Allocations with reflection, waiting time

Abstract

A new scheme of allocating particles with reflection is considered. For one variant of this scheme we present some exact and asymptotic results on the waiting time until all or almost all cells are occupied.

The process of occupation of cells in the scheme of allocation with reflection

“…Moreover, it follows from [1] that Therefore, in this case the transition matrix Ρ = \\PIJ\\Q has the distinct characteristic roots go = 0,<7i,...,qN-i,qN -1, and its powers P n = ||p^|| can be calculated by the method presented in [2,Chapter XVI].…”

“…(this formula for j = 0 was obtained by a different method in [1]). The corresponding integral relation has, obviously, the form…”

“…(13) From (12) and (13) it follows that the investigation of distributions of the process {ξ(η)} can be reduced to a similar problem for the waiting process {i>j*(p), 0 < & < Ν -j} whose structure is highly transparent and therefore its research is sufficiently simple. Namely, it follows from [1] that…”

“…In the paper [1] a new scheme of allocating particles, called the p-scheme, was introduced. In this scheme particles are allocated consecutively, independently of each other and equiprobably into TV cells, but in contrast to the classical scheme, some repeated throwings may be necessary in order to allocate a particle in view of the fact that particles may be reflected by occupied cells.…”

“…One of such characteristics, the waiting time until all or almost all sells are occupied, was considered in detail in [1]. In the present paper, continuing the waiting time theme, we focus our attention on another characteristic which is popular in allocation schemes, namely, on the number of empty cells μο(η) = μο( η > Ν, ρ) or, what is equivalent, on the number of occupied cells £(n) = Ν-μο(η) after allocating η particles.…”