Soundness problem for Resource-Constrained Workflow nets

Resource-Constrained WF-nets (RCWF-nets)

Resource places Pr

Production net Np

with production places Pp

PrPp = A Petri net N with a set of places PpPr is an RCWF-net if its projection on Pp is a workflow net.

i f

Mixable instances vs. independent instances

• Instances in the production net can be independent, e.g. as in handling insurance claims

• Or they can interfere with each other: produce a number of bicycles, all exactly the same, no matter which wheels go to which one…

• For independent instances, we can introduce token id’s, and further reduce the model to classical Petri nets by substituting the production net by a state machine obtained from the reachability graph of the production net.

• So in both cases we can just deal with classical Petri nets

Different types of resources

• Durable: machines, people• Consumable: paper, building materials, etc.• Producible: whatever you produce in your process

• We will focus on durable resources only• They get involved in the process when executing

some tasks and then get released• When all instances terminate, it’s logical to expect

that all resources are back

Resource or not?

Resource or not?

Resource or not?

Resource or not?

Resource or not?

Resource or not?

Resource or not?

Resource or not?

Resource or not?

At any moment of time, the amount of resources cannot exceed the initial amount of them.

Soundness for RCWF-nets

• An RCWF-net N is (k,R)-sound if for any marking m reachable from the initial marking k[i]+R, marking k[f]+R is reachable and mr < R

• N is k-sound if there is an R such that N is (k,R’)-sound for all R’R

• N is R-sound if it is (k,R)-sound for any natural k• N is sound if there is an R such that N is (k,R’)-sound

for any natural k and any R’R

• So we parameterize on the number of instances, the number of resources, or on both.

• Soundness of RCWF-nets only covers the proper termination requirement

An evident necessary condition of soundness

• The production net of a sound RCWF-net is generalized sound, i.e. for every marking reachable from k[i], marking k[f] is reachable.

• For any sound RCWF-net its production net is generalized sound

Redundancy and persistency

no resource place can ever obtain tokens, if it was not marked initially.

every resource place should becomemarked again when the net terminates.

Non-redundancy criterion

Non-persistency criterion

Checking that all resources are back if terminated

• The net should work correctly for all “large” markings,

• i.e. we can always add resource tokens to make a firing sequence that is executable in the production net also executable in the RCWF-net

• That implies (with some extra steps) that, for every sound RCWF-net every transition invariant of the RCWF-net is a transition invariant of its production net and vice versa

Corollary

• The transition invariant check allows us to ensure that if all the instances of an RCWF-net terminate (tokens on f), then all the resources are back

• i.e. if the production net of an RCWF-net is sound and the invariant condition holds, deadlocks and livelocks can happen due to resources only

Still it can go wrong

Reducing the soundness problem to the home space problem

An RCWF-net is sound iff

is a home space of its transformed net.

R0 resources

Checking the home space property

• Partition the space of the reachable markings into

with

The HS property holds for all markings in R’

Goal: to show that it is sufficient to check that it holds for the set F’’ of minimal markings of R’’

We partition R’’ further into

and define

Proof by induction on i

i = 0 is trivial

Checking the home space property (2)

• Take . If the hs-property holds.

• If not, take such that • Note that for some and• Then and

• , hence it contain at least one token on some production place, while

contains none• Thus

Problem solved?

• Well, we can check reachability on an unbounded net – in theory it is decidable, but the check was never implemented…

• Next to investigate: can we apply algebraic methods to solve the problem like we did for generalized soundness? It’s no way straightforward…

to be continued…