HYPENS Manual. Fausto Sessego, Alessandro Giua, Carla Seatzu. February 7, 2008

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1 HYPENS Manual Fausto Sessego, Alessandro Giua, Carla Seatzu February 7, 28 HYPENS is an open source tool to simulate timed discrete, continuous and hybrid Petri nets. It has been developed in Matlab to allow designer and user to take advantage of several functions and structures already defined in Matlab, such as optimization routines, stochastic functions, matrices and arrays, etc. The tool can also be easily interfaced with other Matlab programs and be used for analysis and optimization via simulation. The large set of plot functions available in Matlab allow one to represent the results of the simulation in a clear and intuitive way. Background on Hybrid Petri nets We recall the FOHPN formalism used by HYPENS, following [3]. Net structure: A FOHPN is a structure N = (P, T, P re, P ost, D, C. The set of places P = P d P c is partitioned into a set of discrete places P d (represented as circles and a set of continuous places P c (represented as double circles. The cardinality of P, P d and P c is denoted n, n d and n c. The set of transitions T = T d T c is partitioned into a set of discrete transitions T d and a set of continuous transitions T c (represented as double boxes. The cardinality of T, T d and T c is denoted q, q d and q c. The pre- and post-incidence functions that specify the arcs are (here R + = R+ {}: P re, P ost : P c T R +, P d T N. We require that t T c and p P d, P re(p, t = P ost(p, t, so that the firing of continuous transitions does not change the marking of discrete places. Transitions in T d may either be deterministic or stochastic. In the case of deterministic transitions the function D : T d R + specifies the timing associated to timed discrete transitions. In the case of stochastic transitions D defines the parameter(s of the distribution function corresponding to the timing delay. The function C : T c R + R+ specifies the firing speeds associated to continuous transitions (here R + = R + { }. For any continuous transition t j T c we let C(t j = [V j, V j], with V j V j: V j represents the minimum firing speed (mfs, V j F. Sessego, A. Giua and C. Seatzu are with the Department of Electrical and Electronic Engineering, University of Cagliari, {fausto.sessego,giua,seatzu}@diee.unica.it.

2 HYPENS Manual Rev represents the maximum firing speed (MFS. The incidence matrix of the net is defined as C(p, t = P ost(p, t P re(p, t. The restriction of C (P re, P ost, resp. to P x and T y, with x, y {c, d}, is denoted C xy (P re xy, P ost xy, resp.. A marking is a function that assigns to each discrete place a non-negative number of tokens, and to each continuous place a fluid volume. Therefore M : P c R +, P d N. The marking of place p i is denoted M i, while the value of the marking at time τ is denoted M(τ. The restriction of M to P d and P c are denoted with M d and M c, resp. A system N, M(τ is an FOHPN N with an initial marking M(τ. Net dynamics: The enabling of a discrete transition depends on the marking of all its input places, both discrete and continuous. More precisely, a discrete transition t is enabled at M if for all p i t, M i P re(p i, t, where t denotes the preset of transition t. The enabling degree of t at M is equal to enab(m, t = max{k N M k P re(, }. If t is infinite-server semantics, we associate to it a number of clocks that is equal to enab(m, t. Each clock is initialized to a value that is equal to the time delay of t, if t is deterministic, or to a random value depending on the distribution function of t, if t is stochastic. If a discrete transition is k-server semantics, then the number of clocks that are associated to t is equal to min{k, enab(m, t}. The values of clocks associated to t decrease linearly with time, and t fires when the value of one of its clocks is null (if k clocks reach simultaneously a null value, then t fires k times. Note that here we are considering enabling memory policy, not total memory policy. This means that if a transition enabling degree is reduced by the firing of a different transition, then the disabled clocks have no memory of this in future enabling [, 4]. If a discrete transition t j fires k times at time τ, then its firing at M(τ yields a new marking M(τ such that M c (τ = M c (τ + C cd σ, and M d (τ = M d (τ + C dd σ, where σ = k e j is the firing count vector associated to the firing of transition t j k times. To every continuous transition t j is associated an instantaneous firing speed (IFS v j (τ. For all τ it should be V j v j(τ V j, and the IFS of each continuous transition is piecewise constant between events. A continuous transition is enabled only by the marking of its input discrete places. The marking of its input continuous places, however, is used to distinguish between strong and weakly enabled: if all input continuous places of t j have a not null marking, then t j is called strongly enabled, else t j is called weakly enabled. We can write the equation which governs the evolution in time of the marking of a place p i P c as ṁ i (τ = t j T c C(p i, t j v j (τ where v(τ = [v (τ,..., v nc (τ] T is the IFS vector at time τ. The enabling state of a continuous transition t j defines its admissible IFS v j. If t j is not enabled then v j =. If t j is strongly enabled, then it may fire with any firing speed v j [V j, V j]. If t j We are using an enabling policy for continuous transitions slightly different from the one proposed by David and Alla [4]. See [2] for a detailed discussion.

