RoleofStochasticPetriNet(SPN)inProcessDiscoveryfor ...

Research ArticleRole of Stochastic Petri Net (SPN) in Process Discovery forModelling and Analysis

Shabnam Shahzadi1 Xianwen Fang1 and David Anekeya Alilah 2

1Department of Mathematics and Big Data Anhui University of Science and Technology Huainan China2Department of Mathematics Masinde Muliro University of Science and Technology Kakamega Kenya

Correspondence should be addressed to David Anekeya Alilah aliladavid2010gmailcom

Received 3 June 2021 Accepted 18 June 2021 Published 30 June 2021

Academic Editor Ishfaq Ahmad

Copyright copy 2021 Shabnam Shahzadi et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

For exploitation and extraction of an eventrsquos data that has vital information which is related to the process from the event logprocess mining is used ere are three main basic types of process mining as explained in relation to input and output eseare process discovery conformance checking and enhancement Process discovery is one of the most challenging processmining activities based on the event log Business processes or system performance plays a vital role in modelling analysis andprediction Recently a memoryless model such as exponential distribution of the stochastic Petri net (SPN) has gained muchattention in research and industry is paper uses time perspective for modelling and analysis and uses stochastic Petri net tocheck the performance evolution stability and reliability of the model To assess the effect of time delay in firing the transitionstochastic reward net (SRN) model is used e model can also be used in checking the reliability of the model whereas thegeneralized stochastic Petri net (GSPN) is used for evaluation and checking the performance of the model SPN is used toanalyze the probability of state transition and the stability from one state to another However in process mining logs are usedby linking log sequence with the state and by this modelling can be done and its relation with stability of the model canbe established

1 Introduction

Process mining is an analytical method for finding moni-toring and improving actual processes by extracting in-formation from event logs that are freely available in moderninformation systems It provides targeted facts based onevent logs that help in doing research analysis and im-proving the existing business processes

Presently the process mining has been observed as animportant technology and has been used in the application ofbusiness processes and hence applied successfully in manyorganizations It is a process that is centric (not-data-centric)truly intelligent (read from historical data) and based on facts(event data rather than theory) In addition to process dis-covery the mining process allows automatic discovery of aprocess model from event logs which provides insight into itand enables various types of model-based analysis Process

acquisition models can be extended with information topredict the elimination of functional conditions In particularcapturing of activities and waiting times in the businessprocess are required for the understanding of the processrsquosefficiency In addition these enrich types used as a source forthe prediction algorithms which are used for predicting timeuntil the process is completed To provide an explanation ofthe time forecast a set order that allows for a good balancebetween ldquooverfittingrdquo and ldquounderfittingrdquo is used It is worthnoting that the words ldquooverfittingrdquo and ldquounderfittingrdquo areorthogonal to nonfitting e model is not valid if visual cuesdo not occur according to that model Establishing theremaining run time of the business process and its function isan important administrative function that allows for im-proved resource allocation is also improves the quality ofresults when clients inquire about the status and expectedcompletion of a given business process

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 8699164 7 pageshttpsdoiorg10115520218699164

e three main types of process mining are processdiscovery conformance checking and enhancement In thisresearch work attention is placed on the process discoveryIn process discovery it is known that without using any apriori information event logs produce a process model Incase the event logs have any information about the resourcea model-related resource for example social networkingcan be discovered which can be able to show how differentpeople worked in an organization collectively On the otherhand the process model can be used for analyzing costresource utilization or process performance and automa-tion Models are used for redesigning a process for planningand controlling in order to make decisions in a processere are two types of process model (i) formal model usedfor discussion and documentation and (ii) informal modelused for analysis or enactment In this research informalmodelling is used ere are certain errors that occur duringmodelling (i) when a model defines a prepared version or afact (ii) failure to properly do human conduct and (iii)when the model is at the wrong abstraction level

