Controllability of weakly dependent siphons under elementary-siphon control

Abstract

Monitors are added to problematic siphons to avoid deadlocks. Li and Zhou (2004, 2006a,b, 2008a add monitors to elementary siphons only while controlling the rest-dependent siphons to save on costs. After failing a marking linear inequality (MLI) test, Li and Zhou perform a linear integer programming (LIP) test (NP-hard). We proposed a new MLI test earlier to avoid the LIP and extended it to systems of simple sequential processes with general resource requirements (S 3 PGR 2 ) for strongly dependent siphons (SDSs). The control policy for weakly dependent siphons (WDSs) is rather conservative due to some negative terms in the MLI. This paper shows that WDS and SDS have the same controllability (i.e. MLI). As a result, the control for WDS need no longer be that conservative. We also develop an optimization (by redundancy elimination) of the computation required for the LIP test to ensure deadlock prevention of systems of simple sequential processes with resources (S 3 PR). A favourable result for this policy is that any n-dependent (n . 2) WDS (similar to SDS) needs no monitor and hence the complexity for synthesizing a controller becomes polynomial. Application to a slight variant of a well-known benchmark is illustrated.

Modelling and Control of Resource Allocation Systems within Discrete Event Systems by Means of Petri Nets -- Part 1: Invariants, Siphons and Traps in Deadlock Avoidance

“…Of course, since that time the deadlock theory, and especially the deadlock avoidance one, has been intensively developed -see e.g. [46,78,79,80,81,75,74,73,82,86,87] and especially [63,8,25,55,56,57,59,62,64,28,29,30,31,32,33,34,35,36,37,38,39,40,92,93,94,91,88,89,65,66,67,68,69,58,60,61,47,48,49,50,51,…”

“…Let S ⊆ P be a subset of places of N . The (n × 1) vector S σ σ σ is called [58,51,91] the characteristic P -vector of S if ∀p ∈ S : S σ p = 1; otherwise S σ p = 0. The (m × 1)…”

“…The T -vector S γ γ γ is associated [91] with each siphon S such that S γ(i) is the number of tokens gained in or lost from S by firing the transition t i once. A dependent siphon S 0 strongly depends on elementary siphons…”

“…In [60,62,64] important findings in the area of robust deadlock control of AMS with unreliable resources are published. In [53,50,49,48,47,91] the application of elementary siphons, being topical at present, is broadly investigated. In [55,28,29,30,31,32,33] the very useful iterative solution how to avoid the need of enumerating all the states or siphons using mathematical programming techniques were published.…”

“…The special syntactic characteristics of this class allow to study the modelled systems from a structural perspective. The newer publications [63,8,25,55,56,57,59,62,64,28,29,30,31,32,33,34,35,36,37,38,39,40,92,93,94,91,88,89,65,66,67,68,69,58,60,61,47,48,49,50,51,52,53] investigate also such kinds of AMS/FMS.…”