Ieraoal Joural of Performably Egeerg Vol.7 No. 5 Sepember pp. 44-454. RAS Cosulas Pred Ida Sochasc Per Nes wh Low Varao arx Expoeally Dsrbued Frg Tme P. BUCHHOLZ A. HORVÁTH* ad. TELE 3 Iformak IV TU DormudD-44 Dormud Germay Dparmeo d Iformaca Uversà d Toro I-49 Toro Ialy 3 Deparme of Telecommucaos Techcal Uversy of Budapes H-5 Budapes Hugary (Receved o November ad revsed o ay 8 ) Absrac: arx expoeal (E) dsrbuos wh low squared coeffce of varao (scv) are such ha he desy fuco becomes zero a some pos ( ). For such dsrbuos here s o equvale fe dmesoal PH represeao whch hbs he applcao of exsg mehodologes for he umercal aalyss of sochasc Per es (SPNs) wh hs kd of E dsrbued frg me. To overcome he lmaos of exsg mehodologes we apply he flow erpreao of E dsrbuos ad sudy he rase ad he saoary behavour of sochasc Per es wh E dsrbued frg mes va ordary dffereal ad lear equaos respecvely. The ma resul of hs sudy s a heory sag ha all kds of E dsrbuos ca be used lke phase ype (PH) dsrbuos sochasc Per es ad he umercal compuao of rase or saoary measures s possble wh mehods smlar o hose used for arkov models.. Iroduco eywords: Sochasc Per e phase ype dsrbuo marx expoeal dsrbuo The mehod of exeded arkov cha (EC) [9] s a wdely used aalyss echque for sochasc Per es wh PH dsrbued frg mes. I s based o he geerao of a arkov cha ha descrbes he behavour of he markg process ad addoally he phase processes of he volved PH dsrbuos. The resulg arkov cha ca he be aalyzed wh esablshed umercal echques for rase or saoary aalyss. Followg he geeral resuls [8] was lkely ha a sochasc model E dsrbuos ca be used place of PH dsrbuos ad several resuls wll carry over. There are some resuls o hs dreco bu s o easy o prove resuls he geeral seg because probablsc argumes assocaed wh PH dsrbuos do o loger hold. I [4] has bee show ha marx geomerc mehods ca be appled for quas brh deah processes (QBDs) wh raoal arrval processes (RAPs) [] whch ca be vewed as a exeso of E dsrbuos o arrval processes. To prove ha he marx geomerc relaos hold he auhors of [4] use a erpreao of RAPs ha has bee proposed []. However he resulg proofs are lmed o QBDs. The closes relaed resul cosders a subclass of SPNs wh E dsrbued frg mes [6]. Tha paper proves he applcably of a exeded sysem of dffereal equaos for he rase aalyss ad a exeded sysem of lear equaos for saoary aalyss case of E dsrbuos wh srcly posve desy ( ). Due o he smlary o he EC based soluo we refer o hs soluo mehod as EC-lke *Correspodg auhor s emal: horvar@d.uo. 44
44 P. Buchholz A. Horváh ad. Telek soluo. [6] proofs ha he EC-lke soluo s applcable for SPNs wh E dsrbued frg mes whose desy s srcly posve ( ). I hs paper we exed he resul of [6] for he case of E dsrbued frg mes whose desy mgh be zero ( ). The mporace of hs exeso comes from he fac ha he desy of mpora ad praccally covee E dsrbuos s zero some pos ( ). Some of he mos mpora examples of hese E dsrbuos are he E dsrbuos wh low scv as s dealed below. The mehodology appled hs paper s dffere from he oe used [6]. The proof of [6] s based o he fac ha ay E dsrbuo wh srcly posve desy o ( ) ca be represeed as a PH dsrbuo wh a poeally larger vecor-marx par. Ths approach s o applcable for E dsrbued frg me whose desy mgh be zero o ( ) sce hese E dsrbuos cao be represeed as a PH dsrbuo wh fe dmeso [3]. Isead we provde a proof of he applcably of EC-lke soluo based o he flow erpreao of E dsrbuos provded by Blad ad Neus [5]. The paper s orgased as