For information system analytical evaluation has be-come an integral part for the whole process design Manydiverse model specification techniques have been proposedfor example Petri net BPMN UML function diagramsBPEL and EPC Petri net is widely used in business pro-cesses either as the first model language or as a basis forvalidation e Petri net was first introduced in 1962 by CarlAdam Pete It can be described as a graphical method of theformal definition of logical interaction between componentsor the flow of activities in a complex system PN is par-ticularly well suited for modelling concurrency and conflictsequencing conditional branch and looping synchroniza-tion limited resource allocation and mutual exclusion Itenables the study of logical properties of modelled system foreach individual finite state machine which can be seen aspossible and yet shows an interaction whenever they occurwhen using idea behind the Petri net given by ProfessorPetri Petri net is categorized as being application and theoryof Petri net (ATPN) and Petri net and performance mod-elling (PNPM) which includes stochastic Petri net OriginalPetri net did not have the notation of time and was thereforeused for studying logical properties In order to introducetime duration in Petri net we link event data with transitionthat is important for availability reliability and performancequantification Hence that is the main reason why Petri netwas extended to have time linked with event occurrencegiving rise to the stochastic Petri net

Stochastic Petri net was introduced earlier in 1980 It hada graphical representation that was used for modelling ofdiscrete events and the bipartite graph of the transitions andplaces in SPN is used for knowing event mechanism It is alsoworth noting that the time of firing tokens is considered as arandom variable and is exponentially distributed Reach-ability graph in stochastic Petri net is a continuous firing ratethat is linked with each transition and may be marking-dependent If the history of a given process is known forexample event log with time information it is possible toextract stochastic performance data and insert it into themodel is research paper is therefore based on generalized

stochastic Petri net (GSPN) which does not restrict distri-butions in a particular way GSPN is very useful in the casewhere some events happen in extremely small timeWhereasSPN model handles this situation by introducing immediatetransition in the model which has zero firing time the othertransitions are timed transitions which are exponentiallydistributed In this research stochastic reward net is used inorder to check survivability and reliability SRN is basicallyan extension of GSPN and it was introduced in 1989 SRN isextensively marking-dependent which is used for firing ratesand probabilities by giving priority to the transitionserefore SPN has an advantage in that sense and it analyzesthe probability of state transition and the stability of onestate to another For the process with logs the log sequence isbased on the results of the transition execution and not thecomplete state results Hence this paper links the log se-quence with the state for stability of the model and analysis

e remainder of the paper is organized as follows inSection 2 related work is discussed in Section 3 preliminarydefinitions are presented and Section 4 discusses general-ized distributed transition stochastic Petri net model withexample Section 5 elaborates about the stochastic rewardnet survivability and reliability model Conclusion is pre-sented in Section 6

2 Related Work

Some related literature associated with Petri net model andstochastic overall performance for feature information arepresented in the literature review Hu et al [1] proposed atechnique that is primarily based on SPN model which isexponentially distributed for workflow log and depends ontransition rate of firing Anastasiou et al [2] proposedcompletely unique strategies whereby they centered at thelocation data for generalized stochastic Petri net model forcustomer modelling flows In their research work fortransition intervals they used constant hyper-Erlang dis-tributions which shows waiting and run times and also usedGSPN to upgrade the other corresponding transitionssubnet depicting the same features which the hyper-Erlangdistribution has ey observed at all the transitions inde-pendently which did not cause problems in the sequence butthe similarities within the method especially the manysimilar transitions were not considered in their technique

According to Leclercq et al [3] non-Markovian sto-chastic Petri nets and attempts at eliciting exist ey werelooking at a way of removing a model of normally dis-tributed data In their work they focused entirely on theanticipation algorithm prepared for maximization conver-gence Compared to the method used in this study they areunable to manage lost data and different performanceguidelines e reconstruction of the model parameters ofstochastic structures was also investigated in a study byBuchholz et al [4] where they dealt with the problem ofadjusting the model parameters of the basic stochasticsystem Contrary to this study the distribution of transitionsystem was targeted in advance although the main purposewas to make GDT_SPN version transitions which werecomparable to for example incomplete Wombacher and

2 Mathematical Problems in Engineering

Iacobrsquos [5] statistical estimatesrsquo distribution does not haveinitial process times Rozinat et al [6] checked out how toacquire records for simulation models in an attempt todiscover data dependencies which are mostly taken intoconsideration for discovering optimal standard alignmentbetween model and log which are doing manual replies thatmake decisions mean durations and trendy deviationsbecause before that these were not taken under consider-ation e technique proposed in this paper has successfullydealt with noise in a much far better way through buildingthe notion of alignments which is able to pick out the finestdirection through the model for a noisy trace According tovan der Aalst [7] the available process mining techniquesare used to consider the noise and probability for creatingcontrol flow and the importance of business processmodelling is recognized e best way to represent thesemethods is stochastic Petri net which is the main researchtask of the process mining in business modelling Rogge-Solti et al [8] proposed an algorithm in their research workfor process discovery of each execution and also used dif-ferent raw event data for discovering various classes of SPNIn their study they used notation alignment which is basedon plug-in in process mining framework

