The Application of Phase Type Distributions for Modelling Queuing Systems

Transcription

1 Th Alication of Phas Ty Distributions for Modlling Quuing Systms Eimutis VAAKEVICIUS Dartmnt of Mathmatical Rsarch in Systms Kaunas Univrsity of Tchnology Kaunas, T - 568, ithuania ABSTRACT Quuing modls ar imortant tools for studying th rformanc of comlx systms, but dsit th substantial uuing thory litratur, it is oftn ncssary to us aroximations in th cas th systm is nonmarkovian. Phas ty distribution is by now indisnsabl tool in cration of uuing systm modls. Th uros of this ar is to suggst a mthod and softwar for valuating uuing aroximations. A numrical uuing modl with rioritis is usd to xlor th bhaviour of xonntial has-ty aroximation of srvic-tim distribution. Th rformanc of uuing systms dscribd in th vnt languag is usd for gnrating th st of stats and transition matrix btwn thm. Two xamls of numrical modls ar rsntd a uuing systm modl with rioritis and a uuing systm modl with uality control. Kywords: Quuing Aroximation, Distribution Fitting, Phas Ty distributions, Markov Chain, Numrical Modl. INTRODUCTION Th us of uuing modls is a basic tool for studying systms involving contntion for rsourcs. Major alication aras includ comuting systms, tlcommunication systms, and manufacturing systms. Siml modls involving rstrictiv distributional assumtions oftn rovid ssntial insights into th bhaviour of a uuing systm. Howvr, to obtain mor rcis and dtaild information about systm bhaviour, in gnral on must mloy a modl that allows scification of gratr dtail about th actual inut distributions (.g. th srvic tim and inrarrival-tim distributions of th systm. Us of has-ty (PH distributions is a common mans of obtaining tractabl uuing modls [-4]. Th aroach of this rsarch is to bgin with srvic-tim distribution to b aroximatd. Siml thr-momnt aroximation, along with mor rfind aroximation taking into account distribution sha, is rsntd for original inut distribution. Thn both th original and aroximating distributions ar usd in modlling th uu with siml rioritis. It is known that cration of analytical modls ruirs larg fforts. Us of numrical mthods rmits to crat modls for a widr class of systms. Th rocss of crating numrical modls for systms dscribd by Markov chains consists of th following stags: dfinition of th stat of a systm crating uations dscribing Markov chain comutation of stady stat robabilitis of Markov chain 4 comutation masurs of th systm rformanc. Th most difficult stags ar obtaining th st of all th ossibl stats of a systm and transition matrix btwn thm. In th ar is usd a mthod for automatic construction of numrical modls for systms dscribd by Markov chains with a countabl sac of stats and continuous tim.. APPROXIMATION OF SERVICE TIME DISTRIBUTION t us considr th M/G/c uu. In this multi-srvr modl with c srvrs th arrival rocss of customrs is a Poisson rocss with rat λ and th srvic tim T of a customr has a gnral robability distribution function G(t. It is assumd that ρ λe (S/c is smallr than. Th M/G/c uu with gnral srvic tims rmits no siml analytical solution, not vn for th avrag waiting tim. Usful aroximation can b obtaind by th mixtur and convolutions of xonntial (has-ty distributions. Thn a Markov chain with a countabl sac of stats and continuous tim can rrsnt th volution of th systm. Suos w lt m k, k, k dnot th kth non - cntral momnt (i.. E [ T ], whr T is a random variabl of srvic tim. Dfin a random variabl X in this way: X with rob. X ( X + X with rob., whr X and X ar indndnt random variabls having xonntial distribution with aramtrs and rsctivly +. It is asy to vrify that th dnsity function of X is givn by f ( x x x ( x + ( 8 SYSTEMICS, CYBERNETICS AND INFORMATICS VOUME 5 - NUMBER 6 ISSN:

2 Momnt matching is a common mthod for aroximating distributions, scially in th ara of uuing aroximations. Though two-momnt uuing aroximations ar common, thy may lad to srious rror whn th cofficint of variation, ν (th standard dviation dividd by th man, is high [, 5]. Th first thr momnts of any no dgnrat distribution with suort on [, can b matchd by th distribution (. To obtain th valus of th aramtrs,, and of aroximation, a comlx systm of non-linar uations nds to b solvd:! m! m! m Th solution of th systm is th following [6]: g g mk, g,, k k g! gg + g k + g ± ( g + 4 ( g g ( g g g ( g ( g + ( g + (. (4. NUMERICA MODE OF QUEUING SYSTEM WITH SIMPE PRIORITY First of all th cratd softwar is tstd with a siml systm that has analytical formulas to calculat systm masurs. Suos that thr ar two classs of customrs in a uuing systm. Thir srvic tims follow a lognormal distribution. Th arrival rocss is Poisson with aramtrs λ ir λ rsctivly. W shall suos that class has highr siml riority than class. This uuing modl has l and l waiting ositions for ach class of customrs to await srvic rsctivly. t us calculat th man numbr of customrs in th uu and th man waiting tim of a customr in ach class. t us assum that srvic-tim distribution is aroximatd by xrssion (. Th schm of considring a systm is rrsntd in Fig.. λ λ Fig.. Quuing systm with siml riority A nw customr can not b acctd for th srvicing whil a rvious on has not assd throughout all th hass of srvic. A Markov chain with th countabl sac of stats and continuous tim can dscrib th functioning of such a systm. To construct a numrical modl of th systm th aroach roosd in [7] is alid. Th st of vnts in th systm: {,,, } E 4, 5, whr a customr arrivd from class a customr arrivd from class comltd srvic in th first has with robability 4 comltd srvic in th first has with robability 5 comltd srvic in th scond has. Th st of all fasibl stats of th systm is: {( n, n, n, }, n, l n, l N whr n numbr of customrs from class rsnt in th systm n numbr of customrs from class rsnt in th systm, if th systm is mty n, if a customr from class is bing srvd, if a customr from class is bing srvd, if,if thsystm is mty,if a customr is bingsrvd in thfirst has has. a customris bing srvdin th scond Th man numbr of customrs ( and uu and th man waiting tim ( W and ( ( W in th of a ISSN: SYSTEMICS, CYBERNETICS AND INFORMATICS VOUME 5 - NUMBER 6 9

3 customr in ach class ar givn by th following formulas l j ( j nπ ( n, n, n, n j n, n, n, whr ( n, n, n n ( j ( j W, j,, λ j π, 4 is th stady stat robability of th systm stat. As an xaml, dscrib th vnt in th vnt languag. : IF n 4 if n thn ls nd if if n > thn ls if n > thn ls n n nd if nd IF Rturn Intns END 4 nd if Th cratd softwar, using th dscrition of vnts, gnrats th st of fasibl stats of th systm, th matrix of transition rats btwn thm and th stationary robabilitis of th stats. Alying th obtaind robabilitis, it is ossibl to comut th dsird charactristics of th systm rformanc.. Rsults If th numbr of waiting ositions for srvic in ach class of customrs is unlimitd,.g. l and l, thn valus ( i W and ( i can b calculatd by th analytical formulas : ( ( ( λ + λ E ( X ( + ν X λ W λ ( ( λ W λ ( λ E ( X ( λ + λ E ( X ( + ν ( λ E ( X ( ( λ + λ E ( X ν σ ( X E( X, X X whr E(X and σ ( X ar th man and th standard dviation of th srvic tim. Suos that srvic-tim is distributd according to th lognormal distribution with robability dnsity ( x x ( ln x λ [ / α ], > g x αx π with aramtrs α.9, λ. 5. Th first thr noncntral momnts of th distribution ar th following: m.468, m and m Th rsults of th analytical modl with aramtrs λ., λ.9, l, l ar th following: (.855, (.4. Th rsults of th numrical modl with th following valus of aramtrs: λ., λ.9, ar:.7447, , l 5, l 5 (.855, (.44. As it is sn from th rsults, softwar calibration is succssful and w can mov forward with analyzing uuing systms that do not hav analytical formulas to calculat various systm masurs 4. NUMERICA MODE OF QUEUING SYSTEM WITH QUAITY CONTRO Suos that thr is on flow of customrs and two uuing systms. Thir srvic tim follow a lognormal distribution. Th arrival tim rocss is Poisson with aramtr λ. This uuing modl has l and l waiting ositions bfor ach uuing systm to await srvic. This systm holds uality control that rdircts with robability alrady rocssd customrs to go through both uuing systms again. t us calculat th man numbr of customrs in both uus and th man waiting tim of a customr in ach uu. A nw customr can not b acctd for th srvicing whil a rvious on has not assd throughout both hass of uuing systm. Th st of vnts in th systm: {,,,,,,, } E, , 9 SYSTEMICS, CYBERNETICS AND INFORMATICS VOUME 5 - NUMBER 6 ISSN:

4 whr a customr arrivd to th first uuing systm comltd srvic of th first uuing systm in th first has with robability comltd srvic of th first uuing systm in th first has with robability 4 comltd srvic of th first uuing systm in th scond has 5 comltd srvic of th scond uuing systm in th first has with robability 6 comltd srvic of th scond uuing systm in th first has with robability, and a customr has assd uality control 7 comltd srvic of th scond uuing systm in th first has with robability, and a customr has faild uality control 8 comltd srvic of th scond uuing systm in th scond has, and a customr has assd uality control 9 comltd srvic of th scond uuing systm in th scond has, and a customr has faild uality controlth st of all fasibl stats of th systm is: whr {( n, n, n,, n5, }, n, l, l N, n numbr of customrs in th first uu n n, if th first has of th first uuing systm is mty, if a customr is bing srvd in th first has of th first uuing systm., if th scond has of th first uuing systm is mty, if a customr is bing srvd in th scond has of th first uuing systm. n 4 numbr of customrs in a scond uu n 5 n 6, if th first has of th scond uuing systm is mty, if a customr is bing srvd in th first has of th scond uuing systm., if th scond has of th scond uuing systm is mty, if a customr is bing srvd in th scond has of th scond uuing systm. t us assum that srvic-tim distribution is aroximatd by xrssion (. Th schm of xamining systm is rrsntd in Fig.. λ l l Fig.. Quuing systm with uality control Th man numbr of customrs ( and th man waiting tim ( W of a customr in th first uu ar : IF n > givn by th following formulas if n 5 < and < l thn n 5 n5 + ( j mπ ( n, n, n,, n5, ls n 4 + m n, n, n,, n5, nd if ( j ( j W λ, j,, if n > thn whr π ( n, n, n,, n5, is th stady stat n n n n + n robability of th systm stat. ls n n nd if As an xaml, dscrib th vnt in th vnt languag. n End IF Rturn Intns END ISSN: SYSTEMICS, CYBERNETICS AND INFORMATICS VOUME 5 - NUMBER 6

5 Th cratd softwar in C++, using th dscrition of vnts, gnrats th st of fasibl stats of th systm, th matrix of transition rats btwn thm and th stationary robabilitis of th stats. Alying th obtaind robabilitis, it is ossibl to comut th dsird masurs of th systm rformanc. 4. Rsults Th rsults of th numrical modl with th following valus of aramtrs: ar : λ,.7447, ,.9, l + l ( , ( 7.6 ( W.76, ( W Conclusions A mthod and cratd softwar for automatic construction of numrical modls for systms dscribd by Markov chains togthr with uuing aroximation allows: to analys comlx non-markovian uuing systms alying Markov chains thory, uuing systms with th infinit (countabl sac of stats can b aroximatd by th finit sac of stats. 5. REFERENCES [] W. Whitt, On aroximations for uus, III: Mixturs of xonntial distributions, AT&T Bll abs Tch Journal, Vol. 6, No., 984, [] M.A. Johnson, An mirical study of uuing aroximations basd on has-ty distributions, Commun. Statist.-Stochastic Modls, Vol. 9, No. 4, 99, [] J.A. uhman, M.J. Johnson, Bhaviour of uuing aroximations basd on saml momnts, Alid Stochastic Modls and Data analysis, Vol., 994, [4] M. J. Faddy, Examls of fitting structurd hasty distribution, Alid Stochastic Modls and Data Analysis, Vol., 994, [5] M. A. Jonhson, M. R. Taaff, An invstigation of has distribution momnt matching algorithms for us in uuing modls, Quuing systms, Vol. 8, No., 99, [6] E. Valakvičius, On Aroximation of Non- Markovian Quuing Modls, In: Th Scond Intrnational confrnc Simulation, gaming, training and businss rocss rnginring in orations, Riga, atvia,, SYSTEMICS, CYBERNETICS AND INFORMATICS VOUME 5 - NUMBER 6 ISSN: