To present the strategic, organizational, and operational functions of the management process, we define Workflow [6] [7] . To understand and assess the existing parties in a socio-economic organization, it is crucial to understand the business processes in the organizational structure. They advocate for flexible information system design and deployment. These information systems’ technical foundation is the realization of new products through adaptation to change and market demands. Given the context of managing and guiding businesses, business process management is innovative. According to [5] a business process is any set of actions that include one or more types of input and offer a customer-satisfactory outcome. This definition covers the relationships’ semantics between the inputs and the outputs (pre-condition and post-condition) in a business process. So it is an abstraction of the concept of the process that becomes clear in this definition.

However, a workflow [12] [14] [55] ensures the automation of formal rules to restructure, simulate, and optimize the company’s business processes. According to [56] , the workflow optimization procedure, therefore, concerns the definition of the scheduling of fundamental interdependencies, the representation flexibility, at the time of the action, certain routes integrating the data specific to an instance particular process [18] [56] .

In 1962, Carl Adam Petri innovated a formal modeling tool called the Petri Net [57] . They emerged as a powerful graphical tool for the representation of complex system and sequential mechanisms [57] . They have effectively been used for the specification, verification, and analysis of the behavior of complex systems [58] . With a variety of PNs [59] [60] [61] [62] application fields and their efficient uses to specify and verify the dynamic behavior of complex systems, PNs several classes have appeared in the literature. Using the advantages of the SPN to complex systems analysis and verification, we suggest a model that covers the life-cycle of the manufacturing process workflow. To specify, validate, and predict the execution delay associated with the duration of the tokens propagating within firing transitions, we propose a P-timed oriented extension of the SPN [63] [64] . This proposed model, as well as the tools for specification, verification, and evaluation, is presented in the following section.

The Workflow P-Timed Stochastic Petri Net, denoted WPTSPN, is the 8-uplets:

WPTSPN:

=<P, Tr, A, Sac, Ac, Typ-tk, GTyp-P, CardM>

· P = P b   ∪   P w   ∪   P r   ∪ P f , represents the finished set of places;

- P b : all initial places,

- P w : all workflow places,

- P r : all resource places,

- P f : all final places.

· T r = T r i   ∪   T r d   ∪ T r s , represents the finished set of transitions (non-empty finite set);

- T r i : immediate transitions set,

- T r d : determinstic transitions set,

- T r s : stochastic transitions set.

· A: set of Workflow-actors is non-empty finished;

· Ac: workflow activities finished set;

· Sac: finished set of Sub-activities associated with a Workflow-activity;

· Typ-tk: the representation by our model distinguishes between four token types (activity, sub-activity, actor, and any encapsulation of tokens) then the Typ-tk-function is in charge of this classification.

T y p - t k : t k → { 1,2,3,4 } ;     t k ↦ { 1,     t k ∈ A 2,     t k ∈ S a c 3,     t r ∈ A c 4,     t k ∈ u p l e t (1)

· GTyp-P: the representation adopted by our model requires that we distinguish the ordinary places of a PN, those linked to the semantic workflow, and those endowed by a residence time. So, GTyp-P-function makes it possible to distinguish the model’s places.

G T y p - P : P → { o ,   w ,   r ,   f }

p ↦ G T y p - P ( P ) = ( o ,     ( initialplace ) w ,     ( workflowplace ) r ,     ( resourceplace ) f ,     ( finalplace ) (2)

· CdM: the marking cardinality of a WPTSPN is a family indexed by the set of places P, such that tokens (tk) in place p:M(p) is an element of CdM(M(p)).

The proposed WPTSPN aims to represent (specify) a flexible production business process according to a P-timed orientation, to then check (verify) the model properties that coincide with the domain of application, and finally to predict the delay relating to the execution of an activity (firing a non-immediate transition of the model). So to achieve these ends, we enrich our model with formal functions.

In this subsection, we enriched our model with the functions to increase the analytical coverage according to the workflow semantics, P-timed orientation, and the random residence time assigned to the tokens during a sequence of firing transitions. These <GDensity, GDistrib, TFringT, IRP, MST, GTruncated, MIP, MPO, MIT, MTO> formal tools are defined as follows:

· GDensity: the probability density function of the WPTSPN components (tokens, and stochastic transitions) represents a stochastic probability density (Normal, Log-Normal, or Exponential).

G D e n s i t y : ( A c   ∪ S a c   ∪   T r s ) × { e ,   n ,   l n } → R + t y p =   T y p - d i s t ( k )

( k ,   t y p ) ↦ G D e n s i t y ( k ) = ( t y p = e ,   λ ⋅ e − λ x   ( λ ≥ 0,   x ≥ 0 ) ; t y p = n ,   1 / ( 2 2 π ) ⋅ e − 1 / 2 ⋅ x 2 , ( x ≥ a p ) ; t y p = l n ,   1 / x σ 2 π ⋅ e − ( ( ln x − μ ) 2 / 2 σ 2 ) , ( x ≥ a p ,   σ ∈ R + ,   μ ∈ R ) . a p = min ( I R P p )     ( ∀ t r j ∈ p I )   and   ( G T y p - P ( p ) ∈   { o ,   w } ) . (3)

· GDistrib: each WPTSPN component (tokens, and stochastic transitions) is assigned an arbitrary probability distribution (Normal, Log-Normal, or Exponential). This assignment is performed by the function GDistrib.