follows. Sec. preses he defo of E dsrbuos ad some mpora resuls abou her represeaos. Examples of E dsrbuos wh low scv are repored Sec. 3. I Sec. 4 we roduce SPNs wh E dsrbued frg me ad gve he ecessary elemes for her aalyss. Sec. 5 provdes examples o show applcaos of he approach. Fally Sec. 6 cocludes he paper. arx Expoeal Dsrbuos We quckly recall he basc defo of E [5] ad PH [] dsrbuos for compleeess. Defo Le X be a radom varable wh cumulave dsrbuo fuco (cdf) Ax F ( x) = Pr( X < x) = αe where α s a row vecor of sze A s a marx of sze X ad s he colum vecor of oes of sze. The we say ha X s marx expoeally dsrbued wh represeao α A or shorly E( α A ) dsrbued. Defo If X s a E( α A ) dsrbued radom varable ad α ad A have he properes ha α α = (here s o probably mass a = ) A < A for j A ad A s o-sgular he we say ha X s phase ype dsrbued wh represeao α A or shorly PH( α A ) dsrbued. Usg he oao a = A he probably desy fuco (pdf) ad he momes of Ax X are respecvely f ( x) = αe a ad µ = E( X ) =! α ( A). X. A Covee Subclass of E Dsrbuos Oe of he ma problems of workg wh E dsrbuos s ha he moooe creasg propery of F X (x) (or he o-egavy of f X (x) ) s hard o check. Alhough here are specal subclasses of E dsrbuos whose cosruco esures ha he assocaed PDF s o-egave. Defo 3 The se of E dsrbuos wh pdf f ( ) = a( ) / a( ) d where λ µ a( ) = ( r ( ) + s ( )) e + ( q ( ) + w ( )) e cos ( ω + φ ) ad r ( ) s ( ) q ( ) w ( ) are arbrary fe polyomals of ad λ µ ω φ are j
Sochasc Per Nes wh Low Varao arx Expoeally Dsrbued Frg Tme 443 posve real umbers s referred o as E dsrbuos wh quadrac polyomals. E dsrbuos wh quadrac polyomals are guaraeed o have o-egavy of he pdf. Addoally as s dscussed [7] some mpora exreme E dsrbuos belog hs class: umercal vesgaos suggess ha he order E dsrbuos wh real egevalues ad mmal scv ad he order k + E dsrbuos wh real ad complex egevalues ad mmal scv belog o he class of E dsrbuos wh quadrac polyomals. Based o [7] Table lss he mmal scv of E dsrbuos wh quadrac polyomals of dffere order. To he bes of our kowledge he values preseed Table represe he mmal scv of he whole E class (wh or whou complex egevalues) of he gve order bu here s o proof or couer example (a E wh lower scv) s avalable up o ow whch verfy or desroy hs cojecure. To avod vald saemes below we are gog o alk abou E dsrbuo wh low scv bu we hk ha ca be read as E dsrbuo wh mmal scv. PH dsrbuos of order are kow o have scv greaer or equal o /. Cosequely he / m_scv parameers Table dcae he mmal sze of a PH dsrbuo o approxmae such a low coeffce of varao. Ths propery of he class of E dsrbuos wh quadrac polyomals makes her use very effce for approxmag dsrbuos wh low scv. Table : mal Squared Coeffce of Varao of E Dsrbuos wh Quadrac Polyomals Order m_scv /m_scv m_scv /m_scv Real poles Real ad Complex poles 3.76583 3.6556.9 4.97756 4.9333 5.75.4988 6.675 5.38453 7.66.8643.355 6.863 9.69 7.8677.67.488 3.39 8.776 3.9465 9.6486 6.653.6569 38.39.58365 9.94.44973.63.7494 57.65.397335 5.677 3.3543 8.3766.4696 8.95 4.376 3.599 5.867 34.944.938 7.379. Ierpreao of arx Expoeal Dsrbuos va Flows I [5] he auhors provde a sochasc erpreao of E dsrbuos va flows. Ths erpreao s he followg for E( α A ) of sze. Cosder doubly fe coaers of lqud whose al coes are α...α ad a addoal coaer whose coe s zero ally. Assume ha lquds flow from coaer o coaer j wh j = / j a cosa rae gve by A ( j). Tha meas ha f he h coaer has c amou of lqud a me u he c A ( j) du amou of lqud flows from he h o j h coaer he erval [ u u + du]. Furher from coaer lqud flows oward he + h coaer a cosa rae gve by he h ery of a. Le us deoe by v ( u) + he level of lqud coaer a me u. As show [5] he vecor v( u) = v ( u)... v ( u) referg o he frs coaers follows
