discrete–continuous modeling using hybrid

DISCRETE–CONTINUOUS MODELING USING HYBRID STOCHASTIC PETRI NETS Graham Horton and Markus Kowarschik System Simulation Group Computer Science Department Erlangen University Martensstr. 3, 91058 Erlangen, Germany

fgraham j [email protected]

KEYWORDS

Petri net.

Numerical methods, Performance Analysis, Hybrid simulation, Probabilistic, Stochastic

Genuinely hybrid systems are characterized by having non-trivial interactions between the discrete and continuous parts of the model; in this case a modeling environment is desirable which allows a unified specification of the complete system to be studied. Moreover, we would like the modeling paradigm to be convenient, flexible and intuitive. In this paper we present Hybrid Stochastic Petri Nets (HSPNs) as a candidate for such a modeling environment.

ABSTRACT We present Hybrid Stochastic Petri Nets (HSPNs) as a revision and extension of the Fluid Stochastic Petri Nets (FSPNs). HSPNs can be used for modeling dynamic systems that comprise both discrete and continuous quantities. Discrete state changes are assumed to be Markovian and continuous variables are assumed to be deterministic. HSPNs are designed to improve the clarity and consistency of the graphical representation of the net. This is achieved by the introduction of a new class of transition and by a re–formulation of the enabling and firing rules. By utilizing a different numerical solution technique, we can overcome certain modeling limitations concerning instantaneous changes of state. The technique is based on an approximation of the hybrid system using a discrete–state stochastic process, from which a continuous–time Markov chain (CTMC) can be derived. Examples demonstrate the enhancements contained in the new formalism.

INTRODUCTION A large number of systems in engineering are of a hybrid nature; they contain both discrete and continuous variables. Discrete variables may represent the states of a switching or controlling network or changing environmental states; the continuous variables usually – but not always – represent physical quantities such as temperature or electrical potential. We will consider systems in which the discrete state changes are of a stochastic nature. In some cases, interactions between continuous and discrete variables are fairly simple, and such hybrid systems are modeled by treating the continuous and discrete models separately, using completely different paradigms. One typical example would be to model the continuous variables by a system of differential equations and the discrete system by a (discrete)

In [4], Horton et. al. introduced Fluid Stochastic Petri Nets (FSPNs) as just such a hybrid modeling paradigm. The HSPN described here represent a modification to the original FSPN definition which is intended to improve certain aspects of the syntax and semantics of the graphical net description. In addition, by using an alternate method of deriving the equations of state, we are able to extend the modeling possibilities compared to those of the FSPN. There are several difficulties associated with extending an essentially discrete modeling paradigm like Petri nets to include continuous values. In particular, firing and enabling rules for transitions must be re-defined to allow for the movement of continuous markings. Problems occur where continuous and discrete net elements meet. With hindsight, the FSPN definition does not seem to address these problems adequately. We suggest the alternative definition presented here as a more elegant solution that is more consistent with traditional Petri net usage. The extended modeling possibilities of HSPNs are not inherent to the revised net definition, but are made possible by a different approach to the numerical solution. In the FSPN case, the dynamics of the net are described by a system of partial differential equations for a probability density function, which are then discretized and solved numerically. Unfortunately, some desirable functionality (concerning certain events with zero delay) had to be omitted from the FSPN, in order that the differential equations could be obtained. By relinquishing the necessity of explicitly setting up a differential equation for the numerical solution of the model, these omissions are no longer necessary, and instantaneous changes of state can be generally incorpo-

rated into the model. In addition to genuinely hybrid systems, HSPNs – like their predecessors FSPNs – can be used to model other classes of problem. These include the approximation of models with large discrete states spaces (the so–called fluid approximation, which is well–known in queueing networks), enhanced reward modeling, described for FSPNs in [5], and the implementation of supplementary variables for representing non-exponentially distributed events, described, for example in [3]. The paper is structured as follows: In the next section a specification of HSPNs is given. In the subsequent section a comparison is made between FSPNs and HSPNs, pointing out their differences and the advantages of the new approach. The following section then describes our method of numerically analyzing HSPN. Two examples are then given in the following section. In the final section our conclusions are presented.

state space of the HSPN. This notation is introduced in order to simplify the following parts of our definition. According to the GSPN formalism, the marking

m0 = (~x(0); (0)) 2 M is called the initial marking of the HSPN. As usual, the set of transitions of a HSPN is denoted by At first, we merely consider the immediate transitions Ti and the timed transitions Te , which the GSPN user is already familiar with. We maintain the common convention and represent immediate transitions as bars and timed transitions as rectangles. From now on, we use the term impulse transition to denote a transition belonging to Ti [ Te . A third transition type will be defined below, where we will introduce the mechanism of continuous state changes.

