Stochastic Activity Networks ( SAN )

Sharif University of Technology ,Computer Engineer Department , Winter 2013

Verification of Reactive SystemsMohammad Esmail Esmaili

Prof. Movaghar

IntroductionStochastic activity networks have been

used since the mid-1980s for performance, dependability, and performability evaluation.

Stochastic Activity Networks (SANs) are a stochastic generalization of Petri nets which have been defined for the modeling and analysis of distributed real-time systems.

Activity networksActivity networks are the non-probabilistic

model on which SANs are built, just as in a similar fashion, (un-timed) Petri nets provide the foundation for stochastic Petri nets .

Activity networks are nondeterministic models which have been developed for representing concurrent systems.

The transitions in Petri nets are replaced by the primitives called "activities."

DefinitionsActivities: which are of two kinds: timed

activities and instantaneous activities. Each activity has a non-zero integral number of cases (possible actions).

Timed activities represent the activities of the modeled system whose durations impact the system's ability to perform .

Instantaneous activities, represent system activities that, relative to the performance variable in question, are completed in a negligible amount of time.

Definitions places : as in Petri nets. input gates : each of which has a finite set of

inputs and one output. Associated with each input gate are an n-ary computable predicate and an n-ary computable partial function over the set of natural numbers .

output gates, each of which has a finite set of outputs and one input. Associated with each output gate is an n-ary computable function on the set of natural numbers, called the output function.

DefinitionsP denote the set of all places of the networkIf S is a set of places (S P), a marking of S is

a mapping the set of possible markings of S is the set of

functions : = .an input gate is defined to be a triple, (G, e,

f ), where G P is the set of input places associated with the gate, e : is the enabling predicate of the gate, and f : the input function of the gate .

Definitionsan output gate is a pair, (G, f ), where G P is

the set of output places associated with the gate and f : is an output function of the gate.

DefinitionsAn activity network (AN) is an eight-tuple :

AN = (P,A, I, O,)P is some finite set of places.A is a finite set of activities.I is a finite set of input gates.O is a finite set of output gates.species the number of cases for each activity.T imed; Instantaneous} specifies the type of

each activity.

DefinitionsThe net structure is specified via the

functions and . maps input gates to activities maps output gates to cases of activities.Several implications of this definition are

immediately apparent. First, each input or output gate is connected to a single activity. In addition, each input of an input gate or output of an output gate is connected to a unique place.

Graphical RepresentationTo aid in the modeling process, a

graphical representation for activity networks is typically employed. In fact, for all but the smallest networks, speciation via the tuple formulation presented in the definition is extremely cumbersome.

Not only is the graphical representation more compact, but it also provides greater insight into the behavior of the network.

Graphical RepresentationHere places are represented by circles (A, B, and

C), as in Petri nets. Timed activities (T1 and T 2) are represented as

hollow ovals. Instantaneous activities (I1) are represented by solid bars.

Cases associated with an activity are represented by small circles on one side of the activity (as on T 1).

An activity with only one case is represented with no circles on the output side (as on T 2).

Gates are represented by triangles.

Activity Network BehaviorAn input gate has two components:• enabling function (state) → Boolean; also called

the enabling predicate• input function(state) → state; rule for changing

the state of the modelAn activity is enabled if for every connected input

gate, the enabling predicate is true, and for each input arc, the number of tokens in the connected place ≥ number of arcs.

We use the notation MARK(P) to denote the number of tokens in place P.

Enabling Rule

CasesCases represent a probabilistic choice of an

action to take when an activity completes.

Output GatesWhen an activity completes, an output gate

allows for a more general change in the state of the system. This output gate function is usually expressed using seudo-C code.

Completion RulesWhen an activity completes, the following events take

place (in the order listed), possibly changing the marking of the network:

If the activity has cases, a case is (probabilistically) chosen.

The functions of all the connected input gates are executed (in an unspecified order).

Tokens are removed from places connected by input arcs.The functions of all the output gates connected to the

chosen case are executed (in an unspecified order).Tokens are added to places connected by output arcs

connected to the chosen case.

Definition of a Stochastic Activity NetworkGiven an activity network that is

stabilizing in some specified initial marking, a stochastic activity network is formed by adjoining functions C, F, and G, where C species the probability distribution of case selections, F represents the probability distribution functions of activity delay times, and G describes the sets of “reactivation markings" for each possible marking.

Definition of a Stochastic Activity NetworkA stochastic activity network (SAN) is a five-

tuple :SAN = (AN,,C,F,G)

Where : AN = (P,A, I, O,) is an activity network. is the initial marking and is a stable marking

in which AN is stabilizing.C is the case distribution assignment.F is the activity time distribution function

assignmentG is the reactivation function assignment .

SAN Termsactivation - time at which an activity begins.completion - time at which activity completes.abort - time, after activation but before

completion, when activity is no longer enabled .

active - the time after an activity has been activated but before it completes or aborts.

Illustration of SAN Terms

References[1] Stochastic Activity Networks: Formal

Definitions and Concepts , William H. Sanders and John F. Meyer , Lecture Notes in Computer Science, Volume 2090, 2001, pp 315-343 .

[2] Stochastic Activity Networks: A New Definition and Some Properties , A. Movaghar , Scientia Iranica, Vol. 8, No. 4, pp. 303-311, October 2001.

[3] users.crhc.illinois.edu/nicol/ece541/slides .