444 P. Buchholz A. Horváh ad. Telek he se of ordary dffereal equaos (ODE): dv ( u) / du = v( u) A wh al codo v () = α. The soluo of hese ODEs s v( u) = α exp( Au). The s easy o see ha he followg relaos hold bewee he levels of he lquds he coaers ad a radom varable X dsrbued accordg o E( α A ): FX ( u) = Pr( X > u) = v ( u) = v+ ( u) =.e. he oal amou of lqud prese coaers v...v a me u correspods o he probably ha X s greaer he u ad f X ( u) = v( u) a.e. he pdf of X ca be coeced o he level of he lquds hrough he vecor a ad we wll refer o he quay v ( u) a as he frg poeal. I follows ha he agg of a E dsrbued radom varable ca be capured by he real valued vecor v (u). 3 Examples of E Dsrbuos wh Low Coeffce of Varao 3. Order 3 E Dsrbuos wh Complex Egevalues Frs we cosder he order 3 E srucure wh low scv repored [7]. I s f () = a ω + φ a a ue cos = ue (+ cos( ω + φ)) = e ( u + u cos( φ) cos( ω) u s( φ)s( ω)) Where from ( f ) d = we have u = a( a + ω ) /( a + ω + a cos( φ) aω s( φ)). O he oher had we have he followg real marx represeao a a x A f ( ) = αe ( A) = ( g c + d c d) exp a ω a + ω = ω a a ω a a = ( g c d c + d) exp a ω a + ω = a ω a ω + a a ω a ω = age + ( c d)( a + ω) e e + ( c + d)( a ω) e e = a = e ( ag + ( ac + ωd) cos( ω) ( ad ωc)s( ω)) where from he frs marx represeao o he secod oe a smlary rasformao s appled. Havg a ω ad φ fxed from f ( ) f ( ) we have ag = u ( ac + ωd) = u cos( φ) ( ad ωc) = u s( φ) from whch = u a cos( φ) aω s( φ) aω cos( φ) + a s( φ) g c = d =. a ( a + ω + a cos( φ) aω s( φ)) ( a + ω + a cos( φ) aω s( φ)) Wh a = φ = 3.47863 ad ω =.3593 he mmal scv of hs srucure s obaed ad s.9 ~ /5 [7]. The pdf of hs dsrbuo s depced Fgure ad o dcae he flexbly of hs class of dsrbuos Fgure depcs he pdf obaed a a = φ = ad ω =. 3. Order 3 E Dsrbuos wh Real Egevalues The order 3 E srucure wh real egevalues ad wh mmal scv s [7] a a f ( ) = e (( w + w ) + v ) = e ( w + ( w w + v ) + ) w
Sochasc Per Nes wh Low Varao arx Expoeally Dsrbued Frg Tme 445 3 where w = ( aw a av a )/ esures f ( ) d =. Sarg from a Erlag w ype marx represeao we have a a 3 Ax a a x a x a x3 ( ) = ( ) = ( 3) exp =.!!! f αe A x x x a a e + + a a From he dey of he coeffces of we have w ww + v w x = x = x3 =. 3 a a a The mmal scv wh real poles s obaed a w =.358998 ad s.76583. The pdf of hs dsrbuo s depced Fgure 3..4.5.3....5 3 4 5 Fgure : Probably desy fuco of E3 wh a = φ = 3.47863.5..5..5 3. Fgure : Probably desy fuco of E3 wh a = φ = ω = 3.3 Hgher order E Dsrbuos wh Complex Egevalues k Le he desy of a E dsrbuo be a ω + φ f ( ) = ue cos = = k a ue (+ cos( ω + φ )). If = k + = 7 a = ω =.88499 = φ = 3.963 φ = 3.944 φ3 = 4.869 ad u s se such ha f ( ) d = he he scv of hs dsrbuo s.488 < /3. The aalycal reame of hs case s raher cumbersome bu ca be avoded by a umercal approach o oba he assocaed marx represeao. The momes of he dsrbuo ca be compued from m f ( d = Based o hese = for ) momes a marx represeao of f() ca be obaed a wo seps umercal mehod. I he frs sep we geerae marx A such ha exhbs he same expoeal coeffces.e. egevalues as f (). We have A = a a ω ω a O a kω kω a.e. he egevalues of A are a ad { a + ω a ω} for = k. I he secod sep we oba vecor α by solvg he lear sysem! α ( A) = m for =. We appled hs umercal procedure for = 7 ( k = 3 ) ad obaed he E dsrbuo wh marx represeao ( α A) where α ={9.7787-4.756-6.56599
