Timing Constraint Workflow Nets for Workflow Analysis. Abstract: The analysis of the correctness and rationality of a workflow model plays an

Transcription

1 Vol.33, No., March 03, pp Timing Consrain Workflow Nes for Workflow Analysis JianQiang Li a YuShun Fan a MengChu Zhou b a Deparmen of Auomaion, Tsinghua Universiy, Beijing b Deparmen of ECE, New Jersey Insiue of Technology, Newark, NJ USA lijq99@mails.singhua.edu.cn, fan@cims.singhua.edu.cn, zhou@nji.edu Absrac: The analysis of he correcness and raionaliy of a workflow model plays an imporan role in he research of workflow echniques and successful implemenaion of workflow managemen. This paper poins ou he relevan problems in he verificaion and analysis of a workflow model. I discusses wo imporan properies: schedulabiliy and boundedness of a workflow model considering iming consrains. To specify he iming consrains, WorkFlow ne [] is exended wih ime informaion, leading o Timing Consrain WorkFlow ne (TCWF-ne). This paper presens a model mapping mehod o conver a Direced Nework Graph (DNG) based workflow model, which is buil by a graphic process modeling language [] exended wih ime informaion, ino a TCWF-ne. I hen discusses he schedulabiliy verificaion and synhesis of TCWF-nes. Due o he fac ha here is no ieraion in he TCWF-ne obained hrough model ransformaion, an algorihm o decompose he acyclic and free-choice TCWF-ne ino a se of T-componens is presened. Then, o avoid he run-ime congesion of a workflow model, which is he key o he performance managemen of he business process auomaion a boundedness verificaion mehod is derived. The usefulness of he research resuls is illusraed by an example. Keywords: Workflow model, Peri nes, Boundedness, and Schedulabiliy. Inroducion Workflow managemen is a key echnology in supporing business process reengineering and an effecive means realizing full or parial auomaion of a business process [3]. Despie he abundance of workflow managemen sysems developed for differen ypes of workflow based on differen paradigms [4-7], he lack of rigorous heoreic foundaion and hen effecive model verificaion and analysis mehods has blocked workflow echniques research and applicaion. Workflow specificaions address many issues including process conrol, resource,

2 Vol.33, No., March 03, pp informaion, and funcion perspecives. Hence, we know from research ha he raionaliy and correcness analysis should be carried ou from four aspecs ha are relevan for workflow modeling and workflow execuion: process conrol logic, iming consrain logic, resource dependency logic, and informaion dependency logic. The objecive of correcness analysis of process conrol logic is o avoid he deadlocks or srucural conflics in he execuion of a workflow model because of he errors in is process conrol. Some verificaion and conflic deecion mehods have been discussed in [, 8-9]. Resource dependency logic verificaion focuses on he proof of correcness of he saic or dynamic resource allocaion rules and consisence wih he process conrol logic. The informaion dependency logic, however, indicaes he inernal consisence of a workflow-relaed daa and correc emporary relaion among differen workflow applicaion daa. The iming consrain verificaion and analysis deal wih he emporal aspecs of a workflow model such as deadlines, variable calendar windows, ime scales, and alering mechanisms for overdue acions. I also includs he schedulabiliy analysis of he consiuen aciviies of a workflow model and is boundedness verificaion in he case of muli-workflow-insances running concurrenly, which is discussed in his paper. Noe ha he iming consrain verificaion and synhesis should be conduced afer he process conrol verificaion is done, which means ha he workflow model considered here is free of srucural conflics (deadlock). In order o improve he efficiency and qualiy of a business process, each aciviy and he precedence relaions beween differen consiuen aciviies in a workflow model should have reasonable iming consrains according o he ransacion insance arrival ime, he required service qualiy, and efficien resource managemen. Our invesigaion shows ha mainly wo kinds of iming consrains should be considered: exernal and inernal iming consrains. The former follows implicily from conrol dependency of a workflow schema. I causes he buffer ime ha is handled by a workflow manager or workflow engine. For example, an aciviy can only sar 5 minues afer is preceding aciviies have finished, or i mus sar 0 minues afer anoher specific aciviy ends. To specify his kind of iming consrain, he conceps of Lower Bound Consrain (LBC) and Upper Bound Consrain (UBC) [0] should be inroduced and used in his paper. Inernal iming consrain, which is managed by a resource agen (e.g. a sofware sysem or human) ha is responsible for he enacmen of an aciviy, is embedded in

3 Vol.33, No., March 03, pp he descripion of every individual aciviy. I includes he defined execuion duraion and he execuable ime span of his aciviy. Addressing he issues of how o verify he correcness of a workflow model from he ime dimension and selec raionally he ime parameers a he build-ime is imporan in realizing efficien workflow managemen. When a workflow model is deployed in pracice, he sae of an aciviy in a workflow insance may include iniiaed, enabled, ready, running, suspended, dead, error, and end. However, in he ime dimension verificaion and analysis of workflow models, only he emporal behavior of ask execuion, which is defined a he build ime, is concerned. Thus we consider only he enabled, ready, running, and end saes of aciviies in his paper. The workflow process definiion models he life-cycles of all he ransacion insances. On one hand, he workflow engine inerpres and mainains a workflow insance for each ransacion insance in he running ime environmen. Each aciviy in he workflow insance mus be allocaed enough ime o complee is execuion wih respec o imposed iming consrains. On he oher hand, a workflow model deployed in pracice has a sable ransacion insance inpu rae, which means ha here are many ransacion insances handled concurrenly according o he same workflow model. When he inpu rae of an aciviy is greaer han is oupu rae, he congesion or overflow occurs. To avoid i in he running environmen, he execuion duraions of he coordinaed aciviies in he process should conform o cerain consrains. Therefore, our ime dimension workflow model verificaion and analysis mehod is addressed hrough wo levels: () schedulabiliy verificaion and synhesis of a workflow model; and () boundedness verificaion of a workflow model in he environmen of muli-workflow insances running concurrenly. Peri Ne (PN) originaed from he early work of Carl Adam Peri [] has found many applicaions in compuer science. Because of heir formal semanics, local sae-based sysem descripion, and abundan analysis echniques [], heir use as a mahemaical foundaion for he formal analysis of workflow models is aracive o many researchers of workflow echniques. Since Zisman [30] used PN o model workflow processes for he firs ime in 977, researchers have proposed heir own echniques based on PN o model workflow. Some [3], [37] recognize he adapabiliy problem inheren o workflows, i.e., he frequenly and/or radically changing characer due o changing business rules. Thus, he work [], [4], [4], [3], [3], [33], 3

