An Analysis of the LRE-Algorithm using Sojourn Times

Transcription

1 An Anlysis of the LRE-Algoithm using Sooun Times Nobet Th. Mülle Abteilung Infomtik Univesität Tie D-5486 Tie, Gemny E-mil: Tel: Fx: KEYWORDS Disete event simultion, LRE-lgoithm, onfidene intevl, sooun times ABSTRACT The lssil method of evlution of simultions suely is the bth mens method, see e.g. [Btley et l. 987], giving onfidene intevls to exess the eision of the simultion esults. Unfotuntely, it is not suited fo e event simultion tehniques, s they do not odue the neessy lge bthes. As n ltentive, the LRE-lgoithm hs been intodued in [Sheibe 984], whee the lengthy finl nlysis of its bsis hs ust been given in [Sheibe 999]. Its nme is bsed on the Limited Reltive Eo tht hs been hosen to desibe the eision of the esults. In this e, we esent nothe, muh simle, nlysis of the LRE-lgoithm bsed on sooun times, whih gives bette undestnding of the lgoithm nd lely exhibits etin limittions. Additionlly, it llows to ete onfidene intevls, nd so it is ossible to ome the esults of the two vey diffeent methods. The LRE-lgoithm The bth mens method (e.g. [Btley et l. 987]) ties to onstut indeendent smles fom given time seies by building bthes tht tend to hve lesse utooeltion thn the oiginl seies. So essentilly it ims on the edution of utooeltion. It esults in onfidene intevls fo the men of the simultion esults. In ontst, the LRE-lgoithm, s intodued in e.g. in [Sheibe 984] nd used in viety of es lte on, e.g. [Sheibe 999, Gög nd Sheibe 996], mesues the utooeltion nd ties to dedue the eision of the simultion esults fom this utooeltion (insted of eduing it). In ddition, it does not im t oduing men vlues, but ties to estimte the distibution of the esults. Additionlly, insted of onfidene intevls, the Limited Reltive Eo hs been intodued. In the following, we will biefly ell the LRE-lgoithm, moe eisely: vesion LRE-III fo disete sequenes of el vlued ndom vibles X ; X ; X 3 ; : : :. We ssume tht these vlues e identilly distibuted with F (x) := P (X i x) indeendent fom the index i, but of ouse thee my be signifint utooeltion. Fo simliity, we ust onentte on the estimtion of one oint of the distibution of the X i, s this lso is the stting oint of the oiginl nlysis. So fo one hosen vlue x, we ty to estimte F (x) o, equivlently, the invese distibution funtion G(x) = F (x). The LRE-lgoithm essentilly edues the oiginl time seies X ; X ; X 3 ; : : : to sequene Y ; Y ; Y 3 ; : : : of boolen vlues with Y i =, X i x. Plese note tht the ndom vibles Y i of this new time seies e still identilly distibuted nd still my be utooelted! Fo the exettion E[Y ] of the Y i, we get E[Y ] = P (X x) + 0 P (X > x) = P (X x) = F (x) So ou oiginl question of estimting F (x) hs been edued to the estimtion of E[Y ]. To do this, the following vlues e omuted fom the time seies (Y n ) (i.e. they e mesued duing simultion): quntity = (n) of vlues Y i = fo i n, i.e. the quntity of X i with X i x. quntity = (n) of tnsitions fom Y i = to Y i+ = 0 fo i n, i.e. the quntity of obseved is (X i ; X i+ ) with X i x<x i+. In this hte, n will be teted s fixed, nd we will simly use nd insted of (n) nd (n). Between ny two subsequent tnsitions Y i =! Y i+ = 0 nd Y =! Y + = 0 thee must be thid tnsition Y k = 0! Y k+ =, i < k <. So it is not neessy to mesue the following vlues, s they n be dedued with suffiient eision fom n; ; : quntity v = n of vlues Y i = 0 fo i n, i.e. the quntity of X i with X i > x. quntity of tnsitions fom Y i = 0 to Y i+ = fo i n, i.e. the quntity of obseved is (X i ; X i+ ) with X i >xx i+. quntity b of tnsitions within fx xg, i.e. the quntity of is (Y i ; Y i+ ) with Y i = Y i+ =.