G D i s t r i b : ( A c   ∪   S a c ∪ T r s ) × { e ,   n ,   l n } → R + t y p d = T y p - d i s t ( k ) t y p - k = T y p - t k ( k )

( k , t y p - d , t y p - k ) ↦ G D i s t r i b ( k ) = ( t y p = e ,   1 − e − λ x   ( λ ≥ 0,   x ≥ a p ) ; t y p = n ,   1 / ( 2 2 π ) ⋅ ∫ − ∞ t     e − 1 / 2 ⋅ x 2 d x , ( x ≥ a p ,   t ≥ a p ) ; t y p = l n ,   1 / 2 + 1 / 2 ⋅ e r f − ( ( ln x − μ ) 2 / 2 σ 2 ) , ( x ≥ a p ,   σ ∈ R + ,   μ ∈ R ,   e r f   istheGausserrorfunction ) ; a p = min ( I R P p )     ( ∀ t r j ∈ p I )   and   ( G T y p - P ( p )   ∈   { o ,   w } ) . (4)

· TFringT: Our model assigned three probability distributions (Normal, Log-Normal, or Exponential) to each stochastic transition [66] . This function makes it possible to determine each stochastic transition firing time, with TFringT-function, according to the distribution chosen. We use the gravity center method [48] which consists of taking the abscissa corresponding to the gravity center of the each stochastic density transition.

T F r i n g T : T r s × { e ,   n ,   l n } → R + t y p = T y p - d i s t ( t r s )

( t r s , t y p ) ↦ T F r i n g T ( t r s ) = ( t y p = e , ∫ − ∞ t ( λ ⋅ e − λ x )   ⋅   x d x ∫ − ∞ + ∞ ( λ ⋅ e − λ x ) d x ; t y p = n , ∫ − ∞ + ∞ ( ( 1 / ( 2 2 π ) ⋅ e − 1 / 2 ⋅ x 2 ) ⋅ x ) d x ∫ − ∞ + ∞ ( 1 / ( 2 2 π ) ⋅ e − 1 / 2 ⋅ x 2 ) d x ; t y p = l n , ∫ − ∞ + ∞ ( 1 / x σ 2 π ⋅ e − ( ( ln x − μ ) 2 / 2 σ 2 ) ⋅   x ) d x ∫ − ∞ + ∞ ( 1 / x σ 2 π ⋅ e − ( ( ln x − μ ) 2 / 2 σ 2 ) ) d x ; ( x ∈ R + , λ ∈ R + ,   σ ∈ R + ,   μ ∈ R ) . (5)

· IRP: IRP_i function defines the static residence (stay) interval of the model’s places (initial or workflow). ( ∀ p i ∈ ( P o ∪ P w ) \ G T y p - P ( p i )   ∈ { O ,   w } ). When it leaves the place p i , the output transitions firing becomes equal to b i . After this random time ( b i ), the token will be “dead” and will no longer participate in the transitions validation. Formally IRP is expressed as follows:

I R P : P o ∪ P w → ℚ + ∪ ( ℚ + × + ∞ ) ;

∀ p i ∈ P o ∪ P w \ G T y p - P ( p i ) ∈ { o ,   w } ;

p i → I R P i = [ a i ,   b i ]     with   0 ≤ a i ≤ b i ;

( a i = min ( a i ,   b i )   ( ∀   t r ∈ p I ) ; b i = max ( a i ,   b i ) ;   ( ∀   t r ∈ p I ) ; (6)

· MST: (Mean Stay Time) Taking into account the specificity of the token’s type (activity or sub-activity) in places of our model. So, the MST-function calculates each token’s average stay time in timed locations and which are of type (initial or workflow). We associate time value, framed by the min ( T F r i n g T ( t r ) ) ( ∀ t r ∈ p I ) and the min ( T f i r i n g T ( t r ) ) ( ∀ t r ∈ p O \ ( G T y p - P ( p ) ) ∈ { O ,   w } , T y p - t k ( M ( p ) ) ∈ { 2,   3,   4 } ), taking into account the random distribution assigned to each token in this place. Practically, we use the gravity center method [48] which determines the abscissa corresponding to the density attributed to the place in question.

M S T : ( P ∪ A c ∪ S a c × T r s × { e ,   n ,   l n } ) → R + t y = G T y p - P ( p ) ∈ { o ,   w } t y p = G T y p - d i s t ( t t k ) t t k = T y p - t k ( M ( p ) ) m = min ( T F r i n g T ( t r ) ) ∀ t r ∈ p I M = min ( T F r i n g T ( t r ) ) ∀ t r ∈ p O

( p ,   t y ,   t y p ,   t t k ) ↦ M S T ( p ) = ( t y ∈ { o , w } ,   t y p =   e ,   t t k ∈ { 1,2 } ,   ∫ m M ( λ ⋅ e − λ x )   ⋅   x d x ∫ m M ( λ ⋅ e − λ x ) d x ; t y ∈ { o , w } ,   t y p =   n ,   t t k ∈ { 1,2 } ,   ∫ m M ( ( 1 / 2 2 π ⋅ e − 1 / 2 ⋅ x 2 ) ⋅ x ) d x ∫ m M ( 1 / 2 2 π ⋅ e − 1 / 2 ⋅ x 2 ) d x ; t y ∈ { o , w } ,   t y p =   l n ,   t t k ∈ { 1,2 } ,   ∫ m M ( 1 / x σ 2 π ⋅ e − ( ( ln x − μ ) 2 / 2 σ 2 ) ⋅   x ) d x ∫ m M ( 1 / x σ 2 π ⋅ e − ( ( ln x − μ ) 2 / 2 σ 2 ) ) d x ; ( x ∈ [ m ,   M ] , λ ∈ R + ,   σ ∈ R + ,   μ ∈ R ) . m < min ( I R P ( p ) )   and   M < max ( I R P ( p ) ) . (7)

· GTruncated: We use a truncated density function to estimate and predict the delay related to the residence time of a token in a timed place. And for it to cover the densities allocated to the tokens’ residence time in each timed place. We define the generalized truncated density sojourn probability law of a timed place function as follows:

T r u n c a t e d : ( P ∪ T r s )   →   R + ( t 0 ,   t ) ∈ T D × T D t y p ∈ { e ,   n ,   l n } t t k ∈ { 1,2,4 } ( k , t y p ) ↦ G D e n s i t y ( k ) ( k , t y p ) ↦ G D i s t r i b ( k )

T r u n c a t e d ( k , t t k ) = ( 0,     t ≤ t 0 ,   G D i s t r i b ( t 0 ) ≤ 1 ; G D e n s i t y ( t ) / 1 − G D i s t r i b ( t 0 ) ,   t > t 0 ,   G D i s t r i b ( t 0 ) < 1 ; G D e n s i t y Δ ( t − t 0 ) ,   G D i s t r i b ( t 0 ) = 1 ; (8)

· Let MIP: the places input-transitions matrix:

( P × T r ) → { 0,1 } (9)

Such that:

M I P ( P i ,   t r k ) = 0 , if I P ( P i ,   t r k ) = ∅ , 1 if not

· Let MPO: the places output-transitions matrix :

( P × T r ) → { 0,1 } (10)

Such that:

M P O ( P i ,   t r k ) = 0 , if P O ( P i ,   t r k ) =   ∅ , 1 if not.