4 Vol.33, No., March 03, pp [34] focuses on how process conrol or daa flow is modeled by PN o improve he adapabiliy of a workflow model in a changing marke environmen. Table highlighs several proposed PN classes for workflow modeling. However, despie he populariy of using PN for business process specificaion, Han [3] warns ha PN canno apply direcly for modeling workflows due o heir fixed srucure. Table. Overview of he proposed PN classes for workflow modeling Peri ne class Informaion Conrol Nes (ICN) [4] Workflow-nes (WF-nes) [] Reconfigurable Nes [4] Modular Process Nes [3] Elemen Ne sysem [33] Higher Order Objec Nes (HOON) [3] Predicae Peri nes (PPN) [34] Brief descripion By adding a complemenary daa flow model, generalizing conrol flow primiives and simplifying semanics, ICN is a Colored Peri ne varian inended o represen conrol and daa flow of office workflow. A message-based excepion handling mechanism for ICN is provided o improve he flexibiliy of a workflow sysem. Wih one inpu place (ε) and one oupu place (θ), indicaing he beginning and end of he modeled business procedure, WF-nes require ha every ransiion (place) mus be locaed on a pah from ε o θ. They are suiable no only for he represenaion and valididaion bu also for he (process conrol logic) verificaion [8],[9] of workflow procedures. As an exension of WF-nes, a Reconfigurable Nes consiss of several Peri nes which consiue he differen possible configuraions for some mode of operaion o suppor dynamic changes and realize self-modificaion of a workflow sysem. Modular Process Nes are based on a hierarchical module concep and he consrucs for synchronous and asynchronous communicaion beween inerpreed nes and heir environmen as a framework for flexible workflow modeling and enacmen, and can be described as Elemen Ne sysems wih minimal synacic exension. By decomposing a WFMS ino wo basic componens, namely a WF model and a WF Execuion Model, only a subclass of Elemen Ne Sysem (EN-sysem) are needed o describe a WF model. Thus he WF Execuion Module can be implemened wih enhanced flexibiliy and adapabiliy. Based on Modular Process Nes, he srucures of he organisaion and resource configuraion are explicily embodied in HOON. The ne model and is enviromen are arranged in a clien/server manner. Using iner-ask dependencies o specify inernal srucure of a workflow, PPN capures he relaionships among subransacions wihin a flexible ransacion model. Oher workflow researchers have noiced he imporance of workflow model analysis in supporing business process reengineering and successful workflow managemen. Differen 4

5 Vol.33, No., March 03, pp kinds of PNs are employed for correcness and raionaliy analysis of a workflow model. Because he conrol flow is a he hear of workflow specificaion, many researchers have addressed he correcness verificaion of he process conrol of a workflow model. Some ransformaion rules beween WF-nes are proposed o allow workflow designers o modify workflow srucures while preserving is correcness and consisency [9]. A WF-ne-based verificaion mehod [38] for workflow Task Srucures is presened in [38]. The process verificaion and consisency checking echniques are discussed in [3] based on ordinary P/T nes and he echnique presened in [9]. In [46], an algebraic echnique based on Peri nes is proposed o check he local consisency of a workflow model and, using a composiion heorem, i can be used o verify he iner-organizaion workflow model. More work can be found in [39], [40], [4]. Colored Peri Ne (CPN) is used for he formal specificaion of a workflow process, and simulaion has been used o verify he coordinaion beween workflow aciviies [45]. However, alhough he lieraure on workflow has consisenly sressed he imporance of ime [3-5, 7, 0, 7, 4-43], only a few papers address he applicaion of PN o emporal analysis of a workflow model. Through exending ordinary P/T nes wih an inerval funcion and a imesamp funcion o model absolue as well as relaive ime, exising PN verificaion approaches are employed [3] o es wheher i is feasible o execue a workflow wih specified emporal consrains. In [48], a se of linear reasoning algorihms for commonly used workflow paerns is presened o invesigae he emporal properies of a workflow wih iming consrains of deerminisic inervals. Performance analysis of workflow is research opic ye o be given he imporance i deserve [3, 7, 35-36, 4-50]. All he rouing consrucs of a workflow are mapped ino a higher-level Sochasic PN (SPN), hen hroughpu ime of he process is analyically compued [47]. Based on four performance equivalen formulae, an approximae performance analysis mehod of a workflow is presened in [49]. These wo echniques boh assume he infinie availabiliy of resources in he workflow configuraion. Generalized Sochasic Peri Nes (GSPN) are used o model workflow [35, 44], and hen a mehod based on a coninuous ime Markov chain (CTMC) is used o obain upper bounds of he execuion performance. A simple GSPN, which is a so-called load equivalence aggregaion (LEA) model, has been developed in [36], and hen he model is simulaed using a Coloured GSPN (CGSPN) o obain some performance relaed measures of human resources in a workflow. Commonly, he echniques applying PN in he domain of 5

6 Vol.33, No., March 03, pp workflow exploi he correspondence beween he special kind of ime-relaed PN and he dynamic behavior of workflow sysems, and use exising PN analysis echniques for workflow analysis. For more lieraure on he applicaion of PN o workflow modeling and analysis, readers can refer o he wo workshops on workflow managemen [4], [43, 50]. Considering Han s saemen as menioned above and he fac ha mos of he commercial workflow producs use differen kinds of DNGs insead of PN for heir model specificaion, he mapping from such DNG o PN are worhy o invesigae. Also, o realize sysemaic ime dimension verificaion and analysis of workflow models, i is necessary o incorporae all he relevan iming informaion ino PN-based workflow models. The nex secion inroduces a graphic process modeling language [] exended wih iming informaion and he concep of TCWF-nes by incorporaing iming informaion ino WF-nes. Secion 3 proposes a model mapping mehod from a DNG-based workflow model buil by he exended basic process modeling language o TCWF-ne. Secion 4 presens he schedulabiliy verificaion mehods and heurisic rules for he iming synhesis of a workflow model. Based on ha almos all workflow models have a free-choice characerisic [], [9], Secion 5 provides an effecive algorihm decomposing free-choice and acyclic TCWF-ne ino a se of T componens and a boundedness verificaion mehod. Secion 6 presens a case sudy. Finally, Secion 7 makes conclusions.. Basic conceps A basic process modeling language [] based on a sandard process definiion noaion, which is proposed by Workflow Managemen Coaliion (WfMC) [5], can be used o represen he componens of a workflow in a simple and direc way. In his language, processes are modeled using wo ypes of objecs: node and ransiion. A node is classified ino wo subclasses: ask and choice/merge coordinaor. The ask, graphically represened by a recangle, represens he work o be done o achieve some objecives. I can be used o build implicily sequence, fork, and synchronous srucures. The ask is furher classified ino four ypes: aciviy, sub-process, block, and null ask, which are necessary for he process modeling. However, for simpliciy, all kinds of asks are reaed only as aciviies in our proposed model mapping mehod. The choice/merge coordinaor, graphically represened by a circle, is used o build explicily choice 6

