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Liveness -Enforcing Supervision of Sequential Resource Allocation Systems

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### Liveness-Enforcing Supervision of Sequential Resource Allocation Systems

### The RAS Logical Control Problem:Characterization of the optimal solution and its complexity

Spyros Reveliotis

School of Industrial & Systems Eng.

Georgia Institute of Technology

Talk Outline

- Problem motivation and the abstraction of the Resource Allocation System (RAS)
- Formal characterization of the considered problem, its optimal solution, and the involved complexity
- The current State of Art
- Special RAS structure admitting optimal liveness-enforcing supervision of polynomial complexity w.r.t. the RAS size
- Suboptimal, polynomial-complexity liveness-enforcing supervisors for many of the remaining cases
- A generic methodology for verification and design of efficient suboptimal liveness-enforcing supervisors

The current state of art:Dealing with the considered problem in the 300mm FAB

A modeling abstraction:Sequential Resource Allocation Systems (RAS)

- A set of (re-usable) resource types R = {Ri, i = 1,...,m}.
- Finite capacity Ci for each resource type Ri.
- a set of job types J = {Jj, j = 1,...,n}.
- An (partially) ordered set of job stages for each job type, {pjk, k = 1,...,lj}.
- Aresource requirements vector for each job stage p, ap[i], i = 1,...,m.
- Jobs release their currently held resources only upon allocation of the resources requested for their next stage

Sequential RAS deadlock: A RAS state in which there exists a subset of jobs s.t. every job in this subset in order to proceed requires some resource(s) currently allocated to some other job in this subset.

An Event-Driven RAS Control Scheme

RAS Domain

Feasible

Admissible

Actions

Actions

System State Model

Performance Control

Logical Control

Event

Commanded

Action

Configuration Data

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Finite State Automata (FSA)-based modeling of RAS behavior

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Safe vs. Unsafe Region and

the Optimal Logical Control Policy

Complexity Considerations

- State Safety is an NP-complete problem in sequential RAS

(by reduction of the 3SAT problem)

- State Transition Diagram (STD) size:

- where:
- C = max resource capacity
- Q = max number of stages supported by a resource
- m = number of resource types

Dealing with the non-polynomial complexity

- Special RAS structure admitting an optimal logical control policy of polynomial complexity w.r.t the RAS size
- Polynomial-Kernel (PK-) RAS logical control policies: Sub-optimal one-step-lookahead policies based on state properties that are polynomially verifiable, e.g.,
- RUN (Resource Upstream Neighborhood)
- RO (Resource Ordering)
- Banker’s algorithm
- An analytical framework for
- interpreting the correctness of the above policies, and
- enabling the “automatic” validation and synthesis of new members from this class of policies

Some Major Contributors and Research Groups in this Area

The first attempts, primarily in the computer system context (60’s and 70’s)

- Dijkstra, Havender, Habermann, Coffman, Holt
- Gold, Araki, Sugiyama, Kasami, Okui

The problem revival in the manufacturing context (late 80’s / early 90’s)

- Banaszak& Krogh
- Viswanadham, Narahari & Johnson
- Wysk, Joshi & Smith

The current DES-based community (mid-90’s to present)

- Colom, Ezpeleta & Tricas
- Xie & Jeng
- Zhou and his colleagues
- Fanti & her colleagues
- Roszkowska
- Hsieh
- Reveliotis, Lawley, Ferreira, Park and Choi

Structure of the process sequential logic

Linear: each process is defined by a linear sequence of stages

Disjunctive: A number of alternative process plans encoded by an acyclic digraph

Merge-Split or Fork-Join: each process is a fork-join network

Complex: a combination of the above behaviors

Structure of the stage resource requirement vectors

Single-unit: each stage requires a single unit from a single resource

Single-type: each stage requires an arbitrary number of units, but all from a single resource

Conjunctive: Arbitrary number of units from different resources

A RAS taxonomyRAS admitting optimal logical control of polynomial complexity

- Type 1: The search for a process terminating sequence can be organized in a way that backtracking is not necessary:

Process advancing events can be selected in such a manner that the resource slack capacity is increased monotonically

- e.g., under “nested” resource allocation: resources are released by a process in a sequence that is reverse to that followed for their acquisition
- Type 2: Unsafety Deadlock deadlock is polynomially identifiable.