· Let MIT: the transitions input-places matrix:

( T r × P ) → { 0,1 } (11)

Such that:

M I T ( t r k , P i ) = 0 , if I T r ( t r k , P i ) = ∅ , 1 if not

· Let MTO: the transitions output-places matrix:

( T r × P ) → { 0,1 } (12)

Such that:

M T O ( t r k , P i ) = 0 , if T r O ( t r k , P i ) = ∅ , 1 if not

The firing transitions semantics adopted by the WPTSPN model are defined in the next sections.

The firing semantics adopted by the WPTSPN depend on three factors: the static stay (residence) interval of the places at the transition input, the stochastic firing time, the average stay time of tokens in workflow entry places, and preconditions related to the interaction between the actors responsible for executing the activity represented by the transition. To explain this firing mechanism, we need to apply it to the context of the transition tr₆. According to Table 3, the transition Tr₆ which represents the operation below (“Machine run”).

Figure 1 illustrates the tr₆ transition and the mechanism of division (splitted) into three micro-transitions: { t r 6 c , t r 6 w , t r 6 p } .

The transition tr₆ is stochastic according to the WPTSPN, so we assign a random value of firing time using t y d t = G T y p - d i s ( t r 6 ) ∈ { e ,   n ,   l n } and T F r i n g T ( t y d t ,   t r 6 ) .

We note that according to the representation semantics of the WPTSPN: Let input places: I T r 6 = { P 0 , P 7 , P 8 } and output place: T r 6 O = { P 10 , P 0 , P 8 } .

· P₀: specifies the park of actors, G T y p - P ( P 0 ) = r ;

· P₇: specifies the both a machine bounded input and conveyor output buffer, G T y p - P ( P 7 ) = w ;

· P₈: specifies place of the maintenance sub-process, G T y p - P ( P 7 ) = r ;

· P₁₀: specifies the end of workflow process G T y p - P ( P 7 ) = f ;.

It is assumed that non-workflow places (P₀, P₈), i.e., resources, are not timed, so the tokens they store are always available, such as actors, primary materials, or semi-finished products. The timed place P₇ is a workflow place dedicated to storing an activity or sub-activity type token (noted tk: T y p - t k ( t k ) = ∈ { 2,3 } ). Then we assign a static interval of stay (residence): I R P ( P 7 ) = [ a 7 , b 7 ] .

At the moment when the token named tk such that Gtype(tk) = 2 (i.e. of type Sac) instead of p₇, we assign it a stochastic value of stay using the density function ( τ t k = G d e n s i t y ( t k , t y ) ) where ty varies between the three possible distributions of the model { e ,   n ,   l n } . If there are several (Gtype(tk) = 2) type tokens, an average residence time is used using the M S T ( P 7 ) function.

The firing semantics of a stochastic transition with timed places assumes that a time counter is triggered when a token arrives. Let t be the value of this time counter. We associate a token-life-cycle time named ( τ t k , t ) . We discuss the following three cases:

· τ t k < a i ( P 7 ) ; so the tk token is not available.

· t ∈ [ a 7 , a 7 + τ t k ] and ( a 7 + τ t k < T F i r i n g ( T r 6 ) < b 7 ); During Tr₆ firing, WPTSPN detects the transition type via the function calls Distinct(Tr₆) = 3 and GTyp-dist(Tr₆) = ln. We know that the overall firing time value is determined by the function call TFiringT(Tr₆, ln). The transition firing Tr₆, which is in the sub activity executing charge Sac₁, consists in sequentially firing the three micro-transitions Tr₆^c, T r 6 w and T r 6 p . The transition pre-condition T r 6 c is the uplet <11, time, 1> and the stochastic value ( 0.33 ∗ φ 3 ). The micro-transition firing T r 6 w is influenced by the stochastic value ( 0.67 ∗ φ 11 ). Once the function activity call is done, T 6 w micro-transition firing is achieved. The T 6 p firing is immediate; i.e., the token Sac₁ is moving towards the place P₁₀ and the two tokens A₁ and A₃ will be released to the place P₀ (the original place before firing Tr₆), also a token in place P₈.

· τ k > b 7 ; so the tk token is dead. We note that the tokens which circulate in the three micro-transitions have the same behavior according to the static residence time. Note that each micro transition from the split mechanism inherits a behavior similar to the transition (splitted).

The main properties that generally studied the WPTSPN are verification and checking. First, verification consists in determining the techniques for deciding on each of them. Second, checking, which is based on the dynamic timed evolution to analayze and verify this model properties, is then presented.

The WPTSPN model can be used to qualitatively analyze the basic FMS properties to provide the dynamic behavioral aspects of complex systems. The CdM (marking cardinalities graph) associated with the model at any time in time represents the number and position of tokens concerning the places of the model. Firing a transition moves the model into a new marking cardinality. The marking cardinality of an WPTSPN helps to define which places are occupied and which are free.

The model adopts two verification approaches:

· Checking based on the elapsed time following a firing transitions sequence:

- Reachability (Accessibility): A CdM’ marking cardinality is said to be accessible from CdM if there are sequences of markings cardinalities (CdM₀ to CdM_n) and transitions ( t r 0 to t r n , ∀ ( t r 0 , t r n ) ∈ ( T r × T r ) ) such as C d M i + 1 is immediately accessible from CdM’ ( i ∈ [ 0, n − 1 ] ) with C d M i = C d M 0 and C d M ′ = C d M n . The set of each markings cardinalities accessible from CdM is noted the set of reachability.

So reachability seems to be a tool for finding erroneous states, for practical problems the constructed graph usually has far too many states to compute. This allows us to detect execution failures of a business process (FMS) modeled by a WPTSPN.

- Safeness: An WPTSPN is safe if each of its places ( ∀ p ∈ P ) is safe. So, a place p is safe if it does not contain more than one token. The model transitions if it can fire infinitely often, so if there is always a fixed (necessarily infinite) firing sequence that ensures the execution of a workflow activity.

- Boundeness: A WPTSPN is said to be bounded if it is composed only of bounded places. A place ( ∀ p ∈ P ) is bounded if there is a limit on the tokens maximum number ( C d M ( p ) ≤ K \( ∀ p ∈ P , K > 1 )).

This property indicates that the system represented by a WPTSPN never risks exceeding the storage limits of semi-finished material, finished products, and workers… This property indicates that the system represented by a WPTSPN never risks exceeding the storage limits of semi-finished material, finished products, and workers…

- Conservativeness: A WPTSPN is said to be conservative if the tokens-number in the ∀ C d M ∈ C d M remains constant.

It is obvious that conservativeness is a special case of structural delimitation and that coherence is a case of repetitiveness. Then this property can be partial, consistency and repetitiveness are the relaxations of the circulation of tokens in the places of the WPTSPN.

- Liveness: A transition is said to be alive if for each ( ∀ C d M ∈ ( C d M , W P T S P N ) ) there is a firing sequence which brings the WPTSPN to a CdM in which this tr transition is fired. An WPTSPN is active if each transitions ( ∀ t r ∈ T r ) are active. If the WPTSPN is activated and correct, this indicates that there are no deadlocks in the FMS operations.

- Properness: if the initial marking CdM₀ is accessible from each marking ( ∀ C d M ∈ ( C d M , u ) ) in the accessibility set, the WPTSPN is said to be proper.

This property verifies that the tokens which represent the actors (worker) return to their place of origin before the each firing model’s transition for which they will always be available at the future firing (excution of an activity workflow FMS).

· Checking is based on randomly residence intervals associated with the token in workflow-places and the token life-cycle:

We keep the verification properties related to the firing transitions context, and we add other definitions related to the P-timed SPN contexts that we adopt in this work. Let DT is a finite time domain; Let t: time in finished time domain ( ∀ t ∈ D T ); Let state characterizing the WPTSPN graph situation be defined by a doublet SE = <CdM, Q>, where:

· CdM is the same as the previous definition of the WPTSPN marking function,

· Q is a time-of-stay application which associates with each token k ( ∀ k ∈ { S A C ∪ A C } ) in place p i ( ∀ p i ∈ P   \   T y p - t k ( M ( p i ) ) ∈ { 1,2,4 } ) a real number q i k where q k is the this token lifetime (the time elapsed since its arrival in place p i ). The q i k associated with a token k in the place p i must be less than or equal to bi where [ a i , b i ] is the Randomly residence interval associated with the place p i . This token k in the place p i is involved in the activation and validation of its p^O transitions when q i k age great. The token is considered dead when its lifespan (age) is strictly greater than b i .

Note that the lifetime q i k of a token k ∈ ( A c × S a c ) (given by a local clock associated with it) is relative to this token arrival time in the place p i \ T y p - t k ( M ( p i ) ) ∈ { 1,2,4 } . We suppose that the token k arrives in the place p i at the instant τ (given by a global clock for example), this token time duration is equal to zero. At the absolute instant τ ′ its lifetime is q i k = τ ′ − τ . It participates in the validation of the output transitions of the place ( ∀ t r ∈ p i O which contains it, only from the instant τ ′ = τ   +   a i , and it is dead from the instant τ ′ > τ   +   b i . Then, a token is dead if its lifetime becomes strictly greater than the upper limit of the dynamic interval associated with its host place and if none of its exit transitions is validated at this time. Therefore, a transition can be activated and validated in the semantics of autonomous PNs and not in the WPTSPN’s semantics because of time constraints.

· A transition ( t r ∈ T ) is potentially firable (validated in the semantics of model) from the state S E ( C d M , Q ) if and only if:

- It is validated within the meaning WPTSPNs in this state, that is to say ∀ p i ∈ t I r ;   C d M ( p i )   ≥   C a r d ( I ( p i , t r ) ) ;   T y p - t k ( M ( p ) ) ∈ { 1,2,4 } (Card: function which counts the tokens number in input flow or respectively output flow) ∀ p i ∈ t I r , there is at least I ( p i , t r ) tokens in the place such as: min ( b i   −   q i k ) − max ( 0, max ( a i − q i k ) ) ≥ 0 ( ∀ k ∈ { S A C ∪ A C } ) and [ a i , b i ] is the dynamic interval associated with the place p i . In addition, there are no j tokens (which do not participate in firing the tr transition) such as: ( b i   −   q i k ) ≤ max ( 0, max ( a i   −   q i k ) ) .

- Otherwise, this token is dead. It is, then, associated with this place interval: [ max ( 0 , max ( a i   −   q i k ) ) , min ( b i   −   q i k ) ] .

Figure 2 illustrates the token life-cycle according to the firing semantics of the WPTSPN.

When we perform in the intersection of all these intervals ( ∀ p ∈ t I r , we associate an interval), we select the interval in which the transition remains potentially firable. According to the state SE notion, from a given state, an infinite number of states is generally accessible. There are, therefore, two possibilities of having this new state from an earlier state:

- the time fluidity (time passage);

- the transition firing (the potential firing interval lower limit is equal to zero from this new state).

· The state e’ accessibility from a previous state e ( ( e , e ′ ) ∈ S E × S E ):

- by a τ time-flow: we note e ( C d M , Q ) → e ′ ( C d M ′ , Q ′ ) , is verifiable if: C d M ≠ C d M ′ , ∀ k ∈ ( S A C ∪ A c ) in the p i \ T y p - t k ( M ( p i ) ) ∈ { 1,2,4 } place, q ′ i k = q i k + τ ; where q i k (respectively q ′ i k ) is the token k lifetime in state e (respectively in state e’) and b i is the dynamic interval upper bound associated with the place p i .