7 Vol.33, No., March 03, pp and merge srucures. A ransiion linking wo nodes in a graph is graphically represened by a direced edge and used o specify he execuion order and flow beween is ail and head nodes. Figure shows hree modeling objecs. Because he ieraion srucure is nesed in a ask ha has an exi condiion defined for ieraive purposes, a DNG-based workflow model buil by he basic process modeling language mus be srucurally acyclic. Refer o [] for more deails. ask choice/merge ransiion Figure. Graphic represenaion of hree modeling objecs In order o incorporae he necessary iming informaion ino a workflow model, we need o exend his basic process modeling language by considering is inernal and exernal iming consrains. As menioned before, he inernal iming consrain defined in an aciviy refers o execuion duraion and execuable ime span, where he execuable ime span managed by a responsible resource agen ranges from he ime of ask allocaion o allowable laes compleion ime. Given a workflow model, designers can assign execuion duraion and execuable ime span (during which he aciviy can be execued) o every individual aciviy based on heir experience and expecaion from he pas execuion. In order o specify exernal iming consrains, i.e., he emporal dependency relaions beween differen aciviies, he conceps of LBC and UBC need o be inroduced. Assuming ha A and B represen wo aciviies, LBC(A, B, DL) saes ha he duraion beween source even happened in running ime of aciviy A and desinaion even happened in he running ime of aciviy B mus be greaer han or equal o a lower bound ime value DL. UBC(A, B, DU) demands ha he ime disance beween source even happened in he running ime of aciviy A and desinaion even happened in he running ime of aciviy B mus be smaller han or equal o an upper bound ime value DU. Here, we call A and B as source and desinaion aciviies, respecively. For simpliciy, LBC/UBC used in his paper refers only o he iming consrains beween he end execuion even of he source aciviy and sar enabled even of he desinaion aciviy. Assuming ha aciviy A complees is execuion a ime T 0 and he execuion duraion of aciviy B is D(B), and he iming consrains of LBC(A, B, DL) and UBC(A, B, DU) are imposed on B, we define he enabled ime span of aciviy B as (T 0 +DL, T 0 +D(B)+DU). These 7

8 Vol.33, No., March 03, pp iming consrains are assigned according o he relevan organizaional rules, laws, commimen, echnical demands, and so on. They can also be seleced based on he workflow performance requiremen. A merge node is only used for he descripion of he logic relaion beween is inpu and oupu nodes. Thus is enabled ime span and execuion duraion are boh se o zero. Then B wih LBC(A, B, DL) or UBC(A, B, DU) canno be a merge node. If LBC(A, B, DL) or UBC(A, B, DU) is needed, in which B is a merge node, a dummy aciviy is used o specify his siuaion. I means ha he ransacion insance roues ou as soon as i reaches a merge node. However, he choice node may include acions such as he oupu pah selecion. Then i can be reaed as an aciviy and he iming consrain menioned above can be imposed on i. Also, he choice node s oupu pah selecion can be specified hrough assigning differen iming consrains (execuable ime span) o differen succeeding pahs. For he sake of simpliciy, we assume ha all he ime informaion is given in a same ime uni. A (,3) A (0,5)/4 (0,4)/ Figure. An example of he iming consrains in DNG-based workflow model A simple example in Fig. is used o show he specificaion of ime informaion in a DNG-based workflow model. (LBC, UBC)=(,3) on he edge from A o A specifies he emporal dependency relaion beween aciviies A and A as an exernal iming consrain. A s (0, 4)/ specifies A s inernal ime consrains, i.e., is defined execuable ime span is [T +0, T +4] if i sars enabling a ime T ; and is execuion duraion is. A s inernal ime consrains can be inerpreed similarly. If A complees is execuion a T 0, he sar enabled ime span of A is [T 0 +, T 0 +3], and A s enabled ime span is [T 0 +, T 0 ++3]. In he run-ime environmen, because of he iming consrain imposed by he enabled ime span, A s acual execuable ime span will be [T +0, Min{T 0 ++3,T +4}], where T 0 + T T Peri nes as a design language for he specificaion of a complex workflow, as well as a powerful analysis echnique for he correcness of workflow procedures are discussed in [, 4]. Some basic conceps abou PN [], [7] are given as follows: Definiion : PN = (P, T, F) is a free-choice PN iff, T, φ implies = ; and i is a marked graph iff each place p has exacly one inpu and one oupu ransiion. 8

