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

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.

Appendix: Proof of Lemma 1

Let ψ′ _n  =  min {k: S _k  > n·c}. Then ψ′ _n is a stopping time and, from Lemma 2.3 of Gut (1974), we have that

$$E\left[\left(\sum_{i=1}^{\psi'_n}(X_i-\mu)\right)^r\right] \leq C(r,E[X^r])\cdot E[(\psi'_n)^{r/2}] \label{eq:xx} $$

(63)

where C(r,E[X ^r]) is a constant depending only on r and E[X ^r]. Equation 63 further implies that

$$E\left[n^{-r/2}\cdot\left(\sum_{i=1}^{\psi'_n}(X_i-\mu)\right)^r\right] \leq C(r, E[X^r])\cdot E\left[\left(\frac{\psi'_n}{n}\right)^{r/2}\right] \label{eq:unif_int_1} $$

(64)

From Eq. 64 and Theorem 2.3 of Gut (1974), we get

$$ \sup_{n\geq 1} E\left[n^{-r/2}\cdot\left(\sum_{i=1}^{\psi'_n}(X_i-\mu)\right)^r\right]< \infty $$

(65)

which implies the uniform integrability of $\left\{n^{-r/2}\cdot\left(\sum_{i=1}^{\psi'_n}(X_i-\mu)\right)^r, n\geq 1\right\}$ (Billingsley 1968).

By the definition of the renewal process ψ′ _n ,

$$ n \cdot c = \sum_{i=1}^{\psi'_n}X_i + \left(\sum_{i=1}^{\psi'_n}X_i - n\cdot c\right) $$

(66)

which further implies that

$$ n^{-1/2}\cdot(n \cdot c -\mu\cdot \psi'_n) =n^{-1/2}\cdot \sum_{i=1}^{\psi'_n}(X_i -\mu) + n^{-1/2}\cdot \left(\sum_{i=1}^{\psi'_n}X_i - n\cdot c\right) \label{eq:xxx} $$

(67)

Equation 67 combined with the triangle inequality and the fact that

$$ 0\leq \sum_{i=1}^{\psi'_n}X_i - n\cdot c \leq K $$

(68)

also imply that

$$ |n^{-1/2}\cdot(n\cdot c- \mu\cdot\psi'_n)| \leq |n^{-1/2}\cdot\sum_{i=1}^{\psi'_n}(X_i-\mu)|+ n^{-1/2}\cdot K $$

(69)

and based on the inequality (a + b)^r ≤ 2^r − 1·(|a|^r + |b|^r), a,b ∈ R, we finally get

$$ |n^{-1/2}(n\cdot c- \mu\cdot\psi'_n)|^r \leq 2^{r-1}\cdot\left(|n^{-1/2}\sum_{i=1}^{\psi'_n}(X_i-\mu)|^r+ n^{-r/2}\cdot K^r\right) \label{eq:ineq4} $$

(70)

Hence, the uniform integrability of $\{n^{-r/2}\cdot(\sum_{i=1}^{\psi'_n}(X_i-\mu))^r, n\geq 1\}$ and Eq. 70 imply the uniform integrability of $\{n^{-r/2}\cdot(n\cdot c- \mu\cdot\psi'_n)^r, n\geq 1\}$. Since ψ′ _n  = ψ _n  + 1 we have that

$$ n^{-1/2}\cdot(n\cdot c- \mu\cdot\psi_n) = n^{-1/2}\cdot(n\cdot c- \mu\cdot\psi'_n)+n^{-1/2}\cdot \mu $$

(71)

which gives

$$ n^{-r/2}\cdot|n\cdot c- \mu\cdot\psi_n|^r \leq 2^{r-1}\cdot(n^{-r/2}\cdot |n\cdot c- \mu\cdot\psi'_n|^r+n^{-r/2}\cdot \mu^r) $$

(72)

and implies the uniform integrability of $\{n^{-r/2}\cdot(n\cdot c-\mu\cdot\psi_n)^r,\ n\geq 1\}$.