Declarative approach to cyclic steady state space refinement: periodic process scheduling

Transcription

1 It J Adv Mauf Techl DOI /s ORIGINAL ARTICLE Declarative apprach t cyclic steady state space refiemet: peridic prcess schedulig Grzegrz Bcewicz Zbigiew A. Baaszak Received: 16 April 2012 / Accepted: 13 Octber 2012 # The Authr(s) This article is published with pe access at Sprigerlik.cm Abstract Prblems f cyclic schedulig are usually bserved i flexible maufacturig systems which prduce multitype parts where the autmated guided vehicle system plays the rle f a material hadlig system, as well as i varius ther multimdal trasprtati systems where gds ad/r passeger itierary plaig plays a pivtal rle. The schedulability aalysis f the prcesses executed i the s-called systems f ccurret cyclic prcesses (SCCPs) ca be executed withi a declarative mdelig framewrk. Csequetly, the csidered SCCP schedulig prblem ca be see as a cstrait satisfacti prblem. Such a represetati prvides a uified way fr evaluatig the perfrmace f lcal cyclic prcesses as well as f multimdal prcesses supprted by them. Here, the crucial issue is that f a ctrl prcedure (e.g., a set f dispatchig rules), which wuld guaratee the cyclic behavir f the SCCP. I this ctext, we discuss the sufficiet cditis guarateeig the schedulability f bth lcal ad multimdal cyclic prcesses, ad we prpse a recursive apprach i desigig them. Keywrds Cyclic prcesses. Declarative mdelig. Cstrait prgrammig. State space. Peridicity. Dispatchig rules 1 Itrducti Operatis i cyclic prcesses are executed i sequeces that repeat a idefiite umber f times. I everyday G. Bcewicz (*) Departmet f Electrics ad Cmputer Sciece, Kszali Uiversity f Techlgy, Kszali, Plad bcewicz@ie.tu.kszali.pl Z. A. Baaszak Departmet f Busiess Ifrmatics, Warsaw Uiversity f Techlgy, Warsaw, Plad Z.Baaszak@wz.pw.edu.pl practice, these prcesses arise i differet applicati dmais (such as maufacturig, time sharig f prcessrs i embedded systems, digital sigal prcessig, ad i cmpilers where schedulig lp peratis fr parallel r pipelied architectures takes place) as well as i service dmais, cverig such areas as wrkfrce schedulig (e.g., shift schedulig, crew schedulig), timetablig (e.g., trai timetablig, aircraft rutig ad schedulig), ad reservatis (e.g., reservatis with r withut slack, assigig classes t rms) [7, 10, 11, 14, 15, 19, 21, 23, 24]. Such cyclic schedulig prblems belg t the class f decisi prblems, i.e., prblems aimed at fidig whether r t there exists a sluti satisfyig certai assumed cditis [21]. Mrever, because f their iteger dmais, the csidered prblems belg t a class f Diphatie prblems [9, 18]. This meas that certai classes f cyclic schedulig prblems ca be see as udecidable [1]. Therefre, takig it accut the udecidability f Diphatie prblems, t all the behavirs (icludig cyclic behavirs, i.e., thse that belg t a space f cyclic steady states) are reachable uder the cstraits impsed by the structure f the system. This is als the case with the system behavir that ca be achieved i systems pssessig specific structural cstraits. This meas that sice the system s cstraits determie its behavir, bth the system structure cfigurati ad the desired cyclic schedule must be csidered simultaeusly. Thus, fr the sluti f a cyclic schedulig prblem, the cfigurati f a system structure must be determied i rder t eable schedulig f the prcesses; hwever, the schedulig must be perfrmed s as t devise the system cfigurati. I this ctext, this wrk discusses certai slvability issues ccerig the prblems f dispatchig cyclic prcesses, i particular the cditis guarateeig the slvability f the schedulig f cyclic prcesses. Their examiati may replace exhaustive ad time-csumig searchig fr the slutis satisfyig the required system fuctiig.

2 It J Adv Mauf Techl A umber f mdels ad methds have bee prpsed t slve the cyclic schedulig prblem [13]. Amg them, the mst frequetly applied are the mathematical prgrammig appraches (usually iteger prgrammig r mixed iteger prgrammig [24]), max-plus algebra [16], cstrait lgic prgrammig [2 5, 22], swarm ad evlutiary algrithms [6, 10, 20], ad Petri ets [19]. The majrity f these are rieted at fidig a miimum cycle r maximum thrughput while assumig a deadlck-free flw f prcesses. The appraches aimed at estimatig the cycle time based the cyclic prcess structure ad the emplyed sychrizati mechaism (i.e., redezvus r mutual exclusi istaces) while takig it accut deadlck avidace cstraits are rather uique [8, 22, 25]. I this ctext, apart frm the abve-metied slvability cditis ccerig pririty dispatchig rules guarateeig the reachability f a cyclic steady state f a system, ur mai ctributi csists i prpsig a ew mdelig framewrk fr evaluatig the cyclic steady state f a give system f ccurret cyclic prcesses (SCCP). This paper is ccered with the fllwig questis: ca the assumed system behavir be achieved uder the give cstraits the structure f the system? Ca we fid such a system structure fr which the assumed system behavir ca be achieved? Therefre, the aim f this paper is t prvide the cditis useful fr geeratig cyclic steady states i a system csistig f ccurretly iteractig cyclic prcesses, where mutual exclusi prtcls prvide a sychrizati mechaism which ctrls the access f the prcesses t shared (cmm) resurces. This bjective ecmpasses a rather large class f digital ad/r lgistic etwrks that have cmm prperties, eve thugh their itrisic differeces are sigificat. The mst imprtat prperty is related t differet subetwrk ifrastructures which eable schedulig multimdal prcesses executed thrugh cmm shared sectis f differet lcal etwrks [3]. Fr istace, i the case f a metr etwrk, the itierary f a give passeger, assumig lie chages, is a example f a multimdal prcess, where itierary plaig ca be viewed as a relevat multimdal prcess schedulig. Csequetly, this study aims t preset a declarative apprach t the reachability prblem that ca be emplyed by decisi makers i rder t geerate, aalyze, ad evaluate cyclic steady states reachable i a give SCCP structure. By emplyig the framewrk f the cstrait satisfacti prblem (CSP) [5], we state ur mai prblem regardig the dispatchig rules resultig i the SCCP cyclic schedules f the assumed cycle time ad the Ξ-peridicity f lcal prcesses. A illustrative mdel f the cstrait satisfacti prblem implemeted i the Oz/Mzart laguage is discussed frm the perspective f multimdal prcesses. This ctributi ca be see as a ctiuati f ur frmer wrk [3 5]. Hece, ur apprach ca be viewed as a extesi f the ccept f cstraits sufficiet fr the cyclic behavir f lcal ad multimdal prcesses [5], as well as a extesi f the ccept f the state space itrduced i [3, 4]. This meas that the cditis itrduced earlier which guaratee the cyclic behavir f a SCCP ad the geerati f its state space prvide a frmal framewrk fr the develpmet f reachability cditis guarateeig the geerati f cyclic steady states, i.e., the cditis which allw t distiguish betwee the iitial states ad the cyclic steady states i bth lcal ad multimdal prcesses. The rest f the paper is rgaized as fllws: Secti 2 itrduces systems f ccurretly flwig lcal cyclic ad multimdal prcesses, describes the emplyed tati, ad states the prblem. I Secti 3, we discuss certai issues ccerig the geerati f feasible state space as well as prvide tw methds aimed at refiig cyclic steady states. Tw cases illustratig the implemetati f bth methds are discussed i Secti 4, whereas Secti 5 presets ur cclusis. 2 Systems f ccurret cyclic prcesses 2.1 Declarative mdelig A autmated guided vehicle system (AGVS) with distiguishable vehicles, pick-up/delivery pits (PDPs), ad trasprtati rutes is preseted i Fig. 1. Such a system ca be mdeled i terms f SCCPs, where cyclic multimdal prcesses (represetig trasprtati rutes) are executed alg the parts f cyclic lcal prcesses (represeted by the itieraries f vehicles) which are itercected with each ther thrugh cmm resurces f the AGVS (i.e., PDPs). Figure 2 presets the SCCP mdel frm Fig. 1. Six lcal cyclic prcesses are csidered, viz., P 1,P 2, P 3, P 4, P 5, ad P 6. The prcesses fllw the rutes cmpsed f trasprtati sectrs (distiguished i Fig. 2 by the set f resurces R ={R 1,, R c,, R 17 }, where R c is the cth resurce). Sme f the lcal cyclic prcesses are pipelie flw prcesses, i.e., they ctai streams (represetig vehicles frm Fig. 1) f the prcesses fllwig the same rute while ccupyig differet resurces (sectrs). Fr istace, prcesses P 4 ad P 5 ctai tw streams: P 4 ¼ P4 1; P2 4,adP5 ¼ P5 1; P2 5, respectively, i.e., the prcesses (vehicles) mvig alg the same rute. The remaiig lcal prcesses ctai uique streams: P 1 ¼ P1 1, P2 ¼ P2 1, P 3 ¼ P3 1,adP6 ¼ P6 1. I ther wrds, the streams P1 1, P1 2, P1 3, P1 4, P2 4, P1 5, P2 5,adP1 6 represet the eight vehicles frm Fig. 1. The kth stream f the ith lcal prcess P i is deted as Pi k.