9 Vol.33, No., March 03, pp Definiion : PN = (P, T, F) and PN = (P, T, F ) are wo PN. PN is a subne of PN iff P P, T T and F = F ((P T ) (T P )). PN is generaed by T iff P = T T (where he preses and posses are aken w.r.. F). I is called a T -componen of PN iff PN is he subne generaed by T and, p P : p T p T. A PN modeling he process aspec of workflow is called a WorkFlow ne []. Definiion 3: A Peri ne PN is called WorkFlow ne (WF-ne) if and only if: PN has wo special places: ε and θ. Place ε is a source place: ε = φ; Place θ is a sink place: θ = φ. If we add a new ransiion o PN which connecs place θ wih ε, namely, ={θ}, ={ε}, hen he resuling PN is srongly conneced. A WF-ne has proper erminaion propery if saring from he iniial sae (wih only one oken in place ε), i is always possible o reach he sae wih only one oken in place θ. A WF-ne wih proper erminaion propery is sound if i has no dead ransiions, i.e., for each ransiion, i is possible o reach (saring from iniial sae) a sae where is enabled. If here are no srucural conflics in a workflow model buil by he basic process modeling language menioned above, is corresponding WF-ne mus be sound []. Obviously, a WF-ne gives only process conrol specificaion of a workflow model. To realize he ime dimension verificaion and analysis of a workflow model, is emporal behavior should be specified, and hen some exensions wih ime informaion o he WF-ne are needed. Differen ways exis o inroduce ime ino PN. Timed Peri nes rea a iming consrain as a single delay [8-0]. Time Peri nes rea a iming consrain as a delay pair consising of lower and upper bounds [-3]. Timing Consrain Peri Ne (TCPN) [6], which exends PN by adding minimum, maximum, and duraional iming consrains o places or ransiions, synhesizes all he iming consrains considered in previous wo cases. Considering he iming consrains o be specified in a workflow model, we propose o use TCPN for workflow modeling and analysis. In TCPN, a ime pair [d min (x), d max (x)] is associaed wih node x P T. [d min (p), d max (p)] denoes he period ha p can enable is oupu ransiion afer a oken arrives. The oken enabled ime of p is defined as [K a (p)+ d min (p), K a (p)+ d max (p)], where K a (p) is he oken arrival ime a p. [d min (), d max ()] represens he period ha is fireable afer i is enabled. If p is he only inpu 9

10 Vol.33, No., March 03, pp place of, s fireable duraion is deermined collecively by K a (p), d min (p)/d max (p) and d min ()/d max () [6]. A ransiion is schedulable means ha i is firable and can complee is firing successfully. The formal semanics of TCPN can be used in a workflow model o specify naurally he siuaions such as he excepion handling or selecion of a succeeding rouing pah. In TCPN, all he okens (denoe as TK s) used for enabling a ransiion will be preserved during he ransiion s firing. If all he ransiions enabled by TK s fail o complee heir firing, TK s will be rapped in heir corresponding place. This kind of sae evolving mechanism ogeher wih he relaive and absolue ime mode, as well as he weak firing rules make he TCPN paricularly suiable for he specificaion abou he execuion of a workflow model in he run-ime environmen. Based on he concep of a WF-ne and TCPN, he definiion of Timing Consrain WorkFlow netcwf-neis given below: Definiion 4: A TCWF-ne is a four uple <WF-ne, C, α, M>, where: WF-ne=(P, T, F) is a WorkFlow ne; P ={p, p,, p m } is a se of places represening he sae of a ransacion insance or he condiion of is oupu ransiions; T ={,,, m } is a se of ransiions represening aciviies of workflow; F is a se of direced arcs linking places and ransiions, and used o describe precedence relaions among aciviies; C is a se of non-negaive real number pairs [d min, d max ] relaed o each ransiion or place, which is used o represen he imposed iming consrains of an aciviy or sysem sae; α is a se of firing delays associaed wih ransiions, where α() represens he execuion duraion of ransiion mapped from is corresponding aciviy; M is a se of m-dimensional markings where M(p) denoes he number of okens represening he number of ransacion insances in p. Wha we should noe here is ha a ransiion in he TCWF-ne corresponds o an aciviy of workflow; however, a ransiion defined in he basic process modeling language menioned above is used o specify he execuion order and flow beween is ail and head nodes. In addiion, E F () (L F ()) denoes he earlies fireable beginning ime (laes fireable ending ime) of ; E E () (L E ()) denoes he earlies enable beginning ime (laes enable ending ime) of 0

11 Vol.33, No., March 03, pp ; F begin () (F end ()) denoes he ime a which begins (ends) firing. I should be noed ha a TCWF-ne is used o specify he dynamic behavior of he life-cycle of ransacion insances. For he verificaion and analysis of a workflow model, we have o consider he mapping from a workflow model buil by he exended process modeling language o a TCWF-ne. 3. Model mapping According o he semanic properies of workflow models buil by mos of he workflow modeling ools (see e.g. [], [5]), we know ha mos of hem enjoy he free-choice characerisic. We use Fig. 3 o explain why a pracical workflow model should exhibi his characerisic. Assuming ha a ransacion insance leads o wo okens (branches of he same insance) in nodes D and A respecively in Fig. 3(a). Because a ransacion insance s rouing depends only on he coordinaion beween is aribues and he workflow conrol daa, D can chooses R as is oupu pah and A chooses R 3. Then synchronous aciviy B will no ake place and he insance will no erminae successfully. The same resul follows from Fig. 3(b) if D chooses R and A chooses R 3. Therefore, such non-free-choice srucures as hose in Fig. 3 should no appear in pracical workflow models. Thus, his paper deals wih only workflow models or TCWF-nes wih free-choice semanics. X X D A R R R3 R4 R D R R3 A R4 B (a) B (b) Figure 3. Two non-free-choice srucures Now we inroduce a model mapping mehod from a DNG-based workflow model buil by he exended basic process modeling language o a free-choice TCWF-ne. Each choice/merge node D is mapped ono a place p D. For each of D s oupu pah i, ransiion is creaed. Each aciviy A is mapped o a ransiion A. If i is in a sequence srucure, p A is creaed as A s inpu place. However, if A is a synchronous aciviy, for each of is inpu pah j, a place, whose oupu ransiion is A, is creaed. For he end aciviy E, a sink place θ is added as he oupu place of is mapped ransiion E. Finally, he ransiion in he graphic process modeling i D j p A