T.

DEFINITION OF HSPN HSPN can be most conveniently specified by considering first their instantaneous, and then their continuous state changes.

1 r p1

Instantaneous State Changes The starting point for the definition of our HSPN formalism is the conventional GSPN paradigm [1], which we will extend by integrating the possibility of modeling continuous quantities. In addition to the discrete places we introduce continuous places, which contain non–negative real markings (levels). These real markings for example characterize physical quantities like temperature, time and electrical potential. Continuous places are graphically represented as pairs of concentric circles. They correspond to the fluid places in the FSPN definition. The set P of the places of a HSPN thus consists of its discrete places Pd and its continuous places P . We use the expression pi t to denote the number of tokens which the discrete place pi 2 Pd contains at time t; the marking of the continuous place pk 2 P at time t is denoted by xk t . Note that in the graphical representation of the net, we will use the usual notation P to denote the (discrete or continuous) marking of place P . The numbers of discrete and continuous places are denoted by D and C , respectively.

# ()

()

#

()

According to the two classes of places, the marking m t of a HSPN at time t is given by its discrete part

 = (t) = (#pi (t); 1  i  D) 2 IND 0 and its continuous part

~x = ~x(t) = (xk (t); 1  k  C ) 2 IRC0;+ : We define

M  IND ; IRC; 0

0+




p3

p2

to be the discrete–continuous

* 0.2 2

if

#p1  1

3

Figure 1: A simple HSPN model Figure 1 shows a simple HSPN model. p1 is a continuous place, whereas p2 and p3 are discrete places. Besides, there are two impulse transition: the immediate transition 2 and the timed transition 3 . The set of HSPN arcs is named A. In our definition, we distinguish between the impulse arcs Ai and the flow arcs Af . Impulse arcs are drawn as single lines, flow arcs are drawn as double lines (in order to suggest the analogy to a pipeline). The HSPN model shown in Figure 1 contains four impulse arcs and one flow arc. Flow arcs are defined to enable continuous changes of the fluid–like quantities in the net. Thus, we will explain their meaning below, together with the functionality of the new class of transitions. An impulse arc connects an impulse transition with a place, which may be either discrete or continuous. If the place is discrete, a certain number of tokens is moved along the impulse arc when the impulse transition fires. Otherwise, a specified amount of continuous quantity is instantaneously transported along the impulse arc as the firing event occurs. Therefore, we can generally interpret

Ai to be a weight

mapping: 8 > > >
> > :

[Pd  (Ti [ Te )℄ [ [(Ti [ Te )  Pd℄  M ! IN ; [P  (Ti [ Te )℄ [ [(Ti [ Te )  P ℄  M ! IR ; [ fg:

The firing rate function

0

0+

Depending on both the discrete and the continuous part of the HSPN marking, Ai assigns a weight to each impulse arc of the net. We adopt the following convention, which is already known from the GSPN paradigm: if an impulse arc is not explicitly weighted, we assume the weight 1 by default. It is clear from the context whether this must be considered as an integer or as a real number. In addition, an impulse arc reaching from a continuous place to an impulse transition may be labeled with the symbol  (see the impulse arc connecting place p1 with transition 2 in Figure 1). This means that the whole content of the continuous place is instantaneously removed when the corresponding transition fires. Although it is not possible for the majority of physical quantities to change discontinuously, there are situations where it is necessary to model an instantaneous change concerning the continuous marking ~x of the net (see for example the realization of Cox’ method of supplementary variables at the end of this paper). Our HSPN formalism allows the use of guard functions in the usual sense. Guard functions can be defined for each class of transitions, including the new transitions to be defined below in . All the guard functions belonging to a HSPN model can be comprised as a mapping