3 It J Adv Mauf Techl Fig. 1 Example f a AGVS Leged: - pick-up/delivery pits (PDPs) - Autmated Guided Vehicle (AGV) -prducti rute f the pallet - trasprtati sectr Apart frm lcal prcesses, we csider tw multimdal prcesses (i.e., prcesses executed alg the rutes csistig f parts f the rutes f lcal prcesses): mp 1 ad mp 2. Fr example, the displacemet f the pallet alg the prducti rute depicted by the dashed lie crrespds t the multimdal prcess mp 1 supprted by AGVs, which i tur ecmpass lcal trasprtati streams, P1 1 ad P1 3, ad the als e f tw pipelie-like flwig streams: P5 1 ad P5 2. This meas that the prducti/trasprtati rute specifyig hw a multimdal prcess is executed ca be see as cmpsed f parts f the rutes f lcal cyclic prcesses. Prcesses ca iteract with each ther thrugh shared resurces, i.e., the trasprtati sectrs. The rutes f the csidered lcal prcesses (streams) are as fllws: p 1 1 ¼ ð R 1; R 2 ; R 3 ; R 4 Þ; p 1 2 ¼ ð R 4; R 5 ; R 6 Þ; p 1 3 ¼ ð R 3; R 9 ; R 8 ; R 7 ; R 5 Þ; p 1 4 ¼ p 2 4 ¼ ðr 2 ; R 16 ; R 10 ; R 11 ; R 12 ; R 9 Þ; p 1 5 ¼ p2 5 ¼ ð R 8; R 17 ; R 13 ; R 14 ; R 12 Þ; p 1 6 ¼ ð R 14; R 11 ; R 15 Þ; where R 2 R 5, R 8 R 14,R 16, ad R 17 are the shared resurces, sice each f them is used by at least tw streams, ad R 1, R 6,R 7, adr 15 are the -shared resurces, because they are exclusively used by e stream ly. I geeral, the rute p k i is the sequece f resurces used i rder t execute the Fig. 2 Example f FMS SCCP mdel f a AGVS Leged: - -th resurce f set - uccupied resurce: - resurce ccupied by the stream ad ctrlled by the pririty dispatchig rule ; ( - a set f streams)

4 It J Adv Mauf Techl peratis f the stream Pi k. Nte that the streams p1 4 ad p2 4 which belg t P 4 ad the streams p 1 5 ad p2 5 which belg t P 5 fllw the same rute (these streams crrespd t vehicles mvig alg the same rute). Csider tw cyclic multimdal prcesses mp 1 ad mp 2 which fllw the rutes mp 1 ad mp 2, respectively (see Fig. 2): mp 1 ¼ ðr 1 ; R 2 ; R 3 ; R 9 ; R 8 ; R 17 ; R 13 Þ; mp 2 ¼ ðr 15 ; R 14 ; R 12 ; R 8 ; R 7 ; R 5 ; R 6 Þ: Let us assume that mp 1 ad mp 2 ca be see as fllws: mp 1 ¼ ððr 1 ; R 2 ; R 3 Þ; ðr 9 ; R 8 Þ; ðr 17 ; R 13 ÞÞ; mp 2 ¼ ððr 15 ; R 14 Þ; ðr 12 ; R 8 Þ; ðr 7 ; R 5 Þ; ðr 6 ÞÞ; where: (R 1, R 2, R 3 ), (R 9, R 8 ), (R 17, R 13 ) the parts (subsequeces) f rutes p 1 1, p1 3 ad p1 5 icluded i mp 1 (R 15, R 14 ), (R 12, R 8 ), (R 7, R 5 ), (R 6 ) the parts (subsequeces) f rutes p 1 6, p1 5, p1 3, ad p1 2 icluded i mp 2. Let us assume that multimdal prcesses d t ctai subprcesses, i.e., that each multimdal prcess csists f a uique stream. The class f the csidered SCCPs fllws the cstraits stated belw [5]: A ew perati may start a resurce ly if the curret perati has bee cmpleted ad the resurce has bee released, Lcal prcesses share cmm resurces i the mutual exclusi mde, the perati f a lcal prcess ca ly be suspeded if the ecessary resurce is ccupied, suspeded lcal prcesses cat be released, ad lcal prcesses are -preemptible, i.e., a resurce may t be take frm a prcess as lg as it is used by it, Multimdal prcesses ecmpassig pallet flw cveyed by AGVs fllw lcal trasprtati rutes, ad differet multimdal prcesses ca be executed simultaeusly alg the same lcal prcess, Lcal ad multimdal prcesses execute cyclically with perids Tc ad Tm, respectively; resurces ccur uiquely i each trasprtati rute, I a cyclic steady state, each ith stream must cver its lcal rute the same umber f times Ξ ψ i ; the factrs Ξ ad ψ i are defied belw. A resurce cflict (caused by the applicati f the mutual exclusi prtcl) is reslved with the aid f a pririty dispatchig rule [2], which determies the rder i which streams access shared resurces. Fr istace, i the case f the resurce R 9 ; σ 9 ¼ P3 1; P1 4 ; P2 4, the pririty dispatchig rule determies the rder i which streams ca access the shared resurce R 9, i.e., the resurce is first give t stream P3 1, the t the stream P1 4, the t P2 4 ad the ce agai t P3 1, ad s. The stream Pk i ccurs the same umber f times i each dispatchig rule assciated with the resurces featurig i its rute. Therefre, the SCCP shw i Fig. 2 is specified by the fllwig set f dispatchig rules: Θ ¼ fσ 1 ;...; σ 17 g, as well as f 1 P1 1 ¼ f2 P1 1 ¼ f3 P1 1 ¼ f4 P1 1 ¼ 1, f4 P2 1 ¼ f5 P2 1 ¼ f6 P2 1 ¼ 3, etc., where f c Pi k is the umber f ccurreces f P k i i the cth pririty dispatchig rule. This meas that durig the same perid, the stream P2 1 is repeated three times, while P1 1 ly ce. Thus, the pririty rules determie the frequecies f the mutual appearace f lcal prcesses sharig the same resurce. I geeral, the set f dispatchig rules Θ determies the sequece f relative frequecies f mutual executis f lcal prcesses, ad is deted by Ψ ¼ ðy 1 ; y 2 ;...; y Þ, where y i 2 N, y i ¼jj d js c;d ¼ Pi 1 ; d 2 f 1;...; lpðcþ g jj; 8i 2 f1;...; g; 8σc 2 Θ i ; ð1þ where Θ i is the set f dispatchig rules assciated with the resurces featurig i the rute fllwed by P i, s c,d is the dth elemet i the sequece σ c =(s c,1,, s c,d,, s c,lp(c) ), is the umber f prcesses, ad lp(c) is the legth f σ c. Therefre, the SCCP shw i Fig. 2 is specified by the sequece: Ψ = (1,3,1,1,1,2). This meas that e executi f lcal prcesses P 1,P 3, P 4, ad P 5 (ad their assciated streams) crrespds t three executis f prcess P 2 ad tw executis f P 6. Sice the sequece Ψ f relative frequecies f mutual executis f lcal prcesses des t ecessary ecmpass a cyclic steady state f the SCCP, we itrduced a ew parameter describig the umber f ccurreces f Ψ i a cyclic steady state, deted by Ξ 2 N. Ξ ccurreces f Ψ i e perid is termed Ξ-peridicity. Fr the csidered SCCP, Ξ=2 implies that tw executis f the sequece Ψ = (1,3,1,1,1,2), i.e., tw executis f lcal prcesses P 1,P 3, P 4, ad P 5 crrespd t six executis f the prcess P 2 ad fur executis f P 6. We ca defie the mutual frequecy mψ f multimdal prcesses mp i (e.g., i the case f the executi f multimdal prcesses (see the SCCP i Fig. 2), mψ = (1,1) implies that e executi f the prcess mp 1 crrespds t e executi f mp 2 ). We ca similarly defie mξ, which determies the umber f executis f mψ i a cycle. I geeral, the fllwig tati is used: Asequece p k i ¼ p k i;1 ; pk i;2 ;...; pk i;j ;...; pk i;lrðiþ specifies the rute f the stream f a lcal prcess Pi k (the kth stream f the ith lcal prcess P i ). Its cmpets defie the resurces used i the executi f peratis, where