12 Vol.33, No., March 03, pp language is mapped o he direced arc linking corresponding place and ransiion in he TCWF-ne direcly. We have so far compleed he opology srucure mapping and obained a free-choice TCWF-ne srucure model of a workflow process. Because he workflow model considered here is free of srucural conflics (deadlock), he resuling TCWF-ne mus be sound []. During he model mapping, we mus also consider he imposed iming consrains in a process model besides he srucure mapping. In a TCWF-ne, he sae of a ransacion insance is specified by a place. Hence he iming dependency relaions (LBC and UBC) beween aciviies, which are managed by he workflow manager or workflow engine, are mapped o a ime pair in a place. The ime pair [d min (p), d max (p)], which is used o specify he enabled ime span of is oupu ransiion afer a oken arrives a p, includes he firing delay of is oupu ransiion. However, a UBC, as menioned above, consrains he period ranging from he end of is source aciviy o he sar enabling of is desinaion aciviy. I means ha DU in UBC(A, B, DU) doesn include he execuion duraion of aciviy B. Therefore, he acual [d min (p), d max (p)] of p is he enabled ime span of is corresponding aciviy. For a place p, which is creaed for he aciviy (choice node) B in he LBC(A, B, DL) and/or UBC(A, B, DU), he ime pair [d min (p), d max (p)] specifying he ime dependency relaion is se o [DL, DU+D(B)], [0, DU+D(B)], and [DL, ] according o hree cases: ) LBC and UBC are boh imposed, ) only UBC imposed, and 3) only LBC imposed, respecively. If here is neiher LBC nor UBC, he ime pair for p is se o [0, ]. As menioned above, due o he fac ha he workflow engine handles he rouing in a merge node D, he ime pair for p D is se o [0, 0]. Hence, a ransiion in p D is execuable once i is enabled. When a workflow engine is aking care of he conrol and execuion of a workflow insance, he ask is allocaed o he resource agen ha is responsible for is execuion. The execuable duraion, i.e., he buffer ime ha is handled by he corresponding resource agen (execuor), is ranged from he ask allocaion ime o he allowed laes compleion ime. The ime pair [d min (), d max ()] is se o he corresponding execuable ime span defined for he aciviy in he workflow model. If here is no execuable ime span imposed for he corresponding aciviy, he pair is se o [0, ]. However, he execuable ime span of he ransiion creaed for he merge node is se o [0,0], implying ha a oken roues ou once i reaches a merge place (he execuion duraion of is oupu ransiion mus be zero).

13 Vol.33, No., March 03, pp The execuion duraion of each aciviy can be deerminisic or sochasic. In he deerminisic case, i is mapped o α() direcly. In he sochasic case, α() mapped o a ransiion in TCWF-ne is he average value of is corresponding random execuion duraion. In his siuaion, ime dimension verificaion or analysis is conduced from he viewpoin of expecaion. [0, ] [0,5][4] [,5] [0,4][] p A A p A A Figure 4. A TCWF-ne fragmen Le s use a simple example o illusrae he ime informaion mapping beween he wo kinds of workflow models. Fig. 4 is a TCWF-ne fragmen, which is mapped from he sub-workflow in Fig.. Aciviies A and A are mapped ono ransiions A and A, and hen p A and p A are creaed respecively as inpu places of A and A. A s inernal ime consrains (0, 5)/4 is mapped direcly ono A s [0, 5][4], where [0, 5] and [4] correspond o [d min ( A ), d max ( A )] and α( A ) respecively. [d min ( A ), d max ( A )]=[0, 5] means ha if A, which is enabled a ime T, is said o be firable during he ime period from T +0 o T +5, which corresponds A s execuable ime span. A s iming consrains can be inerpreed similarly. Because A s execuion duraion is, hen he exernal iming consrain (LBC, UBC)=(,3) beween A and A is mapped ono [d min (p A ), d max (p A )]=[, 3+]=[,5]. If A s firing is compleed a T 0 (i.e., K a (p A )=T 0 ), p A can only enable is oupu ransiion A during he ime from T 0 + o T 0 +5, which corresponds o A s enabled ime span. Since here are no exernal iming consrains for A, p A s [d min (p A ), d max (p A )] is se o [0, ]. 4. Workflow model schedulabiliy verificaion and synhesis As defined in [7], a marking M n, is said o be reachable if here is a firing sequence σ =(M 0 M i M i n M n ) or simply (,,, n ) ha ransforms M 0 o M n. Due o iming consrains, o prove ha M n is reachable in TCWF-ne, we have o prove ha all he ransiions in σ are schedulable wih respec o M 0. In oher words, le n be he final ransiion of σ from M 0 o M n, M n is reachable if and only if n and all he ransiions ha occurred prior o n are schedulable. δ k (M n ), o be used below, denoes he collecion of places and ransiions excep he firs 3

14 Vol.33, No., March 03, pp ransiion in he k-h firing sequence from M 0 o M n [6]. Using TCPN scheduabiliy [6] as reference, we give he definiion of he scheduabiliy of a TCWF-ne and hen is corresponding verificaion mehod. I is known ha he arrival ime of a ransacion insance mus be aken ino accoun in he schedulabiliy analysis of a TCWF-ne model, which indicaes a srong characerisic. Thus we do no disinguish he srong and weak schedulabiliies. However, we differeniae local and global schedulabiliies here. Definiion 5: A ransiion in marking M of a TCWF-ne is locally schedulable iff s enabled marking M is reachable from M, d max ()-d min () α() and, p : d max (p)-d min (p)-d min () α(). If can complee is firing successfully in marking M of he TCWF-ne, i is globally schedulable. Obviously, he local schedulabiliy is a necessary condiion of he global one. A schedulable TCWF-ne means ha all is ransiions are globally schedulable. Assuming p i, E E ()/L E () is consrained by oken enabled imes of all inpu places of, i.e., E E ()= Max (K a (p i )+d min (p i )) and L E ()= Min (K a (p i )+d max (p i )). Therefore, assuming ha s is i mapped from he sar aciviy we have he following heorem. i Theorem A locally schedulable ransiion in marking M of TCWF-ne is globally schedulable iff L F () - E F () α(), where L F ()= L E () = Min (K a (p i )+d max (p i )) i = Min {F end ( s )+ Σ d max (p mk )} k p i, p mk δ k (M). E F ()= E E ()+ d min () = Max (K a (p i )+d min (p i ))+ d min () i = Max {F end ( s )+Σ d min ( nk )+ Σα( nk )+Σ d min (p mk )} + d min () k p i, nk, p mk δ k (M). Theorem is derived direcly from he TCPN heory. A rivial modificaion of he proof of relaed heorems in [6] can prove i. Is proof is omied here. However, wo poins need o be noed here. Firs, our TCWF-ne is sound. When he workflow model is insaniaed o a 4