3 HYPENS Manual Rev is weakly enabled, then it may fire with any firing speed v j [V j, V j], where V j V j since t j cannot remove more fluid from any empty input continuous place p than the quantity entered in p by other transitions. Linear inequalities can be used to characterize the set of all admissible firing speed vectors S. Each vector v S represents a particular mode of operation of the system described by the net, and among all possible modes of operation, the system operator may choose the best, i.e., the one that maximize a given performance index J(v [2]. We say that a macro event (ME occurs when: (a a discrete transition fires, thus changing the discrete marking and enabling/disabling a continuous transition; (b a continuous place becomes empty, thus changing the enabling state of a continuous transition from strong to weak; (c a continuous place, whose marking is increasing (decreasing, reaches a flow level that increases (decreases the enabling degree of discrete transitions. Let τ k and τ k+ be the occurrence times of two consecutive ME as defined above; we assume that within the interval of time [τ k, τ k+, denoted as a macro period (MP, the IFS vector is constant and we denote it v(τ k. Then the continuous behavior of an FOHPN for τ [τ k, τ k+ is described by M c (τ = M c (τ k + C cc v(τ k (τ τ k, M d (τ = M d (τ k. Example. Consider the net system in Fig..a. Place p is a continuous place, while all other places are discrete. Continuous transitions t, t 2 have MFS V =, V 2 = 2 and null mfs. Deterministic timed discrete transitions t 3, t 5 have timing delays 2 and.5, resp. Exponential stochastic discrete transitions t 4, t 6 have average firing rates are λ 4 = 2 and λ 6 =.5. The continuous transitions represent two unreliable machines; parts produced by the first machine (t are put in a buffer (p before being processed by the second machine (t 2. The discrete subnet represents the failure model of the machines. When p 3 is marked, t is enabled, i.e. the first machine is operational; when p 2 is marked, transition t is not enabled, i.e. the first machine is down. A similar interpretation applies to the second machine. Assume that we want to maximize the production rates of machines. In such a case, during any MP continuous transitions fire at their highest speed. This means that we want to maximize J(v = v + v 2 under the constraints v V, v 2 V 2, and when p is empty v 2 v. The resulting evolution graph and the time evolution of M, v and v 2 are shown in Fig..b and c. During the first MP (of length both continuous transitions are strongly enabled and fire at their MFS. After one time unit, p gets empty, thus t 2 becomes weakly enabled fires at the same speed of t. At time τ =.5, transition t 5 fires, disabling t 2. 2 HYPENS Tool HYPENS has been developed in Matlab (Version 7.. It is composed of 4 main files. The first two files, make HPN.m and enter HPN.m create the net to be simulated: the former requires input data from the workspace while the latter is a guided procedure. The file simulator HPN.m computes the timing evolution of the net that is summarized in an