- by firing a transition t r i if and only if: t r i can be firing directly (the lifetimes of all the marks which validate t r i are greater than or equal to the dynamic intervals lower bounds associated with their places) from e, ∀ p ∈ P w , C d M ′ ( p ) = C d M ( p ) − C d M ( I ( p , t r i ) ) + C d M ( O ( t r i , p ) ) . When the tokens do not flow and keep the same lifetime in e and e’ (this is assumed to be an immediate transition), then we assume those tokens that are flowed or created have a lifetime (age) equal to zero. We note that the preceding firing rule allows us to determine the states and the state-space accessibility relations. The set of firing transitions’ sequences from the initial state characterizes the dynamic behavior of the WPTSPN.

According to the state analysis, either by time fluidity or by firing a transitions sequence, the WPTSPN verifiable properties are:

· An WPTSPN model’s state e ( C d M , Q ) is tokens-alive if all of the tokens in CdM are alive.

· A WPTSPN is token-alive for an initial CdM₀ marking (the initial state e₀) if all the markings accessible from CdM₀ are token-alive states.

We adopt the hypothesis that if a token, in a cardinality marking a state accessible from e₀, is verified dead, the WPTSPN is dead tokens. Consequently, this dead token no longer participates in the transitions validation. There are two objectives through which we can verify the token death upstream of a synchronization transition tr:

- the tokens on circuits containing it do not arrive at a compatible time in its entry places ( ∀ p ∈ t I r ),

- the tokens, on the transition elementary oriented paths from parallelism to tr, do not arrive at compatible instants upstream of the transition tr.

· A transitions firing sequence (made up of the transitions and the associated firing time ( ( S E ,   t r ) ; ∀ t r ∈ ( T r s ∪ T r d ) ) is called a token-liveness sequence (k-v-sequence) if and only if each of the states accessible ( ∀ e ∈ S E ) by this transition sequence is token-alive states.

· A transition firing sequence (SE, u) is repetitive if, for a state ∃ e   ∈   S E reachable from e₀ (the initial state of graph states), e is accessible from e by the (SE, u) sequence transitions firing.

· A transition firing sequence (SE, u) is complete if it contains each of the transitions ( u ⊆ T r ) composing the WPTSPN model. The WPTSPN model checking P-timed properties definitions help to adopt the tow following assumptions in [30] :

- If an WPTSPN has a repetitive and complete (tok-v-sequence), then this model has at least one operation which results in a steady-state after a finite time.

- If an WPTSPN is boundless, alive, and alive-tokens then each operation of the model results in a steady-state after a finite time.

After verifying the WPTSPN model’s probabilities according to two P- and T-timed orientations, we show our formal mechanism for predicting of the delay relative to the residence time of sac-token in timed places.

In this subsection, we demonstrate the specificity in the delay prediction that is related to the token’ randomly residence time assigned in the model’ timed places. We include the Event-log related to WPTSPN model to provide delay semantics in the activities’ description. Let TD represent the temporal domain associated with the state firing sequence SE.

So, we define an Event Log by LEv = (SE, C, D):

· SE: states’ firing sequence set,

· C: cases finite set where each case specifies an FMS business process instance,

· D: S E → T D . A function to assign each event to time-space and classify it as a time function.

Each deterministic and stochastic transitions in the WPTSPN ( t r ∈ ( T r d   ∪   T r s ) ) are covered by the set SE. And, each SE/CdM states in the event log contain the C set. We can utilize a time capture approach based on a probabilistic aspect to assure the prediction of the delay associated to task execution in the business processes represented by our model. As a result, this model was integrated with an event-log (Eventlog : LEv) to predict remaining durations as well as the probability of execution delays. To induce the elapsed time required by the model, we use the same definition of a sequence of delays { d 0 , d 1 , d 2 ,   ⋯ , d n } , where d ∈ T D is the time elapsed before the occurrence of se₀ and d_i is the elapsed time between occurrences of s e i and ( ∀ s e i   ∈   S E ) The prediction method is based on:

· The enrichment of places with probabilistic delays in order to determine the duration of an activity and the associated stay time.

· To determine the probability of a decision outcome, assign weights to places.

· The probability density function associated with stochastic places determines the randomly residence time distribution.

· Using an adjacent algorithm in the WPTSPN to predict the residence delay based on the elapsed time.

A monitoring system that collects and records the FMS business process steps is required for our WPTSPN to be stochastically verified. According to this assumption, the time since the previous occurrence represents an information source that benefits the preacher in acquiring precise accuracy for the model. With the duration distribution, it is possible to visualize the conditional impacts on the remaining time. The formal representation is as follows:

· t ∈ T D : time space,

· t r ∈ ( T r d ∪ T r s ) : timed transition,

· P i ∈ P w : timed place,

· t k   ∈ A c   ∪ S a c : token,

· G D i s t r i b ( k ) k ∈ ( P ∪ T r d ∪ t r s ) : the duration distribution function,

· G D e n s i t y ( k ) k ∈ ( P ∪ T r d ∪ t r s ) : The differential function G D i s t r i b ( k ) becomes the generalized density function.

· G t r u n c a t e d P : the generalized truncated density function.

In a formal way, we denote that ( t ∈ T D ) represents time, tk token ( T y p - t k ( t k ) ∈ { 2,3,4 } ) and ( P i ∈ P w ) is a timed place with the assigned static residence. Let ( t 0 ∈ T D ; t 0 > 0 ) be tr the current place time sensor. Let Gtruncated_Δ be the truncate delta function which captures the entire probability mass at a single point.

We aim to build an algorithmic prediction solution based on the immediate detection of events. It is assumed that the duration of each stay place is purely isolated from other independent activities. This hypothesis allows us to simplify the forecasting process while remaining within the norms of other prediction approaches.