This kind of results are available for sub-classes of DIS-SU-RAS only.

DC-RAS with “nested” resource allocation

Every process transition corresponds either to a pure allocation or a pure de-allocation.

Resources allocated as a block are also de-allocated as a block. The “scope” of each such allocation is defined by the processing stages that engage the corresponding resource block.

In each path of the process-defining graph that corresponds to a single realization of the process, the “scopes” of two different allocations are either disjoint or one contains the other – this is equivalent to the statement that resource blocks are de-allocated in reverse order of their allocation.

A(R1)

A(R2)

A(R3)

D(R3)

D(R2)

D(R1)

R1

R1+R2

R1+R2+R3

R1+R2

R1

A polynomial algorithm resolving safety in DC-RAS with nested allocations

- Given a state RAS state s, let:
- δi(s) be the slack capacity of resource Ri at s, for all i;
- Sa(s) be the set of “active” processing stages at s;
- <Ajk1, Ajk2, …, Ajkn(jk)> be the resource allocation sequence for the resources occupied by a job instance executing proc. stage Ξjk in Sa(s);
- Q := { Ajkn(jk) | Ξjk in Sa(s) }.
- While Q is not empty:
- Try to find an allocation Ajki in Q that is de-allocateable under the current slack capacities;
- If no such allocation exists, declare s as unsafe and exit.
- O.w.,
- add the resources corresponding to Ajki to the slack varsδi(s);
- remove Ajki from Q and, if i > 1, enter in Q the allocation Ajki-1.
- Declare state ssafe and exit.

An Example Result of the 2nd Type

Theorem 1:In a DIS-SU-RAS where every resource has at least two units of capacity, the optimal logical control policy is polynomially implementable (through one-step lookahead)

Proof: We shall show that for this class of systems,

- unsafety deadlock, and
- deadlock is polynomially identifiable.

A polynomial deadlock detection algorithm for DIS-SU RAS

- Given a state s of a DIS-SU RAS,
- R := the entire set of the system resources;
- DEADLOCK := FALSE;
- While (R is not empty AND not DEADLOCK)
- Try to identify a resource R in R s.t. R is not allocated to capacity in s or it contains a job requesting advancement to a resource not in R or out of the system.
- If successful, R := R\{R} else DEADLOCK:=TRUE;
- Return DEADLOCK
- Algorithm complexity: O(|R|2Cmax)

DEADLOCK

Rk

Rl

Rj

Unsafety DeadlockThe topological relationship of DEADLOCK and UNSAFE spaces / Deadlock-free unsafe states one

step away from deadlock

The absurdity of the existence of a deadlock-free unsafe stateone step away from deadlock

for the considered RAS class

An alternative mechanism for establishing UNSAFETY= DEADLOCK in various sub-classes of DIS-SU-RAS

Basic structure of deadlock-free unsafe states one step away

from deadlockin DIS-SU-RAS

Potential

Deadlock 2

Potential

Deadlock 1

C=1

Potential

Deadlock n

Potential

Deadlock i

Polynomial-Kernel Policies

- Search-based: Confine the system operation to those states from which there exists a terminating sequence that completes one process stage at a time. This sub-class of states are called ordered, and the resulting policy is the renowned (Dijkstra’s) Banker’s algorithm.
- Algebraic: Confine the system operation to those statess that satisfy an inequality of the type:

A·sb

Remark: The system state sis a vector with its components indicating how many jobs execute each processing stage of the considered RAS

R2

R3

J1 :R1®R2®R3

J2 :R3®R2®R1

O(R1) = 1, O(R2) = 2, O(R3) = 1

Example: The RUN (Resource Upstream Neighborhood) Policy for SU-RAS

- A partial resource reservation scheme based on a(partial) ordering of the resource set: A job instance executing on a resource reserves capacity on every downstream resource of order greater than or equal to the order of the currently held resource, unless there is an intermediate resource of higher order than the considered downstream resource.

