Colliding stacks: A large deviations analysis

Abstract

We analyze the performance of a prototypical scheme for shared storage allocation: two initially empty stacks evolving in a contiguous block of memory of size m. We treat the case in which the stacks are more likely to shrink than grow, but with the probabilities of insertion and deletion allowed to depend arbitrarily on stack height as a fraction of m. New results are obtained on the m+ m asymptotics of the stack collision time, and of the final stack heights. The results of Wentzell and Freidlin on the large deviations of Markov chains are used, and the relation of their formalism to the Hamiltonian formulation of classical mechanics is emphasized. Certain results on higher-order asymptotics follow from WKB expansions.

Distributed algorithms with dynamical random transitions

“…Additional results for this model have been proved by Louchard [13]. Maier [16] provided a large deviation analysis of colliding stacks for the more difficult case in which the transition probabilities are nontrivially state-dependent. Flajolet [5] had earlier provived a combinatorial analysis and was able to extend previous results of Yao [20].…”

“…This algorithm is to be compared to another strategy, namely allocating separate zones of size m/ 2 to each stack. This problem of Knuth [11] has been investigated by Yao [20], Flajolet [5], Louchard and Schott [14], Louchard [13], and Maier [16]. The natural formulation of the two stacks problem is in terms of random walks in a triangle in a two-dimensional lattice space: A state is a pair formed by the size of both stacks.…”

Optimal management of two parallel stacks in two-level memory

“…In [6][7][8][9][10][11], a mathematical model of the process was constructed as a two-dimensional random walk in a triangular domain with two reflecting barriers and one absorbing barrier.…”

“…In [8][9][10][11], the asymptotic behaviour of the stack sizes when they collide and the run-time before overflow were studied for 0 p 1=4.…”

Optimal Implementation of Two FIFO-Queues in Single-Level Memory

“…In [2][3][4][5][6] a mathematical model of the process was constructed as twodimensional random walk in a triangle with two reflecting and one absorbing barriers. In [7][8] we consider the problem of one and two stacks managing in two-level memory.…”