4 HYPENS Manual Rev Discrete part Continuous part 2 [ ] T M d = c M v = ( = [ 2] T v p 2 t 3 p 3 t 4.5 t [,] p d = [ ] T [ ] T M p c M v = ( = M.5 p 4 t 5 p 5 t 6 t 2 [,2] t 5 d = [ ] T 2 [ ] T M c M v = (.5 = 2 2 v 2 2 (a (b (c Figure : (a An FOHPN, (b its evolution graph, and (c its evolution in time. array of cells called Evol. Based on this array, the file analysis HPN.m computes useful statistics and plots the simulation results. Function make HPN: [Pre, Post, M, vel, v, D, s, alpha] = make HPN (Pre cc, Pre cd, Pre dc, Pre dd, Post cc, Post cd, Post dd, M c, M d, vel, v, D, s, alpha. This function has the following input arguments. Matrices P re cc, P re cd, P re dc, P re dd, P ost cc, P ost cd, P ost dd ; The initial marking M c and M d of continuous/discrete places. Matrix vel (R + q c 2 specifies, for each continuous transition, the mfs and the MFS. Vector v N q d specifies the timing structure of each discrete transition. The entries of this vector may take the following values: - deterministic; 2 - exponential distribution; 3 - uniform distribution; 4 - Poisson distribution; 5 - Rayleigh distribution; 6 - Weibull distribution; etc. Matrix D (R + q d 3 associates to each discrete transition a row vector of length 3. If the transition is deterministic, the first element of the row is equal to the time delay of transition. If the transition is stochastic, the elements of the row specify the parameters of the corresponding distribution function (up to three, given the available distribution functions. Vector s N q d keeps track of the number of servers associated to discrete transitions. The entries take any value in N: if the corresponding transition has infinite servers; k > if the corresponding transition has k servers. Vector alpha specifies the conflict resolution policy among discrete transitions.

5 HYPENS Manual Rev When alpha N q d then two cases are possible. If all its entries are zero, conflict resolution is solved by priorities that depend on the indices of transitions (the smallest the index, the highest the priority. Otherwise, all its entries are greater than zero and specify the weight of the corresponding transition, i.e., if T e is the set of enabled transitions, the probability of firing transition t T e is π(t = alpha(t/ ( t T e alpha(t. When alpha N 2 q d the first row specifies the weights associated to transitions (as in the previous case while the second row specifies the priorities associated to transitions. During simulation, when a conflict arises, priority are first considered; in the case of equal priority, weights are used to solve the conflict. See [4] for details. Output data of function enter HPN are nothing else than input data, appropriately rewritten so as to constitute the input to function simulator HPN. Matrices P re and P ost are defined as: P re cc NaN P re cd P re = NaN NaN NaN, P ost = P ost cc NaN P ost cd NaN NaN NaN P re dc NaN P re dd P ost dc NaN P ost dd where a row and a column of NaN (not a number have been introduced so as to better visualize the continuous and/or discrete sub-matrices. The initial marking is denoted as M and is defined as a column vector. All other output data are identical to the input data. Function enter HPN: [Pre, Post, M, vel, v, D, s, alpha] = enter HPN. This function creates the net following a guided procedure. The parameters are identical to those defined for the previous function make HPN.m. Function simulator HPN: Evol = simulator HPN(Pre, Post, M, vel, v, D, s, alpha, time stop, simulation type, J. The input data of function simulator HPN coincide with the output data of the previous interface functions, plus three additional parameters. time stop is equal to the time length of simulation. simulation type {2,, } specifies the simulation mode. Mode : no intermediate result is shown but only array Evol is created to be later analyzed by function analysis HPN. Modes and 2 generate on screen the evolution graph: in the fist case the simulation proceeds without interruptions until time stop is reached; in the second case the simulation is carried out step-bystep J R q c is a row vector that associates to each continuous transition a weight: J v is the linear cost function that should be maximized at each MP to compute the IFS vector v. This