15 Vol.33, No., March 03, pp workflow insance by he arrival of a ransacion insance, each inpu place of he enabled ransiion has exacly one oken during he model runime. Then, i is unnecessary o compare he arrival ime of differen okens in he same places. Second, according o he heorems in [6] p can only enable is oupu ransiion during he ime span [K a (p)+ d min (p), K a (p)+d max (p)]. Therefore, if he enabled ime of is oupu ransiion is T 0, is fireable ime-span is [T 0 + d min (), Min{T 0 +d max (),K a (p)+d max (p)}]. If d max (p)>d max (), K a (p)+min{d max (p), d max ()} in formula (4.b) of [6] is equal o K a (p)+d max (). Obviously, i does no make sense. Thus, i should be replaced by K a (p)+d max (p), he raionaliy of which can be found easily. In he following discussions, schedulable refers o globally schedulable. According o Theorem, a workflow modeled by a TCWF-ne is schedulable if and only if all he ransiions in he TCWF-ne are schedulable. To faciliae he use of Theorem, Theorem is deduced. Theorem : A TCWF-ne is schedulable iff all is synchronous ransiions are globally schedulable and all he remaining ransiions are locally schedulable. Proof: A schedulable TCWF-ne implies ha every ransiion is globally schedulable. Thus he necessiy is obvious. If all he synchronous ransiions are globally schedulable, all he ransiions preceding hese synchronous ransiions can complee firing successfully, i.e. globally schedulable. Thus we need o consider only he ransiions behind all he synchronous ransiions along all he possible pahs from he source ε o he sink θ. Consider any non-synchronous ransiion. The number of is inpu places mus be exacly. Assuming p i, we have L F () - E F () = Min (K a (p i )+d max (p i ))-( Max (K a (p i )+d min (p i ))+ d min ()) i = K a (p i )+d max (p i )-( K a (p i )+d min (p i )+ d min ()) = d max (p i )- d min (p i )- d min (). i Since is a locally schedulable ransiion, d max (p i )- d min (p i )- d min () α(). Then L F () - E F () α(). By Theorem, is globally schedulable. Hence, all he ransiions in a TCWF-ne are globally schedulable, or he ne is schedulable. Before he schedulabiliy synhesis of a TCWF-ne from he aspec of ime dimension is discussed, wo relevan conceps of he iming consrains of a ransiion should be inroduced. In marking M i of a TCWF-ne, s schedulable decision span is defined as S()= [E F (), L F ()-α()] 5

16 Vol.33, No., March 03, pp and is schedulable buffer ime Y()=L F ()-α()-e F (). A ransiion compleing is firing successfully mus sar is firing during he absolue ime span S(). Y() represen he inerval one can manage o sar firing a ransiion. Assigning reasonable S() for each ransiion and hen he corresponding aciviy is an imporan issue for leveraging he process performance and resource load. Based on he synhesis mehods of Forward Compuaion and Backward Compuaion [6], we give some heurisic rules abou how o assign S() for each considering is relaive imporance, he load of resources allocaed for s firing, and he influence of is firing on is succeeding ransiions. According o differen opological srucures of a TCWF-ne, several cases are discussed. For ransiions in a sequence, OR_join or AND_spli srucure: If he resource saisfies he requiremen, S() and Y() of he preceding ransiion should be scheduled early and reduced as much as possible respecively o expand he schedulable decision span of he succeeding ransiions. The reason is ha here is a oal ime limi for a workflow model and hen he execuion of preceding ransiions has influence on schedulable decision span of all he succeeding ransiions. If somehing goes wrong wih he firing of a preceding ransiion, he larger Y() and abundan S() for he succeeding ransiions give hem enough ime o handle he excepion. For ransiions in an OR_spli srucure: One can designae muually exclusive fireable period [d min (), d max ()] for he ransiions corresponding o conflic aciviies, hrough which he differen prioriies are se. We can use i for he specificaion of he excepion handling of ime violaion or he selecion of an oupu pah in a conflic srucure. If ransacion insance s rouing in an OR_splic srucure does no depend on is iming aribues, he ransiions in his srucure can be reaed he same as in sequence srucures. For ransiions in an AND_join srucure: In his synchronous srucure, he execuion of a succeeding ransiion canno begin unil all is preceding ransiions complee heir firing. Therefore, in order o reduce he muual waiing ime of individual execuion pahs, he ime consrains for he preceding ransiions mus be seleced o make hem as accordan (according o heir succeeding synchronous ransiion s S() and Y() consrained by differen inpu pahs) as possible. 6

17 Vol.33, No., March 03, pp Boundedness verificaion of a workflow model A workflow model is insaniaed by muliple ransacion insances. Boundedness verificaion of a workflow model concerns wheher congesion or overflow may occur in he environmen of muli-insances running concurrenly. In a TCWF-ne, source place ε wih a okens (ransacion insances) arrival rae λ can be viewed as a srucure of a ransiion (wih no inpu arcs) wih firing rae λ connecing o ε. This semanics cause many okens of muliple insances reside in he same TCWF-ne. A TCWF-ne is bounded if every place is bounded. Then, workflow s boundedness verificaion corresponds o verify if here is a place ha can have an infinie number of okens. To realize he boundedness verificaion of a TCWF-ne, an exended ne PN = ( P, T, F ) [] is defined. PN is he PN obained by adding an exra ransiion which connecs place θ and ε o a TCWF-ne PN, where P =P, T =T {}, and F =F {(θ,),(,ε)}. Based on he TCWF-ne mapped from he DNG-based workflow model, he exended free-choice TCWF-ne is obained by adding a new ransiion beween he inial and end places, i.e., ={θ}, ={ε}. We know each workflow model describes he life-cycles of several kinds of ransacion insances. Corresponding o each kind of ransacion insances, here is a rouing pah in a workflow model. Because he original DNG-based workflow model buil by he basic process modeling language is srucurally acyclic, here is no ieraion srucure in he TCWF-ne obained hrough he model ransformaion. Heurisically, we decompose an exended free-choice TCWF-ne specifying a workflow model o a se of T-componens represening he rouing pah of each kind of ransacion insance. Based on he resuling se of T-componens, a boundedness verificaion mehod of he corresponding TCWF-ne can be deduced. Noe ha he algorihm discussed below focuses only on a TCWF-ne s srucure decomposiion, he corresponding ime informaion of every ransiion or place in is subnes is unchanged. Also, for simpliciy, we use PN= (P, T, F) o represen an exended TCWF-ne. Before he decomposiion algorihm is inroduced, some relevan conceps are given below. Definiion 6: An elemenary pah in PN= (P, T, F), called a pah for shor, is (x, x,, x k ) such ha arc (x i, x i+ ) exiss i k-, and x i =x j implies i=j, i, j k where x i P T. I is called an (elemenary) circui if x i =x j, i, j k implies i= and j=k. PN = (P, T, F ) is a subse of PN= (P, T, F), pah= { 0, p, p m, m } in PN is a ransiion pah of PN iff: I is a pah 0, m T 7

