The Maximum Size of Dynamic Data Structures

This paper develops two probabilistic methods that allow the analysis of the maximum data structure size encountered during a sequence of insertions and deletions in data structures such as priority queues, dictionaries, linear lists, and symbol tables, and in sweepline structures for geometry and Very-Large-Scale-Integration (VLSI) applications. The notion of the "maximum" is basic to issues of resource preallocation. The methods here are applied to combinatorial models of file histories and probabilistic models, as well as to a non-Markovian process (algorithm) for processing sweepline information in an efficient way, called "hashing with lazy deletion" (HwLD). Expressions are derived for the expected maximum data structure size that are asymptotically exact, that is, correct up to lower-order terms; in several cases of interest the expected value of the maximum size is asymptotically equal to the maximum expected size. This solves several open problems, including longstanding questions in queueing theory. Both of these approaches are robust and rely upon novel applications of techniques from the analysis of algorithms. At a high level, the first method isolates the primary contribution to the maximum and bounds the lesser effects. In the second technique the continuous-time probabilistic model is related to its discrete analog--the maximum slot occupancy in hashing

Random trees in queueing systems with deadlines

AbstractWe survey our research on scheduling aperiodic tasks in real-time systems in order to illustrate the benefits of modelling queueing systems by means of random trees. Relying on a discrete-time single-server queueing system, we investigated deadline meeting properties of several scheduling algorithms employed for servicing probabilistically arriving tasks, characterized by arbitrary arrival and execution time distributions and a constant service time deadline T. Taking a non-queueing theory approach (i.e., without stable-stable assumptions) we found that the probability distribution of the random time sT where such a system operates without violating any task's deadline is approximately exponential with parameter λT = 1μT, with the expectation E[sT] = μT growing exponentially in T. The value μT depends on the particular scheduling algorithm, and its derivation is based on the combinatorial and asymptotic analysis of certain random trees. This paper demonstrates that random trees provide an efficient common framework to deal with different scheduling disciplines and gives an overview of the various combinatorial and asymptotic methods used in the appropriate analysis