ρ (assume a = m Problem 1 Assignment 4 Thomas Adam, Stephan Brumme, Haik Lorenz June 11 th, 2003 master, 1 st semester, , ,
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1 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Problem Istall the OMet dscrete evet smulato pacage the latest verso o your computer ad ru the test smulatos. You ca fd the software at Read the Maual. Set up a M/M/ smulato. Vary the load as.;.;.;.4;.5;.6;.7;.8 (assume a servce rate of oe customer per secod ad vary accordgly). For each create customers (the tal umber of customers the system s zero). determe the mea system respose tme for all the customers (sample mea) loo every. secods at the system observe the umber of customers the system at each samplg pot ad cout how ofte exactly customers are foud ( ). Plot the relatve frequeces a hstogram. Compare your smulato results wth aalytcal results. Submt your code (C) your smulato results ad the correct aalytcal results. Eve though OMet was orgally desged for Ux/Lux systems the ew staller (latest verso s.b dated March 7 th ) set up the system flawlessly o our Wdows ad XP maches. The relatoshp whle ad m e.g. we have to evaluate eght dfferet systems. m leads to.;.4;.6;.8;.;.;.4;.6 Frst we aalyze the problem a mathematcal maer to get a dealzed result. The so-called probablty of queueg better ow as Erlag s C formula turs out to be: σ P P ( m jobs) ( jobs) m 4 ( ) m ( m) ( ) Performace Evaluato Techques summer term
2 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester We eed to ow before performg ay further calculatos. Accordg to lterature: m ( m) ( ) m m 4 ( ) ( m) Usg the gve sequece of we obta: 9 σ Lttle s Law gves: E [ r] σ m σ 99 ( ) 5 4 ( ) The pcture below shows the steady-state queue legth as computed by Excel: Aalytcal steady-state queue legth Probablty w w w 6 w 9 w w 5 w 8 w w 4 w 7 w w w 6 Legth of Queue 5 Performace Evaluato Techques summer term
3 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester The OMet sute comes alog wth a few comprehesve demos explag a M/M/ queue detal ( the subdrectory samples/ffo). We used the provded code ad modfed t to a M/M/ queue. Most of the source code fles have bee geerated so we avod to prt all of them. evertheless the core fuctoalty ca be foud ffo.cpp: // // fle: ffo.cc // (part of Ffo - a OMeT demo smulato) // #clude "ffo.h" #clude <fstream> usg amespace std; vod FFAbstractFfo::actvty() msgservced ULL; msgservced ULL; edservcemsg ew cmessage("ed-servce-server-"); edservcemsg ew cmessage("ed-servce-server-"); recordhstogram ew cmessage("record-hstogram"); queue.setame("queue"); hst.setrage(. 5.); hst.setumcells(5); // sed record request scheduleat( smtme() recordhstogram ); for(;;) cmessage *msg receve(); f (msgedservcemsg) // server has fshed a job edservce( msgservced ); f (queue.empty()) msgservced ULL; else // ew job avalable > catch t msgservced (cmessage *) queue.pop(); smtme_t servcetme startservce( msgservced ); scheduleat( smtme()servcetme edservcemsg ); else f (msgedservcemsg) // server has fshed a job edservce( msgservced ); f (queue.empty()) msgservced ULL; else // ew job avalable > catch t msgservced (cmessage *) queue.pop(); smtme_t servcetme startservce( msgservced ); scheduleat( smtme()servcetme edservcemsg ); else f (msgrecordhstogram) // record curret queue legth hst.collect(queue.legth()); // resubmt message for ext record.s later scheduleat( smtme(). recordhstogram ); else f (msgservced) // ew job arrval ad server s dle > start there arrval( msg ); Performace Evaluato Techques summer term
4 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester msgservced msg; smtme_t servcetme startservce( msgservced ); scheduleat( smtme()servcetme edservcemsg ); else f (msgservced) // ew job arrval ad server s actve server s dle > start o server arrval( msg ); msgservced msg; smtme_t servcetme startservce( msgservced ); scheduleat( smtme()servcetme edservcemsg ); else // ew arrval but o dlle server > queue t arrval( msg ); queue.sert( msg ); vod FFAbstractFfo::fsh() ev << "*** Module: " << fullpath() << "***" << edl; ev << "Stac allocated: " << stacsze() << " bytes"; ev << " (cludes " << ev.extrastacforevr() << " bytes for evromet)" << edl; ev << "Stac actually used: " << stacusage() << " bytes" << edl; ofstream of(par("hst_fle") os::app); t samples hst.samples(); t ; t s ; of << edl << edl << "*********** ew ru ***********" << edl << edl << "cout: " << samples << edl << "bucet cout rel. perc" << edl; whle ((s<samples) && (<hst.cells())) of << << " " << (t) hst.cell() << " " << hst.cell()/samples << edl; shst.cell(); ; /* FILE *f; ffope(par("hst_fle")"a"); hst.savetofle(f); fclose(f);*/ // Defe_Module( FFPacetFfo ); smtme_t FFPacetFfo::startServce(cMessage *msg) ev << "Startg servce of " << msg->ame() << edl; retur par("servce_tme"); vod FFPacetFfo::edServce(cMessage *msg) ev << "Completed servce of " << msg->ame() << edl; sed( msg "out" ); Remar: The complete program s cluded the accompaed Zp archve cl. a Wdows bary. Performace Evaluato Techques 4 summer term
5 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester The aalyss of the data geerated by our OMet program was doe usg Mcrosoft Excel for a secod tme: Lambda. 8 Probablty 6 4 w w w w w4 w5 w6 Legth of Queue 4 5 calculated ru 4 5 calculated av. SRT w w w w 47E-5 6E-6 64E-5 w4 64E-6 w5 64E-7 w6 64E-8 Lambda. 8 Probablty 6 4 w w w w w4 w5 w6 w7 w8 Legth of Queue 4 5 calculated ru 4 5 calculated av. SRT w w w w E w4 95E-5 855E E-5 9E-5 85E-5 w5 9E-5 E-5 E-5 64E-5 785E-6 7E-5 w6 4E-6 w7 68E-7 w8 7E-7 Performace Evaluato Techques 5 summer term
6 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Lambda. 8 Probablty 6 4 w w w w w4 w5 w6 w7 w8 w9 w w Legth of Queue 4 5 calculated ru 4 5 calculated av. SRT w w w w 4 E w w w6 89E-5 474E-5 79E E-5 w7 585E-6 768E-5 7E-5 E-5 w8 6E-5 66E-6 w9 9E-6 w 57E-7 w 7E-7 Performace Evaluato Techques 6 summer term
7 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Lambda Probablty w w w w w4 w5 w6 w7 w8 w9 w w w w w4 w5 Legth of Queue 4 5 calculated ru 4 5 calculated av. SRT w w w w E w w w w7 68E-5 55E w8 77E E-5 w E-5 6E-5 w 56E-5 44E-5 w 8E-6 575E-6 w 64E-5 E-6 w 9E-7 w4 68E-7 w5 47E-7 Performace Evaluato Techques 7 summer term
8 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Lambda Probablty 5 4 w w w w w4 w5 w6 w7 w8 w9 w w w w w4 w5 w6 w7 Legth of Queue 4 5 calculated ru 4 5 calculated av. SRT w w w w 864 E w w w w w w E w 97E w 985E E-5 w 8 47E-5 w 4 E-5 w4 E-5 w5 59E-6 w6 54E-6 w7 7E-6 Performace Evaluato Techques 8 summer term
9 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Lambda Probablty 4 w w w w w4 w5 w6 w7 w8 w9 w w w w w4 w5 w6 w7 w8 w9 w Legth of Queue 4 5 calculated ru 4 5 calculated av. SRT w w w w 77 95E w w w w w w w w w 47E w w4 9E w5 9E-5 846E-5 w E-5 w7 4 5E-5 w8 8E-5 w9 E-5 w 658E-6 Performace Evaluato Techques 9 summer term
10 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Lambda Probablty 4 w w w w w4 w5 w6 w7 w8 w9 w w w w w4 w5 w6 w7 w8 w9 w w w w w4 Legth of Queue 4 5 calculated ru 4 5 calculated av. SRT w w w w E w w w w w w w w w w w E w w6 549E-5 96E w7 5 96E w8 75E w w 7 8 w E-5 w 676E-5 w 47E-5 w4 E-5 Performace Evaluato Techques summer term
11 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Lambda Probablty Legth of Queue 4 5 calculated ru 4 5 calculated av. SRT w w w w 867 7E w w w w w w w w w w w w w w w w w w w w 87E w4 484E w w E E-5 4 w E w8 484E w9 646E-5 479E-5 4 w w w 44 w 7 9E-5 w E-5 w5 69E-5 577E-5 w6 46E-5 w7 69E-5 w8 95E-5 Performace Evaluato Techques summer term
12 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester average System Respose Tme 5 Secods lambda 4 5 calculated ru 4 5 calculated Performace Evaluato Techques summer term
13 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Problem Use the Pollacze-Khtche mea value formula to show that a M/M/ system has twce the expected umber of customers the system as the M/D/ system as ( < ). The Pollacze-Khtche mea value formula: E [ ] ( ) C b ( ) That formula ca be appled to both the M/M/ ad the M/D/ case. The term C b dffers the M/M/ case: C b Ad for M/D/: C b We fd that the expected umber of customers a M/M/ system s determed by: E [ ] M / M / ( ) ( ) ( ) Smplfyg the Pollacze-Khtche formula of a M/D/ system: E [ ] M / D / ( ) ( ) ( ) ( ) Performace Evaluato Techques summer term
14 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Let s compute the lmt: E lm E M / M / M / D / [ ] [ ] ( ) ( ) ( ) ( ) ( ) Ideed a M/M/ system has twce the expected umber of customers a system as the M/D/ system for < ). large ( Performace Evaluato Techques 4 summer term
15 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Problem Cosder the M/M// loss system ( ) wth arrval rate ad beg the rate of a sgle server. Draw the state dagram Gve the geerator matrx Q Fd the steady-state vector (... ) Usg ths show that wth Pr [" Customer loss" ] Assume 5 : (ths s the Erlag loss formula) ad evaluate Pr[ " loss" ] How mght a telephoe compay use ths formula? Customer for.... If ay of the servers the M/M// queue s dle the arrvg job s servced mmedately. If all server are busy the arrvg job wats a queue. The state of the system s represeted by the umber of jobs the system. Thus the state trasto dagram loos as follows:... - (-) The correspodg ( ) ( ) above: geerator matrx Q ca be derved statly from the dagram show Q M ( ) ( ) ( ) L O ( ( ) ) ( ) ( ( ) ) M Performace Evaluato Techques 5 summer term
16 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Performace Evaluato Techques 6 summer term The steady-state vector ca be ferred from Q sce each row equals zero. Therefore: Q Ad more detal: ( ) ( ) ( ) ( ) 4 K Solvg these equatos depedg o yelds: K A closer loo to the dagram reveals that a M/M// queue s a brth-death process. Therefore: Accordg to the ormalzato codto:
17 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Performace Evaluato Techques 7 summer term We solve for : Sce for the sum ca be exteded to: Usg : Substtutg ad : That way we proved Erlag s loss formula. Smlarly we get the steady-state vector: 6 L
18 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester A moder math sute computes the customer loss very qucly. We wrote a short program for Maple 8 Tral verso. Its two les are: rho:5: erlag:->rho^/(*sum((rho^)/..)); The program s targeted at hgh precso computatos ad therefore outputs chuy ratos. However we also wated to obta rouded values: result:seq(erlag()..); result : resultf:evalf(result); resultf : I a M/M// system queueg s ot ecessary sce there are as may servers as customers. So f a customer s able to actually eter the system the he/she wll be served too. Otherwse he/she wll be rejected ad does ot eter the system at all. The capacty of a telephoe etwor (a bacboe) ought to be as hgh as eeded order to master the case that each customer calls/arrves at the same tme (up to calls). I realty that case was ever observed hece the compaes reduce ther capactes to the typcal case order to cut lots of costs. The telecommucato compaes am to serve as may customers as possble ad to refuse as few as feasble. Due to log-term observatos t s approxmately ow advace how may ew calls per tme have to be served ad how log they last average. ow the compaes defe some percetages they le to acheve.e. there are ew calls per mute they last about mutes each ad oly.% should be rejected. The: The fal step s to fd the smallest where Pr [" loss" ] <.% Customer accordg to Erlag s loss formula. That the deotes the lowest capacty eeded to acheve the proposed servce avalablty. Performace Evaluato Techques 8 summer term
19 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester If oe has to estmate the loss usg a gve capacty throughput ad arrval rate he ca utlze Erlag s loss formula too. As a example a telephoe compay gets a chace to approxmate how may calls are lost ( average) wth ther curret telephoe exchage (or swtch). If that umber s too hgh the compay should calculate how may of the lost calls may have bee served by stallg a addtoal exchage. Hece the outcome of all these calculatos are future vestmets (the ew telephoe exchage) ad upcomg eargs (served calls). The resposble maager the ca decde upo these facts whether he admts to the ew vestmets or ot. Performace Evaluato Techques 9 summer term
20 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Problem 4 Cosder the fully Marova queueg etwor show ths fgure: queue queue.5.5 Fd a stablty codto for ths system. Fd the mea tme for a customer to proceed through the system. From the dagram we fer that the gve etwor s a ope etwor ad apply Jacso s theorem. The system s routg probabltes ca be descrbed as: p p.5 p p p.5 p p p.5 The traffc equatos: e e p p p p e p e p e e Reduced: e e.5 e e.5 e Solved: e e 4 e e 4 Queue has to process the tass three tmes faster tha they eter the system whle queue has to be eve qucer: t must be able to hadle four tmes the umber of arrvals. ow we exame both M/M/ queues separately. Each ode has to fulfll the stablty codto <. Due to we fd > 4. The mea tme for a customer to proceed through the system s the outcome of the equato: E [ t] e E[ T ] e E[ ] T Performace Evaluato Techques summer term
21 Assgmet 4 Thomas Adam Stepha Brumme Ha Lorez Jue th master st semester Utl ow the vst ratos are ow but we eed to further vestgate the estmated tme each ode: Hece: [ ] E T [ ] E T 4 4 E[ t] ( ) ( 4) Performace Evaluato Techques summer term
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