3 Preliminaries

Here concepts and some techniques are presented which areused throughout the paper Primarily our main focus istowards event logs and PN SPNGSPN SRN and logsequence

31 Event Log It is a set of collected cases LsubeC so that eachevent performs once in the whole log ie for any c1 c2ϵLsuch that c1 ne c2 zset(1113954c1)cap zset(1113954c2) empty If the event logcontains time stamps tracing order must respect these timestamps at is for any c isin L i and j such that 1le iltjle |1113954c| time(1113954c(i))le time(1113954c(j))

32 State It means a particular condition that a system usesfor some specific purpose at specific time Let c (a ai a0

d i o) be c minus net s b(a times a) represents state space of c s isin S

is a state which shows a multiset of pending obligations

33 Sequence e most natural way to present the traces inthe event log is by sequence At a point when we need toexplain the functional semantics of PN and transition of howa system is made and the performance is also modelled interms of sequences some of the useful operators on se-quences are presented If A is a set thenAlowast represents a set ofall finite sequences over A A finite sequence over A withlength n is a mapping σ isin 1 2 n ⟶ A In this casethe sequence is represented in string form that isσ isin a1 a2 an1113864 1113865 where ai σ(i) for 1le ile n |σ| showslength of the sequence that is |σ| n σ oplus aprime langa1 a2 an aprime rang represents sequence with element aprimeSimilarly σ1oplus σ2 adds sequences σ1 and σ2 causing a se-quence of length |σ1| + |σ2|

34 Petri Net Petri nets were initially presented by C APetri in 1962 Formally a Petri net (PN) is a 5-tuple PN

(P T F W M) where

(i) P p1 p2 pn1113864 1113865 is a finite set of places (ascircles)

(ii) T t1 t2 tn1113864 1113865 is a finite set of transitions (asbars)

(iii) Fsube(P times T)cup (T times P) is a set of arcs connectingP andT

(iv) W F⟶ 1 2 3 is a weighted function(v) M P⟶ 0 1 2 3 marking represents

number of tokens (black dots) in place P and initialmarking is represented as M0

PN is a graphical representation of nodes based on placesand transitions e arc that connects places by transition iscalled the input transition arc and the arc connected fromtransition to places is called the output transition arc and thepositive number is connected to each arc Places connectedto the transition with the input arc are called input placesand vice versa Each place may have zero or more tokensTransitions are enabled if each of the input places has at leastas many tokens corresponding to the input arc Enabledtransitions can be fired When we have fired token thenumber of tokens that are equal to the input arcs multiplicityis transferred from each input place whereas the number oftokens which are equal to the output arc multiplicity isdeposited in the output places Petri net transformation ofone marking into another which is done by using firingtransitions in reachability set M0 shows the initial markingand is defined as the sequences of the firing transitionsfollowed by marking and always starts through initialmarking It is known that a PN is a 5-tuplePN (P T F W M)

Figure 1 describes a Petri net model as a marking vectorthat calculates the number of tokens in each net

n1 n2 np1113872 1113873 withp representing the number of places

(1)

Transition is enabled in case the input places have therequired number of tokens

Figure 2 shows that the enabled transitions can be firedby taking a certain number of tokens from the input placesand then placing them in the output places for example

In real world Petri net can be used in many situationsbased on sequences concurrency synchronization andconflict We present in Figure 3 concurrencyparallelism ofstochastic Petri net model in Figure 4 we present the in-dependency and Figure 5 shows the synchronization of thestochastic Petri net model Reachability analysis of stochasticPetri net is present in Figure 6 However in computernetworking Petri net describes communication protocolsIn original definition of PN concept of time is not includedfor the performance evaluation of dynamic system there-fore it is important to present time delay that is linked witheach transition in Petri net modelis concept of time delayin transition firing emerges in the SPN

