Petri Net Modeling for dynamic MM composite Object

Composite MM object The first text and image shall start simultaneously. The video must start 10 seconds after the first image and first text have been displayed. The second text must be displayed simultaneously with the audio and the video. The Salesman picture is available from the start to the end of the presentation.

10 Sec Image 1 Text 1 Audio Video Text 2 Image 2 10 Sec Audio Video Text 2 Image 2 RTSM [0,20] [0,10] [10,20]

Objective Modeling user navigation with respect to synchronization constraints ( Fast Forward, Rewind, Pause)

UIPN definition UIPN is an eleven-tuple {S,E,T f,T u,A,D,R,M,F i,C,G,}where S={s 1,s 2,…,s m } finite set of Regular places, single circles, m  0; E={e 1,e 2,…e s } finite set of Enforced places, double circles, s  0; P= S  E All places T f ={t 1, t 2,…,t m } a finite set of forced transition, m  0; T u = {t 1,....t m } a finite set of user-initiated transitions, m >_ 0; T = T u  T F, T u  T f = , T  P=  A:{T X P}È {Pt} Directed arcs ( B: output, F: Input) D:p → Real number - predefined Incremental Time duration of places R: p → {r 1,r 2,…r k } Type of media ( modality) F i : p → {f 1,…..f k } Fidelity constraints C = P → R* is a mapping from places to clock values, where R* is a set of non-negative real numbers. G C :p* → Global Clock place. With a real number that start ticking with the initial place and end ticking with the end place. M: p → {0,1,2} state of place (no token, blocked, unblocked}

Execution semantic of UIPN A user-initiated transition tj is associated with it an event which is unpredictable (e.g., pressing of a button). Such a transition is fired if the associated event occurs. An output arc (pi, tj) Є A is enabled at time Τ if at time T if Global clock value is affected by user interaction and ά  c(pi)  β, α  β, ( α, β)= A ( ti, pi) We say that an output arc a from a place pi to a transition tj is enabled if a token is in pi and the value of c(pi) is within the range specified by the interval associated with the arc a. A transition tj Є T is enabled at time T if at time T, (pi, tj) is enabled for all i such that (pi, tj) Є A‘. That is, a transition tj is enabled if all the output arcs to it are enabled. A transition tj Є T is fired at time T if tj is enabled at time T ; The event associated with tj occurs at T. the value of c(pi) is within the range specified by the interval associated with the arc a. Upon firing, a set of backtracking rules is exercised to remove tokens from their input places. Meanwhile transition ti adds a token to its output places

Computation of Fireability domain For a given UIPN, if the clock upper and lower value are known, then it’s possible to select all appropriate places. All fireability domains of the transition that are enabled by current marking are considered in computation. The following rule must be satisfied: ά  c(pi)  β, α  β, ( α, β)= A ( ti, pi) So, for every output arc from the transition to a place, the new clock value must range in between it’s lower and upper predefined duration

10 Sec Image 1 Text 1 Video Text 2 Image 2 Audio Global Clock Global Clock [3 sec = 8 sec] t1t1 t1t1 User initiated transition FF for 5 sec [0,10]

State description A state S of a UIPN is defined as S = ( M, Đ). Here, M is the marking and Đ is a vector of clock values associated with places. So when a token arrive in a place, the clock in the place is activated at a given value, and it remains activated until the token is removed. As long as the clock is active, the clock value increase. In the User interaction case, if a user FF for 5 sec, the Gc is adjusted and t1 is fired. 10 Sec Image 1 Text 1 Video Text 2 Image 2 Audio Global Clock Global Clock

firing rules All places that are connected to the transition must be active ( must contain a token). All arcs connected to the transition must be enabled. In case of user interaction: transition tj Є T is enabled at time T if at time T, the value of c(pi) is within the range specified by the interval associated with the arc B. A transition tj Є T is fired at time T if tj is enabled at time T ; The event associated with tj occurs at T. the value of c(pi) is within the range specified by the interval associated with the arc B

Computing next state Reachability can be determined by the firing rules. If Transition t j fired at time x from state S. Then S’ = ( M’, Đ’) can be computed as: M’(p) = M(p) – B (t i,P) + F (P, t i ) All places that keep their tokens, remain active. The clock of the place from which transition take place become inactive. The Global clock pass the token created from the user to the designated place, after changing it’s own value. The clock of the place that receives the token is either activated or adjusted based on the token value