G : T  M ! f0; 1g: Depending on the current marking of the net, each transition is assigned a boolean value specifying whether the transition is enabled or not. We maintain the convention that unspecified guard functions always return 1 (true). For example, there is a guard function assigned to the immediate transition 2 in Figure 1: 2 is not enabled if the content x1 of the continuous place p1 is less then the value 1 . The weight function

W : Ti  M ! IR

+

assigns a marking–dependent value to each immediate transition of the HSPN. If an immediate transition  2 Ti is enabled by the HSPN marking m 2 M, its firing probability is computed as

W (; m) ; W ( ; m)

P

0

 0 2Ei (m)

( )

if there are several immediate transitions enabled by a marking m.

where Ei m denotes the set of all immediate transitions that are enabled by m. In the case of an unspecified value of W we assume the weight 1 by default. It is important to introduce this weight function W . It specifies the behavior of the HSPN,

F : Te  M ! IR

+

;

is used to assign to each timed transition a marking–dependent exponential firing rate. The dynamics of the discrete HSPN part correspond exactly to the GSPN behavior. The GSPN firing rule is extended to the continuous places of the HSPN in a natural way: enabled impulse transitions cause the HSPN markings to be changed according to the weights of the impulse arcs, which are defined by the mapping Ai . There is no fundamental distinction between the change of discrete and continuous markings. Therefore, we modify the definition of an enabled impulse transition: in an HSPN model, not only the discrete but also the continuous input places of an impulse transition must contain enough tokens and a high enough level of the continuous quantity, respectively, in order that the transition can be enabled. Formally, this can be expressed as follows: an impulse transition  2 Ti [ Te is enabled at time t by the (hybrid) marking mt ~x t ;  t , if

( ) = ( ( ) ( ))

1. all its input places contain an adequate marking according to the definition of the weight mapping Ai , and 2.

G (; (~x(t); (t))) = 1 holds.

This extension is entirely compatible with the classical idea of Petri nets. In order that an impulse transition can fire, it is necessary that all of its input places satisfy a condition concerning their contents. Now, it becomes clear why we decided to call a transition belonging to Ti [ Te an impulse transition: its firing causes an instantaneous (impulse–like) change of the complete HSPN marking.

Continuous State Changes As we have already mentioned above, the definition of continuous (fluid–like) flows of continuous quantity is based on the introduction of a new class of transitions, which we name pump transitions. In our graphical representations pump transitions are drawn as rectangles with crossed diagonals (see transition 1 in Figure 1, for example). The set of pump transitions of a HSPN is denoted by Tp , so that we eventually obtain three classes of HSPN transitions: T T i [ Te [ Tp .

=

Pump transitions may only be connected with continuous places, using flow arcs. This is expressed formally by the following relation:

Af  (P  Tp ) [ (Tp  P ): A pump transition can only be disabled using a guard function. Formally,  2 Tp is enabled by the HSPN marking

m(t)

= ( ( ) ( ))

(

( )) = 1

~x t ;  t at time t, if G ; m t holds. Thus, a pump transition with an unspecified guard function is always enabled.

R : Af  M ! IR ;

0+

;

( ( ))

It is possible to assign to a continuous place pk 2 P a finite capacity Bk according to the rules of the conventional Petri net formalism. By default, we assume an infinite capacity for each continuous place. The introduction of finite capacities, however, does not increase the modeling power of HSPN, since the same dynamic behavior can be achieved by defining appropriate guard functions, too. It just makes the formalism more convenient. We can define the marking–dependent potential rate

rk (m) at which the level xk of the continuous place pk 2 P changes: for every HSPN marking m 2 M, the value rk (m) is defined to be the difference of all the continuous flows entering pk and all those leaving it. Formally, this can be written as follows:

R ((; pk ); m)

 2Tp : G (;m)=1

X

R ((pk ;  ); m) :

 2Tp : G (;m)=1

( ) ( )

The resulting effective rate rkeff m does not necessarily correspond to the potential rate rk m , because neither has rk m < any effect if pk is already empty in marking m, nor has rk m > any effect if pk is defined to have a finite capacity Bk which is already reached in marking m. Similar considerations can be found for example in [4] for FSPN and in [2] for an ATM switch model.