5 It J Adv Mauf Techl p k i;j 2 R (the set f resurces R ¼ R f 1; R 2 ;...; R c ;...; R m g) detes the resurces used by thekth stream f the ith lcal prcess i the jth perati. I the rest f the paper, the jth perati executed the resurce p k i;j i the stream Pk i will be deted by k i;j ; lr(i) is the legth f the cyclic prcess rute (all streams f P i are f the same legth). Fr example, the rute p 1 2 ¼ ð R 4; R 5 ; R 6 Þf the stream P2 1 (Fig. 2)is the sequece p 1 2 ¼ p1 1;1 ; p1 1;2 ; p1 1;3, where the first elemet p 1 1;1 is equal t R 4, whereas the secd, p 1 1;2 ¼ R 5,ad the third, p 1 1;3 ¼ R 6. x k i;j;q ðlþ 2N is the mmet whe the perati k i;j begis its qthexecuti i the lth cycle. ti k ¼ ti;1 k ; tk i;2 ;...; tk i;j ;...; tk i;lrðiþ specifies the perati times f lcal prcesses, where ti;j k detes the time f executi f perati k i;j (fr the SCCP i Fig. 2, see Table 1). mp i ¼ mpr k 1 i 1 ða i1 ; b i1 Þ; mpr k 2 i 2 ða i2 ; b i2 Þ;...; mpr ky i y a iy ; b iy specifies the rute f the multimdal prcess mp i, where 8 < p k mpri k i;a ða; bþ ¼ ; pk i;aþ1 ;...; pk i;b a b p k i;a ; pk i;aþ1...; pk i;lrðiþ ; pk i;1 ;...; pk i;b1 ; ; : pk b > a i;b a; b 2 f1;...; lrðiþg; is the subsequece f the rute p k i ¼ p k i;1 ; p k i;2 ;...; p k i;j ;...; pk i;lrðiþ Þ ctaiig elemets frm pk i;a t pk i;b. I ther wrds, the trasprtati rute mp i is a sequece f parts f rutes f lcal prcesses. Fr istace, the rute fllwed by mp 1 (see Fig. 2) isas fllws: mp 1 =((R 1, R 2, R 3 ), (R 9, R 8 ), (R 17, R 13 )), where mpr1 1ð1; 3Þ ¼ ð R 1; R 2 ; R 3 Þ, mpr3 1ð2; 3Þ ¼ ð R 9; R 8 Þ,adm pr5 1ð2; 3Þ ¼ ð R 17; R 13 Þ. I the rest f the paper, the jth perati executed i the prcess mp i will be deted by m i,j. mx i;j;k ðlþ 2N is the mmet whe the perati m i,j begis its kth executi i the lth cycle. Table 1 Lcal perati times f SCCPs (Fig. 2) Streams i,k t k i;1 t k i;2 t k i;3 t k i;4 t k i;5 t k i;6 P 1 1 1, P 1 2 2, P 1 3 3, P 1 4 4, P 2 4 4, P 1 5 5, P 2 5 5, P 1 6 6, Θ ¼ fσ 1 ; σ 2 ;...; σ c ;...; σ m g is the set f pririty dispatchig rules, where σ c =(s c,1,, s c,d,, s c,lp(c) ) are sequece cmpets which determie the rder i which the prcesses ca be executed the resurce R c, where s c,d H (H is the set f streams,.e.g., fr Fig. 2, H ¼ P1 1; P 2 1 ; P 1 3 ; P 1 4 ; P 2 4 ; P 1 5 ; P ; P 6 ). Usig the abve tati, a SCCP ca be defied as a tuple: SC ¼ ððr; SLÞ; SMÞ; ð2þ where: R ={R 1, R 2,, R c,, R m } the set f resurces, where m is the umber f resurces SL =(ST L, BE L ) the structure f lcal prcesses, i.e. ST L =(U, T) the variables describig the layut f lcal prcesses U ¼ p 1 1 ;...; plsð1þ 1 ;...; p 1 ;...; plsðþ the set f rutes f lcal prcess where ls(i) is the umber f streams belgig t the prcess P i ad is the umber f lcal prcesses T ¼ t1 1 ;...; tlsð1þ 1 ;...; t 1 ;...; tlsðþ the set f sequeces f perati times i lcal prcesses. BE L ¼ ðθ; < ; ΞÞ the variables describig the behavir f lcal prcesses Θ ¼ fσ 1 ; σ 2 ;...; σ c ;...; σ m g the set f pririty dispatchig rules < ¼ ðy 1 ; y 2 ;...; y i ;...; y Þ the sequece f relative frequecies f mutual executis f lcal prcesses Ξ the umber f ccurreces f Ψ i a cyclic steady state. SM =(ST M, BE M ) the structure f multimdal prcesses, i.e. ST M =(M, T) the variables describig the layut f the level f a multimdal prcess, where M ={mp 1,, mp i,, mp w } is the set f rutes f a multimdal prcess ad w is the umber f multimdal prcesses mp i mt ={mt 1 mt w } the set f sequeces f perati times i multimdal prcesses. BE M =(mψ,mξ) the variables describig the behavir f multimdal prcesses m< ¼ ðmy 1 ; my 2 ;...; my i ;...; my w Þ the sequece f relative frequecies f mutual executis f multimdal prcesses mξ the umber f ccurreces f mψ i a cyclic steady state. The SCCP mdel (2) ca be see as a multilevel mdel, cf. Fig. 3, i.e., a mdel cmpsed f a R level

6 It J Adv Mauf Techl Fig. 3 Multilayered mdel f the behavir f a SCCP (see the SCCP i Fig. 2) A ptetial higher level level level The level f multimdal prcesses : SL level The level f lcal prcesses : Rlevel the set f resurces (resurces), a SL level (lcal cyclic prcesses), ad a SM 1 level (multimdal cyclic prcesses), as well as a SM i level (the ith meta-multimdal prcess). The SL level determies the structure f trasprtati rutes f lcal prcesses U, as well as the parameters Θ,Ψ, ad Ξ which specify the required behavir f the system. I tur, the SM 1 level takes it accut multimdal, prcesses, as well as meta-multimdal prcesses (SM 2 level) cmpsed f multimdal prcesses frm the SM 1 level. I ther wrds, we assume that the variables describig SM i are the same as i the case f SM, whereas the rutes f the multimdal prcess f the ith level remai cmpsed f the prcesses frm the (i 1)th level. The preseted mdel is a exteded versi f a simplified mdel limited t R ad SL levels, which is itrduced i [3, 5]. Therefre, i geeral, the SC ¼ ððr; SLÞ; SMÞmdel ca be see as cmpsed f i levels: SC i ¼ ðr; SLÞ; SM 1 ; SM 2 ;... ; SM i : ð3þ Nte that the cyclic behavir f SC i 1 implies the peridic behavir f SC i. 2.2 Prblem frmulati Let us csider a SCCP specified by a give set R f resurces, dispatchig rules Θ, the rutes f lcal ad multimdal prcesses U ad M, respectively, ad a iitial allcati f prcesses. The mst imprtat issue here is the peridicity f the SCCP, i.e., des there exist a cyclic executi f lcal prcesses? Ad if s, what is the perid Tc? Further questis are ccered with cyclic executi f multimdal prcesses. I rder t fid the aswers t the abve questis, mre detailed questis must be aswered first. Which allcatis f iitial prcesses are admissible (i.e., which AGV dckigs are pssible)? Which dispatchig rules Θ assure the peridicity f a give SCCP (i the lcal ad multimdal sese), while retaiig the assumed frequecy (Ψ, mψ) f the executi f prcesses withi the glbal perid (lcal Tc ad multimdal Tm)? I geeral, hwever, apart frm the abve frmulatis f frward prblems, iverse prblems ca be csidered as well. Fr istace, des there exist a SCCP structure f a lcal (SL) ad/r multimdal layer (SM), such that the assumed steady cyclic state ca be achieved? I the rest f the paper, the fllwig frward prblem is csidered: Give a SCCP defied by SC (2), i.e., R, SL =(ST L, BE L ), SM =(ST M, BE M ), is it pssible t attai cyclic behavir f SC (i.e., the ecmpassig cyclic steady states f lcal ad multimdal prcesses)? I ther wrds, assumig the declarative apprach, we are searchig fr the CSP prvidig a frmal framewrk aimed at prttypig dispatchig rules fr the csidered SCCP.