Mathematical Problems in Engineering 3

35 Stochastic Petri Net In SPN data perspective is usedbecause flow of the information between the tasks can bedescribed by it and each transition is exponentially dis-tributed with firing time SPN can be described as aworkflow net if (i) there is only one place to start and (ii)one place to end and (iii) every node is on track from thestart to the end of the task For a generalized stochasticPetri net (GSPN) the transitions are either timed (firingtime represented by rectangular box) or immediate (zerofiring time represented by black bar) Priority is alwaysgiven to the immediate transition over timed transitionfor firing Usually we use probability mass function inorder to complete the firing and break the tie betweenimmediate transitions In GSPN marking is completedwhen at least an immediate transition enables the markingto be visible Inhibitor arcs are also introduced in theGSPN which are used to connect places to transition Atthe terminating point of arrow there is a small hollowcircle which indicates that it is an inhibitor arc If inputplaces of the inhibitor arcs contain more tokens than themultiplicity arcs then at that point transition cannot befired with the inhibitor arc

It has been proven that the condition of limited numberof transitions is related to continuous time Markov chain(CTMC) with GSPN that can be fired in a limited time withnonzero probability When stochastic Petri net is used incomputer network for performance evaluation places aredenoted with packets or cell in buffer or active user or flow inthe system whereas their arrival and departure are repre-sented with transitions

4 Generalized Distributed TransitionStochastic Petri Net

It is based on seven-tuple p τ P W F M0 D1113864 1113865 whereas thebasic Petri net has P T F M01113864 1113865 where

(i) T is a set of transitions which is equal to Ti cupTt

based on immediate and rimed transitions

(ii) P TrArrN+0 is the priority to transition where

forallt isin Ti P(t)ge 1 and forallt isin Ti P(t) 0

(iii) W TirArrR+ represents the probability weight toimmediate transition

P1

P2

T1

T1

P3

Figure 3 Concurrencyparallelism

P1T1

P2

P3T3

P4

Figure 4 Independency

P1T1

T2

P2

P3T3

P4

Figure 5 Synchronization

P1T1

T4

P2

Figure 6 Reachability analysis

Token Output

InputP1

T1P2

Figure 1 General firing sequence of Petri net

P1T1

P2

Figure 2 Enabled transitions


(iv) D TtrArrD is an arbitrary probability distribution D

to timed transition which reflects duration ofcorresponding activities

In Figure 7 we present the generalized distributedtransition stochastic Petri net model for two parallelbranches and the above model shows the conflict betweentransitions Tc andTd

41 Case-Based Alignment Case-based alignment method ismuch more powerful than the naıve replays of logs in themodel because it ensures that global availability best penalizesasynchronous part of replay Since there are two executiontraces tr1 and tr2 of the above model it is assumed thatimmediate transitions are invisible whereas the timed tran-sitions are visible In this study the focus is geared towards theinvisible transitions in the model alignment which is denotedby τ whereas tr2 is not fitted in the model So to overcomethis optimal alignment between model and log is found outusing the method proposed by Adriansyah et al [7] whichgives sequence of replay movements in the trace and model

erefore these movements can be either log movessynchronous moves or model moves Table 1 shows the ex-tension traces of themodel which are thematching subscript ofeach event in the transition net Table 2 gives us perfectalignment for tr1 based on synchronous or invisible modelmoves Also multiple alignments for trace tr2 are presented inTable 3 Hence Table 4 shows the event logs which are based onthe optimal alignment of the model and log e symbol ≫shows no progress on either side Cost-based alignmentprovides the unnecessary moves which are penalized and thehigh costs are not included in the optimal alignments

5 Stochastic Reward Net

(SRN) basically SRN is an extension of the GSPN Re-ward rate in stochastic reward net is linked with eachvisible marking erefore there are many ways ofmeasuring performance It also gives us many other waysthat make specification convenient which are as follows

(i) Every transition can have enabling function (alsoreferred to as security guard) which only makes thetransitions enabled if their marking dependencyfunction is correct is specification gives a su-perior way to enhance the graphical representationand makes SRN easy to understand

(ii) Repeated arcs marking is permitted is charac-teristic can be used in a case when transformation oftotal number of tokens depends on the presentmarking

(iii) Marking of dependent firing rate is allowed isspecification enables the firing value of the transi-tion to be specified as a token number in any Petrinet

(iv) Transitions can be provided with different prioritiesand transitions are enabled only if no other high-value transition is enabled