A× s £ b

Example: The Policy-Admissible Region

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Proving RUN Correctness

- It suffices to show that for every policy-admissible state, other than the empty state, there is at least one loaded job that can advance.
- If there exists a job that needs to advance to a resource of order higher than or equal to the order of the currently held resource, then, this job does not enter a new resource neighborhood upon its advancement. Therefore, (i) it has already reserved capacity on the requested resource and (ii) it can advance without violating the policy.
- If every loaded job requests advancement to a resource of lower order than the order of the currently held resource, consider a minimal order resource containing jobs. Then, (i) the resource requested by any of these jobs has free capacity. Furthermore, (ii) any new neighborhoods entered by these jobs upon their advancement, are empty (since they must belong to even lower-order resources). Therefore, any of these jobs can advance without violating the policy.

Rc

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NH(Rl)

Case 2 in the proof of RUN correctness- Rcis a minimum-order resource containing jobs
- Then,by case assumptions,
- o(Rn) < o(Rc) Rnempty
- Also,
- for any resource Rh such that st(Rn) NH(Rh)and o(Rh) o(Rc):
- st(Rn) NH(Rh) st(Rc) NH(Rh)
- for any resource Rl such that st(Rn) NH(Rl) ando(Rl) < o(Rc):
- Ri, Ri NH(Rl) o(Ri) o(Rl) < o(Rc) Ri empty NH(Rl) empty

T10

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J1 :R1®R2®R3

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Petri Net-based modeling of RASP20

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Siphon-based characterization of RAS liveness: Single Unit-RASS = {R1, R2, P12, P23}

S* = {T10, T22, T11, T21,

T12, T23}

*S = {T11, T23, T12, T22}

*S S*

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marking

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Siphon-based characterization of RAS liveness: Conjunctive RASt20

- Generalizing empty siphon:
- Siphon S is deadly marked ifft*S, t is disabled by some pS

A key result

Theorem 2: Consider a process-resource net N where:

I. every process subnet Ni is

- quasi-live for M0(pi0) = 1,
- reversible for every initial marking M0(pi0), and
- “acyclic”, i.e., strongly connected with every cycle containing pi0;

II. Resources are re-usable, i.e., for every resource Rk, p-semiflow yRk s.t.

- yRk(rk) = 1,
- p sup(Rk), yRk(p) = # units of Rk required for the execution of stage p,
- yRk(p) = 0, o.w.

III. Each process sub-net when augmented with the required resource places is quasi-live (i.e., the process-resource net is “well-marked”).

Then,

- N is liveiff~ resource-induced deadly marked siphon in the modified reachability space.
- Liveness Reversibility
- If N is PT-ordinary, liveness ~ empty siphon in the reachability space.

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Modeling an algebraic PK policy as a set of fictitious resourcesT10

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Computing the maximal empty siphonRemove Marked Places

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Computing the maximal empty siphon (cont.)

Remove enabled transitions and places that will be marked by their firing.; repeat.

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A sufficiency condition for non-existence of reachable empty siphons in structurally bounded Petri nets

Theorem 3: A structurally bounded Petri net N=(P,T,F, M0) has no reachable empty siphons if C(N) = |P|, where

s.t.

Practical Implications

- Theorems 2 and 3 provide the basis for the development of verification tests for
- RAS liveness and
- algebraic PK policy correctness

that take the form of a Mixed Integer Programming formulation with polynomial number of variables and constraints in terms of the size of the underlying RAS.

- Embedded in a search process, these tests can support the design of optimized algebraic PK policies – This is essentially a combinatorial optimization problem and constitutes ongoing research.

Some Additional Developments and Future Work

- An algebraic theory for interpreting the functionality of algebraic PK policies through siphon dependencies and the notion of “basic” / “elementary” siphons.
- A methodology for designing optimized (maximally permissive) algebraic PK policies through non-blocking supervisory control theory and the theory of regionsfor Petri net synthesis from their reachability space.
- A generalization of the concept of algebraic PK policy in order to encompass the potential nonlinearity of the maximally permissive supervisor, based on results from pattern recognition / classification theory, and extension of the correctness verification tests to these policies.
- Future work: Integrate the presented results on the RAS logical control problem with the time-based performance control / scheduling problems arising in these environments.
- The proposed framework: Markov Decision Processes and Approximate Dynamic Programming.

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