Exact Controllability for Dependent Siphons in S3PMR

Abstract

This papers reports controllability results of compound siphons in an S3PMR - a generalization of S3PR by allowing a job place to use more than one resource and when entering the next operation place, it may not release the resource. We categorize dependent siphons into Types I and II. The former (resp. latter) shares resources between the same (resp. different) set of processes. There are two types of Type I dependent siphons: strong and weak. In an earlier paper, we showed that the dependent condition may be relaxed so that an elementary siphon (actually a compound siphon), while requiring a monitor previously, may be controlled after some elementary siphons get controlled. However, the result may not be optimal so that further linear integer programming (as in Li & Zhou’s approach to reduce the total number of monitors from exponential to linear) is needed to determine whether to adjust control depth variables. We derive the exact (both sufficient and necessary) controllability for a Type I strongly 2-dependent siphon and show that any n-dependent (n>3) needs no monitor. Thus, the total time complexity to verify controllability for all Type I strongly relaxed dependent siphons is reduced from exponential to linear. Furthermore, we derive the time complexity for controllability verification of Type II dependent siphons to be |∏E||∏D| (product of total number of elementary siphons and total number of dependent siphons).