7 It J Adv Mauf Techl 3 Refiemet f the space f cyclic steady states 3.1 Geerati f state space Let us csider the fllwig defiiti f a SCCP state, which describes the allcati f bth lcal ad multimdal prcesses: S k ¼ Sl r ; MA k ; ð4þ where: Sl r the rth state f lcal prcesses, crrespdig t the kth state f multimdal prcesses, Sl r ¼ ða r ; Z r ; Q r Þ; ð5þ where: A r ¼ a r 1 ; a r 2 ;...; a r r ð c ;...; a m Þ allcati f prcesses i the rth state, i which a r c 2 P [ fδg; a r c ¼ Pi k, the cth resurce R c, is ccupied by the lcal stream Pi k ad a r c ¼ Δ, the cth resurce R c, is uccupied. Z r ¼ z r 1 ; z r 2 ;...; z r r ð c ;...z m Þ the sequece f semaphres crrespdig t the rth state, i which z r c ¼ Pi k is the ame f the stream (specified i the cth dispatchig rule σ c, allcated t the cth resurce) which was allwed t ccupy the cth resurce, e.g., z r c ¼ Pi k meas that stream Pi k is curretly allwed t ccupy the cth resurce Q r ¼ q r 1 ; q r 2 ;...; q r r ð c ;...; q m Þ the sequece f semaphre idices, crrespdig t the rth state, i which q r c determies the psiti f the semaphre z r c i the pririty dispatchig rule σ c ; z r c ¼ s c; ð qc rþ; q r c 2 N. Fr istace, q r 2 ¼ 2 ad z r 2 ¼ P1 2 crrespd t the semaphre z r 2 ¼ P1 2 takig the secd psiti i the pririty dispatchig rule σ 2. MA k the sequece f allcatis f multimdal prcesses: MA k ¼ ma k 1 ;...; mak u, i which ma k i is the allcati f the prcess mp i, i.e., ma k i ¼ ma k i;1 ; ma k i;2 ;...; ma k k i;c ;...; ma i;m ; ð6þ where m is the umber f resurces R, ma k i;c 2 fmp i ; Δg, ma k i;c ¼ mp i which meas that the cth resurce R c is ccupied by the ith multimdal prcess P i, ad ma k i;c ¼ Δ, thecth resurce R c,isreleasedbytheith multimdal prcess mp i. I this ctext, the state S k is feasible [5] whe: Semaphres f ccupied resurces idicate the streams allcated t thse resurces, Each lcal/multimdal stream is alltted t a uique resurce due t a relevat lcal/multimdal prcess rute. The itrduced ccept f the kth state S k eables t create aspacesf feasible states. T illustrate this, let us csider the state space f a SCCP cmpsed f six resurces ad three lcal cyclic prcesses supprtig e multimdal prcess (see Fig. 4). The bserved behavir is twfld, i.e., the levels f lcal SL ad multimdal SM prcesses ca be distiguished. I the case f the level f lcal prcesses, the states Sl i are deted by filled circles, ad i the case f the level f multimdal prcesses, the relevat states S i 2 S are deted by ufilled circles. States Sl j 2 Sl cabecsideredasa part f assciated states S i 2 S, i.e., the states that are elevated versis f relevat states S i. The trasitis likig feasible states S k ; S l 2 S, while fllwig the cstraits f preempti ad mutual exclusi, are deted by S k S l, ad they ecmpass the ext-state fucti δ: S l = δ(s k ), the defiiti f which [3] leads t the fllwig prperty: Each S i 2 S ca be preceded by a subset f states SP i, SP i S (als SP i ¼;), i.e., 8S k 2 SP i, S i = δ(s k ) but ca result ly i a uique state S j 2 S, i.e., there exists at mst e S j 2 S; S j ¼ dðs i Þ. The deadlck state S * 2 S resultig i a SCCP blckade is free frm ay descedet state. I this ctext, tw types f steady state behavirs ca be csidered: a cyclic steady state ad a deadlck state. The set msc * ¼ S k 1 ; S k 2 ; S k 3 ;...; S k v, msc * S is called the reachability state space f multimdal prcesses geerated by a iitial state S k 1 2 S, if the fllwig cditi hlds: S k i1 1! S k vi1 i! S k v! S k i ; ð7þ where S a! i S b the trasiti defied i [5], S k 1! i S k iþ1 S k 1! S k 2! S k 3!...! S k iþ1 : The set msc ¼ S k i ; S k iþ1 ;...; S k v, msc msc *,is called the cyclic steady state f multimdal prcesses (i.e., a cyclic steady state f a SCCP) with a perid Tm = msc, Tm > 1. I ther wrds, the cyclic steady state ctais such a set f states, where startig frm ay selected state, it is pssible t reach the remaiig states ad fially reach this selected state agai: 8 Sk 2mSc S k Tm1! S k : ð8þ The cyclic steady state Sc specified by the perid Tc f the executi f lcal prcesses is defied i a similar way. Graphically, the cyclic steady states Sc ad msc are described by cyclic ad spiral digraphs, respectively (see Fig. 4). Mrever, sice a iitial state S k 1 2 S either leads t msc r t a deadlck state S *, i.e., S k i1 1! S k vi1 i! S k v! S *, multimdal prcesses ca als reach a deadlck state, deted by circles with a crss i the middle i Fig. 4.

8 It J Adv Mauf Techl Fig. 4 Space f feasible states ecmpassig behavir f the SCCP frm Fig. 2 time [u.t.] The level f multimdal prcesses cyclic the ele trasiet perids deadlck deadlck deadlck deadlck cyclic deadlck deadlck The level f lcal prcesses Leged: - i-th lcal state - trasiti - i-th multimdal state - trasiti I this ctext, the prblem frmulated i Secti 2.2 ca be stated as fllws: Give a SCCP defied by SC (2), i.e., R, SL = (STL, BEL, SEL), M = (STM, BEM, SEM), des there exist a cyclic steady state msc i the state space S f the give SCCP? Nte that the abve questi gives rise t the fllwig questi: Des there exist a iitial state S0 that wuld geerate the cyclic steady state msc? This meas that searchig fr a cyclic steady state msc i a give SCCP ca be see as a reachability prblem where fr a assumed iitial state S0 (i.e., fr selected allcatis f lcal ad multimdal prcesses), we 0 i k Tm 1 lk fr a state S such that S! S! S hlds. Refiig the space f cyclic steady states frm a give space f feasible states des t pse difficulty. Hwever, the prblem f geeratig the space f feasible states is NP-hard. The majrity f states either ed i deadlcks r lead t deadlck states. Therefre, i rder t avid the geerati f the etire space f feasible states, let us fcus a alterative apprach aimed at geeratig cyclic steady states. k k 3.2 Geerati f the space f cyclic steady states Sice the parameters describig a SCCP are usually discrete, ad the relatis betwee them ca be see as - cyclic steady state f lcal prcesses - deadlck state cyclic steady state f multimdal prcesses cstraits, the cyclic schedulig prblems that ivlve them ca be preseted i the frm f a CSP [2, 5, 17]. Mre frmally, CSP is a framewrk fr slvig cmbiatrial prblems specified by pairs: a set f variables ad assciated dmais ad a set f cstraits restrictig the pssible cmbiatis f variable values. Thus, i the case f SC (2), the CSP is defied as fllws: CS ðsc Þ ¼ ððfx ; Tc; mx ; Tmg; fdx ; DTc ; DmX ; DTm gþ; C Þ; ð9þ where: X,Tc, mx,tm the decisi variables, where Tc ad Tm are the lcal ad multimdal peridicities; lsð1þ lsðþ X ¼ X11 ;... ; X1 ;... ; X1 ;... ; X is the set f k k k k sequeces f Xi, ad Xi ¼ xi;1;1 ;... ; x i ;lrðiþ;1 ;... ; xki;1;ξ ;... ; xki;lrðiþ;ξ Þ where xki;j;q is the mmet whe the perati ki;j (lcal prcess) begis i the first cycle, whereas xki;j;q ad xki;j;q ðlþ are liked by xki;j;q ðlþ ¼ xki;j;q þ l Tc; l 2 Z,Tc ¼ xki;j;q ðl þ 1Þ xki;j;q ðlþ. Aalgusly, mx = {mx1, mx2, mxi,, mxw} is the set f sequeces f mxi, ad mxi ¼ ðmxi;1;1 ;... ; m