(v) e expectation of traditional withdrawal measuresfound in GSPN as the inclusion of a transition andthe mean number of tokens in a place the mostdifficult reward work can now be defined

Primarily availability of assessment approaches dependson modelling and measurement methods System avail-ability model-based approach is more effective and inex-pensive for analysis and comparison than measurement-based approach Discrete-event simulation can be used forsystem modelling whereas for analytical modelling bothapproaches can be used Analytical modelling can be cate-gorized into four main parts

(1) Nonstate-space model (reliability graph)(2) Reliability block diagram (RBD)(3) Fault tree (FT)

(4) State-space models (Markov chain stochastic Petrinet among others)

P3 P4

P7

P5

P6

P8P2

T1T2 T4

T3

Tc

Tb

Td

Figure 7 Generalized distributed transition stochastic Petri netmodel for two parallel branches conflict between Tc and Td

Table 1 Small logtr1 (a b d c b)tr2 (b d d)

Table 2 tr1 perfect alignmentLog a ≫ b d ≫ ≫ ≫ c b ≫ ≫

Model a τ b d Τ τ τ c b τ τta t1 tb td t2 t4 t1 tc tb t2 t3

Table 3 tr2 two possible alignmentsLog ≫ ≫ B d d ≫ ≫

Model A τ B d ≫ τ τta t1 tb td t2 t3

Table 4 Event log-based alignmentLog ≫ ≫ b d ≫ ≫ ≫ c ≫ ≫ ≫

Model A τ b d τ τ τ c b τ τtd t1 tb td t2 t4 t1 tc tb t2 t3


e hierarchical models fixed iterative point modelsand nonstate paradigms provide a quick overview of thesystem basic metrics (reliability availability and MTTF)with their proper scanning and the architecture of thesystem However state-space models capture complexfunctionality and system performance is approach canalso be able to manage failure or fix dependencies andcomplex connection between the system components Inorder to avoid a wide range of problems at the point statelevel modelling approach for nonstate-space and state-spacemodels for the other points can also be used

Availability a(t) shows the probability of correct stateat any instant t in operating system without taking intoconsideration the interval (0 t) e instantaneousavailability a(t) related to system reliability is computedby

a(t) r(t) + 1113946t

0r(t minus x)q(x)dx (2)

r(t) shows instantaneous reliability at time t and can bedefined as follows

r(t) 1113946infin

tf(x)dx (3)

f(x) is the probability density function for randomvariable x which represents the system lifetime or time tofailure q(x) represents the renewal process rate in the in-terval (0 t)

q(x) f(x) + 1113946t

0q x

aminus( 1113857f(v)dv (4)

w(x)dx represents probability of renewal processcycle that will be completed in the time interval (x x +

dx) middot r(t minus x) shows the probability of the system whichworks properly in the remaining time interval (t x)r(t minus x)w(x)dx shows the probability of the case wherefault has occupied and repair or renewal will reassumefunctioning with no further faults e concept of a(t) issimilar to the reliability r(t) in the case where the systemis not repairable

For a long running time we have steady-state availability(SSA) where at initial state we have limiting value of a(t)

decreasing by 1 erefore

a a(t) MTTF

MTTF + MTTR

a a(t) μ

μ + λ

(5)

where λ represents the time failure rate and system and μwhich is the repair time is determined as average of numberof repairs over maintenance tie

MTTF (mean time to failure) represents the expectedtime in which a system functions correctly before its firstfailure Mean time to repair (MTTR) shows the expected

time in which a system is used for repair At the point whereboth mean time to failure (MTTF) and mean time to repair(MTTR) are exponentially distributed the arithmetic in-version of failure and the repair rate of the system are asfollows

MTTF 1λ

MTTR 1μ

(6)

In hardware and software of industrial process sto-chastic reward net is an appropriate modelling tool InFigure 8 we present SRN availability model which canspecify the system operations by using transitions arcs andplaces which are the main parts Stochastic reward net foravailability framework is based on three stages

(1) Requirements specifications(2) SRN-based system modelling(3) System analysis

51 Survivability and Reliability Model for PerformanceChecking Survivability and reliability model can be used tocheck the performance of transient behavior after the oc-currence of failure attack and disaster among others It canbe seen that a generalization of recovery time means howmuch effect can be put during the recovery time Surviv-ability can check out the performance of the model by usingldquosystem average interpretation duration indexrdquo in a shortform called SAIDI model

e predictive model of SAIDI is

SAIDI 1113944c

i1emptyiE Xi( 1113857

E Mi Xi( 11138571113858 1113859

E Di Xi( 11138571113858 1113859 (7)

where E(Xi) shows the availability model andE[Mi(Xi)]E[Di(Xi)] shows survivability In short it can besaid that SAIDI is combination of availability and surviv-ability model

