an integrated approach for the specification and analysis of stochastic real time systems

An Integrated Approach for the Speci cationMario Bravetti 1 Dipartimento di Scienze dell'Informazione, University of Bologna, Mura Anteo Zamboni7, 40127 Bologna, Italy Abstract A formala

An Integrated Approach for the Speci cation

Mario Bravetti

1

Dipartimento di Scienze dell'Informazione,

University of Bologna, Mura Anteo Zamboni7, 40127 Bologna, Italy

Abstract

A formalapproach for the speci cationand analysis of concurrent systems is

pro-posed which integrates two dierent orthogonal aspects of time: (i) real-time,

concerning the expression of time constraints and the veri cation of exact time

properties, and (ii) probabilistic-time, concerning the probabilistic quanti cation

of durations of system activities via exponential probability distributions and the

evaluation of system performance We show that these two aspects, that led to

dierent speci cationparadigmscalled timedautomata andMarkovianprocess

al-gebras, respectively,can be expressedinan integrated wayb a singlelanguage: a

processalgebracapableofexpressingactivitieswithgenerallydistributeddurations

Inparticular,weconsiderthecalculusofInteractive GeneralizedSemi-Markov

Pro-cesses (IGSMPs) and we present formal techniques for compositionally deriving,

from an IGSMPspeci cation, (i) apurereal-time model(called Interactive Timed

Automaton), b considering the support of general distributions, and (ii) a pure

probabilistic-time model (called Interactive Weighted Markov Chain), b

approxi-mating generaldistributionswithphase-typedistributions

1 Introduction

Theimportanceofconsideringthebehaviorofconcurrentsystemswithrespect

totimeduringtheirdesignprocesshasbeenwidelyrecognized[17,3,9,2,20,21]

Inparticulartwodierentapproachesforexpressing andanalyzingtime

prop-erties of systems have been developed which are based on formaldescription

paradigms

A rstapproachisdevotedtotheevaluation ofthe performanceof

concur-rentsystems(seee.g [17,3,15]) Accordingtothisapproachthe timespentby

asysteminacertainactivityisexpressed probabilisticallythrough a

distribu-tion of duration Performance measures ofsystems can then be evaluatedvia

1

Email: bravetti@cs.unibo.it

c Publishedb ElsevierScienceB.V

Fig 2 Real-TimeActivity

mathematicalorsimulativetechniques Thisapproachhasledtothede nition

of stochastic process algebras, an extension of standard process algebras [19]

(concurrentspeci cationlanguageswhichallowustorepresentconcurrent

sys-tems compositionally by specifying the behavior of individual processes and

the wa they interact) where a distribution of duration is associated with

each action of a process In most cases, as in [3], the expressiveness of such

algebras is limited to exponential distributions of time, because this causes

the passage of time to be \memoryless" As a consequence it is possible to

completelya oid explicitlyrepresenting durations insemantic models

More-o er the limitation to exponential distributions allows for a straightforward

transformation of the semantic model of a system into a Continuous Time

Markov Chain (CTMC), a stochastic process which is easily mathematically

analyzableforderiving performance measures Forthisreasonthey are called

Markovian process algebras It is worth noting that the limitation imposed

o erdurations is very strongbecause noteven deterministic( cationparadigmcapableofexpressing bothaspectsoftimeshould

beable of expressing both time constraintsand a probabilistic quanti cation

for the possible durations which satisfy such constraints We obtain such

an expressive power by considering stochastic models capable of expressing

generalprobabilitydistributionsforthedurationofactivities Inthiswa time

constraintsareexpressibleviaprobabilitydistributionfunctionsthatassociate

probabilitygreaterthanzeroonlytotimevaluesthatarepossibleaccordingto

the constraints Technically,theset ofpossibletime valuesforthedurationof

anactivityisgivenbythesupportoftheassociateddurationdistribution This

ideaofderivingreal-timeconstraintsfromdistributionsupports,thatwehave

introducedin[6], wassubsequently appliedalsoin[10]and [12] Forinstance,

inFig.4wedepictanactivitywithadistributionwhosesupportistheinterval

of Fig 2 Note that with this approach we can also represent deterministic

durations via trivial distribution functions that give all the probability to a

with phase−type distributions

approximation of general dist.

