# Optimal Node Visitation in Acyclic Stochastic Digraphs with Multi-threaded Traversals and Internal Visitation Requirements

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## Abstract

The original definition of the problem of optimal node visitation (ONV) in acyclic stochastic digraphs concerns the identification of a routing policy that will enable the visitation of each *leaf* node a requested number of times, while minimizing the expected number of the graph traversals. The original work of Bountourelis and Reveliotis (2006) formulated this problem as a Stochastic Shortest Path (SSP) problem, and since the state space of this SSP formulation is exponentially sized with respect to the number of the target nodes, it also proposed a suboptimal policy that is computationally tractable and asymptotically optimal. This paper extends the results of Bountourelis and Reveliotis (2006) to the cases where (i) the tokens traversing the graph can “*split*” during certain transitions to a number of (sub-)tokens, allowing, thus, the satisfaction of many visitation requirements during a single graph traversal, and (ii) there are additional visitation requirements attached to the *internal* graph nodes, which, however, can be served only when the visitation requirements of their successors have been fully met. In addition, the presented set of results establishes stronger convergence properties for the proposed suboptimal policies, and it provides a formal complexity analysis of the considered ONV formulations. From a practical standpoint, the extension of the original results performed in this paper enables their effective usage in the application domains that motivated the ONV problem, in the first place.

## Keywords

Optimal node visitation Acyclic stochastic digraphs Stochastic shortest path problems Stochastic scheduling Fluid relaxation## Notes

### Acknowledgement

This work was partially supported by NSF grants DMI-MES-0318657 and CMMI-0619978.

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