18 Vol.33, No., March 03, pp p j P, j m, and j T, j < m Symmerically, pah= {p 0,, p m, p m } in PN is a place pah of PN iff: I is a pah p 0, p m P j T, j m, and p j P, j < m A ransiion pah defined here has he same semanic as a nice pah in [4]. Definiion 7: A place p in PN = (P, T, F) is a choice place iff p. Suppose ha a pah or circui PN = (P, T, F ) is a subne of PN = (P, T, F) and p' is he firs or source place. Given a choice place p P, is choice degree is defined as he number of choice places in he pah from place p' o p in PN. s s s s s A p A A p A0 p A p A0 p A0 p A0 p A p A0 p A p A p A3 A0 A A p B A3 A0 A0 A0 A A0 A A4 B A4 A4 A4 A4 A5 A5 A5 A5 A5 p C p C p C p C p C C C C C C C p A6 p A7 p A6 p A7 p A6 p A7 A6 A7 A6 A7 A6 A7 p D p D p D p D p D D D D D D p E p E p E p E p E E E E E E (a) (b) (c) (d) (e) 8

19 Vol.33, No., March 03, pp s s s s A p A p A A A p A p A A p A0 A0 p A A p A A p B p A0 A0 p A A p B p A3 A3 p A0 A0 p A A p A A p B p A0 A0 p A A p B p A3 A3 A4 A4 A4 B A4 B B B A5 A5 A5 A5 p C p C p C p C C C C C p A6 p A6 p A7 p A7 A6 A6 A7 A7 p D p D p D p D p E D p E D p E D p E D E E E E (f) (g) (h) (i) Figure 5. The exended TCWF-ne and is decomposiion resul Definiion 8: Suppose ha R is a circui (pah) se. A choice place p is proper iff circui (pah) in R, p is no on i or p s choice degree is he leas. Le s use an example o explain Definiions 6-8. PN = (P, T, F ) in Fig. 5(b) is a circui subne of PN= (P, T, F) in Fig. 5(a). For pah { s, p A, A,, A4 } in PN, s, A4 T and is oher elemens do no belong o PN. Thus i is a ransiion pah of PN. However, because p A4 and A5 belong o PN, pah { s, p A, A,, A4,, A5 } is no a ransiion pah of PN. Similarly, {p C, C, p A7, A7, p D } is a place pah of PN. 9

20 Vol.33, No., March 03, pp s s s s A p A p A A A p A p A A p A p A3 p A p A3 A A A3 A A A3 p B p B p B p B B B B B A5 A5 A5 A5 p C p C p C p C C C C C p A7 p A7 p A6 p A6 A7 A7 A6 A6 p D p D p D p D D D D D p E p E p E p E E E E E (a) (b) (c) (d) Figure 6. A differen decomposing process of he TCWF-ne in Fig. 5(a) If a oken reaches a choice place, i mus choose an oupu pah (ransiion). In Fig. 5(a), p A and p C are wo choice places. To decide heir choice degree, le s consider circui PN in Fig.5(b). Since only p C in he pah from ε o p C is a choice place, p C s choice degree in PN is. Now consider PN = (P, T, F ) of Fig. 6(a). Since here are wo choice places (p A and p C ) in he pah from ε o p C, p C s choice degree in PN becomes. Suppose ha he circui in Fig. 5(b) is he only elemen of R. Then p C is a proper choice place. However, if he circui of Fig. 6(a) is also in R, he proper choice place is no p C bu p A. The reason is ha alhough p C s choice degree is he minimal in he circui of Fig. 5(b), i is no he minimal in he circui of Fig. 6(a). Since p A belongs o only he circui of Fig. 6(a), and is choice degree is he minimal in ha circui, i is a proper choice place in R. Based on hese definiions, we discuss in deail abou he workflow model decomposiion. I has been proved in [4] ha each live and safe free-choice PN is covered by a se of srongly conneced T-componens. Also, an algorihm (Algorihm 7.4 in [4]) is defined o consruc a T-componen from an arbirarily seleced ransiion. Obviously, each T-componen of a TCWF-ne is corresponding o he rouing pah of a specific ransacion insance. As menioned before, he resuling exended TCWF-ne considered in his paper is sound. Hence, one can 0

21 Vol.33, No., March 03, pp assure is liveness and safeness. Nex, we exend he algorihm in [4] o decompose a sound free-choice exended TCWF-ne PN= (P, T, F) ino a se of T componen as follows. Algorihm (Decomposiion) Sep : Corresponding o an arbirarily seleced ransiion in p ={,,, n }, where p P C ={p p }, an circui PN = (P, T, F ) passing is consruced. In addiion, P S = φ and R = {PN }, where P S is a choice place se and R is a subne se. Sep : Repea he following seps unil PN j =(P j, T j, F j ) R, Ψ j ={p p P j p p P S } is empy. Denoe he resuling subne se as R=R ={PN, PN,, PN n }..: R = R, PL=φ;.: For every PN j = (P j, T j, F j ) R, if Ψ j ={p p P j p p P S } is nonempy, choose he choice place p Ψ j whose choice degree is he leas one in Ψ j and PL= PL {p};.3: For every place p k PL, if p k generaed from Ψ k of PN k R belongs o Ψ m (m k) of PN m R and he choice degree of p k is no he leas one in Ψ m, hen PL= PL\{p k };.4: P S = P S PL, and for every place p l PL do For every PN j = (P j, T j, F j ) R, if p l P j, here mus be a nonempy ransiion se η={ p l T j }. Then for every ransiion in η, an arbirary circui PN ={P, T, F } which goes down from source ε o p l along PN j and hen passes is consruced and added o R ; Sep3: For every PN k ={P k, T k, F k } R, repea he following exhausively: If here is T k wih a nonempy se {p p p P k }, hen for every p \P k, an arbirary ransiion pah pah = { 0, p, p m, m } is consruced and merged ino PN k, where = 0, m T k. However, if here are choice places in he merged pah for he new PN k, R ={PN k }, he following seps are repeaed unil PN j = (P j, T j, F j ) R, he se Ψ j ={p p p P j p P S } is empy. Then he resul R is merged ino R. 3.: R = R, PL =φ; 3.: For every PN j = (P j, T j, F j ) R, if Ψ j ={p p P j p p P S } is nonempy, each p mus belong o a pah from 0 o m. Choose he choice place p Ψ j whose choice degree in he pah from 0 o m is he leas one in Ψ j, and le PL = PL {p }; 3.3: For every place p r PL, if p r generaed from Ψ r of PN r R belongs o Ψ s (r s) of PN s R and he choice degree of p r is no he leas one in he pah from 0 o m