( )

0 ( ) 0

F

F

which assigns to each flow arc a flow rate, that may depend both on the discrete and the continuous part of the HSPN marking. For reasons of better handling, we assume R a; ~x;  to be at least piecewise continuous in any parameter xk for each flow arc a 2 Af and every discrete part  of the HSPN marking. For the HSPN shown in Figure 1, this means that transition 1 constantly pumps continuous quantity into place p1 at rate r.

X

A

A

We define the flow rate function

rk (m) =

aspects of the syntax and semantics of the graphical net description.

With this definition we obtain a Petri net formalism which, in our opinion, achieves our goals of modeling hybrid systems both elegantly and conveniently.

COMPARISON WITH FSPN In this section we compare HSPNs with FSPNs, considering first the differences in syntax and semantics, and second the extended modeling possibilities offered by the alternate approach to the numerical analysis.

Net Semantics And Syntax HSPNs were developed on the basis of the existing FSPNs. This development was motivated by the wish to improve certain

T

FSPN

T

if (#A >= 1)

HSPN

Figure 2: Example of equivalent FSPN/HSPN syntax. In the original FSPN definition, the movement of continuous tokens was regulated by the (discrete) timed transitions. In the FSPN shown in Figure 2 (left), fluid leaves the fluid place F continuously if the timed transition T is enabled. However, the fluid marking of the fluid place F has no effect on the enabling of transition T . This has the disadvantage of conflicting with standard Petri net semantics. It also prohibits the possibility of allowing fluid tokens to move instantaneously at the moment of firing of the timed transition in a manner that is consistent with the movement of the discrete tokens. For HSPNs, these observations led to the classification of both immediate and timed transitions as impulse transitions, since they both effect an instantaneous marking change. Continuous marking changes are regulated by the the new pump transition in HSPNs. The same model in HSPN notation is shown in Figure 2 (right). Continuous changes in continuous markings occur solely on the basis of the state (enabled/disabled) of a pump transition, which is determined by a guard function. Furthermore, continuous changes in continuous markings are associated uniquely with continuous arcs. Similarly, instantaneous changes in markings are associated with (discrete) timed and immediate transitions and with impulse arcs. Any (discrete or continuous) place which is connected via an impulse arc to a timed or immediate transition experiences an instantaneous change in marking when the transition fires. In Figure 5, transition Discharge is enabled when a token (or more) is present in place On and the continuous marking in place Q is at least QC . Furthermore, when the transition fires, the continuous marking of the continuous place is reduced by QC (i.e. it is emptied). The semantics of such a situation in an HSPN correspond to the traditional interpretation of a Petri net. Figure 8 (at the end of this paper) compares FSPN and HSPN implementations of the method of supplementary variables for modeling non–exponential discrete state changes. Here, the HSPN model is both easier to understand and more

compact.

equations which describe the underlying stochastic process

Extensions To Modeling Capability HSPNs offer two extensions of the modeling capability of FSPNs. These extensions are not a property of the HSPN per se, but result from the decision to perform the numerical solution of the net without reference to a differential equation. The model properties which prevent the formulation of the differential equation are summarized in Figure 3.

P

1

if (#P1 > 1.0) Immediate guard dependent on continuous marking

Instantaneous change of continuous marking

Figure 3: “Forbidden” FSPNs. The first extension concerns guards on immediate transitions (Figure 3, left). For FSPNs, these could refer only to a discrete marking (i.e. they were not allowed to refer to a continuous marking). Had this been allowed, then the discrete markings which otherwise enable the immediate transition would no longer have been uniquely tangible or vanishing. In consequence the elimination of the vanishing markings, and thus the numerical solution itself would no longer have been possible. The second – and more important – extension concerns instantaneous changes in continuous markings (Figure 3, right). These were forbidden in FSPNs, since they would have prevented the derivation of a differential equation for the probability density function describing the dynamics of the net. If we are prepared to dispense with the density equation, these changes become possible and are represented by the use of standard (single) arcs to connect continuous places with impulse transitions. As a consequence of allowing immediate changes in continuous markings and guard functions on discrete transitions that depend on continuous markings, the dynamics of an HSPN can no longer be represented by a system of differential equations describing a probability density. Their analysis can be performed by simulation, or by the method described in the next section.