446 P. Buchholz A. Horváh ad. Telek.7583.4449.395 -.888} ad A s defed by s srucure..5.7..5..5.6.5.4.3.. 4 6 8 4 6 8 Fgure 3: Probably Desy Fuco of Fgure 4: Probably Desy Fuco of he E3 wh real poles ad w =.358998 order 5 E wh scv=.938 The approach works for = ( k = 5 ). Wh a = φ = 3.667 φ = 3.3369 φ 3 = 4.678 φ 4 = 4.96537 φ 5 = 5.856 ω =.8546 ad u se such ha f ( ) d = he scv of hs dsrbuo s.7494 < /57. The al vecor of s marx represeao s α ={5.3-4.8666-3.963-7.498 8.9355 3.7778.8656 -.4874 -.9385 -.976547.8795}. The same approach works also for = 5 ( k = 7 ). Wh a = φ = 3.47395 φ = 4.456 φ = 3.593 φ = 3.9857 φ = 5.6454 φ = 5.7343 φ = 6.449 3 4 ω =.7459 ad u se such ha f ( ) d = he scv of hs dsrbuo s.938 < /7 ad s pdf s depced Fgure 4. The al vecor of s marx represeao s α ={53.77 5.455-55.47-8.5444 8.5684 7.384.88 4.43-3.6775 -.43 -.6 -.5864.38466.53449.4463}. 4 Sochasc Per Nes wh E Dsrbued Frg Tmes I hs seco we roduce sochasc Per es whch he frg mes of he rasos are E dsrbued. We cosder frs Per es ad her reachably graph. Aferwards he reachably graph s expaded by cosderg dealed sae formao o descrbe he age of he eabled rasos. We sar by brefly preseg some basc defos ad resuls for Per es followg [6]. Defo 4 A Per e s a fve uple PN = ( P T I O ) where P s a se of places T s a se of rasos such ha P T = I : P T N s he pu fuco O : T P N s he oupu fuco ad : P N s he al markg. We assume a orderg he se of rasos such ha for T wh eher < or > holds. Deoe by = { p p P I( p ) > } ad = { p p P O( p) > } he pu ad oupu bag of raso respecvely. A markg s a vecor of legh P whose elemes represe he oke populao of each place. ( p) deoes he p -h eleme of hs vecor. arkg defes he al oke populao. Traso s eabled markg f ad oly f ( p) I ( p ) for all p. If s eabled markg ad fres he a ew markg wh ( p) = ( p) I( p ) + O( p) s geeraed. For hs eve we use he oao. We assume ha mples. The exeso o s sraghforward bu requres a more complcae oao. The se of markgs avalable 5 6 7
Sochasc Per Nes wh Low Varao arx Expoeally Dsrbued Frg Tme 447 from wh repeaed applcao of relao defes he reachably se RS of he Per e. The reachably graph RG s a dreced ad labeled graph wh verex se RS ad a arc labeled wh bewee RS f ad oly f. Furher assumpos abou RS ad RG lke feess or srog coecvy wll be made laer whe ecessary. Le Ea( ) = { T ad for all p P : ( p) I ( p )} be he se of eabled rasos markg. The cocep uderlyg our defo of ewly eabled rasos s deoed as eablg memory []. The geeral approach s applcable for age memory polcy as well. Oly he srucure of he sae descrpor ad he defo of he reseg or maag he memory () has o be modfed ha case. Furhermore we assume sgle server semacs for all rasos. 4. Flow Ierpreao of SPN wh E Dsrbued Frg Tmes Hereafer we show ha he behavour of a PN wh E mgs ca be descrbed hrough he behavour of he levels of he lquds assocaed wh he E dsrbued rasos of he e. Ths s doe by assocag each markg wh a vecor v ( u ) provdg a me u he jo sae (.e. he jo lqud levels) of he E dsrbuos of he rasos ha are eabled markg. We deoe by α A ad a = ( A ) he sze he al vecor he geeraor ad he closg vecor of he E dsrbuo assocaed wh raso. Usg hese oaos we ca prese he ma heorem of he paper. Theorem v ( u ) sasfes he vecor dffereal equao dv( u ) = v( u ) A + v( u ) R ( ) () du