieves asymptotic optimality, depends only on the weights γ s , which are the parameters of the objective (as opposed to system parameters). One of the key qualitative insights of [15] was the surprising fact that an algorithm as simple as GRAND can be asymptotically optimal. The fact that an appropriately generalized, but still extremely simple, version of GRAND is optimal for in a heterogeneous system, is still more surprising.…”