NUMERICAL HSPN ANALYSIS The method we propose for analyzing HSPN models is based on the idea that it is not necessary to derive mathematical

f(~x(t); (t)) ; t  0; (~x(t); (t)) 2 Mg of the HSPN exactly. Instead, we define a finite–volume discretization of the discrete–continuous state space M of the HSPN and approximately represent its dynamic behavior using a discrete–state semi–Markov process (SMP), whose state space is denoted by S . As a consequence, our method can be seen as being equivalent to the analysis of an approximating GSPN: the resulting discrete–state SMP can for example be transformed into a CTMC by eliminating the vanishing states in S . Afterwards, the well–known algorithms for transient and steady– state CTMC analysis [6, 7, 1] can be applied in order to quantify the dynamic behaviour of the HSPN. Since we do not explicitly derive mathematical equations first, we use the term direct discretization to denote our solution approach. Since the HSPN is being replaced by an approximating GSPN for the purposes of the numerical analysis, it could be argued that HSPNs are superfluous, and that the equivalent GSPN model could be directly constructed by the modeler. Against this there are two main counterarguments. First, an abstract model should be as similar as possible to the system being modeled in terms of the latter’s most significant properties. Hence it is natural to use a hybrid modeling tool for modeling hybrid systems. Second, difficult decisions and tedious details concerned with performing the approximation should be hidden from the user. We continue by explaining the discretization of the HSPN state space M and describing the approximation of its dynamics.

Discretization Of The State Space In order that the state space S of the SMP is finite, a maximal level xmax must be chosen for each continuous place k 2 pk 2 P . Our method is based on the assumption that the actual content xk t of pk never exceeds xmax k .

IR

()

In addition, the granularity of the discretization must be specified for each continuous place pk by defining the number Gk of discretization intervals within the range  xk  xmax k .

0

Therefore, we obtain the discretization step width


xk = xGkk

max

for each continuous place pk . Each discrete variable therefore represents, in general, a hypercuboid of dimension C . Its value is equal to the probability of the stochastic process taking on a value inside this hypercuboid. The resulting set of hypercuboids forms the state space S of the SMP. From now on, we assume each of these to be represented by the HSPN marking located in its centre. Figure 4 illustrates our discretization ap) and proach for a HSPN with two continuous places (C two discrete states 0 and 1 .

=2

x2

(0; xmax 2 )

=P ( )

must be added:  i i . Then, the approximate continuous change of the content of each continuous place pk can be computed as rkeff ~x;  =, which has to be respected when determining the following state s0 of s.

max (xmax 1 ; x2 )

0 (0; 0)

When choosing the numbers Gk of discretization intervals for the levels xk of the continuous places, two conditions have to be satisfied in order to achieve a high accuracy:

(xmax 1 ; 0)

x1

x2

(0; xmax 2 )

max (xmax 1 ; x2 )

1.

change of the continuous marking causes a state transition of the SMP.

1

2.

(xmax 1 ; 0)

(0; 0)

x1

Figure 4: Discretization of the hybrid HSPN state space using finite volumes, here: G1 , G2 .

=8

=4


xk must be small enough, so that every impulse–like

M

In order to achieve a high accuracy of the approximation, it is necessary to use small enough values for xk . This issue will be discussed below.




Approximation Of Dynamic Behaviour First, we describe the approximation of the impulse–like changes of the HSPN marking caused by firing events of impulse transitions. Consider the discrete–state SMP to be in state s 2 S and assume m ~x;  to be the central marking of s, i.e. the marking which represents s. We have to distinguish two cases.