where R I α ( ) = a aα Ea( ) : oherwse T f ad Ea( ) Ea( ) f Ea( ) ad Ea( ) f Ea( ) ad Ea( ) f = ad Ea( ) f = ad Ea( ) where I s he dey marx ad s he colum vecor of oes of legh. The al codo s v( ) = α ad v( ) = for. Ea( ) Proof. To prove he heorem we prese he scalar equaos goverg he sysem behavour. Uforuaely requres he roduco of complcaed oaos referrg o he elemes of complcaed muldmesoal vecors ad marces. We deoe by he umber of acve E dsrbuos markg ad by α A ad ( A ) he sze ad he descrpors of he h acve E dsrbuo markg a =. The eres of α A ad a order o avod heavy subscrpg wll be dcaed parehess.e. for example he j h ery of α as α ( ) ad he ery j ()
448 P. Buchholz A. Horváh ad. Telek ( j k) of A as A ( j k). The dex of raso markg wll be deoed by p.e. f he h acve E dsrbuo markg s he p =. The vecor v ( u ) s of legh Ea ) ad s eres are orgased accordg o ( lexcographcal order (also referred o as he mxed-base scheme). Ths order s aurally geeraed by he roecker produc operao of he vecors represeg he level of he coaers assocaed wh he acve rasos markg a me u. For he elemes of he vecor he lexcographcal order meas ha havg a vecor of dces l = l l... l wh l defyg a coaer for each eabled raso of markg he ery of v ( u ) ha descrbes he jo sae of hese coaers s poso (...(( l ) + l ) 3...) + l = ( l k ) k = (where for = k+ k smplcy of oao he empy produc equals o ). A gve ery of he vecors v ( u ) wll be a coaer self ad he vecors v ( u ) provde he expaded sae space of he coaers of he dvdual rasos. The ery of v ( u ) correspodg o he vecor of coaers l l... wll be deoed by v u l... ). l ( The eres of v ( u ) wll be such ha he level of he j h coaer of he h eabled raso of markg ca be recovered by he sum + v + l = l = l = l + = l = ( u j) = L L v( u l l.. l j l... l ). (3) Furher he probably of markg a me u wll be gve by he oal amou of lqud prese v ( u ) as π ( u ) = L v( u l... l l = l = The al codo for he vecors v ( u ) s gve as ). v( ) = α ad : v( ) =. (4) Ea( ) wh whch s easy o see ha l ( α v j) = ( j) j : j ad v( j) = j : j.e. (4) provdes correc al seg of he levels of he lquds. I order o descrbe correcly he evoluo of he PN he evoluo of v ( u ) ad u has o be such ha he level of he j h coaer of he h eabled raso of gve by v ( u j) j ad compued accordg o (3) sasfes he followg codos.. The level v ( u j) s decreased a rae A ( j ). j. There s a exchage of lquds from coaers v ( u k) k = / j o coaer v ( u j) wh rae A ( k ). j 3. For : = /
Sochasc Per Nes wh Low Varao arx Expoeally Dsrbued Frg Tme 449 - he frg poeal of a me u has o be equal o v u p j) a ( j) ( j = ad accordgly he erval [ u u + du] he amou of lqud flowg from he coaers of v ( u ) o he coaers of v ( u ) s du v( u p j) a ( j) ; j = - f Ea( ) he he flow from he coaers of v ( u ) o he coaers of v ( u ) has o be dsrbued amog he levels v( u p ) accordg o α ; - f Ea( ) ad Ea( ) he he flow from v ( u ) o v ( u ) has o be dsrbued amog he levels ( u p ) as was dsrbued amog he levels v u p v ( ).e. he sae (age) of has o be maaed; - f Ea( ) ad Ea( ) he he lqud flowg from v ( u ) o v ( u ) has o be dsrbued amog he levels v ( u p ) accordg o α.e. he sae of has o be alsed; - f Ea( ) he has o mpac o he flow from v ( u ) o v ( u ). I he followg we provde a se of ODEs whch descrbes he evoluo of each coaer of v ( u ) for every markg of he PN. The fac ha hese ODEs sasfy he codos lsed above ca be see by sraghforward bu cumbersome algebrac seps based o he summao