9 It J Adv Mauf Techl x i;lmðiþ;1 ;...; mx i;1;ξ ;...; mx i;lmðiþξ; Þ where mx i,1,1 is the mmet whe the perati m i,j (f a multimdal prcess) begis i the first cycle mx i;j;k ðlþ ¼mx i;j;k þ l Tm, ad Tm ¼ mx i;j;k ðl þ 1Þmx i;j;k ðlþ. The fllwig dmais f decisi variables are csidered: D X, D mx the family f sets f admissible etry values X i, x k i;j;q 2 Z ad mx i; mx i;j;k 2 Z D Tc, D Tm the dmais f the variable Tc 2 N ad Tm 2 N. C cstraits are specified by bth: ep k i;j;qð ST L; BE L Þthe set f cstraits (equatis) likig ST L (structure f lcal prcesses) ad BE L (behavir f lcal prcesses). Each ep k i;j;qð ST L; BE L Þ describes the tempral relati (i accrdace with the cditis preseted abve [5]) betwee the mmets whe the perati begis durig its qth executi: i ¼ 1; ::; ; j ¼ 1;...; lrðiþ; k ¼ 1; ::; lsðiþ; ad q ¼ 1;...; Ξ. eq i,j,k (ST M, BE M ) the set f cstraits (equatis) likig ST M (structure f multimdal prcesses) ad BE M (behavir f multimdal prcesses). Each eq i,j,k (ST M, BE M ) describes the tempral relati betwee the mmets whe the multimdal peratis begi durig their kth executi:i ¼ 1; ::; w; j ¼ 1;...; lmðiþ; ad k ¼ 1;...; mξ. I ther wrds, the prblem discussed i Secti 2.2 bils dw t the fllwig: Give a SCCP described by CS(SC)(9) (i.e., the cstrait satisfacti prblem determied by SC), d there exist X,Tc, mx, ad Tm whse values satisfy all the cstraits C? The sluti t the prblem (9) prvides us with sets f sequeces X ad mx, whse values guaratee the required cyclic behavir f the SCCP while keepig the set f cstraits C satisfied. The cstraits C that ca be see as cditis sufficiet fr the cyclic behavir f the SCCP (resultig i a cllisi-free ad deadlck-free [12] executi f prcesses) are frmulated usig the peratr max, which takes it accut the pipelie ature f the flw f lcal prcesses. The applicati f the max peratr ca be see as a extesi f the max cstrait ccept itrduced i [5]. Cstraits lcal prcesses I rder t explai hw the cstraits f lcal prcesses are desiged, let us csider a example f a SCCP shw i Fig. 5. The perati 1 1;3 (executed by P1 1 the resurce R 5) ca be started (i.e., begi its first executi; q=1) ly if the precedig perati 1 1;2 (executed by P1 1 R 3) has bee cmpleted x 1 1;2;1 þ t1 1;2 ad the resurce R 5 has bee released, i.e., if the streamp3 1 ccupyig the resurce R 5 starts its subsequet perati at x 1 3;3;1 þ 1. Thus, the csidered relati ep1 1;3;1 ðst L ; BE L Þ ca be specified by the fllwig frmulae: x 1 1;3;1 ¼ max x1 3;3;1 þ 1 ; x 1 1;2;1 þ t1 1;2 ; ð10þ where x k i;j;q is the mmet whe the perati k i;j frms, the Pk i stream begis i its qth executi. Table 2 ctais the remaiig cstraits describig the lcal prcesses f the SCCP frm Fig. 5. Fr all cstraits, the fllwig priciple hlds: the mmet whe the perati k i;j begis is calculated as a maximum f the cmpleti time f perati k i;j1 precedig k i;j, ad the release time f the resurce pk i;j awaitig fr k i;j executi. Fig. 5 SCCP with dispatchig rules: σ 1 ¼ P ; P 3 ; σ3 ¼ P ; P 1 ; σ5 ¼ P ; P 2, ad Ψ = (1,1,1, Ξ =1 Leged: resurce ctrlled by the pririty dispatchig rule ; mmet whe the perati begis its executi i the stream mmet whe the perati begis its executi i the prcess

10 Table 2 Cstraits f lcal prcesses determiig the mmets x k i;j;q fr the SCCP frm Fig. 5 x 1 1;1;1 ¼ max x1 1;3;1 þ t1 1;3 Tc ; x 1 1;3;1 þ 1 Tc x 1 2;2;1 ¼ max x1 2;1;1 þ t1 2;1 ; x 1 3;1;1 þ 1 x 1 3;1;1 ¼ max x1 3;3;1 þ t1 3;3 Tc ; x 1 3;3;1 þ 1 Tc x 1 3;3;1 ¼ max x1 2;3;1 þ 1 Tc ; x 1 3;2;1 þ t1 3;2 It J Adv Mauf Techl x 1 1;2;1 ¼ max x1 1;1;1 þ t1 1;2 ; x 1 2;2;1 þ 1 x 1 2;1;1 ¼ max x1 2;3;1 þ t1 2;3 Tc ; x 1 1;3;1 þ 1 Tc x 1 2;3;1 ¼ max x1 2;2;1 þ t1 2;2 ; x 1 2;2;1 þ 1g x 1 3;2;1 ¼ max x1 3;1;1 þ t1 3;1 ; x 1 1;1;1 þ 1 ep k i;j;qð ST L; BE L Þ : mmet whe the perati k i;j begis i its q th executi ¼ ¼ max mmet whe p k i;j is released; ; mmet whe the perati k i;j1 ; cmpletes i ¼ 1; ::; ; j ¼ 1;...; lrðiþ; k ¼ 1; ::; lsðiþ; q ¼ 1;...; Ξ ð11þ Therefre, fllwig the cstraits (11) guaratees a deadlck-free executi f cyclic prcesses at the SL level (see Fig. 3). Nte that the cstraits (11) idirectly take it accut the assumed dispatchig pririty rules Θ ad rutes f streams U. This meas that the SCCP beig free f deadlck depeds the dispatchig rules Θ determiig the peratis executed the shared resurce, i.e., the satisfiability f cstraits (11). This is because the mmet whe the perati k i;j1 cmpletes is determied by the rute f the stream p k i ad, i tur, the mmet whe p k i;j is released depeds the perati executed befre k i;j by pk i;j. Cstraits multimdal prcesses The cstraits determiig the executi f lcal cyclic prcesses have already bee discussed i depth i [3 5]. Multimdal prcesses, hwever, have t yet bee discussed i a similar way. Therefre, fr the sake f simplicity, let us assume that multimdal prcesses are cllisi- ad deadlck-free. At the same time, multimdal prcesses ca ccupy the same resurce ad may use the same lcal prcess fr their executi. The research preseted i [3, 4] fcuses cstraits determiig cyclic steady state behavir ad particularly the cditis behid the ccept f the ext-state fucti. The lack f guaratee that ay cyclic steady state is reachable frm a give iitial state was the mai disadvatage f the results btaied s far. Mrever, because the umber f ptetial iitial states grws expetially with the umber f lcal prcesses, ay real-life implemetati f the results is rather limited. I tur, the cditis btaied i [5], which guaratee that a iitial state belgs t a cyclic steady state, were limited t lcal cyclic prcesses ly. The cstraits describig the relatiship betwee the mmets whe successive peratis begi are shw usig the example f mp 1 frm Fig. 5. The csidered prcess executes the set f the fllwig resurces: R 2,R 3,R 5, ad R 4. The lcal cyclic prcess P2 1 supprts the executi f mp 1 betwee R 2 ad R 3, whereas P1 1 supprts the executi f mp 1 betweer 3,R 5, ad R 4. This meas that the prcess mp 1 ca execute its peratis ly whe the relevat peratis frm lcal prcesses are perfrmed (i.e., P1 1 r P2 1 ). Fr istace, the perati m 1,2 executed R 3 depeds P2 1. The startig mmet mx 1,2,1 f the perati m 1,2 cicides with the startig mmet x 1 2;2;1 þ a Tc, i.e., with the begiig f the perati 1 2;2 ad is executed after the cmpleti f the precedig perati m 1,2 (i.e., the perati m 1,1 cmpleted at the mmet mx 1,1,1 + mt 1,1,1 ). The cstrait specifyig this relatiship has the fllwig frm: mx 1;2;1 ¼ mi x 1 2;2;1 þ a Tc ja 2 Z; x 1 2;2;1 þ a Tc mx 1;1;1 þ mt 1;1;1 ; where: ð12þ x k i;j;q the mmet whe the perati i;j k begis i the qth executi f a lcal prcess mx i,j,q the mmet whe the perati m i,j begis i the qth executi f a multimdal prcess Tc the peridicity f lcal cyclic steady states. Table 3 ctais the remaiig cstraits describig the multimdal prcesses f the SCCP frm Fig. 5. Fr all the cstraits, the fllwig priciple hlds: the mmet whe the perati m i,j begis is the earliest mmet whe the perati f the lcal prcess ca start (f curse the e the multimdal prcess mp i requires), temprally fllwig the mmet whe the perati m i,j 1 begis. Mre frmally: eq i;j;k ðst M ; BE M Þ : mmet whe the perati m i;j begis ¼ ¼ mifset f mmets whe the perati k i;j1 begis; temprally fllwig the mmet whe the previus perati m i;j1 cmpleesg; i ¼ 1; ::; w; j ¼ 1;...; lmðiþ; k ¼ 1;...; mξ: 4 Schedulig f peridic prcesses ð13þ Let us csider the AGVS frm Fig. 1 mdeled i terms f SCCPs (see Fig. 2). Takig it accut (2), it ca be described as fllws:

11 It J Adv Mauf Techl Table 3 Cstraits determiig the mmets mx i;j;k fr the SCCP frm Fig. 5 Multimdal prcess mp 1 mx 1;1;1 ¼ mi x 1 2;3;1 þ a Tc ja 2 Z; x 1 2;3;1 þ a Tc mx 1;4;1 þ mt 1;4;1 Tm ; mx 1;2;1 ¼ mi x 1 2;2;1 þ a Tc ja 2 Z; x 1 2;2;1 þ a Tc mx 1;1;1 þ mt 1;1;1 ; mx 1;3;1 ¼ mi x 1 1;3;1 þ a Tc ja 2 Z; x 1 1;3;1 þ a Tc mx 1;2;1 þ mt 1;2;1 ; mx 1;4;1 ¼ mi x 1 1;1;1 þ a Tc ja 2 Z; x 1 1;1;1 þ a Tc mx 1;3;1 þ mt 1;3;1 Multimdal prcess mp 2 mx 2;1;1 ¼ mi x 1 1;1;1 þ a Tc ja 2 Z; x 1 1;1;1 þ a Tc mx 2;4;1 þ mt 2;4;1 Tm ; mx 2;2;1 ¼ mi x 1 1;2;1 þ a Tc ja 2 Z; x 1 1;2;1 þ a Tc mx 2;1;1 þ mt 1;2;1 ; mx 2;3;1 ¼ mi x 1 2;2;1 þ a Tc ja 2 Z; x 1 2;2;1 þ a Tc mx 2;2;1 þ mt 2;2;1 ; mx 2;4;1 ¼ mi x 1 3;1;1 þ a Tc ja 2 Z; x 1 3;1;1 þ a Tc mx 2;3;1 þ mt 2;3;1 ; md {Tm,Tc}= 0 Give SC ¼ ððr; SLÞ; SMÞ, where: R ={R 1,R 2,, R 18 } the set f resurces, SL =(ST L,BE L ) the structure f lcal prcesses ST L =(U,T) U ¼ p 1 1 ; p1 2 ; p1 3 ; p1 4 ; p2 4 ; p1 5 ; p2 5 ; p6 1 p 1 1 ¼ ð R 1; R 2 ; R 3 ; R 4 Þ; p 1 2 ¼ ð R 4; R 5 ; R 6 Þ; p 1 3 ¼ ð R 3; R 9 ; R 8 ; R 7 ; R 5 Þ; p 1 4 ¼ p2 4 ¼ ð R 2; R 16 ; R 10 ; R 11 ; R 12 ; R 9 Þ; p 1 6 ¼ ð R 14; R 11 ; R 15 Þ; p 1 5 ¼ p2 5 ¼ ð R 8; R 17 ; R 13 ; R 14 ; R 12 Þ T ¼ t1 1; t1 2 ; t1 3 ; t1 4 ; t2 4 ; t1 5 ; t2 5 ; t1 6, where the values f the elemets f the sequeces ti k are give i Table 1 BE L =(Θ, Ψ, Ξ) Θ={σ 1, σ 2,, σ 17 } σ 1 ¼ P1 1 ; σ2 ¼ P1 1; P1 4 ; P2 4 ; σ3 ¼ P1 1; P1 3 ; σ 4 ¼ P1 1; P1 2 ; P1 2 ; P1 2 ; σ 5 ¼ P2 1; P1 2 ; P1 2 ; P1 3 ; σ6 ¼ P2 1; P1 2 ; P1 2 ; σ 7 ¼ P3 1 ; σ8 ¼ P5 1; P2 5 ; P1 3 ; σ 9 ¼ P3 1; P1 4 ; P2 4 ; σ10 ¼ P4 1; P2 4 ; σ 11 ¼ P4 1; P2 4 ; P1 6 ; P2 6 ; σ 12 ¼ P4 1; P2 4 ; P1 5 ; P2 5 ; σ13 ¼ P5 1; P2 5 ; σ 14 ¼ P5 1; P2 5 ; P1 6 ; P2 6 ; σ15 ¼ P6 1; P2 6 ; σ 16 ¼ P4 1; P2 4 ; σ17 ¼ P5 1; P2 5 ; Ψ = (1,3,1,1,1,2) Ξ = 1 SM = (ST M, BE M ) the structure f multimdal prcesses ST M =(M,mT) M={mp 1,mp 2 } mp 1 ¼ ðr 1 ; R 2 ; R 3 ; R 9 ; R 8 ; R 17 ; R 13 Þ; mp 2 ¼ ðr 15 ; R 14 ; R 12 ; R 8 ; R 7 ; R 5 ; R 6 Þ: mt= {mt 1,mt 2 }, where the values f all elemets mt i are equal t 1 uit f time BE M =(mψ, mξ) mψ=(1, 1) mξ=1 The respse t the fllwig questi is sught: Des there exist a cyclic behavir f the SC (i.e., resultig i cyclic steady states f lcal ad multimdal prcesses)? The apprach aimed at geeratig the space f feasible states (see Fig. 4) is time csumig (it is a NP-hard prblem) ad results maily i a deadlck r i states leadig t a deadlck. A alterative apprach, based the CSP frmulati f the CS (9), eables t fcus (ad, if pssible, t geerate) the dedicated cyclic steady states f bth lcal ad multimdal prcesses. 4.1 Schedulig f repetitive prcesses Csider a give SCCP ad its SL level. Searchig fr a pssible cyclic steady state f lcal prcesses frmulated i terms f a CSP ca be stated as the fllwig cstrait satisfacti prblem: CSðSCÞ ¼ ððfx ; Tcg; fd X ; D Tc gþ; CÞ; ð14þ