6 HYPENS Manual Rev optimization problem is solved using the subroutine glpkmex.m of Matlab. Output data are saved in an array of cells called Evol, with the following entries (here K is the number of ME that occur during the simulation run. Type {, 2, 3}: (2, 3 if the net is continuous (discrete, hybrid. M Evol (R + n (K+ keeps track of the marking of the net during all the evolution. In particular, an n dimensional column vector is associated to the initial time instant and to all the time instants in which a different ME occurs, each one representing the corresponding value of the marking at that time instant. IFS Evol (R + q c (K+ keeps track of the IFS vectors during all the evolution. In particular, a q c dimensional column vector is associated to the initial configuration and to the end of each ME. P macro and Event macro Evol are (K + -dimensional row vectors and keep track of the ME caused by continuous places. If the generic r-th ME is due to continuous place p j, then the (r+ th entry of P macro is equal to j; if it is due to the firing of a discrete transition, then the (r+ th entry of P macro is equal to NaN. The entries of Event macro Evol may take values in {,,, NaN}: means that the corresponding continuous place gets empty; means that the the continuous place enables a new discrete transition; means that the continuous place disables a discrete transition. If the generic r-th ME is due to the firing of a discrete transition, then the (r+-th entry of Event macro Evol is equal to NaN as well. The first entries of both P macro and Event macro Evol are always equal to NaN. firing transition is a (K + dimensional row vector that keeps track of the discrete transitions that have fired during all the evolution. If the r-th ME is caused by the firing of discrete transition t k, then the (r + th entry of firing transition is equal to k; if it is caused by a continuous place, then the (r + th entry of firing transition is equal to. Note that the first entry of firing transition is always equal to NaN. timer macro event is a (K + dimensional row vector that keeps into memory the length of ME. The first entry is always equal to NaN. τ is equal to the total time of simulation. Q Evol is a ((K + q d dimension array of cells, whose generic (r +, j-th entry specifies the clocks of transition t j at the end of the r-th MP. P c P d T c T d N 4 is a 4-dimensional row vector equal to [n c n d q c q d ]. Function analysis HPN: [ P ave, P max, Pd ave t, IFS ave, Td ave, Pd freq, Md freq ] = analysis HPN (Evol, static plot, graph, marking plot, Td firing plot, up marking plot, Pd prob plot, Pd ave t plot, IFS plot, up IFS plot, IFS ave plot, Td freq plot. This function computes useful statistics and plots the results of the simulation run contained in

7 HYPENS Manual Rev Evol. The other input parameters are the following. statistic plot {, }: if two histograms are created showing for each place, the maximum and the average value of the marking during the simulation. graph {, }: if the evolution graph is printed on screen. marking plot: is a vector used to plot the marking evolution of selected places in separate figures. As an example, if we set marking plot= [x y z], the marking evolution of places p x, p y and p z is plotted. If marking plot= [ ], then the marking evolution of all places is plotted. Td firing plot {, }: if a graph is created showing the time instants at which discrete transitions have fired. up marking plot is a vector used to plot the marking evolution of selected places in a single figure. The syntax is the same as that of marking plot. Pd prob plot {, }: means that as many plots as the number of discrete places will be visualized, each one representing the frequency of having a given number of tokens during the simulation run. Pd ave t plot is a vector used to plot the average marking in discrete places with respect to time. A different figure is associated to each place, and the syntax to select places is the same as that of marking plot. IFS plot (resp., up IFS plot: is a vector used to plot the IFS of selected transitions in separate figures (resp., in a single figure. The syntax is the same as that of marking plot and up marking plot. IFS ave plot {, }: if an histogram is created showing the average firing speed of continuous transitions. Td freq plot {, }: if an histogram is created showing the firing frequency of discrete transitions during the simulation run. These are the outputs of function analysis HPN.m. P ave (P max R n : each entry is equal to the average (maximum marking of the corresponding place during the simulation run. Pd ave t R n d K : column k specifies the average marking of discrete places from time τ = to time τ k when the k-th ME occurs. IFS ave R q c : each entry is equal to the average IFS of the corresponding continuous transition. Td ave R q d: each entry is equal to the average enabling time of the corresponding discrete transition. Pd freq R n d (z+ specifies for each discrete place the frequency of a given number of tokens