provded (3). For a vecor l = l l... l of markg ad a dex we wll deoe by f ( l ) he se of vecors whch dffers from l a mos poso.e. f l ) = { l... l k l... l : k k = / l }; oe ha l f ( l ). ( + Wh a gve markg lquds flow o he coaer l... from aoher l coaer k... of markg f : k f ( l ) ad a rae A ( k l ). Lqud k flows away sead from coaer l... of markg a rae A ( l l ). l = The coaers of oher markgs from whch lquds flow oward coaer l l... of markg ca be defed as follows. From a coaer = l k = k... k codos hold of markg flud flows o l... of markg f he followg l. : ;. = / : f Ea( ) Ea( ) he we mus have = k ; p 3. = / : f Ea( ) ad Ea( ) he we mus have α ( ) = / ; 4. f Ea( ) he we mus have a ( ) = / ; k 5. f Ea( ) he we have mus a ( ) = / ad α ( ) = /. k l l p Codo () smply saes ha here mus be a raso ha akes he sysem from markg o markg. As descrbed by codo () f s o he raso ha fres ad s eabled boh ad he he lqud descrbg he sae of he E dsrbuo of raso flows from a coaer of o he correspodg oe such a way ha he age (sae) of he raso s maaed. The remag hree
45 P. Buchholz A. Horváh ad. Telek codos have a effec also o he rae a whch lqud flows from l... of l markg o k... of markg. I parcular as codo (3) saes f s k o eabled bu s eabled he lqud flows oly oward hose coaers of ha correspods o local coaers of ha have o be alsed o a o-zero level. The effec of o he rae of he flow s gve by α ( ). If raso s o l p eabled markg he corbues o he flow accordg o codo (4) oly f s local coaer has a flow oward s fcous coaer represeg he ermao of he acvy assocaed wh. The assocaed rae s a ). Codo (5) saes ha f ( k raso s eabled he corbues o he flow f s local coaer k has a flow oward s fcous coaer ad s local coaer l s wh o-zero al lqud level. The assocaed rae s a k ) α ( l ) =. ( / We deoe by g ( l) he se of couples ( k) = / for whch here s a flow from coaer k = k... k of markg o coaer l = l... l of markg. Based o he above descrpo we ca wre he chage of he level of he lqud prese coaer l = l... l of markg as v( u l... l du ) = v( u l... l ( = k f ( l ) ( k ) g ( l) ) A ( l l ) + = v u k... k ) A ( k l ) + (5) ( v( u k.. k ) a ( k ) ( α ( l )) α ( l Ea( ) = / Ea ( ) Ea ( ) where ( α ( l )) gves Ea ( ) α ( l ) f Ea( ) ad oherwse. I marx oao (5) ca be wre as () whch complees he proof. A posve cosequece of Theorem s ha he rase behavour of a PN wh E dsrbued frg mes ca be aalysed based o smlar dffereal equaos as case of arkova PN models bu due o he more geeral srucure of E dsrbuos he cosa coeffces of he dffereal equao do o obey sg resrcos. 4. Saoary Behavour Le us deoe he saoary soluo by w( ) = lmu v( u ) Theorem w ( ) sasfes followg balace equao = w( ) A + w( ) R ( ) Ea( ) : T. where R ( ) s defed (). The proof of he heorem s obaed smply from he rase dffereal equao by akg he u lm. Equao () s a sysem of lear equaos where he cosa marx obaed from A ad R ( ) ca coa posve ad egave elemes a ay poso. The ormalzed soluo of () s uque f s a uque egevalue of he cosa marx hs case he saoary markg dsrbuo ca be obaed as he p )) (6)