Hence emptyi means number of failures at section ldquoirdquo afterfailure of section ldquoirdquo we have

(i) Xi time up to full recovery(ii) Di(Xi) energy demanded up to full recovery(iii) Mi(Xi) energy not supplied up to full recovery

It can also be used to predict the recovery matrix aftercoupling with the availability model the generalized SAIDIform is as follows

SAIDI Customer interrputaion duration

Total number of customers (8)


6 Conclusion

In conclusion it is observed that the performance of thesystem or business process plays an important role inmodelling analysis and prediction Nowadays memorylessmodel such as exponentially distributed stochastic Petri net(SPN) has gained much attention in research and industryis paper is based on time perspective for modellinganalysis and use of stochastic Petri net to check the per-formance evolution stability and reliability of the modelTo know the effect of time delay in firing the transition weuse stochastic reward net (SRN) model Stochastic rewardnet (SRN) model can also be used to check the reliability ofthe model Generalized stochastic Petri net (GSPN) is usedfor evaluation and checking of the performance of themodel It is known that stochastic Petri net is used to analyzethe probability of state transition and the stability from onestate to another whereas in mining process logs are used bylinking log sequence with the state which enables modellingto be done and relates it with stability of the model Gen-eralized distributed transition stochastic Petri net model isused for checking the performance of the model Case-basedalignment is used to check the optimal alignment betweenthe traces SAIDI model is used to check the survivabilityand reliability of the generalized stochastic Petri net modelis paper deals with mathematical and theoretical workthat shows how checking the importance of stochastic Petriis done in order to know the performance of the discoveryprocess model and analysis Further work can be done on itspractical application

Data Availability

e data used to support the findings of the study are in-cluded within the article

Conflicts of Interest

e authors declare that there are no conflicts of interest

References

[1] H Hu J Xie and H Hu ldquoA novel approach for miningstochastic process model from workflow logsrdquo Journal ofComputational Information Systems vol 7 no 9 pp 3113ndash3126 2011

[2] N Anastasiou T Horng and W Knottenbelt ldquoDerivinggeneralized stochastic petri net performance models fromhigh-precision location tracking datardquo in Proceedings of the 5thInternational ICST Conference on Performance EvaluationMethodologies and Tools pp 91ndash100 Paris France May 2011

[3] E Leclercq D Lefebvre and S Ould El Mehdi ldquoIdentificationof timed stochastic petri net models with normal distributionsof firing periodsrdquo IFAC Proceedings Volumes vol 42pp 948ndash953 2009

[4] R Buchholz C Krull and G Horton ldquoReconstructing modelparameters in partially-observable discrete stochastic systemsrdquoin Analytical and Stochastic Modeling Techniques and Appli-cations pp 159ndash174 Springer Berlin Germany 2011

[5] A Wombacher and M E Iacob ldquoStart time and durationdistribution estimation in semi-structured processesrdquo Tech-nical report Centre for Telematics and Information Tech-nology University of Twente Enschede Netherlands 2012

[6] A Rozinat R S Mans M Song and W van der AalstldquoDiscovering simulation modelsrdquo Information Systems vol 34no 3 pp 305ndash327 2009

[7] A Adriansyah B F van Dongen and W M P van der AalstldquoConformance checking using cost-based fitness analysisrdquo inProceedings of the 2011 IEEE 15th International EnterpriseDistributed Object Computing Conference pp 55ndash64 HelsinkiFinland August 2011

[8] A Rogge-Solti W van der Aalst and W Mathias DiscoveryStochastic Petri Nets with Arbitrary Delay Distribution fromEvent Log Springer Berlin Germany 2014

