An Infinite Server 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 input flows of different type customers, with a customer mean service time depending on its type. There is an infinite number of servers. A server-packing configuration is the vector k = {ki}, where ki is the number of type i customers the server “contains.” Packing constraints must be observed; namely, there is a fixed finite set of configurations k that are allowed. Service times of different customers are independent; after a service completion, each customer leaves its server and the system. Each new arriving customer is placed for service immediately; it can be placed into a server already serving other customers (as long as packing constraints are not violated), or into an idle server. We consider a simple parsimonious real-time algorithm, called Greedy, that attempts to minimize the increment of the objective function [Formula: see text], α > 0, caused by each new assignment; here Xk is the number of servers in configuration k. (When α is small, [Formula: see text] approximates the total number [Formula: see text] of occupied servers.) Our main results show that certain versions of the Greedy algorithm are asymptotically optimal, in the sense of minimizing [Formula: see text] in stationary regime as the input flow rates grow to infinity. We also show that in the special case when the set of allowed configurations is determined by vector-packing constraints, the Greedy algorithm can work with aggregate configurations as opposed to exact configurations k, thus reducing computational complexity while preserving the asymptotic optimality.