INTRODUCTION TO QUEUING MODELS

Transcription

1 UNIVERSITY OF MARIBOR FACULTY OF LOGISTICS Wokg ae INTRODUCTION TO QUEUING MODELS Deja Daga Bout Jeeb Celje, Jue 3

2 CONTENT BASICS OF MODELING AND SIMULATION...4. Descto of the system...7. Defto of the model ad smulato.3 Relatosh betwee the system ad the model...4 Modelg ad smulato...5 Modelg methodology.6 Model classfcato. 3.7 Mathematcal modelg 4.8 Theoetcal ad exemetal modelg..5.9 Comute smulato methodology..7. Modelg ad smulato teatve ocedue..9. The classfcato of smulato.... The methodology of system dyamcs DISCRETE EVENT SIMULATION..4. Itoducto of tme...4. Ite-aval tmes dstbuto Posso dstbuto 6.3 Dstbuto of the umbe of avals Itoducto to Queueg systems.8.4. Basc chaactestcs of queueg system 3.4. Queueg temology ad basc aametes Tyes of queueug systems 39.5 Some obablty bascs of smulato Radom geeatos 43

3 3 THEORY OF STOCHASTIC PROCESSES Defto of stochastc ocesses ad basc oetes Makov ocesses Makov chas Posso ocesses Devato of dstbuto of the umbe of evets Devato of dstbuto of the tmes betwee evets Bth ocesses Death ocesses Bth-Death ocesses 65 4 INTRODUCTION TO BASIC QUEUEING MODELS.7 4. Sgle chael queueg models Basc model M/M/ Model M/M/ wth lmted watg sace Model M/M/ (gadually "scaed" o "balked" customes) Model M/M/ wth lmted umbe of customes Model M/M/ wth addtoal seve fo loge queues Multle chael queueg models Basc model M/M/ Model M/M/ wth lmted watg sace Model M/M/ wth lage umbe of seves Model M/M/ (watg sace s ot allowed) Model M/M/ wth lmted umbe of customes Cocluso about the queueg models... LITERATURE

4 BASICS OF MODELING AND SIMULATION The buldg of models by meas of obsevatos ad study of model oetes ae the ma comoets of the mode sceces. Models ca have moe o less fomal chaacte ("hyotheses", "laws of atue", "aadgms", etc) ad ae tyg to ute obsevatos to a atte, whch has the same chaactestcs as the obseved system [Ljug]. The system s cofed aagemet of mutually affected ettes (ocesses) that fluece oe aothe, whee the ocess dcates the coveso ad/o tasot of mateal, eegy ad/o fomato [Isema]. Whe we teact wth a system, we eed some cocet of how ts vaables elate to each othe. Wth a boad defto, we call such a assumed elatosh amog obseved sgals a model of the system. A model of the system s ay exemet, whch ca be aled to ode to aswe some questos about that system. Ths mles that a model ca be used to aswe questos about a system wthout dog exemets o the eal system. Istead of ths, we athe efom smlfed exemets o the model, whch tu ca be egaded as a kd of smlfed system that eflects oetes of the eal system. I the smlest case a model ca just be a ece of fomato that s used to aswe questos about the system [Ftzso]. Models, just lke systems, ae heachcal atue. We ca cut out a ece of a model, whch becomes a ew model that s vald fo a subset of the exemets fo whch the ogal model s vald. Natually, thee ae dffeet kds of models deedg o how the model s eeseted, lke metal models, vebal models, hyscal models, mathematcal models, etc. Metal models hels us aswe questos about esos' behavou, vebal models ae exessed wods, hyscal models ae the hyscal objects that mmcs some oetes of the eal system, ad the mathematcal models ae a descto of the system whee the elatoshs betwee vaables of the system ae exessed mathematcal fom [Ftzso]. I may cases, thee s a ossblty of efomg exemets o models stead of o the eal systems coesodg to the models. Ths s actually oe of the ma uses of models, ad s deoted by the tem smulato, fom the Lat "smulae", whch meas to eted. We usually defe a smulato as follows: "A smulato s a exemet efomed o a model". 4

5 The smulato allows to make the eeated obsevato of the model, whee oe o may smulatos ca be doe. Afte that, the aalyss ca be efomed, whch ads the ablty to daw coclusos, vefcatos ad valdatos of the eseach, ad make ecommedatos based o vaous teatos o smulatos of the model. Ths way, the modelg ad smulato gve us the stog oblem-based dscle, whch allows the eeated testg of a hyothess [Sokolowsk]. Smulato, as a way of solvg of motat ad comlex oblems, s the vey old eseach dscle. Fo examle, the ces ad ules the yeas befoe ou cout, efomed the dffeet ossble stateges of the eemes ad the aswes, whle dog the mltay execses. Also, may motat comlex ad tecoected dustes, the moe tutve tha scetfc smulato methods wee used the boadest sese. Whe the fst comutes have occued, the smulato s becomg a scetfc dscle ad the at of the system theoy aoach. Nowadays, the age of alcato of smulato models s wdely ad exteds to all aeas of scece, esecally ogazatoal, dustal, ecoomcs, tasotato, techcal ad othe motat sceces. Oe of the ma uoses of the smulato s also the aalyss of the esoses of a system the futue, o the cease of udestadg of the system ude cosdeato. I ths way, costs fo exemets o the eal systems ad otetal hazads ca be avoded. I addto, smulato s used whe the teated oblem s vey comlex ad ca ot be solved by othe methods. Smulato models ae ot eally exclusvely bouded oly to the comute, but the use of comute smulato s owadays so exteded, that the wod "comute smulato" became the syoymous of smulato as a techque of oblem solvg [Kljajč]. Fo the wde use of smulato busess evomets, the smle ad effcet solutos ae eeded. They must be coodated wth the eal data ad should ot demad too hgh level of secalzed comute kowledge. Ths s esecally motat fo the easo that the busess smulato based data ae dedcated mostly to maages. The latte must take quck decsos, whch should be based o elable ad u to date fomato. Acceleatg the deloymet of busess systems based smulato s the cosequece of moved commucato betwee the huma ad the mache, whch eabled moe fequet use of smulato models by maages also, esecally whe the busess models ae coceed [Kljajć]. 5

6 Nowadays, the hghly sohstcated gahcal tefaces eables much ease model buldg tha befoe. Thus, smulato, desg of exemets, testg of dffeet sceaos ad the aalyss of system behavo s ossble fo almost eveybody wth the basc kowledge of fomatcs. Cosequetly, the busess smulato s actually tasfeed fom hghly secalzed laboatoes to the desks of commo uses [Kljajč]. Dffeet eseaches the wold ad the exstg lteatue [Djk, Klajč, Saltma, Tug] show that the combato of smulato ad decso suot systems eable decsos of hghe qualty. Smulato aled by addtoal use of amato, whch shows the oeatos of modeled system, ca hel the uses to lea the secfcs of the system wokg mechasm vey quckly. Eve moe, the eseach [Djk] t s cofmed that the decso makes bette udestad the smulato esults, whe they ae eeseted wth amato also. Thus, the combed smulato ad amato motvate the decso makes to seach fo a ew solutos, sce the testg of sceaos, mossble the eal wold, s eabled the smulated wold [Klajč]. Hstocally, scetsts ad egees have cocetated o studyg atual heomea whch ae well-modeled by the laws of gavty, classcal ad o-classcal mechacs, hyscal chemsty, etc. I so dog, we tycally deal wth quattes such as the dslacemet, velocty, ad acceleato of atcles ad gd bodes, o the essue, temeatue, ad flow ates of fluds ad gases. These ae cotuous vaables the sese that they ca take o ay eal value as tme tself cotuously evolves. Based o ths fact, a vast body of mathematcal tools ad techques has bee develoed to model, aalyze, cotol ad smulate the systems (cotuous smulato). It s fa to say that the study of oday ad atal dffeetal equatos cuetly ovdes the ma fastuctue fo system aalyss ad cotol. But the today lfe of ou techologcal ad ceasgly comute-deedet wold, we otce two thgs. Fstly, may of the quattes we deal wth ae dscete, tycally volvg coutg tege umbes (how may ats ae a vetoy, how may laes ae o a uway, how may telehoe calls ae actve). Secodly, may of the ocesses deed o stataeous evets such as the ushg of a butto, httg a keyboad key, o a taffc lght tug gee. I fact, much of the techology we have veted s evet-dve, lke commucato etwoks, maufactug facltes, o the executo of a comute ogams [Cassadas]. 6

7 I ths case, we ae talkg about the dscete ad/o evet dve smulato, whee the states of the system ae chagg the dscete tme momets. These chages ae called the dscete evets, whch ae haeg eodcally secfc tme momets, o asychoously the deedece of codtos, detemed by values of state vaables [Zuačč]. Dscete evet smulato, amog othe fuctoaltes, eables the detemato of effcecy of exstg techology ad the detfcato of bottleecks, the detemato of tme deedece fo suly of oducts vetoy cotol, the desg of the model fo oeatoal lag of oducto, the aalyss of the system the futue, etc [Kljajč]. I the ast, the comlexty of the model costucto ad smulato exemets lmted us to use the smulato the busess evomets. Nowadays, ths oblems ae mostly bdged, esecally, whe the vsual teactve modelg s ossble. The latte eables us to develo the model ad do smulatos the teactve evomet, whch smlfes the model desg ad move the eceto of the system efomace [Kljajć].. Descto of the system The system s a gou of elemets o uts, whch ae coected a ceta whole. Each elemet have ceta oetes, o attbutes ad actvtes, o actvtes. Fgue shows the basc classfcato of systems. Fgue : The basc classfcato of systems. 7

8 Natually, thee ae moe o less comlex systems the eal wold. Fo examle, fgue shows a elatvely comlex system: a house wth sola-heated wam ta wate, togethe wth clouds ad sushe [Ftzso]. Fgue : A house wth sola-heated wam ta wate Fgue 3 shows a less comlex secod-ode systems: a mechacal system ad a electcal system. Fgue: A mechacal system ad a electcal system I geeal, the system ca be descbed by (c.f. fgue 4): Iut vaables X = { X }, =,,3,..., m States of the system Z = { Z }, =,,3,..., Oututs fom the system Y = { Y }, =,,3,..., l 8

9 Fgue 4: System Table shows the examles of dffeet dystems, the elemets, oetes, ad actvtes. Table : The examles of dffeet dystems, the elemets, oetes, ad actvtes State of the system s defed as a value of state vaables at ceta tme. It ca be descbed by the elemets of the system, the oetes ad actvtes. Pocess s the chage of the state of system, duced by the fluece of ut vaables o teal evets the system [Kljajč]. The behavo of the system ca be defed as a esose (eacto) of the system to ut sgals (stmul). Fgue 5 shows the esose y ( t ) of the system to ste ut sgal x( t ) [Zuačč, eg]. Fgue 5: The esose y ( t ) of the system to ste ut sgal x( t ) 9

10 . Defto of the model ad smulato Model of the system s smlfed vsualzato (cocet) of the eal system. System smulato s the methodology of oblem solvg by meas of exemetato o the comute model, whee the ma uose s to aalyze the oeatos of the whole system, o the oeatos of atcula ats of the system ude ceta codtos [Kljajč]. Smulato s a dyamc vsualzato of the system behavo fo the followg uoses [Kljajč]:. The descto of the system o ts ats,. The exlaato of the system behavo the ast, 3. The edcto of the system behavo the futue, ad 4. The udestadg of the system cles. Whe teatg the comute smulato, the modelg ocedue cossts of the followg stes [Kljajč]: defto of the oblem, detemato of objectves, a daft of study, ceato of the mathematcal model, ceato of the comute ogam, valdato of the model, eaato of the exemet (smulato sceaos), smulato ad aalyss of the esults.

11 .3 Relatosh betwee the system ad the model Relatosh betwee the system ad the model ca be defed the followg way (c.f. fgue 6) [Kljajč]:. The system s detemstc ad the model s detemstc. Examle of ths ae the models of smle mechacal systems, lke fo stace the secod ode dffeetal equato:... x+ B x+ C x = U t (.). The system s detemstc ad the model s stochastc. Examle of ths s fo stace the methodology of smlfcato of comlcated fuctos by meas of Mote Calo method. 3. The system s stochastc ad the model s detemstc. Examle of ths ae fo stace the coguetal geeatos of adom umbes. 4. The system s stochastc ad the model s stochastc. Examle of ths ae fo stace the comlex ogazatoal systems, whee the solvg by use of system smulato s eeded. Fgue 6: Relatosh betwee the system ad the model.4 Modelg ad smulato Modelg eesets the elato betwee the smulated system ad the model, whle the smulato eesets the elato betwee the model ad comute ocess. Wth ths hlosohy, the followg ca be eucated [Klajč]: Object X smulates object Y oly, f: a) X ad Y ae both systems, b) Y eesets the smulated system (system), c) X eesets the smlfcato of smulated system (model), d) Valdty of the X deedece of Y s ot ecessay comleted.

12 The followg ca be also stated: Smulato s the ocess of geeato of model behavo. By othe wods, t eesets the exemetg o the model [Kljajč]..5 Modelg methodology Whe the oblem as a subject of study s ecsely defed, the modelg ocedue ca beg. Wth ths famewok, all motat vaables ad the tecoectos must be defed at fst. We focus oly o elevat data, whch ae domat fo the chose asect of the system study. At ths stage, we ca ot defe some secfc ules, how to aoach to ths matte. The oly exceto ae the atual systems, whee the cosevato laws ad cotuty ae stadadzed. The basc ocedue of modelg s show o fgue 7 [Kljajč]. Fgue 7: The basc ocedue of modelg Reseache ca make ceta smlfcatos wth esect to the objectves ad kowledge of the system. He ca deteme the stuctue of the system, collect all the data ad the buld the sutable model wth the famewok of exstg theoy. I sequel, he ca study the oetes of the eal system by use of model, o tes to choose the most aoate stategy of fluecg o the eal system. Vew of the modelg ad smulato ocedue fom the ceta agle s show o fgue 8 [Kljajč]. Fgue 8: The modelg ad smulato ocedue

13 Wth esect to the way of descto ad the sequece of fomato, the followg classfcato of models ca be toduced [Kljajč]: Vebal models, whee the cles of the obseved system ae descbed the atual laguage. Physcal models, whch ae usually the matue mages of the obseved system. The ca be vey useful fo ths kd of eseaches, whee the exemetg wth the ogal system could be exesve o eve dageous. Mathematcal o fomal models, whch ae the most ecse desctos of ceta system..6 Model classfcato Thee ae may classfcatos of models. Oe of the ossble classfcato was toduced by [Foeste] ad s show o fgue 9. Fgue 9: Classfcato of models by [Foeste] The choce of model deeds o the system, whch s take to the cosdeato. Fom fgue 9 t s evdet that the models ca be dvded to the hyscal ad the abstact models as a ossble way of studyg of systems. Natually, evey chose model s deedet o the 3

14 theoy, whch was used to costuct t. Wth esect to the abstacto of the used theoy, we ca talk about exact o vebal theoes, whch deeds o heomeo, whch we wat to teet. Physcal models ae usually smlfed ad educed eal systems, whch ecsely defe the behavo of the eal system, esecally ts moe motat oetes the ceta wokg evomet ad ude secfc codtos. Statc hyscal models ae fo stace the llustato of ubaty o achtectoc solutos the fom of models, whch eables vsual magatos of the ceta satal fom. Dyamcal hyscal models ae fo examle aeodyamc tuels fo vestgato of the acaft oetes, whle hydodyamc chaels ae used fo the vestgato of hydodyamc oetes of shs. Good oetes of dyamcal hyscal models ae the cleaess ad tasaecy. But they have also bad oetes, sce they ae usually too bg, flexble ad ofte do ot eflect causalty deedecy betwee ceta heomeo ad vaables..7 Mathematcal modelg Mathematcal model s the abstact vsualzato of the ceta system ad s useful fo vestgato of system oetes. It eables some coclusos about the chaactestcs ad behavo of the system. Mathematcal models ae moe o less homomohc to the eal system. Fom the level of model sutablty t s deedet, f the acheved esults ad coclusos ae elable ad sgfcat eough. At ths lace t must be oted, that the actce ad develoes exeeces ae vey cucal attemts to desg a adequate model, whch suffcetly eflects the oetes of the system. The mathematcal models ca be classfed to vey dffeet categoes, lke: gahs, tables, logcal symbols, etc, whch eeset ceta state of the system ad ts behavo. Due to the ecso exessve owe ad ossblty of aalyss of futue behavo of the systems quattatve fom, they ca be vey teested fo the System theoy. By the hel, the behavo of the systems ca be aalyzed ad the decso makg o system cotol ca be aled. But we must be awae that t s ot always ossble to fd a aoate mathematcal model. 4

15 .8 Theoetcal ad exemetal modelg Fo the devato of mathematcal models of dyamc systems, oe tycally dscmates betwee theoetcal ad exemetal modelg [Isema]. Fo the theoetcal aalyss, also temed theoetcal modelg, the model s obtaed by alyg methods fom calculus to equatos as e.g. deved fom hyscs. Oe tycally has to aly smlfyg assumtos coceg the system, as oly ths wll make the mathematcal teatmet feasble most cases. I geeal, the followg tyes of equatos ae combed to buld the model [Isema]:. Balace equatos: Balace of mass, eegy, mometum. Fo dstbuted aamete systems, oe tycally cosdes ftesmally small elemets, fo lumed aamete systems, a lage (cofed) elemet s cosdeed.. Physcal o chemcal equatos of state: These ae the so-called costtutve equatos ad descbe evesble evets, such as e.g. ductace o the secod Newtoa ostulate. 3. Pheomeologcal equatos: Descbg evesble evets, such as fcto ad heat tasfe. A etoy balace ca be set u f multle evesble ocesses ae eset. 4. Itecoecto equatos accodg to e.g. Kchhoff s ode ad mesh equatos, toque balace, etc. By alyg these equatos, oe obtas a set of oday o atal dffeetal equatos, whch fally leads to a theoetcal model wth a ceta stuctue ad defed aametes f all equatos ca be solved exlctly. I may cases, the model s too comlex o too comlcated, so that t eeds to be smlfed to be sutable fo subsequet alcato. Eve f ths s ot ossble, the dvdual model equatos may cases stll gve us a motat hts coceg the model stuctue ad thus stll ca be useful [Isema]. I case of a exemetal aalyss, whch s also temed detfcato, a mathematcal model s deved fom measuemets [Isema]. Hee, oe tycally has to ely o ceta a o assumtos, whch ca ethe stem fom theoetcal aalyss o fom evous (tal) exemets. Measuemets ae caed out ad the ut as well as the outut sgals ae subjected to some detfcato method ode to fd a mathematcal model that descbes the elato betwee the ut ad the outut [Isema]. 5

16 The theoetcally ad the exemetally deved models ca also be comaed f both aoaches ca be aled. If the models do ot match, the oe ca get hts fom the chaacte ad the sze of the devato, whch stes of the theoetcal o the exemetal modelg have to be coected [Isema]. The system aalyss ca tycally ethe be comletely theoetcal o comletely exemetal. To beeft fom the advatages of both aoaches, oe does aely use oly theoetcal modelg (leadg to so-called whte-box models) o oly exemetal modelg (leadg to so-called black-box models), but athe a mxtue of both leadg to what s called gay-box models (c.f. fgue [Isema]). Deste the fact that the theoetcal aalyss ca cle delve moe fomato about the system, ovded that the teal behavo s kow ad ca be descbed mathematcally, exemetal aalyss has foud eve ceasg atteto ove the ast 5 yeas. The ma easos ae the followg [Isema]: Theoetcal aalyss ca become qute comlex eve fo smle systems. Mostly, model coeffcets deved fom the theoetcal cosdeatos ae ot ecse eough. Not all actos takg lace sde the system ae kow. The actos takg lace caot be descbed mathematcally wth the equed accuacy. Some systems ae vey comlex, makg the theoetcal aalyss too tme-cosumg. Idetfed models ca be obtaed shote tme wth less effot comaed to theoetcal modelg. The exemetal aalyss allows the develomet of mathematcal models by measuemet of the ut ad outut of systems of abtay comosto [Isema]. Oe majo advatage s the fact that the same exemetal aalyss methods ca be aled to dvese ad abtaly comlex systems. By measug the ut ad outut oly, oe does howeve oly obta models goveg the ut-outut behavo of the system,.e. the models wll geeal ot descbe the ecse teal stuctue of the system. These ut-outut models ae aoxmatos ad ae stll suffcet fo may aeas of alcato [Isema]. 6

17 Fgue : Dffeet kds of mathematcal models agg fom whte box models to black box models [Isema].9 Comute smulato methodology As metoed the evous chate, the aalytcal soluto of the dffeetal equatos, whch descbe a ceta dyamcal system, ca be foud oly fo the smlest ad most dealzed systems. I the case of moe comlcated systems of dffeetal equatos, the use of umecal methods s usually the oly ossble way to fd the solutos. Wth ths famewok, the comute smulato s deftely the most oula aoach. I last two decades, thee wee may oblem oeted smulato laguages develoed. Natually, the coesodg modelg demad vey well kow kowledge of the oblem, whch we wat to vestgate by meas of comute smulato. Fgue shows the basc modelg aoach fo the uose of comute smulato. 7

18 Fgue : The basc modelg aoach fo the uose of comute smulato. The basc modelg aoach fo the uose of comute smulato cossts of the followg stages [Kljajč]: a) Defto of the oblem: Defe the level ad the objectve of the modelg, the volume of the teated system, etc. b) Defe the vaables, the feedback loos, ad teactos betwee vaables ad the ats of the system. c) Aalyze the oblem the wde fame, whch establshes the coecto betwee the obseved system ad cocete solutos fom the moe geeal ot of vew. d) Costuct mathematcal detals of the system wth coesodg equatos, sutable fo the chose smulato laguage. e) Reeat all the stages utl the satsfactoy soluto s ot foud. 8

19 . Modelg ad smulato teatve ocedue I ths chate, let us descbe the modelg ad smulato ocedue moe ecsely. As metoed befoe, evey eal object s obseved by obsevato of avalable data ad exemets, whch ae ossble to be doe o ths object. Mathematcal (cocetual) models ae the costucted o the bass of data aalyss. As we kow, they eeset the mtato of the eal object ad behave smlaly fo the uoses fo whch they seve. Whe buldg the cocetual mathematcal model, the followg ssues must be also take to the cosdeato [Zuačč]: The uose of the modelg must be clealy defed, The costats ad the lmtatos of the model must be also defed, The attbutes of the object, whch wll be cluded the model, must be chose, whee sgfcat detals wll be eglected, The assumtos about ceta eal cles must be dealzed, The stuctue ad aametes of the model must be defed, whch eeset the coectos betwee atcula attbutes the system (fo examle dffeetal o dffeece equatos). Whe the cocetual model s costucted, as much as ossble fomato about ts behavo must be collected. We kow that oly smle cases the aalytcal solutos ca be also deved. But geeal, ths s ot ossble ad the smulato model must be costucted, whch s based o the cocetual model. The data about the model behavo ca be collected by use of deducto (aalytcal teatmet) of cocetual model o by exemetg wth smulato model. Aftewads, the valdato of the model must be aled, whee the aalyss of fttg of eal data ad smulated data s doe. Natually, the vefcato of smulato model must be also executed, whee t s tested, f the smulato model eflects the oetes of cocetual model the oe way (comute smulato ogam s wthout ay faults). The aalyss of the system, the costucto of the model, ts exemetg, vefcato ad valdato must be usually eeatedly doe, utl the demaded esults ae ot acheved. Thus, [Neelemkavl] defes smulato as a slow, teatve ad exemetally oeted techque (c.f. fgue ). 9

20 Fgue :Iteatve ocedue of modelg ad smulato [Neelemkavl]. The classfcato of smulato Smulato as a methodology fo aalyss ad desg of systems ca be classfed to cotuous, dscete ad combed smulato [Zuačč], f the classfcato s elated to the tye of the used model. Wth esect to used tool o techque (tye of comute), by whch t s executed, the smulato ca be classfed to aalog, dgtal, o hybd smulato [Zuačč]. Cotuous smulato ca be aalog o dgtal, dscete smulato s always dgtal, whle combed smulato ca be dgtal o hybd. Fgue 3 shows the classfcato of smulato. [Zuačč]. The seed of smulato executo wth esect to eal system tme detemes the way, how the model ca be smulated: slowe tha eal system tme, eal system tme, faste tha eal system tme.

21 Fgue 3: The classfcato of smulato [Zuačč] Smulato, whch s ot executed eal system tme, s the most commo ad ca be aled o geeal-uose comutes. If the executo s faste o slowe tha eal system tme, deeds o tme costats of eal system ad o caablty of smulato tool, how fast ca efom the smulato. If the smulato s doe eal system tme, the comute must be usually coected to the eal system. Effcet smulato the eal system tme s ossble oly the case, f we have the secfc softwae ad hadwae (mode smulato wokg statos, aalog-hybd comutes) [Zuačč]. Cotuous smulato eables to smulate the systems, whch ca be descbed by lea o olea oday (ODE) o atal (PDE) dffeetal equato wth costat o vayg coeffcets. The codto fo ths kd of smulato s that the state vaables ad devatves ae cotuous though the ete smulato executo, whee the deedet vaable s usually tme. I dscete smulato, the states of the system ca be chaged oly dscete tme stats. These chages ae called the dscete evets, whch ae haeg eodcally secfc tme momets (usually the theoy of automatc cotol [Zuačč]), o asychoously the deedece of codtos, detemed by values of state vaables [Zuačč]. Classcal examle of the asychoous smulato s the ost offce wth oe seve ad watg le (queue). The customes ave adomly to the system ad ae seved accodg to the FIFO (fst, fst out) dscle. I ths case, the comute ca smulate the avals of the customes to the system ad wokg mechasm sde the queueg system. I ths way,

22 vey comlex systems ca be teated ad smulated, whee the aalytcal solutos usually could ot be foud at all. Accodg to the [Celle], the combed o hybd smulato s the smulato, whch ca be descbed o the whole obsevato teval by meas of dffeetal equatos, whee at least oe state vaable o ts devatve s ot cotuous quatty. By ths kd of defto, we ca see that ealty almost evey smulato of the eal system s actually combed cotuous-dscete smulato.. The methodology of system dyamcs Thee ae seveal equvalet ways of the vsualzato of the system, sutable fo comute smulato. I the smulato of busess systems the so-called methodology of system dyamcs has foud the most advaced osto, whch was oosed by [Foeste]. I ths methodology, the so-called block dagams fo dscete evet smulato also foud the lace. The methodology of system dyamcs s ot oly the geeato of equatos, whch descbe the dyamcs of busess ocesses, but t s the whole methodology of solvg of dyamcal oblems (c.f. fgue 4) [Foeste]. Fgue 4:The cocet of oblem solvg the methodology of system dyamcs [Foeste]

23 As t ca be otced fom fgue 4, the aows show the ste by ste solvg ocedue, whee the ext ste ca teatvely fluece o the evous ste, whch meas that the coesodg stes ae mutually deedet. The methodology of system dyamcs ca be vey useful fo develomet of decso suot systems busess evomets, whee the followg ssues ca be acheved [Kljajč]: Leag of the behavou of tegal smulato system, whch ca suot the busess decsos, stategcally lag ad aalyss of ogazatoal systems, Imovemet of the ocess of lag ad decso makg, Acqusto of ew kowledge about the behavou ad maagemet of comlex systems, ad Educato of ofessoal esoel fo lag ad govemet of the comay. 3

24 DISCRETE EVENT SIMULATION The focus of dscete evet smulato s o evets, whch ca fluece o the system. These evets ca [Kljajč]: cause the chage of value of ceta system vaable, tgge o dscoect the system vaable, actvate o deactvate the ceta ocess. Dscete evets ca be teated fom the two ots of vew [Kljajč]: I the case of elemets oetato (atcle oetato), the system elemets eeset the statg ot fo smulato aalyss, I the case of evets oetato, the system evets eeset the statg ot fo smulato aalyss.. Itoducto of tme Tme s eeseted wth a teal smulato clock, whe the smulato of dscete evets s teated. Relatosh betwee the smulato tme ad the eal tme deeds o the atue of obseved system, whee the geeato of tmes of avals of elemets (tasactos) deeds o system codtos. The basc tems, whe we study the ways of watg le fomatos, ae [Kljajč]: The sequece of avals of elemets (tasactos) (aval attes), The ocessg of system elemets (tasactos) (sevce ocess), The ways of watg le fomatos (queueg dscles). The ocessg (seve sevce) of the system elemets (tasactos) ca be descbed by the sevce tme ad the caacty of ocessg (caacty of seve, sevce caacty). The ocessg tme (tme of sevce) s the tme eeded fo the ocessg of the system elemet (tasacto, dyamc etty). The caacty of ocessg (caacty of seve sevce) eesets the umbe of system elemets (tasactos), whch ca be oceeded smultaeously. 4

25 Whe modelg the system, the obablty dstbutos of the tmes betwee two cosecutve elemets avals (te-aval tmes) ad the sevce ocessg tme must be gve. Thee ae seveal ossbltes of watg le fomatos (dscles), lke FIFO, LIFO, adom, etc, whch wll be moe ecsely defed the sequel.. Ite-aval tmes dstbuto The avals of the system elemets to the system ae usually descbed by te-aval tmes. I the eal systems, the umbe of dffeet te-aval tmes dstbutos s actcally defte. By meas of theoetcal dstbutos, the eal dstbutos ae tyg to be descbed. Natually, the theoetcal dstbutos ae oly moe o less accuate aoxmatos of the eal dstbutos. The most fequet theoetcal dstbutos, whch ca be used fo the uose of smulato, ae [Kljajč]: ufom dstbuto, exoetal dstbuto, omal dstbuto, ad Elag dstbuto. Fo the descto of elemets (tasactos, customes) avals to the system, the followg aametes ae usually used [Wsto]: T a... Tme teval betwee two cosecutve avals, E ( T ) = = a t f t dt...the mea o aveage te-aval tme, =... the aval ate (uts of avals e tme ut). E ( T a ) Natually, the T a s the adom vaable ad f(t) s suosed to be the obablty desty fucto of adom vaable T a. 5

26 .. Posso dstbuto Suose we ae dealg wth the adom vaable T a, whch eesets the tme teval (, t ] betwee the tme og ad the fst followg evet (aval). I the case of Posso dstbuto of the umbe of avals, t ca be show that the obablty fo T a to take the value betwee t ad t + dt, follows the exoetal law [Daga, Wsto, Taha, Hlle]: Ta t, f t = f t = e t > (.) whee f(t) s the exoetal obablty desty fucto of adom vaable T a, ad s ostve costat. If the te-aval tmes have a exoetal dstbuto, tha t ca be show that they also have the so-called o-memoy oety [Wsto]. Ths fdg s vey motat, because t mles that f we wat to kow the obablty dstbuto of the tme utl the ext aval, the t does ot matte how log t has bee sce the last aval [Wsto]. Fgue 5 shows fou examles of exoetal dstbuto fo aval ates =,,.5,.5. Fgue 5: Fou examles of exoetal dstbuto fo aval ates =,,.5,.5 6

27 Fgue 5 eesets the fomato about the chace, that the ew evet ot yet haeed utl the tme t. Fom fgue 5 we ca cocluded that t s vey ulkely to have vey log te-aval tmes [Wsto, Daga]. The loge the te-aval tme s, the smalle s the chace that the ew evet ot yet haeed..3 Dstbuto of the umbe of avals It ca be show that thee s a stog coecto betwee the Posso dstbuto of the umbe of avals, ad the exoetal dstbuto of the te-aval tmes. If the teaval tmes ae exoetal, the obablty dstbuto of the umbe of avals occug ay tme teval of legth t s gve by the followg motat theoem [Wsto]: Ite-aval tmes ae exoetal wth aamete f ad oly f the umbe of avals to occu a teval of legth t follows a Posso dstbuto wth aamete t. I geeal, a dscete adom vaable N has a Posso dstbuto wth aamete, f the obablty dstbuto fucto has a fom [Wsto]: [ ] P N = = e, =,,,... (.)! If we defe N ( t ) to be the umbe of avals to occu dug ay tme teval of legth t, fom the evous theoem we ca aly the followg exesso [Wsto]: ( t) t P N t = = e, =,,,... (.3)! Sce N ( t ) has a Posso dstbuto wth aamete exectato ad vaace ae [Wsto]: t, t ca be show that the E N t = VAR N t = t (.4) It follows that a aveage of t avals occu dug a tme teval of legth t, so may be thought as the aveage umbe of avals e tme ut, o the aval ate [Wsto]. 7

28 Fgue 6 shows sx examles of Posso dstbuto fo aval ates =.5,,, 3, 5,. Fgue 6: Sx examles of Posso dstbuto fo aval ates =.5,,, 3, 5,..4 Itoducto to queueg systems Queues (watg les) ae a at of eveyday lfe. We all wat queues to buy a move tcket, make a bak deost, ay fo gocees, mal a ackage, obta food a cafetea, stat a de a amusemet ak, etc. We have become accustomed to cosdeable amouts of watg, but stll get aoyed by uusually log wats [Hlle]. Howeve, havg to wat s ot just a etty esoal aoyace. The amout of tme that a ato s oulace wastes by watg queues s a majo facto both the qualty of lfe thee ad the effcecy of the ato s ecoomy. Fo examle, befoe ts dssoluto, the U.S.S.R. was otoous fo the temedously log queues that ts ctzes fequetly had to edue just to uchase basc ecesstes. Eve the Uted States today, t has bee estmated that Amecas sed 37,,, hous e yea watg queues. If ths tme could be set oductvely stead, t would amout to ealy mllo eso-yeas of useful wok each yea! [Hlle] 8

29 Eve ths staggeg fgue does ot tell the whole stoy of the mact of causg excessve watg. Geat effceces also occu because of othe kds of watg tha eole stadg le. Fo examle, makg maches wat to be eaed may esult lost oducto. Vehcles (cludg shs ad tucks) that eed to wat to be uloaded may delay subsequet shmets. Alaes watg to take off o lad may dsut late tavel schedules. Delays telecommucato tasmssos due to satuated les may cause data gltches. Causg maufactug jobs to wat to be efomed may dsut subsequet oducto. Delayg sevce jobs beyod the due dates may esult lost futue busess [Hlle]. Let us codese some most tycal eal cases of queueg systems [Daga]: Watg fo sevce estauats, baks, shos, o at the docto's offce, Watg fo tasfe by bus, ta, o lae, Watg fo a tcket to the cema, theate o game, Watg of cas at a gas stato, Watg of acafts o takeoff o ladg, Watg of machey to be eaed, Watg of the mateal the waehouse fo sale, Watg fo a telehoe call to establsh a coecto, ad so o. The lmted umbe of seves s usually the easo fo watg les, whch ae fomed, whe all the customes ca ot be seved smultaeously. But same cases, the watg les ae also fomed as a cosequece of tme lmtatos, whe the sevce s ossble oly ceta tme tevals. Fo examle, ass though the tesecto wth a taffc lght s oly ossble whe the lght s gee. The ettes that queue fo sevce ae called customes, o uses, o jobs, deedg o what s aoate fo the stuato at had [Stewat]. Customes who eque sevce ae sad to ave at the sevce faclty ad lace sevce demads o the esouce. At a docto s offce, the watg le cossts of the atets awatg the tu to see the docto; the docto s the seve who s subjected to a lmted esouce, ths case tme. The laes may be vewed as uses ad the uway vewed as a esouce that s assged by a a taffc cotolle. The esouces ae of fte caacty meag that thee s ot a fty of them o ca they wok ftely fast. Futhemoe avals lace demads uo the esouce ad these demads ae uedctable the aval tmes ad uedctable the sze. Take 9

30 togethe, lmted esouces ad uedctable demads mly a coflct fo the use of the esouce ad hece queues of watg customes [Stewat]. Ou ablty to model ad aalyze systems of queues hels to mmze the coveece, to maxmze the use of the lmted esouces ad to eable the ossble movemets of the exstg system. A aalyss may tell us somethg about the exected tme that a esouce wll be use, o the exected tme that a custome must wat. Ths fomato may the be used to make decsos as to whe ad how to ugade the system: fo a ovewoked docto to take o a assocate o a aot to add a ew uway, fo examle [Stewat]. Fo the effcet aalyss ad otmzato of queueg systems, the coesodg models must be bult. They ca be used to aswe questos lke the followg [Wsto]: What facto of the tme s each seve dle? What s the exected umbe of customes eset the queue? 3 What s the exected tme that a custome seds the queue? 4 What s the obablty dstbuto of the umbe of customes eset the queue? 5 What s the obablty dstbuto of a custome s watg tme? Queueg theoy s the study of watg systems [Hlle]. It uses queueg models to eeset the vaous tyes of queueg systems (systems that volve queues of some kd) that ase actce. Fomulas fo each model dcate how the coesodg queueg system should efom. Theefoe, they ae vey helful fo detemg how to oeate a queueg system the most effectve way. Povdg too much sevce caacty to oeate the system volves excessve costs. But ot ovdg eough sevce caacty esults excessve watg ad all ts ufotuate cosequeces. So the models eable fdg a aoate balace betwee the cost of sevce ad the amout of watg [Hlle]. 3

31 .4. Basc chaactestcs of queueg system The basc chaactestcs of queueg systems ae [Kljajč]: The dstbuto of te-aval tmes of customes The dstbuto of sevce tmes The umbe of seves The caacty of queueg system The dscle of queue The umbe of sevg levels. I sequel, we ae gog to look at the chaactestcs of the queueg systems moe closely. The Basc Queueg Pocess The basc ocess assumed by most queueg models s the followg [Hlle]. Customes equg sevce ae geeated ove tme by a ut souce. These customes ete the queueg system ad jo a queue. At ceta tmes, a membe of the queue s selected fo sevce by some ule kow as the queue dscle. The equed sevce s the efomed fo the custome by the sevce mechasm, afte whch the custome leaves the queueg system. Ths ocess s dected Fgue 7 [Hlle]. Fgue 7: The Basc Queueg Pocess. Iut souce: Oe chaactestc of the ut souce s ts sze. The sze s the total umbe of customes that mght eque sevce fom tme to tme,.e., the total umbe of dstct otetal customes. Ths oulato fom whch avals come s efeed to as the callg 3

32 oulato. The sze may be assumed to be ethe fte o fte (so that the ut souce also s sad to be ethe ulmted o lmted). The fte case s moe dffcult aalytcally because the umbe of customes the queueg system affects the umbe of otetal customes outsde the system at ay tme. Howeve, the fte assumto must be made f the ate at whch the ut souce geeates ew customes s sgfcatly affected by the umbe of customes the queueg system [Hlle].. Queue: The queue s whee customes wat befoe beg seved. A queue s chaactezed by the maxmum emssble umbe of customes that t ca cota. Queues ae called fte o fte, accodg to whethe ths umbe s fte o fte. The assumto of a fte queue s the stadad oe fo most queueg models, eve fo stuatos whee thee actually s a (elatvely lage) fte ue boud o the emssble umbe of customes, because dealg wth such a ue boud would be a comlcatg facto the aalyss. Howeve, fo queueg systems whee ths ue boud s small eough that t actually would be eached wth some fequecy, t becomes ecessay to assume a fte queue [Hlle]. 3. Queue Dscle: The queue dscle efes to the ode whch membes of the queue ae selected fo sevce. Fo examle, t may be fst-come-fst-seved, adom, accodg to some oty ocedue, o some othe ode. Fst-come-fst-seved usually s assumed by queueg models, uless t s stated othewse [Hlle]. 4. Sevce Mechasm: The sevce mechasm cossts of oe o moe sevce facltes, each of whch cotas oe o moe aallel sevce chaels, called seves [Hlle]. If thee s moe tha oe sevce faclty, the custome may eceve sevce fom a sequece of these (sevce chaels sees). At a gve faclty, the custome etes oe of the aallel sevce chaels ad s comletely sevced by that seve. A queueg model must secfy the aagemet of the facltes ad the umbe of seves (aallel chaels) at each oe. Most elemetay models assume oe sevce faclty wth ethe oe seve o a fte umbe of seves. The tme elased fom the commecemet of sevce to ts comleto fo a custome at a sevce faclty s efeed to as the sevce tme (o holdg tme). A model of a atcula queueg system must secfy the obablty dstbuto of sevce tmes fo each seve, although t s commo to assume the same dstbuto fo all seves. The sevce-tme dstbuto that s most fequetly assumed actce, s the exoetal dstbuto. Othe 3

33 motat sevce-tme dstbutos ae the degeeate dstbuto (costat sevce tme) ad the Elag (gamma) dstbuto [Hlle]. 5. Seves aallel ad seves sees: Ths kd of seves classfcato s most usual the queueg theoy [Wsto]. Seves ae aallel f all seves ovde the same tye of sevce ad a custome eed oly ass though oe seve to comlete sevce. Fo examle, the telles a bak ae usually aaged aallel; ay custome eed oly be sevced by oe telle, ad ay telle ca efom the desed sevce. Seves ae sees f a custome must ass though seveal seves befoe comletg sevce. A assembly le s a examle of a sees queug system. 6. Fte souce models ad heomea of balkg: These ae two tycal stuatos, whch ca hae ealty. I the fst case, the avals ae daw fom a small oulato, fo examle we have the maches, whch ae watg to be eaed (lmted umbe of customes). I the secod case, the aval ocess deeds o the umbe of customes, whch ae aleady eset the system. I ths case, the ate at whch customes ave to the faclty, deceases whe the faclty becomes too cowded. Fo examle, f you see that the bak akg lot s full, you mght ass by ad come aothe day. If a custome aves but fals to ete the system, we say that the custome has balked. Hee we ca dstgush betwee two dffeet stuatos. I fst stuato, the aval ate gadually deceases, whch meas that as bgge the quatty of aleady eset customes the system s, moe balked the ew otetal custome wll become. I the secod stuato, the aval ate s costat, utl the watg sace becomes fully loaded. At that momet, the aval ate mmedately falls dow to ad ew otetal customes wll deftely go away u-seved (examle of lmted watg sace)..4. Queueg temology ad basc aametes Accodg to Kedall, each queueg system s descbed by sx chaactestcs: / / 3 / 4 / 5 / 6, whch ae [Wsto]: - The atue of the aval ocess, - The atue of the sevce tmes, 3 - The umbe of aallel seves, 33

34 4 - The queue dscle, 5 - The maxmum allowable umbe of customes the system (cludg customes who ae watg ad customes who ae sevce), 6 - The sze of the oulato fom whch customes ae daw. Aothe veso of the descto of the queueg system ca be gve as follows [Kljajč]: whee the symbols ae: Notato = A / B / X / Y / Z (.5) A The dstbuto of te aval tmes B The dstbuto of sevce tmes X The umbe of aallel seves Y The lmtatos of sevce caacty Z The dscle of queue (.6) Table shows the meag of symbols (.6), whch deote the basc chaactestcs of queueg systems [Kljajč]. Table : The basc chaactestcs of queueg systems 34

35 Examle: Let us have the followg queueg system: Notato = M / D // 5 / PRI (.7) I ths case, the te-aval tmes ae exoetally dstbuted, the sevce tme s detemstc, we have oe seve wth the total caacty fve ad the sevce has a oty mode of sevg. Eve smle veso of the descto of the queueg system ca be eseted at gve caacty, f the default s FIFO dscle [Wsto, Hlle]: Notato = Ite-aval dstbuto/sevce tmes dstbuto/umbe of seves (.8) The followg stadad abbevatos ae used (ethe fo te-aval tmes o sevce tmes) [Wsto]: M: Posso adom ocess [Wsto] wth the eksoetal dstbuto of tmes, G: Some geeal dstbuto of tmes, D: Detemstc dstbuto of tmes, Ek: Elags dstbuto of tmes wth shae aamete k. If we ae dealg, fo stace, wth the system M/M/, the abbevatos mea the followg: The ut aval ocess s the Posso ocess, the sevce tmes ae dstbuted exoetally, ad the umbe of seves s. O the othe had, f we ae dealg, fo stace, wth the system M/G/, t meas that the ut aval ocess s the Posso ocess, the sevce tmes ae dstbuted wth esect to some geeal dstbuto, ad the umbe of seves s. Let us cout some othe tycal queueug systems: G/M/, M/D/, G/G/, etc. The queueg systems ca be teated as so-called bth-death systems [Wsto, Daga], whee customes the system eeset the obseved oulato, avals ae the bths, ad the deatues ae the deaths. The bth-death ocesses ae the stochastc ocesses, whch wll be moe ecsely exlaed chate 3. 35

36 The systems of tye M/M/ has so-called Makova oety (aleady metoed o-memoy oety), whee the theoy of Makov ocesses ca be aled. O the cotay, the othe tyes of queueg systems do ot have ths oety ad must be teated by use of secal, moe dffcult aoaches [Wsto, Daga]. The basc quattes, whch deseve a lot of atteto the theoy of queueg systems, ae [Schaums, Daga, Wsto]: The umbe of the customes the system ad the queue, ad The customes tme of beg the system ad the customes watg the queue. The basc aametes, whch defe the oety of queueg chael, ae [Kljajč, Daga, Hlle, Wsto]: E ( N ) = L...aveage umbe of the customes the system (cludg those oe, whch ae aleady the sevce ocess), ( q ) E N E N S = L... aveage umbe of the customes the queue, q = L... aveage umbe of the customes the sevce ocess, s E(W )...aveage customes tme of beg the system (sum of the watg tme the queue ad the tme beg the sevce ocess), E( W q )...aveage customes watg tme the queue, E( W s )...aveage customes tme beg the sevce ocess. whee»e«deotes the exectato. (.9) 36

37 Let us defe some othe quattes, whch ae also motat the queueg theoy [Kljajč, Hlle]: State of the system...umbe of the customes the system, Queue legth... umbe of the customes watg fo sevce to beg, N ( t )...umbe of the customes the system at tme t, t...obablty of exactly customes queueg system at tme t,...umbe of the seves queueg system,...mea aval ate (exected umbe of avals e tme ut) of ew customes, whe customes ae aleady the system..mea sevce ate fo oveall system (exected umbe of customes comletg sevce e tme ut), whe customes ae aleady the system. (.) Whe s costat fo all, ths costat s deoted by. Smlaly, whe the mea sevce ate s costat fo all, ths costat s deoted by [Hlle]. Quatty ca also be teated as the seed of avals of tasactos to the system, whle the quatty ca also be teated as the seed of the sevce, fo examle oe chael system the seed of the sgle seve [Kljajč]. I the case of sgle chael system, we ca toduce the followg utlzato facto [Hlle]: ρ = (.) whch eesets the facto of the system s sevce caacty ( ), that s beg utlzed o the aveage by avg customes ( ) [Hlle]. 37

38 I the case of multle chael system, we ca toduce the followg utlzato facto [Hlle]: ρ= (.) Ceta otato also s equed to descbe steady-state esults [Hlle]. Whe a queueg system has ecetly begu oeato, the state of the system (umbe of customes the system) wll be geatly affected by the tal state ad by the tme that has sce elased. The system s sad to be a taset codto. Howeve, afte suffcet tme has elased, the state of the system becomes essetally deedet of the tal state ad the elased tme The system has ow essetally eached a steady-state codto, whee the obablty dstbuto of the state of the system emas the same (the steady-state o statoay dstbuto) ove tme. Queueg theoy has teded to focus lagely o the steady-state codto, atally because the taset case s moe dffcult aalytcally [Hlle]. L L E W E W : Relatoshs betwee, q,, ( q ) It has bee oved, that a steady-state the so-called Lttle's fomula ca be aled [Hlle]: L = E W (.3) Also, we ca toduce the followg exesso: L q ( q ) = E W (.4) ad: E ( W ) = E ( W q ) + (.5) These elatoshs ae extemely motat because they eable all fou of the fudametal quattes ( L, L, W, W ) to be mmedately detemed as soo as oe s foud aalytcally. q q Ths stuato s fotuate, because some of these quattes ofte ae much ease to fd tha othes whe a queueg model s solved fom basc cles [Hlle]. 38

39 .4.3 Tyes of queueug systems I ths chate, let us shotly toduce some tycal queueg systems [Kljajč]. Fgue 8 shows the smle queueg system wth oe queue ad sgle seve. Fgue 8: The smle queueg system wth oe queue ad sgle seve Fgue 9 shows the smle queueg system wth oe queue ad multle aallel seves. Fgue 9: The smle queueg system wth oe queue ad multle aallel seves Fgue shows the closed queueg system of mataace of machey. Fgue : The closed queueg system of mataace of machey 39

40 Fgue shows moe comlex queueg system wth may queues ad may chaels, whee the oty logc s also volved. Fgue : Moe comlex queueg system.5 Some obablty bascs of smulato As metoed befoe, the behavo of comlex ogazato systems must be descbed by meas of stochastc fuctos. Wth ths famewok, the followg stuatos ae ossble [Kljajč]:. The kowledge about the system s comlete, so the system must be descbed adomly wth the aoate dstbuto fucto,. The descto of the system s vey comlcated, thus t must be aoxmated by use of secfc stochastc fucto. 3. The behavo of the system has a sgfcatly stochastc chaacte. Sce the comute smulato eesets exemetg o the comute, the adom evets must be geeated fo the uts ad oututs of the system model. The vaable, whch eesets the outcome of the adom evet, s called the adom (stochastc) vaable. Fo the evets, whch ae descbed by adom vaable, the stock of the values ad the obablty dstbuto s kow oly. Based o these two chaactestcs, the mea (exected) value ad the vaace ca be also foud. If the stock of the values ad the obablty dstbuto ae 4

41 dscete, tha we ae talkg about dscete adom vaables, othewse we ae talkg about the cotuous adom vaables. Let us have some cotuous adom vaable X, whch has the obablty desty fucto f ( x ). The obablty, that the adom vaable X takes some value the teval a X b, ca be deved as follows (c.f. fgue ): b P a X b = f x dx (.6) a whee the followg exesso s always vald: ( x) dx = f (.7) Fgue : Illustato of obablty P ( a X b) Patculaly motat fo the descto of the adom vaable X s the so-called cumulatve dstbuto fucto, whch ca be exessed as follows (c.f. fgue 3): x F x = P X x = f t dt (.8) 4

42 Fgue 3: Illustato of cumulatve fucto We ca gve some addtoal oetes, whch ae vald fo the adom vaable X [Daga]:. Sce the F( x) s mootocally cea s g, t follows : f ( x) df = dx. F( ) = P( X ) = f t dt = = ( ) 3. F b F a P a X b (.9) The exectato of the adom vaable ca be defed as: = E X x f x dx (.) ad the vaace ca be defed as [Daga]: VAR X E X E X x f x dx x f x dx = = (.) 4

43 I the case of dscete adom vaable X, whee stock of values s: x, x, x3,... x ad the coesodg obabltes ae:,, 3,..., wth oety exessos ca be gve: = =, the followg F x = P X x = x = = x x x x = x E X = x x = x x = x x = = VAR ( X ) = E ( X ) E ( X ) = x x = : = : (.).6 Radom geeatos Fo the uose of comute smulato, the so-called adom geeato s used [Kjajč, Wsto]. It seves fo geeatg of adom umbes, whch eeset the tme stats fo avals of customes to the system, ad fo the geeato of tmes elated to the sevce wth esect to some obablty law. Oe of the tycal adom geeatos s so-called oulette adom geeato, whee the behavo smla to oulette behavo s aled. The latte also has the fgets, why the famous smulato Mote Calo method has got ts ame [Wsto]. The ocedue of segmetato ad usg a oulette wheel s equvalet to geeatg tege adom umbes ceta teval, fo examle betwee ad 99 [Wsto]. Ths follows fom the fact that each adom umbe the sequece has a equal obablty of showg u (fo examle, the case of teval betwee ad 99 the obablty s.). I addto, each adom umbe s deedet of the umbes that ecede ad follow t. The defto of adom umbe geeato s [Hlle]: A adom umbe geeato s a algothm that oduces sequeces of umbes that follow a secfed obablty dstbuto ad ossess the aeaace of adomess. 43