To perform the forecast on a state case, we build the forecast algorithm with six entries:

· business process model, its states Graph representative or markings cardinalities graph,

· current moment noted t₀,

· minimum time limit: t m = M i n ( T F i r i n g T ( t r ) ) ( ∀ t r   ∈   I P i ) ,

· maximum time limit: t M = M i n ( T F i r i n g ( t r ) ) ( ∀ t r   ∈   P i O ) ,

· Nbt: activity Cases total number,

· activity current trace c which represents all the events observed up to the moment t₀.

The organizational chart illustrates the prediction process, which begins with a case activity search taking place in the model or one of its representative SE states. It’s a SE graph in the WPTSPN that specifies all of the observed events in this scenario.

Notation: we use the same concept which is an alignment method for localizing the current state in the business process model by adjusting the (place/transitions). The next step is to arrange the results of the probabilistic simulation (t_m, t_M, MST(P), se, Nbr: iterations number). Based on the WPTSPN firing sequence states, the simulated case probability distribution type is established for each iteration. Despite the fact that their represented place’s probability density is the same, all cases are thought to be independent. The simulation operation is responsible for simulating the trace sequences in the restricted sampling model using conditioned truncated distribution densities without analyzing the transition source GDistrib(P). Then, the conditioned truncated distributions must be raised one more G T r u n c a t e d ( t ) ( ∀ t   ≥   M S T ( P ) ) samples.

We must notice that the remaining durations are estimated using the empirical estimation method.

We adopt a conversion manual method of the WF-Net (represented by BPMN) to WPTSPN using the terms adapted by [71] [72] [73] [74] .

· The application field is an FMS-Workflow.

· The set of A   =   { A 1 ,   A 2 ,   A 3 ,   A 4 ,   A 5 } : the actors set representing human resources allocated to a workflow.

· The workflow activities set named AC represents the finished product manufacture via an FMS business process.

· The set S A C = { s a c 1 , s a c 2 , ⋯ , s a c n } the sub-activities to be executed sequentially or in parallel according to a work scheduling strategy on the production multiple lines.

· Places Specification:

The set P = { P 0 ,   P 1 ,   P 2 ,   P 3 ,   P 4 ,   P 5 ,   P 6 ,   P 7 ,   P 8 ,   P 9 ,   P 10 }               ∪ { P 11 ,   P 12 ,   P 13 ,   P 14 , P 15 ,   P 16 ,   P 17 ,   P 18 ,   P 19 ,   P 20 ,   P 21 ,   P 22 } .

The model places, without taking into account the intermediate places, are explained by the Table 2.

The { P 11 ,   P 12 ,   P 13 ,   P 14 ,   P 15 ,   P 16 ,   P 17 ,   P 18 ,   P 19 ,   P 20 ,   P 21 , P 22 } places set represents intermediate places set represents the intermediate places (micro-places) when we adopt the splitting approach for non-immediate transitions (like in [7] ). Such micro-places’ role is to avoid the loss of tokens during the transitions splitting as explained in [7] . In specifying the FSM, P₁ is deemed as an initial place and P₁₀ is a final place.

· Transitions Specification:

Like in [7] , the WPTSPN is liable to distinguish between the three transition types (immediate, deterministic and stochastic). Each stochastic or deterministic model transition is divided into three micro-transitions (Tr^c, Tr^w, and Tr^p). The sub-model transition specification and the associated probability values are well described in Table 2.

Each model transition, whether stochastic or deterministic, is divided into

three micro-transitions (Tr^c, Tr^w, and Tr^p). Table 3 and Table 4 contain a detailed description of the sub-model transition specification as well as the corresponding probability values. Table 3 describes the transition Tr₆ (which corresponds to Op₆ in Table 1) as follows: (“Machine run operation”).

· Tokens Specification:

Three different token types are adopted by our WPTSPN, which is an FMS business process representation:

- A firing sequence triggered by the presence of a SAC-type token named sac₁ specifies the finished product cycle, from its primary form to its packaging. In other words, the FMS workflow represented by the WPTSPN, started by P₁ is considered as begin place contains sac₁.

- To ensure the execution of the semi-automatic operations of the FMS workflow modeled by WPTSPN, we represent five types of tokens A having the execution agents’ role in the production section or the maintenance workflow. Place P₀ is considered the workers’ park.

- An encapsulation tokens type comes from the places-entries of a non-immediate transition. This new token role guaranteed the non-loss of the tokens-entries during the division and the sequential micro-transitions firing.

In this subsection, with the development of the model WPTSPN representative

of the FMS business process. The specification phase is completed by this step which summarizes the previous subsection steps.

From the FMS workflow specified by our WPTSPN model representing an extension of model in [7] , we consider an abstraction with two levels:

· The first level presents the graphical PN representation via places, tokens randomly residence (stay), transitions, arc, tokens, and firing pre-conditions (see Figure 4(a)).

· According to a micro-level life cycle, the second level explains the micro places, micro-transitions, and their firing pre-conditions (tokens consumption phase, execution phase, and tokens production phase) (liken in [7] ).

To illustrate the micro-places and the micro-transitions, we present the abstraction-second level applied to the transition Tr₆ in Figure 4(b). The Cardinality Marking Graph (Figure 5), State Graph based (Figure 6) on the timed places, Randomly residence (stay) time tokens, and tokens’ life-cycle shown earlier are helpful tools to interpret and verify the P-timed properties following a transitions firing sequence.

The rules for conversion from a markings cardinalities graph CdM to a state graph G-SE are:

· each CdM is transformed into a state e;

· in case c d M i comes from a c d M i − 1 by firing an immediate transition of the WPTSPN, then a new state will not be considered and we associate a reversible arc in state e i − 1 .

Table 5 illustrates the correspondence between SE states and cdM markings and their generating transitions.

According to the markings cardinalities graph and the states graph associated with the WPTSPN, we notice:

· According to the workflow semantics of the FMS system modeled by WPTSPN, the arrival of a sac₁ type token in the place considered initial. Then the state e₀ and cdM₀ are interpreted as the start of the FMS management process.