P1 P2

Repair

1MTTF

1MTTR

T2

T1C

Figure 8 Stochastic Petri net availability model


e three main types of process mining are processdiscovery conformance checking and enhancement In thisresearch work attention is placed on the process discoveryIn process discovery it is known that without using any apriori information event logs produce a process model Incase the event logs have any information about the resourcea model-related resource for example social networkingcan be discovered which can be able to show how differentpeople worked in an organization collectively On the otherhand the process model can be used for analyzing costresource utilization or process performance and automa-tion Models are used for redesigning a process for planningand controlling in order to make decisions in a processere are two types of process model (i) formal model usedfor discussion and documentation and (ii) informal modelused for analysis or enactment In this research informalmodelling is used ere are certain errors that occur duringmodelling (i) when a model defines a prepared version or afact (ii) failure to properly do human conduct and (iii)when the model is at the wrong abstraction level

For information system analytical evaluation has be-come an integral part for the whole process design Manydiverse model specification techniques have been proposedfor example Petri net BPMN UML function diagramsBPEL and EPC Petri net is widely used in business pro-cesses either as the first model language or as a basis forvalidation e Petri net was first introduced in 1962 by CarlAdam Pete It can be described as a graphical method of theformal definition of logical interaction between componentsor the flow of activities in a complex system PN is par-ticularly well suited for modelling concurrency and conflictsequencing conditional branch and looping synchroniza-tion limited resource allocation and mutual exclusion Itenables the study of logical properties of modelled system foreach individual finite state machine which can be seen aspossible and yet shows an interaction whenever they occurwhen using idea behind the Petri net given by ProfessorPetri Petri net is categorized as being application and theoryof Petri net (ATPN) and Petri net and performance mod-elling (PNPM) which includes stochastic Petri net OriginalPetri net did not have the notation of time and was thereforeused for studying logical properties In order to introducetime duration in Petri net we link event data with transitionthat is important for availability reliability and performancequantification Hence that is the main reason why Petri netwas extended to have time linked with event occurrencegiving rise to the stochastic Petri net

Stochastic Petri net was introduced earlier in 1980 It hada graphical representation that was used for modelling ofdiscrete events and the bipartite graph of the transitions andplaces in SPN is used for knowing event mechanism It is alsoworth noting that the time of firing tokens is considered as arandom variable and is exponentially distributed Reach-ability graph in stochastic Petri net is a continuous firing ratethat is linked with each transition and may be marking-dependent If the history of a given process is known forexample event log with time information it is possible toextract stochastic performance data and insert it into themodel is research paper is therefore based on generalized

stochastic Petri net (GSPN) which does not restrict distri-butions in a particular way GSPN is very useful in the casewhere some events happen in extremely small timeWhereasSPN model handles this situation by introducing immediatetransition in the model which has zero firing time the othertransitions are timed transitions which are exponentiallydistributed In this research stochastic reward net is used inorder to check survivability and reliability SRN is basicallyan extension of GSPN and it was introduced in 1989 SRN isextensively marking-dependent which is used for firing ratesand probabilities by giving priority to the transitionserefore SPN has an advantage in that sense and it analyzesthe probability of state transition and the stability of onestate to another For the process with logs the log sequence isbased on the results of the transition execution and not thecomplete state results Hence this paper links the log se-quence with the state for stability of the model and analysis

e remainder of the paper is organized as follows inSection 2 related work is discussed in Section 3 preliminarydefinitions are presented and Section 4 discusses general-ized distributed transition stochastic Petri net model withexample Section 5 elaborates about the stochastic rewardnet survivability and reliability model Conclusion is pre-sented in Section 6

2 Related Work

Some related literature associated with Petri net model andstochastic overall performance for feature information arepresented in the literature review Hu et al [1] proposed atechnique that is primarily based on SPN model which isexponentially distributed for workflow log and depends ontransition rate of firing Anastasiou et al [2] proposedcompletely unique strategies whereby they centered at thelocation data for generalized stochastic Petri net model forcustomer modelling flows In their research work fortransition intervals they used constant hyper-Erlang dis-tributions which shows waiting and run times and also usedGSPN to upgrade the other corresponding transitionssubnet depicting the same features which the hyper-Erlangdistribution has ey observed at all the transitions inde-pendently which did not cause problems in the sequence butthe similarities within the method especially the manysimilar transitions were not considered in their technique