8 HYPENS Manual Rev during the simulation run. Here z is the maximum number of tokens in any discrete place during the simulation run. Md freq R (n d+ K where K denotes the number of different discrete markings that are reached during the simulation run. Each column of Md freq contains one of such markings and its corresponding frequency as a last entry. 3 A numerical example Let us consider again the FOHPN system in Fig.. For simplicity, we assume that input data using function make HP N.m are entered in a file net.m where we define the following matrices: P re CC = [ ] ; P re CD = [ ] ; P re DC = [ ] ; P re DD = [ ] ; P ost CC = [ ] ; P ost CD = [ ] ; P ost DC = [ ] ; P re DD = [ ] ; M C = [ ] ; M D = [ ] ; vel = [ 2] ; D = [ 2 NaN NaN.5 NaN NaN.5 NaN NaN] ; alpha = [ ] ; v = [ 2 2] ; s = [ ] ; We save the above input data in an array of cells net array = {P re CC, P re CD, P re DC, P re DD, P ost CC, P ost CD, P ost DC, P ost DD, M C, M D, vel, v, D, s, alpha} Now, input should be rewritten in an appropriate form so as to provide the input data of function simulator HP N.m. Two different syntax are possible:. correct net = make HP N (P re CC, P re CD, P re DC, P re DD, P ost CC, P ost CD, P ost DC, P ost DD, M C, M D, vel, v, D, s, alpha

9 HYPENS Manual Rev correct net = make HP N (net array This also allow to verify if input data are consistent. If such is the case, then output data of function make HP N are saved in an array of cells called correct net; if input data are not consistent error messages are displayed. If data are consistent, in the prompt of Matlab the following information are visualized: transition t --> msf= MFS= transition t2 --> msf= MFS=2 transition t3 --> server inf; weight ; transition t4 --> server inf; weight ; transition t5 --> server inf; weight ; transition t6 --> server inf; weight ; correct_net = Columns through 6 [6x7 double] [6x7 double] [5x double] [2x2 double] [x4 double] [4x3 double] Columns 7 through 8 [x4 double] [x4 double] In particular, it is summarized the minimum firing speeds and the maximal firing speeds of continuous transitions, and the number of servers and the weight associated to discrete transitions. Then, it is visualized that correct net is an array of eight cells, and the dimension of each cell is explicitly shown. If celldisp(correct net is executed then the following data are shown: correct net{} = P re = [ NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN ] correct net{2} = P ost = [ NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN ]

10 HYPENS Manual Rev correct net{3} = M = [ ] ; correct net{4} = vel = [ 2] ; correct net{5} = v = [ 2 2] ; correct net{6} = D = [ 2 NaN NaN.5 NaN NaN.5 NaN NaN] ; correct net{7} = s = [ ] ; correct net{8} = alpha = [ ] ; To simulate the net behavior we use function simulator HP N, input data are correct net plus three additional parameters that we assume as follows: time stop = 7 (the time length of simulation is assumed equal to seven simulation tipe = (specifies that we want visualized the evolution graph without interruptions until time stop is reached J = [ ] (the objective of function we want maximize for each Macro Period is equal to V + V 2 where V and V 2 are the IFS of continuous transitions t and t 2 Now, we execute Evol = simulator HP N (correct net, 7,, [ ] The evolution graph is generated on the screen: The current continuous marking Istantaneous firing speeds t strongly enabled t2 2 strongly enabled The current discrete marking

11 HYPENS Manual Rev Residual times of clocks Q t3 2. t4 NaN t5.5 t6 NaN Enabling vector e Place p gets empty after a time interval. (time tot= The current continuous marking Istantaneous firing speeds t strongly enabled t2 weakly enabled The current discrete marking Residual times of clocks Q t3. t4 NaN t5.5 t6 NaN Enabling vector e The discrete transition that fire at time.5 (time tot=.5 is t5 ( time The current continuous marking Istantaneous firing speeds t strongly enabled t2 not enabled The current discrete marking