22 Vol.33, No., March 03, pp (inψ s ), hen PL = PL \{p r }; 3.4: P S = P S PL, and for every place p l PL do For every PN j =(P j, T j, F j ) R, if p l P j, a nonempy pos ransiion se η ={ p l (in PN) T j } exiss. Then for every ransiion η, here mus be a place pah pah={p l,, p } in which only p l and p belong o he ransiion pah from 0 o m in PN j. Use pah o replace he corresponding par from p l o p of he ransiion pah from 0 o m in PN j o obain and add a new subse PN o R ; In Algorihm, Sep is used o consruc a circui PN (subne of PN) passing one arbirary oupu ransiion of an arbirary choice place in PN. Obviously, i mus conain source and sink places. For example, Fig. 5(b) and Fig. 6(a) can be viewed as wo circuis consruced respecively from choice place p C s oupu ransiion and choice place p A s oupu ransiion A. The choice place se P C conains all he choice places in PN. In he TCWF-ne of Fig. 5(a), P C ={p A, p C }. P S represens he choice place se in which all he choice places have been invesigaed and seleced o propagae new subnes. Sep finds all he circuis generaed from PN. In each ieraion of Sep, some proper choice places of P C \P S are invesigaed and seleced o be merged ino P S. Seps. and.3 selec all he proper choice places in P C \P S ino PL. Nex, for each circui (of curren R ) in which one (only one) proper choice place p PL is locaed, Sep.4 propagaes p - (p represens p s oupu ransiions se in PN) circuis corresponding o p - new place pahs of p and merges hem ino R (R of nex ieraion). Obviously, each choice place can only be seleced o be a proper choice once, which guaranees ha each arbirarily consruced circui in Sep.4 is new for R. During each ieraion of Sep, all he proper choice places of P C \P S in curren R are merged ino P S. They are pu ino P S in an ascending sequence of heir choice degrees. Then, when a choice place p is pu ino P S, he choice places wih less choice degree in PN j mus have been in P S and p - circuis are creaed for each circui PN j conaining p. Therefore, if p carries he maximal choice degree in every PN j R, hen all he circuis passing p and is preceding choice places are included in R. Oherwise, Sep will coninue is ieraion. Based on Algorihm 7.4 [4] o grow a T-componen from a given single ransiion, Sep 3 exends every subne in R from is fork ransiions (each of which has muliple oupus in PN), and obains he corresponding T-componens. Is validiy is demonsraed in [4]. However, i is possible ha he new merged ransiion pahs in Sep 3 include new choice places ha do no belong o P S. In his siuaion, here mus be new subnes ha are derived from he choice places in he new merged ransiion pahs of PN j. Then Seps (he proper choice place is in pah) C

23 Vol.33, No., March 03, pp similar o Seps.-.4 (he proper choice place is in circui) are needed o consruc and merge all hese new subnes ino R. Panagos and Rabinovich [4] demonsraed he exisence of a ransiion pah pah corresponding o place p in Sep 3. In fac, according o he characerisics of a sound free-choice exended TCWF-ne (every closed loop mus cover source place ε and sink place θ), every pah saring from PN i can reurn o PN i and he join node mus be a ransiion. Oherwise, here mus be a join place and i is no safe, hen he TCWF-ne is no sound. Symmerically, i can also be proved in Sep 3.4 (Sep.4 is similar o Sep 3.4) ha he place pah saring from p l mus reurn o anoher place p in pah. The reason is ha if a join place is no in pah, hen he reurn node of pah o PN i is a place or a ransiion disinc from m. These wo siuaions will desroy he soundness of an exended TCWF-ne. If he reurn node is in pah bu no a place, i also conflics wih he soundness of he TCWF-ne. Hence, he conclusion of he exisence of he place pah pah is derived. In his algorihm, each place in every ransiion generaing ne is reaed only once, i.e., eiher in a circui or ransiion pah. Therefore, we can assure ha each place has exacly one inpu and one oupu arc, and each elemen in he resuling se R is a marked graph, hen a T-componen. Algorihm 7.4 in [4] focuses on how o generae a T-componen from an arbirarily seleced ransiion. Algorihm demonsraes how o obain all he T-componens conained in a TCWF-ne. From he analysis menioned above we can draw he conclusion: The algorihm can decompose every free-choice exended TCWF-ne PN={P, T, F} o R, a se of all T-componens ha can be derived from PN. Obviously, each T componen ne in R corresponds o a processing procedure of a kind of ransacion insance, i.e., a rouing pah of a specific ransacion insance in a workflow model. Obviously, he wors case is ha he number of possible T componens could grow exponenially as he number of choice places increases in a TCWF-ne. Then he decomposiion algorihm mus be of exponenial ime complexiy. However, he sum of he ieraion number in Seps and 3 is a mos P C, hen he algorihm can erminae in a polynomial number of arihmeic operaions. On he oher hand, Seps and 3 in our brue force algorihm can be carried ou concurrenly, which means each circui generaed in Sep can be processed by Sep 3 immediaely. If muli-processors are used, his propery can improve he ime performance of he algorihm. Now we demonsrae how o verify he boundedness of a TCWF-ne PN specifying a workflow process based on he resuling T-componens. Le s suppose ha PN including k ransiions is mapped from a workflow model buil for he processing of n kinds of ransacion insances {I, I,, I n }, and each kind of ransacion insance I i accouns for a cerain proporion 3