Necessary and Sufficient Conditions for Stability of a Bin-packing System

Objects of various integer sizes, o,, o , are to be packed together into bins of size N as they arrive at a service facility. The number of objects of size o, which arrive by time t is A , where the components of A' = (A , * , A ')' are independent renewal processes, with A'/t A as t --oo. The empty space in those bins which are neither empty nor full at time t is called the wasted space and the system is declared stabilizable if for some finite B there exists a bin-packing algorithm whose use guarantees the expected wasted space is less than B for all t. We show that the system is stabilizable if the arrival processes are Poisson and A lies in the interior of a certain convex polyhedral cone A. In this case there exists a bin-packing algorithm which stabilizes the system without needing to know A. However, if A lies on the boundary of A the wasted space grows as O(V\t) and if A is exterior to A it grows as O(t); these conclusions hold even if objects may be repacked as often as desired. ONLINE ALGORITHMS; RANDOM WALKS; STOCHASTIC ALGORITHMS 1. Problem description and results In a one-dimensional bin-packing model we suppose that objects of various integer sizes in the set O = {o1, , o *, n} are to be packed together into bins of integer size N as they arrive at a service facility. The number of objects of size o, which arrive by time t is A', where the components of A' = (A, * , A t)' are independent renewal processes, with A'/t-A =(A,' , An)' as t -o. The empty space in those bins which are neither empty nor full at time t is called the wasted space and this is denoted by W'. The system is declared stabilizable if for some finite B there exists a bin-packing algorithm whose use guarantees that the expected wasted space is less than B for all t. Of course this is equivalent to the notion that the expected number of partially full bins remains uniformly bounded as time increases. Received 26 July 1985; revision received 29 October 1985. * Postal address: AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA. ** Postal address: Queens' College, Cambridge CB3 9ET, UK.