12 HYPENS Manual Rev Residual times of clocks Q t3.5 t4 NaN t5 NaN t Enabling vector e The discrete transition that fire at time.5 (time tot=2. is t3 ( time The current continuous marking.5 Istantaneous firing speeds t not enabled t2 not enabled The current discrete marking Residual times of clocks Q t3 NaN t t5 NaN t Enabling vector e The discrete transition that fire at time (time tot=4.279 is t6 ( time The current continuous marking.5 Istantaneous firing speeds t not enabled t2 2 strongly enabled The current discrete marking

13 HYPENS Manual Rev Residual times of clocks Q t3 NaN t4.65 t5.5 t6 NaN Enabling vector e Place p gets empty after a time interval.25 (time tot= e+ The current continuous marking Istantaneous firing speeds t not enabled t2 weakly enabled The current discrete marking Residual times of clocks Q t3 NaN t4.4 t5.25 t6 NaN Enabling vector e The discrete transition that fire at time.4 (time tot=4.929 is t4 ( time The current continuous marking Istantaneous firing speeds t strongly enabled t2 weakly enabled The current discrete marking

14 HYPENS Manual Rev Residual times of clocks Q t3 2. t4 NaN t5.849 t6 NaN Enabling vector e The discrete transition that fire at time.849 (time tot=5.779 is t5 ( time The current continuous marking Istantaneous firing speeds t strongly enabled t2 not enabled The current discrete marking Residual times of clocks Q t3.5 t4 NaN t5 NaN t Enabling vector e The discrete transition that fire at time.5 (time tot=6.929 is t3 ( time The current continuous marking.56 Istantaneous firing speeds t not enabled t2 not enabled The current discrete marking

15 HYPENS Manual Rev Residual times of clocks Q t3 NaN t4.443 t5 NaN t Enabling vector e Evol = Columns through 6 [3] [5x double] [2x double] [x double] [x double] [x double] Columns 7 through [x double] [x double] {x cell} [x4 double] Output data of function simulator HP N are saved in an array of then cells called Evol that provides the input of function analysis HP N. If we execute the following data are shown in the screen: E[M(p]=.367 E[M(p2]=.429 E[M(p3]=.57 E[M(p4]=.57 E[M(p5]=.429 max(p=.5 max(p2=. max(p3=. max(p4=. max(p5=. Prob[M(p2=]=.57 Prob[M(p2=]=.429 Prob[M(p3=]=.429 Prob[M(p3=]=.57 Prob[M(p4=]=.429 Prob[M(p4=]=.57 analysed net = analysis HP N(Evol

16 HYPENS Manual Rev Prob[M(p5=]=.57 Prob[M(p5=]=.429 E[v]=.57 E[v2]=.55 E[t3]=.286 E[t4]=.43 E[t5]=.286 E[t6]=.43 ****************************************** Md(=[ ] f(=.4286 ****************************************** Md(2=[ ] f(2=.9277 ****************************************** Md(3=[ ] f(3=.2358 ****************************************** Md(4=[ ] f(4= ****************************************** Md(5=[ ] f(5=.357 ****************************************** Md(6=[ ] f(6= ****************************************** Md(7=[ ] f(7=.7 ****************************************** analysed_net = [x5 double] [x5 double] [x2 double] [x6 double] [5x2 double] [] [6x7 double] Now, let us the function analysis HP N may also have different inputs that enable us to specify the plots we want visualize. The following examples clarify this. analysed net = analysis HP N(Evol, statistic plot = : the maximum and the average value of the marking during the simulation are displayed (see Figure 2;