Sochasc Per Nes wh Low Varao arx Expoeally Dsrbued Frg Tme 45 properly ormalzed soluo of hs se of lear equaos. 5 Two SPN Examples The frs e a SPN wh sychrosed acves s eded o compare Erlag ad E dsrbuos modelg low coeffces of varao. The basc e s show Fgure 5. We assume ha he al markg s gve by pug okes a he places p ad p. Frs we cosder a cofgurao where raso has a expoeally dsrbued frg me wh mea. ad he rasos ad 3 have decally dsrbued E or Erlag dsrbued frg mes. We apply he E dsrbuos wh 3 7 ad 5 phases whch have bee defed above ad compare hem wh Erlag dsrbuos wh he same umber of phases. I all cases we assume ha he mea frg me of he dsrbuos s ad he coeffce of varao s as low as possble. Table coas he umber of saes ad he umber of o-zero elemes he overall geeraor marx. The umber of saes depeds oly o he umber of phases ad o o he o-zero srucure of he marces for he dsrbuos. The umber of o-zero elemes he resulg geerag marx depeds o he umber of o-zero elemes he marx ad he al vecor of he dsrbuos. Sce he E dsrbuos have more o-zero elemes s vecor ad marx he geeraor marx becomes more dese whe usg E sead of Erlag dsrbuos. Table : Number of Saes ad No-zero Elemes for p3 Dffere Number of Phases for Dsrbuos of ad 3. p Erlag E phases saes o zeros o zeros p 3 96 365 839 p4 3 7 54 977 877 Fgure 5: SPN wh 3 4674 98 sychrosed acves 5 8 86597 7366 As he measure of eres we cosder frs he oke dsrbuo place p 3. For deermscally dsrbued frg mes of he rasos ad 3 wh he same mea he model s equvale o a /D// queueg model wh mea er-arrval ad mea servce me equal o. Fgures 6 ad 7 coa he resuls for dffere cofguraos ad clude for comparso he resuls for a /D// queue ad for he same e wh expoeally dsrbued frg mes. I ca be clearly see ha wh he same umber of phases E dsrbuos approxmae deermsc dsrbuos much beer ha Erlag dsrbuos do. I parcular he E dsrbuos wh ad 5 phases ha have a very small coeffce of varao approxmae he oke dsrbuo of he model wh deermsc dsrbuos que well oly for populao 9 ad he probables are uderesmaed.
45 P. Buchholz A. Horváh ad. Telek Fgure 6: Toke Dsrbuo a place p 3 for he Erlag ad E dsrbuos wh 3 ad 7 phases Fgure 7: Toke Dsrbuo a place p 3 for he Erlag ad E dsrbuos wh ad 5 phases Addoally we aalyse he rase behavour of he e whe all rasos have E or Erlag dsrbued frg mes wh 5 phases ad mea. The resuls s show Fgure 8. The E dsrbuo resuls a good approxmao of a sep fuco whch would occur a deermsc sysem. p p p 9 p p9 p3 8 p4 4 p5 6 p3 p6 p4 p p7 8 p8 3 Fgure 9: SPN odel of Produco Cells Fgure 8: Trase Toke Populao place p 3 The secod example we cosder a SPN model of produco cells s from [] ad ca be see Fgure 9. The e descrbes wo cosecuve produco cells wh wo ypes of maeral o be processed. The maches are subjec o falures (rasos ) ad repars ( 9 ). For furher explaaos of he model we refer o []. We assume ha he rasos 9 ad have expoeally dsrbued frg mes wh rae. for rasos 9 ad ad rae. for rasos ad. The remag rasos have E frg mes wh mea. for 6 ad mea.5 for 7 ad 8. I Fgure he rase populao of place p (ha s he mea umber of okes a p ) he erval [] s show for E dsrbuos wh 3 ad 5 phases. Furhermore he e has bee aalysed wh expoeally dsrbued frg mes a all rasos. I ca be see ha wh expoeally dsrbued frg mes for all rasos he populao coverges quckly owards he seady sae whereas he E dsrbuos show a cyclc behavour whch s smoohed by he frg of he expoeal dsrbuos ad.e. by falures durg he processg sep. For he E dsrbuo wh 5 saes he rase populao coas peaks ha descrbe cycles whou falures ad hose wh a sgle falure. (The probably of wo falures durg a cycle s so small ha here s o a correspodg peak he rase probables.) Ths behavour s o vsble f expoeal dsrbuos or wh E dsrbuos wh 3 phases are used. 7
