On the Analysis of a Random Interleaving Walk–Jump Process with Applications to Testing
Anis Yazidi, Ole‐Christoffer Granmo, B. John Oommen · Sequential Analysis · 2011
Although random walks (RWs) with single-step transitions have been extensively studied for almost a century as seen in Feller (1968 Feller , F. ( 1968 ). An Introduction to Probability Theory and Its Applications , vol. 1 , third edition, New York : Wiley . [Google Scholar]), problems involving the analysis of RWs that contain interleaving random steps and random “jumps” are intrinsically hard. In this article, we consider the analysis of one such fascinating RW, where every step is paired with its counterpart random jump. In addition to this RW being conceptually interesting, it has applications in testing of entities (components or personnel), where the entity is never allowed to make more than a prespecified number of consecutive failures. The article contains the analysis of the chain, some fascinating limiting properties, and simulations that justify the analytic steady-state results. Some simulation results for the chain's transient behavior are also included. Finally, a comparative testing against a hidden Markov model shows that within the testing framework, the results of our model are competitive, if not superior. As far as we know, the entire field of RWs with interleaving steps and jumps is novel, and we believe that this is a pioneering article in this field.