Computing Traversal Times on Dynamic Markovian Paths

TL;DR

This paper develops methods to compute the expected traversal time for paths in dynamic, stochastic graphs, addressing the complexity introduced by random edge availability and dependencies.

Contribution

It introduces a quadratic-time algorithm using probability generating functions for calculating ETT in initial configuration scenarios, along with linear bounds.

Findings

Quadratic-time algorithm for ETT computation in initial configuration cases

Linear upper and lower bounds on ETT

ETT can be efficiently approximated in dynamic Markovian paths

Abstract

In source routing, a complete path is chosen for a packet to travel from source to destination. While computing the time to traverse such a path may be straightforward in a fixed, static graph, doing so becomes much more challenging in dynamic graphs, in which the state of an edge in one time slot (i.e., its presence or absence) is random, and may depend on its state in the previous time step. The traversal time is due to both time spent waiting for edges to appear and time spent crossing them once they become available. We compute the expected traversal time (ETT) for a dynamic path in a number of special cases of stochastic edge dynamics models, and for three edge failure models, culminating in a surprisingly challenging yet realistic setting in which the initial configuration of edge states for the entire path is known. We show that the ETT for this "initial configuration" setting can be computed in quadratic time, by an algorithm based on probability generating functions. We also give several linear-time upper and lower bounds on the ETT.