According to Leclercq et al [3] non-Markovian sto-chastic Petri nets and attempts at eliciting exist ey werelooking at a way of removing a model of normally dis-tributed data In their work they focused entirely on theanticipation algorithm prepared for maximization conver-gence Compared to the method used in this study they areunable to manage lost data and different performanceguidelines e reconstruction of the model parameters ofstochastic structures was also investigated in a study byBuchholz et al [4] where they dealt with the problem ofadjusting the model parameters of the basic stochasticsystem Contrary to this study the distribution of transitionsystem was targeted in advance although the main purposewas to make GDT_SPN version transitions which werecomparable to for example incomplete Wombacher and



3 Preliminaries








(P T F W M) where









(1)














P1

P2

T1

T1

P3


P1T1

P2

P3T3

P4


P1T1

T2

P2

P3T3

P4


P1T1

T4

P2


Token Output

InputP1

T1P2


P1T1

P2


















P3 P4

P7

P5

P6

P8P2

T1T2 T4

T3

Tc

Tb

Td












a(t) r(t) + 1113946t



r(t) 1113946infin

tf(x)dx (3)


q(x) f(x) + 1113946t

0q x






a a(t) MTTF

MTTF + MTTR

a a(t) μ

μ + λ

(5)




MTTF 1λ

MTTR 1μ

(6)





SAIDI 1113944c


E Mi Xi( 11138571113858 1113859

E Di Xi( 11138571113858 1113859 (7)








6 Conclusion


Data Availability




References









P1 P2

Repair

1MTTF

1MTTR

T2

T1C




3 Preliminaries








(P T F W M) where









(1)














P1

P2

T1

T1

P3


P1T1

P2

P3T3

P4


P1T1

T2

P2

P3T3

P4


P1T1

T4

P2


Token Output

InputP1

T1P2


P1T1

P2


















P3 P4

P7

P5

P6

P8P2

T1T2 T4

T3

Tc

Tb

Td












a(t) r(t) + 1113946t



r(t) 1113946infin

tf(x)dx (3)


q(x) f(x) + 1113946t

0q x






a a(t) MTTF

MTTF + MTTR

a a(t) μ

μ + λ

(5)




MTTF 1λ

MTTR 1μ

(6)





SAIDI 1113944c


E Mi Xi( 11138571113858 1113859

E Di Xi( 11138571113858 1113859 (7)








6 Conclusion


Data Availability




References









P1 P2

Repair

1MTTF

1MTTR

T2

T1C












P1

P2

T1

T1

P3


P1T1

P2

P3T3

P4


P1T1

T2

P2

P3T3

P4


P1T1

T4

P2


Token Output

InputP1

T1P2


P1T1

P2


















P3 P4

P7

P5

P6

P8P2

T1T2 T4

T3

Tc

Tb

Td












a(t) r(t) + 1113946t



r(t) 1113946infin

tf(x)dx (3)


q(x) f(x) + 1113946t

0q x






a a(t) MTTF

MTTF + MTTR

a a(t) μ

μ + λ

(5)




MTTF 1λ

MTTR 1μ

(6)





SAIDI 1113944c


E Mi Xi( 11138571113858 1113859

E Di Xi( 11138571113858 1113859 (7)








6 Conclusion


Data Availability




References









P1 P2

Repair

1MTTF

1MTTR

T2

T1C


















P3 P4

P7

P5

P6

P8P2

T1T2 T4

T3

Tc

Tb

Td












a(t) r(t) + 1113946t



r(t) 1113946infin

tf(x)dx (3)


q(x) f(x) + 1113946t

0q x






a a(t) MTTF

MTTF + MTTR

a a(t) μ

μ + λ

(5)




MTTF 1λ

MTTR 1μ

(6)





SAIDI 1113944c


E Mi Xi( 11138571113858 1113859

E Di Xi( 11138571113858 1113859 (7)








6 Conclusion


Data Availability




References









P1 P2

Repair

1MTTF

1MTTR

T2

T1C





a(t) r(t) + 1113946t



r(t) 1113946infin

tf(x)dx (3)


q(x) f(x) + 1113946t

0q x






a a(t) MTTF

MTTF + MTTR

a a(t) μ

μ + λ

(5)




MTTF 1λ

MTTR 1μ

(6)





SAIDI 1113944c


E Mi Xi( 11138571113858 1113859

E Di Xi( 11138571113858 1113859 (7)








6 Conclusion


Data Availability




References









P1 P2

Repair

1MTTF

1MTTR

T2

T1C



6 Conclusion


Data Availability




References









P1 P2

Repair

1MTTF

1MTTR

T2

T1C



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