(time bounds are lost)

Stochastic compositional mapping:

(prob quantification lost)

derivation of time bounds from support of general distributions Real−Time compositional mapping:

Stochastic Process Algebra with General Distributions

Pure Real−Time evaluation of

performance measures via mathematical analysis of CTMCs

verification of

model checking

of Timed Automata

real−time properties via

derivation of the minimization, event simulation,

underlying GSMP

discrete Integrated Stochastic Real−Time

Pure Stochastic Time

system specification via a

via a Markovian

via (nets of)

Fig 5 Stochastic Real-TimeIntegratedApproach

single value oftime

1.2 An Integrated Approach

Representing the real-time and probabilistic-time in a single speci cation

paradigmallowsustomodelaconcurrentsystemmorepreciselybyexpressing

and analyzing the relationships between the two aspects of time Moreover,

the capability of expressing general distributions gives the possibility of

pro-ducing much more realisticspeci cations of systems System activities which

have an uncertain duration could be represented probabilistically by more

adequate distributions than exponential ones (e.g Gaussian distributions or

experimentallydetermined distributions)

The price to pay by using general distributions is the complexity of the

stochastic process representing the system behavior: a Generalized

Semi-Markov Process (GSMP) Only for very restricted cases we can derive

per-formance measures from aGSMP by means of exact mathematical analysis

Asaconsequenceitisimportantthat,besides developinganewstochastic

real-timespeci cation languageby usinggenerally distributedtime and some

new (usuallycomplexandlimitedinpower) analysismethodologiesfor sucha

language, we alsodevelop formalautomatizableprocedures forderiving,from

an integrated stochastic real-timespeci cation, a traditionalpure

stochastic-time speci cation and a traditionalpure real-timespeci cation

More in the details,in Fig 5 weshow how process algebrawith generally

distributed time can oer the possibility of such an integrated approach for

themodelingandanalysisofStochasticReal-Timeconcurrent/distributed

sys-tems Speci cations(termsofsuchaprocessalgebra)canbedirectlyanalyzed

through standard discrete event simulation (see e.g [13]), state space

mini-mization (via a e.g a notion of bisimulationbased congruence), and deriv

a-tion ofthe underlyingperformance modelinthe formofaGSMP.Besides the

possibility of performing direct analysis, we can have formal techniques for

compositionallyderiving, froma system speci cation:



A pure stochastic-time (Markovian) speci cation in the form of a term of

a Markovian process algebra, by approximating general distributions with

combinations of exponential distributions (the so called phase-type

distri-butions) A consequence of this transformation is that all duration values

fordelaysgetprobabilitygreaterthan0 Hence theinformationabouttime

constraints(related tothe real-timebehavior of the system)is lost



A pure real-timespeci cation in the form of a net (aparallel composition)

of Timed Automata, by considering the support of general distributions,

i.e the set of time values that are given probability (density) greater than

0, and by turning probabilistic choices into non-deterministic choices As

aconsequence the informationrelated tothe probabilistic-timebehavior of

the system is lost

Inthiswa wheneverauserisinterestedinevaluatingsystempropertieswhich

are relatedtothestochastic-timeorreal-timeaspect onlyofthe speci ed

sys-tem,theanalysiscanbedoneautomaticallybyderivingthespeci ctraditional

pure(stochastic-timeorreal-time)modelandbyanalyzingit Thisisvery

im-portant fromapractical viewpoint inthat it givesthe opportunity ofreusing

existing techniques and tools already developed for performance evaluation

and model-checking of non-probabilistic real-time properties Moreover, the

advantage of deriving a traditional pure stochastic-time and real-time model

from the same initial integrated speci cation (w.r.t generating them

inde-pendently)isthat theyareguaranteedtobeconsistent,inthattheyrepresent

dierentaspects of the same initialsystem speci ... relatedtothestochastic-timeorreal-timeaspect onlyofthe specied

sys-tem,theanalysiscanbedoneautomaticallybyderivingthespecictraditional

pure (stochastic- timeorreal -time) modelandbyanalyzingit...

distributed time can oer the possibility of such an integrated approach for

themodelingandanalysisofStochasticReal-Timeconcurrent/distributed

sys-tems Specications(termsofsuchaprocessalgebra)canbedirectlyanalyzed... actions), and derivation of

the underlyingperformance modelinthe formof aGSMP forIGSMPs which

are complete both fromthe interactiveand fromthe performance viewpoints

Asfarasthestochastic-timeandreal-timeprojectionsofFig.5areconcerned,