17 HYPENS Manual Rev The maximum marking of places The average marking of places wrt time Max(p i Average number of tokens p i p i Figure 2: analysed net = analysis HP N(Evol, analysed net = analysis HP N(Evol, [ ], graph = : this parameter is only relative to discrete nets, thus it has no effect in our example regardless of the value it assumes; analysed net = analysis HP N(Evol, [ ], [ ], marking plot = : the marking evolution of all places is shown (see Figure 3 in two different figures. The first one shows the marking evolution of places p, p 2, p 3, p 4, and the second one shows the marking evolution of places p 5 2 ; analysed net = analysis HP N(Evol, [ ], [ ], [ ], T d firing plot = : a graph showing the time instants at which discrete transition have fired is shown (see Figure 4; analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ 3] Up marking plot = [ 3]: the marking evolution of places p and p 3 are plotted in the same figure (see Figure 5; analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ ], P d prob plot = : the frequency of having a given number of tokens during the simulation run is shown for each discrete place (see Figure 6; analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ ], [ ], P d ave t plot = : this parameter is only relative to discrete nets, thus it has no effect in our example regardless of the value it assumes; 2 By default if the number of places is greater than 4, then the graphs are shown in different figures

18 HYPENS Manual Rev M(p The evolution of marking M(p wrt time The evolution of marking M(p wrt time M(p time The evolution of marking M(p 3 wrt time time The evolution of marking M(p 4 wrt time.8 M(p M(p time time The evolution of marking M(p 5 wrt time.8 M(p time Figure 3: analysed net = analysis HP N(Evol, [ ], [ ],

19 HYPENS Manual Rev transitions t i Macro Events: disc. trans. t i (i or cont. place ( tempo Figure 4: analysed net = analysis HP N(Evol, [ ], [ ], [ ], M(p M(p 3 Markings wrt time Markings time Figure 5: analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ 3]

20 HYPENS Manual Rev Frequency of a given number of tokens in p 2 Frequency of a given number of tokens in p Prob[M(p 2 ] Prob[M(p 3 ] M(p M(p 3 Frequency of a given number of tokens in p 4 Frequency of a given number of tokens in p Prob[M(p 4 ] Prob[M(p 5 ] M(p M(p 5 Figure 6: analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ ],

21 HYPENS Manual Rev IFS of t wrt time 2 IFS of t 2 wrt time.8.5 IFS(t.6.4 IFS(t time time Figure 7: analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ ], [ ], [ ], 2.5 IFS wrt time IFS(t IFS(t 2 IFS(t i time Figure 8: analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ ], [ ], [ ], IF S plot = : the IFS of all continuous transitions are shown (see Figure 7; analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], up IF S plot = : the IFS of all continuous transitions are shown in the same figure (see Figure 8; analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ ], [ ], [ ],, [ ], [ ], IF S ave plot = : an histogram showing the average firing speed of continuous transitions is created (see Figure 9; analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ ], [ ], [ ],, [ ], [ ], [ ], T d freq plot = : an histogram showing the firing frequency of discrete transitions is created (see Figure ;

22 HYPENS Manual Rev Average firing speed of continuous transitions.5 Mean speed vel t i Figure 9: analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], Mean number of firing Firing frequency of discrete transitions ti Figure : analysed net = analysis HP N(Evol, [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ],

23 HYPENS Manual Rev References [] M. Ajmone Marsan, G. Balbo, G. Conte, S. Donatelli, and G. Franceschinis. Modelling with Generalized Stochastic Petri Nets. Wiley, 995. [2] F. Balduzzi, A. Giua, and C. Seatzu. Modelling and simulation of manufacturing systems with first-order hybrid Petri nets. Int. J. of Production Research, 39(2: , 2. [3] F. Balduzzi, G. Menga, and A. Giua. First-order hybrid Petri nets: a model for optimization and control. IEEE Trans. on Robotics and Automation, 6(4: , 2. [4] R. David and H. Alla. Discrete, Continuous and Hybrid Petri Nets. Springer, 24.