=( )

If there is an immediate transition  2 Ti enabled by m, then s is a vanishing state of the SMP. The firing probability of  is computed by using the weight function W , as we have explained in the previous section. If  fires, the resulting state s0 2 S is determined by evaluating the mapping Ai , which has also been introduced above. If the marking m does not enable any immediate transition, s is said to be a tangible state of the SMP. Assume the timed transition  2 Te to be enabled by m. Its firing rate  is determined by the firing rate function F , which has been defined in the previous section, too. However, there is an important aspect which has to be respected in this situation. During the exponentially distributed firing time of a timed transition, the continuous part of the HSPN marking may change due to the functionality of pump transitions. If  F ; ~x;  is the firing rate of  in state s, the mean firing time of  is =. During this time interval, the content xk of the continuous place pk 2 P changes by approximately rkeff ~x;  =. This must be considered when determining the following state s0 .

= ( ( )) 1 ( )

Note that if there are several timed transitions i 2 Te enabled by the tangible state s and the firing of each of them yields the same succeeding state s0 (without respecting the continuous changes), then these transitions have to be considered together. At first, their exponential firing rates i F i ; ~x; 

= ( ( ))


xk must be small enough, so that continuous changes of

the fluid–like quantities, which occur during the exponentially distributed firing times of the times transitions, are represented by the SMP.

The second part we have to explain is the approximation of the continuous changes of the HSPN marking. As defined in the previous section, pump transitions cause a continuous transport of fluid–like quantity into and out of the continuous places of the HSPN. This dynamical behaviour must also be modeled by the SMP. We proceed as follows. Assume two states s; s0 2 S , which are represented by the HSPN markings m ~x;  and m0 ~x0 ;  . Note that the discrete parts of the markings are the same. Furthermore, assume that the finite volume corresponding to state s is located on the left side of the one corresponding to s0 . Let xk x0k xk be the distance between the central markings of these finite volumes and assume mM ~xM ;  with ~xM =
x~0 ~x to be the marking in 0 the middle of m and m .

=(

=(

=( )

)


= =1 2 (

)

)

There are three cases to be distinguished.

( ) 0 ( )
eff 2. rk (~xM ; ) < 0: if s is not vanishing, we introduce a transition from s to s. Its exponential rate is chosen as jrkeff (~xM ; )j=
x. 3. rkeff (~xM ; ) = 0: there is no continuous flow of fluid–like 1. rkeff ~xM ;  > : if s is not vanishing, we introduce a transition from s to s0 . Its exponential rate is chosen as rkeff ~xM ;  = x. 0

0

quantity from one finite volume to the other. Thus, an additional exponential transition is not necessary.

Note that our solution method is based on the stochastic approximation of the continuous transport of fluid–like quantities, which is by definition a deterministic process depending on the non–deterministic behaviour in the discrete part of the HSPN.

EXAMPLES As a simple example we consider the light in a public hallway which automatically switches off after a specified time. A

∆x

movement sensor in the hallway causes the light to be switched on. At the same time a capacitor begins to charge. When the charge at the capacitor reaches a preset threshold, the light is switched off and the capacitor is instantaneously discharged. We assume that movement in the hallway occurs with exponentially distributed delays.

Q - c ∆x 0

Q 0 - (n-1) c ∆x

Q 0 - 2 c ∆x

Q

0

We can represent the system consisting of the state (charge) of the capacitor and the state of the light (On or Off) by the HSPN shown in Figure 5. The charge on the capacitor is denoted by Q. A guard function enables the pump transition Charge when the light is on. The capacitor charges at the rate Q0 Q, where Q0 is the maximum possible charge and the constant depends on the capacitance and the series resistance, which are chosen to control the time the light remains on. When the charge on the capacitor reaches a value of QC (and the light is on), transition Discharge fires, the capacitor is instantaneously discharged and the light is switched off. Charge

C

λ Q - n c ∆x 0

(Q=0, Off)

Figure 6: Discretized state space of the light model. T2 P

1

IF (On) T1

Q0 - cQ

On

Q

P

2

QC

λ

Figure 7: Discrete non–exponential model. Discharge

Movement

The firing rate of the replacement exponential transition T1exp then becomes a function of the continuous marking in place P5 :  P5 .

