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

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 depen...

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Published in:	Queueing systems Vol. 79; no. 2; pp. 117 - 143
Main Authors:	Stolyar, Alexander L., Zhong, Yuan
Format:	Journal Article
Language:	English
Published:	Boston Springer US 01.02.2015 Springer Nature B.V
Subjects:	Algorithms Arrivals Asymptotic methods Asymptotic properties Business and Management Cloud computing Clouds Computer Communication Networks Control Customer services Customers Greedy algorithms Networks Operations Research/Decision Theory Optimization Packaging Packing problem Probability Theory and Stochastic Processes Queuing theory Servers Simulation Studies Supply Chain Management Systems Theory United States > US Fluid limit Cloud computing Infinite server system Greedy random algorithm 90B15 Virtual machine Queueing networks 60K25 Stochastic bin packing
ISSN:	0257-0130, 1572-9443
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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( a Z ), where a ≥ 0 is a parameter, is such that the number of zero-servers at any given time t is a Z ( t ) , where Z ( t ) is the current total number of customers in the system. We prove that GRAND( a Z ) 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 steady-state performance on the number of zero-servers.
AbstractList	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( a Z ), where a ≥ 0 is a parameter, is such that the number of zero-servers at any given time t is a Z ( t ) , where Z ( t ) is the current total number of customers in the system. We prove that GRAND( a Z ) 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 steady-state performance on the number of zero-servers. 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 customers 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 steady-state performance on the number of zero-servers. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).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(...), where ... is a parameter, is such that the number of zero-servers at any given time ... is ..., where ... is the current total number of customers in the system. We prove that GRAND(...) with ... is asymptotically optimal, as the customer arrival rates grow to infinity and ..., 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 steady-state performance on the number of zero-servers.
Author	Stolyar, Alexander L. Zhong, Yuan
Author_xml	– sequence: 1 givenname: Alexander L. surname: Stolyar fullname: Stolyar, Alexander L. email: stolyar@research.bell-labs.com organization: Bell Labs, Alcatel-Lucent – sequence: 2 givenname: Yuan surname: Zhong fullname: Zhong, Yuan organization: Columbia University
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Cites_doi	10.1239/jap/1175267175 10.1137/080736831 10.1287/opre.1110.0960 10.2307/1426976 10.1239/jap/1014842828 10.1023/B:QUES.0000046581.34849.cf 10.1214/aoap/1177005872 10.1145/1120582.1120583 10.1287/opre.2013.1184 10.2307/1427309 10.1017/CBO9780511804441 10.1109/INFCOM.2013.6566847
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Keywords	Fluid limit Cloud computing Infinite server system Greedy random algorithm 90B15 Virtual machine Queueing networks 60K25 Stochastic bin packing
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SubjectTerms	Algorithms Arrivals Asymptotic methods Asymptotic properties Business and Management Cloud computing Clouds Computer Communication Networks Control Customer services Customers Greedy algorithms Networks Operations Research/Decision Theory Optimization Packaging Packing problem Probability Theory and Stochastic Processes Queuing theory Servers Simulation Studies Supply Chain Management Systems Theory
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Title	Asymptotic optimality of a greedy randomized algorithm in a large-scale service system with general packing constraints
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