12 It J Adv Mauf Techl Table 4 Cstraits determiig the mmets x k i;j;q whe peratis begi fr the SCCP frm Fig. 2 Lcal prcess P1 1 Lcal prcess P2 1 x 1 1;1;1 ¼ max x1 1;4;1 þ t1 1;4 Tc ; x 1 1;4;1 þ 1 Tc x 1 2;1;1 ¼ max x1 2;3;3 þ t1 2;3 Tc ; x 1 1;1;1 þ 1 þ Tc x 1 1;2;1 ¼ max x1 1;1;1 þ t1 1;1 ; x 2 4;2;1 þ 1 x 1 2;2;1 ¼ max x1 2;1;1 þ t1 2;1;1 ; x 1 3;1;1 þ 1 x 1 1;3;1 ¼ max x1 1;2;1 þ t1 1;2 ; x 1 3;1;1 þ 1 x 1 2;3;1 ¼ max x1 2;2;1 þ t1 2;2;1 ; x 1 2;1;1 þ 1 x 1 1;4;1 ¼ max x1 1;3;1 þ t1 1;3 ; x 1 2;2;3 þ 1 x 1 2;1;2 ¼ max x1 2;3;1 þ t1 2;3;1 ; x 1 2;2;1 þ 1 x 1 2;2;2 ¼ max x1 2;1;2 þ t1 2;1;2 ; x 1 2;3;1 þ 1 x 1 2;3;2 ¼ max x1 2;2;2 þ t1 2;2;2 ; x 1 2;1;2 þ 1 x 1 2;1;3 ¼ max x1 2;3;2 þ t1 2;3;2 ; x 1 2;2;2 þ 1 x 1 2;2;3 ¼ max x1 2;1;3 þ t1 2;1;3 ; x 1 2;3;2 þ 1 x 1 2;3;3 ¼ max x1 2;2;3 þ t1 2;2;3 ; x 1 2;1;3 þ 1 Lcal prcess P3 1 Lcal prcess P4 1 x 1 3;1;1 ¼ max x1 3;5;1 þ t1 3;5 Tc ; x 1 1;4;1 þ 1 x 1 4;1;1 ¼ max x1 4;6;1 þ t1 4;6 Tc ; x 1 1;3;1 þ 1 x 1 3;2;1 ¼ max x1 3;1;1 þ t1 3;1 ; x 2 4;1;1 þ 1 x 1 4;2;1 ¼ max x1 4;1;1 þ t1 4;1 ; x 2 4;3;1 þ 1 Tc x 1 3;3;1 ¼ max x1 3;2;1 þ t1 3;2 ; x 2 5;2;1 þ 1 x 1 4;3;1 ¼ max x1 4;2;1 þ t1 4;2 ; x 2 4;3;1 þ 1 Tc x 1 3;4;1 ¼ max x1 3;3;1 þ t1 3;3 ; x 1 3;5;1 þ 1 Tc x 1 4;4;1 ¼ max x1 4;3;1 þ t1 4;3 ; x 1 6;2;2 þ 1 Tc x 1 3;5;1 ¼ max x1 3;4;1 þ t1 3;4 ; x 1 2;3;3 þ 1 x 1 4;5;1 ¼ max x1 4;4;1 þ t1 4;4 ; x 2 5;2;1 þ 1 x 1 4;6;1 ¼ max x1 4;5;1 þ t1 4;5 ; x 1 3;3;1 þ 1 Lcal prcess P4 2 Lcal prcess P5 1 x 2 4;1;1 ¼ max x2 4;6;1 þ t2 4;6 Tc ; x 1 4;2;1 þ 1 x 1 5;1;1 ¼ max x1 5;5;1 þ t1 5;5 Tc ; x 1 3;4;1 þ 1 x 2 4;2;1 ¼ max x2 4;1;1 þ t2 4;1 ; x 1 4;3;1 þ 1 x 1 5;2;1 ¼ max x1 5;1;1 þ t1 5;1 ; x 2 5;3;1 þ 1 Tc x 2 4;3;1 ¼ max x2 4;2;1 þ t2 4;2 ; x 1 4;4;1 þ 1 x 1 5;3;1 ¼ max x1 5;2;1 þ t1 5;2 ; x 2 5;4;1 þ 1 Tc x 2 4;4;1 ¼ max x2 4;3;1 þ t2 4;3 ; x 1 4;2;3 þ 1 x 1 5;4;1 ¼ max x1 5;3;1 þ t1 5;3 ; x 1 6;1;1 þ 1 x 2 4;5;1 ¼ max x2 4;5;1 þ t2 4;5 ; x 1 4;6;1 þ 1 x 1 5;5;1 ¼ max x1 5;4;1 þ t1 5;4 ; x 2 4;6;1 þ 1 x 2 4;6;1 ¼ max x2 4;5;1 þ t2 4;5 ; x 1 4;1;1 þ 1 þ Tc Lcal prcess P5 2 Lcal prcess P6 1 x 2 5;1;1 ¼ max x2 5;5;1 þ t2 5;5 Tc ; x 1 5;2;1 þ 1 x 1 6;1;1 ¼ max x1 6;3;2 þ t1 6;3 Tc ; x 2 4;5;1 þ 1 x 2 5;2;1 ¼ max x2 5;1;1 þ t2 5;1 ; x 1 5;3;1 þ 1 x 1 6;2;1 ¼ max x1 6;1;1 þ t1 6;1 ; x 1 6;3;2 þ 1 Tc x 2 5;3;1 ¼ max x2 5;1;1 þ t2 5;1 ; x 1 5;4;1 þ 1 x 1 6;3;1 ¼ max x1 6;2;1 þ t1 6;2 ; x 2 5;5;1 þ 1 x 2 5;4;1 ¼ max x2 5;3;1 þ t2 5;3 ; x 1 5;5;1 þ 1 x 1 6;1;2 ¼ max x1 6;3;1 þ t1 6;3 ; x 1 6;2;1 þ 1 x 2 5;5;1 ¼ max x2 5;4;1 þ t2 5;4 ; x 1 5;1;1 þ 1 þ Tc x 1 6;2;2 ¼ max x1 6;1;2 þ t1 6;1 ; x 1 6;3;1 þ 1 x 1 6;3;3 ¼ max x1 6;2;2 þ t1 6;2 ; x 1 6;4;1 þ 1

13 It J Adv Mauf Techl Table 5 Values f the mmets x k i;j;q whe peratis begi fr the SCCP frm Fig. 2 Startig mmets: x 1 1;1;1 ; x1 2;1;1 ; x1 1;2;1 ; x1 3;1;1 ; x1 2;2;1 ; x2 4;4;1 x 1 2;3;1 ; x1 1;3;1 x 1 2;1;2 x 1 3;3;1 x 1 3;5;1 ; x2 5;1;1 ; x1 5;3;1 ; x1 4;4;1 ; x1 4;5;1 ; x1 3;2;1 x 1 5;2;1 ; x1 6;2;1 x 1 6;3;1 x 2 5;2;1 ; x2 4;3;1 x 1 6;1;1 x 1 6;1;2 Values: Crrespdig Sl 0 Sl 1 Sl 2 Sl 3 Sl 4 Sl 5 Sl 6 Sl 7 Sl 8 Sl 9 states: Startig mmets: x 1 2;2;2 ; x1 4;1;1 ; x1 2;3;2 ; x1 4;3;1 ; x1 2;1;3 ; x1 5;5;1 x 1 2;2;3 ; x1 1;4;1 ; x1 2;3;3 ; x 1 4;5;1 ; x2 4;5;1 x 1 3;4;1 ; x2 4;1;1 ; x2 4;2;1 ; x2 5;4;1 x 1 5;1;1 x 2 5;5;1 x 1 4;2;1 ; x2 4;2;1 ; x2 5;3;1 x 2 4;6;1 ; x1 6;3;2 x 1 6;2;2 Values: Crrespdig states: Sl 10 Sl 11 Sl 12 Sl 13 Sl 14 Sl 15 Sl 16 Sl 17 Sl 18 where: X ¼ X1 1; X 2 1; X 3 1; X 4 1; X 4 2; X 5 1; X 5 2; X 6 1 X1 1 ¼ x1 1;1;1 ; x1 1;2;1 ; x1 1;3;1 ; x1 1;4;1 the mmets whe peratis executed alg p 1 1 ¼ ð R 1; R 2 ; R 3 ; R 4 Þbegi X2 1 ¼ x1 2;1;1 ; x1 2;2;1 ; x1 2;3;1 ; x1 2;1;2 ; x1 2;2;2 ; x1 2;3;2 ; x1 2;1;3 ; x1 2;2;3 ; x 1 2;3;3 Þ; X3 1 ¼ x1 3;1;1 ; x1 3;2;1 ; x1 3;3;1 ; x1 3;4;1 ; x1 3;5;1 ; X 1 4 ¼ x1 4;1;1 ; x1 4;2;1 ; x1 4;3;1 ; x1 4;4;1 ; x1 4;5;1 ; x1 4;6;1 ; X4 2 ¼ x2 4;1;1 ; x2 4;2;1 ; x2 4;3;1 ; x2 4;4;1 ; x2 4;5;1 ; x2 4;6;1 ; X5 1 ¼ x1 5;1;1 ; x1 5;2;1 ; x1 5;3;1 ; x1 5;4;1 ; x1 5;5;1 ; resurces time [u.t.] Leged: (allcatis) (semaphres) (semaphre idices) - executi f stream - suspesi f the stream - -th lcal state f SCCP Fig. 6 Gatt chart illustratig the cyclic steady state f lcal prcesses fr the SCCP frm Fig. 2, where Ψ = (1,3,1,1,1,2) ad Ξ =1