(# )

Off

Figure 5: HSPN model of the lighting example. Figure 6 shows the discretized state space of the model. One unknown represents the probability ^ Q , the others represent the probabilities of the stochastic process being in x for x ; x; : : : ; QC x each of the volumes x; x while the light is on. The state space is a Markov chain and can be solved by any appropriate method.

(

+
)

O =0


=0




Our next example shows how an HSPN can be used to model a non–exponential transition using the method of supplementary variables. Figure 7 shows an non-exponential transition T1 which competes with an exponential transition T2 . The method of supplementary variables replaces the non– exponential transition T1 with an exponential transition T1exp , whose firing rate is equal to the hazard rate function  of the non–exponential transition. The implementation of the method with the old and the new net notation is shown in Figure 8. First, we require a means to measure the elapsed time since transition T1 was enabled. With an HSPN this is easily achieved using a continuous place P5 , which starts filling at the rate 1 when T1 is enabled.

In the case of the older notation (Figure 8, left), additional places P3 and P4 and transitions T3 and T4 are necessary to empty the continuous place when T1 or T2 fires. By contrast, the new notation (Figure 8, right) is more compact, requiring only the addition of the continuous place P5 and the pump transition T5 to fill it. When transition T1 fires or is disabled by the firing of T2 , the continuous place is simultaneously emptied. With the newer notation the method of supplementary variables is easily illustrated.

CONCLUSIONS In this paper we have presented a specification for Hybrid Stochastic Petri Nets, which builds on and extends the previous Fluid Stochastic Petri Nets. The new formalism is designed to achieve syntax and semantics of the net which are more consistent with traditional Petri net usage. By differentiating between transitions and arcs associated with instantaneous and with continuous marking changes, we achieve greater simplicity and readability of the nets. For the numerical analysis of the HSPN, we suggest a method which does not describe the net dynamics by a partial differential equation for a probability density function. This allows us to discretize the stochastic process at an earlier stage

P3

T2

P

1

T3

T1

P1

exp

T1

1.0

λ(#P 5) P4

* P5

T1exp

* *

P5

λ(#P 5) 1.0

*

P2

T5 IF (#P5 = 1.0)

T4 P2 Previous syntax

New syntax

Figure 8: Old and new implementations of the non–exponential model. during the solution process, which in turn allows instantaneous state changes to be modeled. This important facility is not available if density equations are to be derived. There are several important problems associated with HSPNs which we have not yet addressed. The choice of discretization step size is critical, and depends on the maximum rate of any enabled transition in the net, which may not be known a priori. If capacities are not given for the continuous places, then the state space is infinite, and a truncation position must be determined (the so–called far–field boundary).

References [1] G. Ciardo, A. Blakemore, P.F. Chimento, J.K. Muppala, and K.S. Trivedi. Automated Generation and Analysis of Markov Reward Models Using Stochastic Reward Nets. In C. Meyer and R. J. Plemmons, editors, Linear Algebra, Markov Chiains and Queueing Models, volume 48 of IMA Volumes in Mathematics and its Applications. Springer, 1993. [2] A.I. Elwalid and D. Mitra. Statistical Multiplexing with Loss Priorities in Rate–Based Congestion Control of High–Speed Networks. IEEE Transactions on Communications, 42(11):2989– 3002, November 1994. [3] R. German and C. Lindemann. Analysis of Stochastic Petri Nets by the Method of Supplementary Variables. Performance Evaluation, 20:317–335, 1994. [4] G. Horton, V.G. Kulkarni, D. Nicol, and K.S. Trivedi. Fluid Stochastic Petri Nets: Theory, Applications, and Solution Techniques. Technical Report 96–5, ICASE, NASA Langley Research Center, Hampton, Virginia, 1996. [5] G. Horton and K.S. Trivedi. Computation of the Distribution of Accumulated Reward with Fluid Stochastic Petri–Nets. Technical Report 96–29, ICASE, NASA Langley Research Center, Hampton, Virginia, 1996. [6] B. Philippe, Y. Saad, and W.J. Stewart. Numerical Methods in Markov Chain Modeling. In Operations Research, volume 40, No. 6, pages 1156–1179. Operations Research Society of America (ORSA), 1992. [7] W.J. Stewart. Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton, New Jersey, 1994.