· The state e₄ and cdM₆ are interpreted as the end of the FMS workflow.

· According to the hypothesis of state stability during the firing of an immediate transition [63] , we observe that two tr₃ and tr₄ generate a state of temporal stability despite the change in the marking cardinality based on ordinary circulation tokens.

· The firing of the tr₇ transition generates, according to the semantics of the workflow [6] , a change in the state of the sub-activity sac₁ from in-execution to suspended.

· Each firing sequence comprising the two immediate transitions tr₇ and tr₈ generate a return to the (cdM₅, e₂), so it is a firing circuit.

· The study of markings cardinalities graph allows us to notice that it is possible to fire three transition sequences: s 1   = { t r 1 ,   t r 2 ,   t r 3 ,   t r 4 ,   t r 5 ,   t r 7 ,   t r 8 ,   t r 5 ,   t r 6 } ; s 2 = { t r 1 ,   t r 2 ,   t r 5 ,   t r 7 ,   t r 8 ,   t r 5 ,   t r 6 } ; s 3   = { t r 1 ,   t r 2 ,   t r 5 ,   t r 6 } . The s₃ firing sequence ends with a transition providing the finished product (the sac₁ token instead

P₁₀), which contains no transition circuit, so its firing is assumed to be purely sequential.

The model’s properties are checked according to verification semantics based on T-timed:

· The WPTSPN associated with the FMS workflow for the firing sequence transition s₃ is 9-bounded.

· The WPTSPN is not pure.

· The WPTSPN has no source transition.

· The transitions are quasi-live so the WPTSPN model is almost alive.

· Note that the WPTSPN has an accessibility problem when it reaches cdM₆.

· We see that cdM₅ and cdM₈ are equivalent to what a firing circuit shows.

· The marking cardinality cdM₅ has no successor. So it’s a deadlock.

· The WPTSPN is quasi-live, non-conservativeness, not proper, not balanced, and not sound.

The model properties specificity verifying based on the P-timed semantics and the tokens cycle life carried following a behavioral simulation of the FMS business process. So, the following subsection is dedicated to simulating the tokens’ randomly timed behavior in the places at the WPTSPN sequence s₃.

A cardinality marking graph/state graph interpretation verifies that cdM₆/e₄ is the final state of the FMS business process. We select the transitions in three sequences s 1   = { t r 1 ,   t r 2 ,   t r 3 ,   t r 4 ,   t r 5 ,   t r 7 ,   t r 8 ,   t r 5 ,   t r 6 } , s 2   = { t r 1 ,   t r 2 ,   t r 5 ,   t r 7 ,   t r 8 ,   t r 5 ,   t r 6 } , and s 3   = { t r 1 ,   t r 2 ,   t r 5 ,   t r 6 } lead to the final state.

The s₃ sequence is preferred to be simulated since it does not contain circuits. In our firing and randomly residence time analysis, we try to apply the method adopted by the WPTSPN.

· Let { e 0 ,   e 1 ,   e 2 ,   e 3 ,   e 4 } : the states’ firing sequence corresponding to s₃.

· Let { P 0 ,   P 1 ,   P 2 ,   P 3 ,   P 4 ,   P 5 ,   P 6 ,   P 7 ,   P 8 ,   P 10 } : the places’ subset is involved in the s₃ firing sequence.

· Let MIP₃: the matrix of the places input-transitions involved in the transitions sequence s₃.

· Let MPO₃: the matrix of the places output-transitions involved in the transitions sequence s₃.

M I P 3 = ( 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 ) M P O 3 = ( 1 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 )

· Let MIT₃: the matrix of the transitions input-places involved in the transitions sequence s₃. Let MTO₃: the matrix of the transitions output-places involved in the transitions sequence s₃.

M I T 3 = ( 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 )

M T O 3 = ( 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 )

Thus, the simulated system evolution depends on different situations according to the densities types (Exponential, Normal, and Log-Normal). To calculate the firing time of each transition as well as the tokens’ random stay in each timed place, we target the transitions firing sequence S₃. A data collection specifying the parameters of our simulation solution must be provided as an input {places, MIP₃, MPO₃}, {transitions, MIT₃, MTO₃, times at the latest, and firing rates}, and the parameters related to the probability distribution type ( λ   =   0.6 , μ   = 0.6 and σ   =   0.7 ). Table 6 illustrates our simulation transitions basic data.

After retrieving the input data, our simulation process follows the following steps:

· Step 1: The gravity center method [48] [50] which is used by the T F i r i n g T ( t r j ) ( ∀ t r j   ∈   s 3 ) function, is used to determine the firing time for each transition in the firing sequence (s₃/se₃). This function returns three-time values, one for each density type (exponential, normal, and log-normal). The outcomes of this stage are presented in Table 7: (sorting time with the gravity center method according to the three types of distribution and their cumulative values).

We note:

- EGFT: Exponential Gravity Center Transition Firing Time;

- NGFT: Normal Gravity Center Transition Firing Time;

- L-NGFT: Log-Normal Gravity Center Transition Firing Time;

- CEGFT: Cumulative Exponential Center Transition Firing Time;

- CNGFT: Cumulative Normal Center Transition Firing Time;

- CL-NGFT: Cumulative Log-Normal Gravity Center Transition Firing Time.

· Step 2: In this simulation phase, we calculate the cumulative firing time ( S F T =   ∑ ( T F i r i n g T ( t r j ) ) ) for each transition t r j in its firing sequence order (s₃/se₃). Also, for each transition, the three cumulative firing time values (see (Table 7)) are the basis for determining token dwell times. Then, for each timed place applied to the sequence, the lower bounds of the static stay intervals are determined from the matrix MIP₃. For each density type, the MEFIP₃, MNFIP₃, and ML-NFIP₃ matrix are constructed to select each

min-value row corresponding to the lower sac-token boundary at the current timed place (random residence).

