Linear probing with lazy deletions, parking problem with exponential sojourn time, receiver release Cambridge ring and other related problems
Résumé
In this paper we intend to give an exact analytical evaluation of some parameters arising in a general bin-packing problem. There is an infinite sequence of cells ranked from the left to the right. A cell is either empty, either busy, i.e. contains an item. Items are generated over the cells according to independent Poisson processes of l item per cell and per time unit. When an item is generated on an empty cell it enters the cell which therefore becomes busy. When an item is generated on a busy cell, it tries the next cell on the right and so forth, until it finds an empty cell which it enters. Each stored item has a sojourn time in the cell which is supposed to be exponential of mean one. When the item leaves a cell it disappears completely, i.e. quits the system, the cell immediately becomes empty and available for a next item. We give an exact evaluation of the steady state of the system in terms of a Taylor expansion with respect to variable l.