We study the distribution of the sandpile group of random d-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the p-Sylow subgroup of the sandpile group is a given p-group P , is proportional to | Aut(P )| −1 . For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.This answers an open question of Frieze and Vu whether the adjacency matrix of a random regular graph is almost surely invertible. Note that for directed graphs this was recently proved by Huang. It also gives an alternate proof of a theorem of Backhausz and Szegedy. MSC classes: 05C80, 15B52, 60B20 Theorem 2. Let Γ n be the sandpile group of D n . For any finite Abelian group V we have lim n→∞ E| Sur(Γ n , V )| = 1.Let Γ n be the sandpile group of H n . Let V be a finite Abelian group. If d is odd, then lim n→∞ E| Sur(Γ n , V )| = | ∧ 2 V |, 1 The rank of a group is the minimum number of generators.

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Recent developments of matrix analytic methods make phase type distributions (PHs) and Markov Arrival Processes (MAPs) promising stochastic model candidates for capturing traffic trace behaviour and for efficient usage in queueing analysis. After introducing basics of these sets of stochastic models, the paper discusses the following subjects in detail: (i) PHs and MAPs have different representations. For efficient use of these models, sparse (defined by a minimal number of parameters) and unique representations of discrete time PHs and MAPs are needed, which are commonly referred to as canonical representations. The paper presents new results on the canonical representation of discrete PHs and MAPs.(ii) The canonical representation allows a direct mapping between experimental moments and the stochastic models, referred to as moment matching. Explicit procedures are provided for this mapping. (iii) Moment matching is not always the best way to model the behavior of traffic traces. Model fitting based on appropriately chosen distance measures might result in better performing stochastic models. We also demonstrate the efficiency of fitting procedures with experimental results.

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