14 It J Adv Mauf Techl Table 6 Cstraits determiig the mmets mx i,j,k whe peratis begi fr the SCCP frm Fig. 2 Multimdal prcess mp 1 mx 1;1;1 ¼ mi x 1 1;1;1 þ a Tc ja 2 Z; x 1 1;1;1 þ a Tc mx 1;7;1 þ mt 1;7;1 Tm ; mx 1;2;1 ¼ mi x 1 1;2;1 þ a Tc ja 2 Z; x 1 1;2;1 þ a Tc mx 1;1;1 þ mt 1;1;1 ; mx 1;3;1 ¼ mi x 1 1;3;1 þ a Tc ja 2 Z; x 1 1;3;1 þ a Tc mx 1;2;1 þ mt 1;2;1 ; mx 1;4;1 ¼ mi x 1 3;2;1 þ a Tc ja 2 Z; x 1 3;2;1 þ a Tc mx 1;3;1 þ mt 1;3;1 ; mx 1;5;1 ¼ mi x 1 3;3;1 þ a Tc ja 2 Z; x 1 3;3;1 þ a Tc mx 1;4;1 þ mt 1;4;1 ; mx 1;6;1 ¼ mi x 1 5;2;1 þ a Tc ; x 2 5;2;1 þ a Tc ja 2 Z; x 1 5;2;1 þ a Tc mx 1;5;1þ mt 1;5;1 ; x 2 5;2;1 þ a Tc mx 1;5;1 þ mt 1;5;1 ; mx 1;7;1 ¼ mi x 1 5;3;1 þ a Tc ; x 2 5;3;1 þ a Tc ja 2 Z; x 1 5;2;1 þ a Tc mx 1;5;1 þ mt 1;5;1 ; x 2 5;2;1 þ a Tc mx 1;6;1 þ mt 1;6;1 Multimdal prcess mp 2 mx 2;1;1 ¼ mi x 1 6;3;1 þ a Tc ; x 1 6;3;2 þ a Tc ja 2 Z; x 1 6;3;1 þ a Tc mx 2;7;1þ mt 2;7;1 Tm; x 1 6;3;2 þ a Tc mx, 2;7;1 þ mt 2;7;1 Tm ; mx 2;2;1 ¼ mi x 1 6;1;1 þ a Tc ; x 1 6;1;2 þ a Tc ja 2 Z; x 1 6;1;1 þ a Tc mx 2;1;1 þ mt 2;1;1 ; x 1 6;1;2 þ a Tc mx 2;1;1 þ mt 2;1;1 mx 2;3;1 ¼ mi x 1 5;5;1 þ a Tc ; x 2 5;5;1 þ a Tc ja 2 Z; x 1 5;5;1 þ a Tc mx 2;2;1 þ mt 2;2;1 ; x 2 5;5;1 þ a Tc mx 2;2;1 þ mt 2;2;1 mx 2;4;1 ¼ mi x 1 5;1;1 þ a Tc ; x 2 5;1;1 þ a Tc ja 2 Z; x 1 5;1;1 þ a Tc mx 2;3;1 þ mt 2;3;1 ; x 2 5;1;1 þ a Tc mx 2;3;1 þ mt 2;3;1 mx 2;5;1 ¼ mi x 1 3;4;1 þ a Tc ja 2 Z; x 1 3;4;1 þ a Tc mx 2;4;1 þ mt 2;4;1, mx 2;6;1 ¼ mi x 1 3;5;1 þ a Tc ja 2 Z; x 1 3;5;1 þ a Tc mx 2;5;1 þ mt 2;5;1, mx 2;7;1 ¼ mi x 1 2;3;1 þ a Tc ; x 1 2;3;2 þ a Tc ; x 1 2;3;3 þ a Tc ja 2 Z; x 1 2;3;1 þ a Tc mx 2;6;1 þ mt 2;6;1 ; x 1 2;3;2 þ a Tc mx 2;6;1 þ mt 2;6;1 ; x 1 2;3;3 þ a Tc mx 2;6;1 þ mt 2;6;1 X5 2 ¼ x2 5;1;1 ; x2 5;2;1 ; x2 5;3;1 ; x2 5;4;1 ; x2 5;5;1 ; X6 1 ¼ x1 6;1;1 ; x1 6;2;1 ; x1 6;3;1 ; x1 6;1;2 ; x1 6;2;2 ; x1 6;3;2 ; C the set f cstraits that (due t (11)) csists f the cstraits frm Table 4. The sluti f the CS(14) implemeted i the Oz/Mzart platfrm ( a Itel Cre Du 3.00 GHz Table 7 Values f the mmets mx i,j,k whe peratis begi fr the SCCP frm Fig. 2 Startig mmets: mx 1,1,1, mx 1,2,1 mx 1,3,1 mx 2,3,1 mx 2,4,1 mx 1,4,1 mx 2,2,1 Values: Crrespdig lcal states: Sl 0 Sl 2 Sl 7 Sl 15 Sl 17 Sl 6 Crrespdig multimdal states: S 0 S 2 S 7 S 15 S 17 S 25 Startig mmets: mx 1,5,1 mx 2,5,1 mx 1,6,1, mx 1,7,1 mx 2,7,1 mx 2,1,1 mx 2,6,1 Values: Crrespdig lcal states: Sl 9 Sl 12 Sl 0 Sl 2 Sl 6 Sl 17 Crrespdig multimdal states: S 28 S 31 S 38 S 40 S 40 S 51

15 It J Adv Mauf Techl with 4.00-GB RAM ad btaied i less tha 1 s) is shw i Table 5. Therefre, the perid Tc f the btaied cyclic steady state is equal t19. The mmets whe peratis begi i the lcal prcesses are shw i Table 5. A Gatt chart f the cyclic steady state behavir is shw i Fig. 6. Nte that each uit time crrespds t a system state, i.e., the allcati f prcesses t resurces. Thus, the cyclic steady state csists f 19 states: Sc={Sl 0,Sl 1,, Sl 18 }. Similarly t Fig. 4, the states Sl ecmpassig the succeedig allcatis are deted by filled circles (see Fig. 5). 4.2 Schedulig f multimdal prcesses Let us csider the cyclic steady state Sc f the lcal prcesses f a SCCP. What is the cyclic steady state f the multimdal prcesses msc executed i this system? Let us assume tw multimdal prcesses mp 1 ad mp 2. Therefre, we must slve the relevat CSP, i.e., CS(SC) defied as (9), which ca be see i terms f the prblem (14) augmeted by variables mx ={mx 1,mX 2 }, where: mx 1 ¼ mx 1;1;1 ; mx 1;2;1 ; mx 1;3;1 ; mx 1;4;1 ; mx 1;5;1 ; mx 1;6;1 ; mx 1;7;1 mx 2 ¼ mx 2;1;1 ; mx 2;2;1 ; mx 2;3;1 ; mx 2;4;1 ; mx 2;5;1 ; mx 2;6;1 ; mx 2;7;1 ; as well as by the cstraits (stated i (13)) specified i Table 6. The btaied sluti csists f the sluti (see Table 5) t the prblem (14) that has already bee btaied, as well as f the mmets mx 1 ad mx 2 whe peratis begi (see the Table 7). The Gatt diagram f the cyclic steady state f behavir f multimdal prcesses (cmpsed f 57 states Sc ¼ S 0 ; S 1 ;...; S 57 )isshwifig.7. The btaied perid is equal t Tm=57 (i.e., the multiple f the perid Tc=19). This meas that withi that perid, the multimdal prcesses cmplete e perid, whereas the lcal prcesses cmplete three. The btaied cyclic steady states Sc ad msc are shw i Fig. 8 (they are deted as Sc 1 ad msc 1, respectively). Apart frm btaiig feasible slutis ecmpassig a cyclic behavir resurces time [u.t.] -see Fig. 6 Leged: - executi f stream - suspesi f the stream - -th lcal state f SCCP - executi f multimdal prcess - executi f multimdal prcess - -th multimdal state f SCCP Fig. 7 Gatt chart f the cyclic steady state f multimdal prcesses f the SCCP frm Fig. 2, where Ψ = (1,3,1,1,1,2), Ξ =1,mΨ (1,1), ad mξ =1

16 It J Adv Mauf Techl Fig. 8 Space f cyclic steady states f the SCCP frm Fig. 2 time [u.t.] Leged: - k-th lcal state - trasiti f the csidered SCCP, we preset ather sluti that fllws a ew dispatchig rule σ8 ¼ P31 ; P51 ; P52 (previusly, the dis patchig rule assciated with R8 was σ8 ¼ P51 ; P52 ; P31 ). Nte that a chage i the dispatchig rule des t chage the perid f lcal prcesses (eve thugh it is differet frm the previus Sc2). Hwever, the perid f the cyclic steady state f the multimdal prcess msc2 des chage: Tm=113. Mrever, the mutual prprtis chaged as well: mψ = (3,2). This meas that durig three executis f the prcess mp1, the prcess mp2 executes twice. The Gatt chart f the sluti f the CS (9) prblem (btaied i less tha 1 s) is shw i Fig. 9. The prvided examples demstrate the cmputatial efficiecy f the apprach based the CSP ccept (i.e., SC (2) ad CS(SC)) aimed at geeratig a cyclic steady state f the SCCP ad determied by U,T,Θ,Ψ,Ξ,M,mT,mΨ, ad mξ. The btaied cyclic steady state behavirs are free frm trasiet perids, i.e., trasiet perids as shw i Fig k-th multimdal state - trasiti This prperty is crucial fr prblems regardig switchig betwee pssible cyclic behavirs f a SCCP [3, 4]. It shuld be ted, hwever, that fr certai cases f CS (SC), cyclic slutis d t exist at all. This meas that the cstraits emplyed i the CS ca be see merely as feasible cditis guarateeig cyclic behavir. I geeral, the lack f ay cyclic behavir implies that the cditis are ctradictry. Hwever, the existece f ctradictig cditis des t ecessarily imply the lack f cyclic behavir. Mrever, it is crucial that the time-csumig geerati f the state space that eables the refiemet f cyclic steady states [3, 4] ca be replaced by direct geerati f cyclic states. Frm the perspective f cmputatial cmplexity, this meas that the expetial grwth f the umber f pssible iitial states (a pssible cyclic steady state may be reachable frm each f them) ca be replaced by the examiati f a umber f dispatchig rules that als grw expetially albeit with a smaller prefactr. This is because ur

17 It J Adv Mauf Techl resurces time [u.t.] resurces time [u.t.] Leged: - executi f stream - suspesi f the stream - -th lcal state f SCCP - executi f multimdal prcess - executi f multimdal prcess - -th multimdal state f SCCP Fig. 9 The Gatt chart illustratig the cyclic behavir f the SCCP frm Fig. 2, takig it accut the ew dispatchig rule σ 8 ¼ P 1 3 ; P ; P 5,Ψ = (1,3,1,1,1,2), = 1, mψ (3,2), ad mξ =1