M E F I P 3 = ( 0 0 0 0 0 0 0 51.80 0 0 0 0 1.12 0 0 0 0 0 16.71 0 0 5.02 0 0 0 5.02 0 0 0 0 16.72 0 0 0 0 51.80 0 0 0 51.80 )

M N F I P 3 = ( 0 0 0 0 0 0 0 55.30 0 0 0 0 0.79 0 0 0 0 0 18.17 0 0 3.17 0 0 0 3.17 0 0 0 0 18.17 0 0 0 0 55.30 0 0 0 55.30 )

M L - N F I P 3 = ( 0 0 0 0 0 0 0 67.47 0 0 0 0 4.99 0 0 0 0 0 29.98 0 0 14.98 0 0 0 14.98 0 0 0 0 29.98 0 0 0 0 67.47 0 0 0 67.47 )

· Step 3: In this third phase, we fix a random residence interval upper limit, which is assigned to a token in each timed place occurring in the sequence s s₃/se₃. For each density type, the MEFIT₃, MNFIT₃, and ML-NFIT₃ matrices are elaborated to select the minimum value of each column corresponding to the upper limit of the token’s stay.

M E I T 3 = ( 0 1.12 0 0 0 0 0 0 0 0 0 0 0 5.02 5.02 0 0 0 0 0 0 0 16.71 0 0 16.71 0 0 0 0 0 0 0 0 0 0 0 51.80 51.80 0 )

M N I T 3 = ( 0 0.79 0 0 0 0 0 0 0 0 0 0 0 3.07 3.17 0 0 0 0 0 0 0 18.17 0 0 18.17 0 0 0 0 0 0 0 0 0 0 0 55.30 55.30 0 )

M N L I T 3 = ( 0 4.99 0 0 0 0 0 0 0 0 0 0 0 14.99 14.99 0 0 0 0 0 0 0 29.98 0 0 29.98 0 0 0 0 0 0 0 0 0 0 0 67.47 67.47 0 )

After determining the randomly residence limits for each sac-token in WPTSPN’ timed place, we clarify that stochastically using the density functions according to the three distributions. Figure 7 represents the sac-token’ randomly residence distributions in the timed place P₇.

· The last step is summed up in determining for each place its three means of the stay time according to the gravity center method using the function M S T ( P i ) , ( ∀ P i ∈ s 3 ).

Table 8 illustrates the three randomly residence of all sac-token in the place P₇, its average residence times according to the three densities (exponential, Normal, and Log-normal), and a token sac₁ cycle-life according to the time horizon.

Figure 8 specifies the process of determining the stay dynamic interval according to the place P₇ and the decision about the token’s life-cycle, according to its time sensor value. The token sac₁ arriving in the place P₇ following the transition tr₅ firing is equipped with an automatic time sensor. This time sensor can activate a perception operation of the Randomly residence and compare its value, triggered from the first moment in the place, with the MST(P₇).

The following subsection shows our delay detection and prediction process related to the randomly residence time in a firing sequence.

The verification and simulation, allow us to detect a delay problem of the execution time relating to the operation op6 of the FMS business process, knowing that the latter is specified by the transition tr₆. So, we determine the cumulative

transition sequence firing time (s₃/se₃), after that, we check this delay indicated in the Figure 9. This figure illustrates the cumulative firing time, according to the three densities, for each transition making up the sequence. The delay detection step launches the prediction process linked to the tokens random residence in timed places. We apply the same prediction process to the micro-places, which are obtained following the division mechanism adopted by the model WPTSPN).

The model prediction approach requires converting the cardinality markings graph and/or the state sequence graph to Event-log: Lev, matching each cdM/se in Event. According to our FMS workflow simulation and verification results, firing sequence transitions denoted s₃/se₃ create a delay risk in FMS execution. Therefore, we propose a delay prediction procedure based on the algorithmic aspect illustrated in Figure 8. Knowing that the delay causes the token death in timed places during the transitions firing sequence. So, we use a prediction process based on a density truncation with the three possible distributions. This last process is illustrated by Algorithm 1.

An illustrative example of the truncated randomly residence densities is applied to sac-token in P₇ and shown in Figure 10. According to our model of transitions division approach (like in [7] ), it is possible to apply the truncating process at the second level (micro-places), as illustrated in Figure 11 and Figure 12.

Knowing that the two places (P₁₇, P₁₈) are considered as two micro-places preventing the splitting mechanism of the tr₆ transition. As our model based on the no interpretation hypothesis of immediate transitions, then t r 6 P does not temporally generate interpretable micro-states.

So, the immediate micro-transition T r 6 p densities’ truncation is not performed (like in [7] [48] ). The delay margin calculating the operation is based on subtraction between the time corresponding to the gravity center stay of the

truncated density and the truncation time. After that, a statistical set is generated by observing the truncation operations on the densities. The empirical mean and empirical variance are determined for each density, with the confidence interval forming the aim around the truncation times delayed estimations. Since the statistical population number is reduced, we adopt the empirical estimate method in [7] and [75] .

Table 9 illustrates the prediction results based on our truncation process applied to P₇, P₁₇ and P₁₈ micro-places. Thus, we determine the delay margins for truncated density. To explore the numerical values meaning in Table 9, we note:

· LST: Lower Limit of Stay Interval;

· TT: Truncated Time or Gravity Center Stay Time;

· EM: Empirical Mean of Truncated Densities;

· EMS: Empirical Margin of Stay;

· EL: Empirical Likelihood;

· CM: Empirical Confidence Margin.

The temporal analysis of tokens’ random residence in timed places adopted by our WPTSPN, shows that if we use an exponential density, then the stay-delay tends towards stability. So, Figure 10 shows a uniformly distributed truncated stay density on the time horizon. According to Table 9, we notice that the stay margin distributed with the log-normal density gratitude is compared to that distributed with the normal one. Also, in the FMS business process evolution, the tr₆ transition favors the end of production because of a maintenance sub-process. The proposed model applies the prediction tool to the micro-places which result from the transitions division mechanism. We note that the prediction process, oriented towards the P-timed context, requires retrospection on the past input transitions firing time, and a future vision (estimation) of the transitions firing time output.