Sochasc Per Nes wh Low Varao arx Expoeally Dsrbued Frg Tme 453 Fgure : Trase Toke Populao Fgure : Sojour Tme Desy of a Place p Sgle Processg Sep Fally he desy of he sojour me for a sgle ru of he e s compued. The ru sars he al markg ad eds whe boh okes reached he places p 7 ad p 8. Fgure shows he desy fuco whch s smooh for expoeally dsrbued frg mes. For he E dsrbuo wh 5 phases we ca aga observe he wo modes wh ad whou falure. 6 Coclusos I hs paper we dscussed a mehodology o evaluae he rase ad he saoary parameers of sochasc Per es wh frg mes whose scv s low. I parcular we proposed he use of E dsrbuos. We preseed E dsrbuos wh low scv ad proved he wdespread cojecure ha SPNs wh E frg mes ca be solved applyg he exeso approach developed for SPNs wh PH frg mes. We demosraed hrough umercal examples he beefs of usg E dsrbuos wh low scv sead of usg Erlag dsrbuos whch are he PH dsrbuos wh lowes possble scv. Refereces [] Ajmoe-arsa. G. Balbo A. Bobbo G. Chola G. Coe ad A. Cuma. The Effec of Execuo Polces o he Semacs ad Aalyss of Sochasc Per Nes. IEEE Trasacos o Sofware Egeerg 989; 5(7): 83-845. [] Asmusse S. ad. Blad. Po Processes wh Fe-dmesoal Codoal Probables. Sochasc Processes ad her Applcao 999; 8:7-4. [3] Asmusse S. ad C. A. O'Cede. arx-expoeal Dsrbuos :Dsrbuos wh a Raoal Laplace Trasform. Ecyclopeda of Sascal Sceces 997; New York Joh Wley & Sos 435-44. [4] Bea N. G. ad B.F. Nelse. Quas-brh-ad-deah Processes wh Raoal Arrval Process Compoes. Sochasc odels ; 6(3):39-334. [5] Blad. ad. F. Neus. arx-expoeal Dsrbuos: Calculus ad Ierpreaos va Flows. Sochasc odels 3; 9():3-4. [6] Buchholz P. ad. Telek. Sochasc Per Nes wh arx Expoeally Dsrbued Frg Tmes. Performace Evaluao ; 67:373-385. [7] Éleö T. ad S. Rácz ad. Telek. mal Coeffce of Varao of arx Expoeal Dsrbuos. d adrd Coferece o Queueg Theory adrd Spa 6; Absrac. [8] Lpsky L. Queueg Theory: A Lear Algebrac Approach. Sprger 8. [9] Scarpa.. No-arkova Sochasc Per Nes Wh Cocurre Geerally Dsrbued Trasos. Ph.D. Thess Deparme of Compuer Scece Uversy of Tur 999. [] Neus. F.. arx-geomerc Soluos Sochasc odels: A Algorhmc Approach. Dover 98. [] Vcaro E. ad L. Sassol ad L. Careval. Usg Sochasc Sae Classes Quaave
454 P. Buchholz A. Horváh ad. Telek Evaluao of Dese-Tme Reacve Sysems. IEEE Tras. Sofware Eg. 9; 35(5):73-79. Peer Buchholz holds a Dploma degree compuer scece (Dpl. -Iform. 987) a Docoral degree (Dr.rer.a. 99) ad a Hablao degree (996) all from he Uversy of Dormud where he s currely a professor for modelg ad smulao. Hs research eress clude echques for performace ad fucoal aalyss of dscree eve dyamc sysems especally he worked o he developme of umercal aalyss echques for large arkov chas. Furhermore he developed sofware ools for he qualave ad quaave aalyss of complex sysems ad appled he aalyss echques ad ools o applcaos from varous areas cludg commucao sysems ad logsc eworks. Adrás Horváh receved he. Sc. degree Compuer Scece from he Budapes Uversy of Techology ad Ecoomcs 998. From 998 o he was a Ph.D. sude a he same uversy. From 3 he s a researcher a he Uversy of Tur (Ialy). Hs research eress are he area of sochasc processes cludg performace aalyss of o-arkova sysems ad modelg ssues of commucao eworks. klós Telek receved he. Sc. degree Elecrcal Egeerg from he Techcal Uversy of Budapes 987. Sce 99 he has bee wh he Deparme of Telecommucaos of he Techcal Uversy of Budapes where he s a full professor ow. He receved he C.Sc. ad D.Sc. degree from he Hugara Academy of Scece 995 ad 4 respecvely. Hs curre research eres cludes sochasc performace modelg ad aalyss of compuer ad commucao sysems.