2 quntity d v of tnsitions within fx > xg, i.e. the quntity of is (Y i ; Y i+ ) with Y i = Y i+ = 0. Plese note tht the oximtions fom bove my ll be wong by t most. So in the following we will simly use =, b = nd d = v = n. The following oximtion is immedite: F (x) = E[Y ] =n () To estimte the eision of this oximtion, the sequene (Y i ) is teted like two node hin. Then the entl ssumtion of the LRE-lgoithm is s follows: (-LRE) Assume the nodes 0 nd to be memoyless, i.e. tet the system like disete homogenous Mkov hin with ust the two nodes 0 nd! In [Gög nd Sheibe 996, Sheibe 999] this hin is lled the F (x)-equivlent -Node Mkov hin. This hin is hteized by its one ste tnsition obbilities: i = Pob(tnsitions stting in i led to ) whee i; f0; g. Resulting fom this, thee e the stedy stte obbilities 0 nd fo being in stte 0 es. in stte. So the entl ssumtion of the LRE lgoithm is F (x) = nd F (x) = 0 fo these vlues deived fom the Mkov hin. Lte in this e we will onside the imlitions of this entl ssumtion nd lso disuss its vlidity. But fist we ell the esults fom [Sheibe 999] fo this -Node hin. The following gh ontins the tnsition obbilities s well s the stedy stte obbilities togethe with the mesued (o deived) vlues: b The following estimtes e obvious: v=n- 00 = E[Y ] = P (Y i = ) n () 0 = E[Y ] = P (Y i = 0) v n = n n (3) 0 = P (Y i = 0 Y i = ) (4) = P (Y i = Y i = ) d b (5) 0 = P (Y i = Y i = 0) v n (6) 00 = P (Y i = 0 Y i = 0) d v n n (7) The nlysis of the LRE lgoithm fom [Sheibe 999] is bsed on n -osteioi gument fo the distibution of, given the mesued vlues n; ; : Stting oint e the densities f 0 nd f 0 of ossible vlues 0 nd 0 tht fit to n; ; : f 0 (x) = ( + ) f 0 (x) = (v + ) v x ( x) b (8) x ( x) d (9) Using these densities, it is shown tht the set of vlues tht fit to n; ; is oximtely noml distibuted fo suffiiently lge vlues of n; ; v; ; b: with N ( n ; ) (0) v n v 3 ( n ) () Insted of defining onfidene intevls, the limited eltive eo (LRE) is oosed. Beuse =n 0 + =, lso is n eo mesue fo the oximtion v=n of 0. If =n is signifintly lge thn v=n, then the eltive eo is bigge thn. In [Gög nd Sheibe 996, v=n =n Sheibe 999], the uthos suose to efom simultion until both vlues e below 0:05. Fo the vlidity of the oximtions (tht e deived vi n lition of the entl limit theoem), they give only vey simle lge smle onditions : n > 000; ; v > 00; ; b; ; d > 0. Estimtion of the sooun times Unfotuntely, the nlysis in [Sheibe 999] is stitly onentted on the sttistis of -Node Mkov hin nd so it does not give hints how good the oximtions e in se tht ondition (-LRE) does not hold. In the following we esent new nlysis of the LRE lgoithm, whee we will onentte on onfidene intevls insted of the LRE s eo mesue. Ou nlysis is bette suited to undestnd the os nd ons of the LRE oh. The time seies Y ; Y ; Y 3 ; : : : defines two sequenes (G () i ) nd i of sooun times in stte es. stte 0 suh tht G () G () Y Y Y 3 : : : = : : : 0 : : : 0 : : : 0 : : : 0 : : : if the initil stte hens to be Y = o G () G () Y Y Y 3 : : : = 0 : : : 0 : : : 0 : : : 0 : : : : : : if the initil stte is Y = 0. When onsideing the LRE-lgoithm, we see tht using the -Node Mkov hin to model the behvio of the simultion ontins thee bsi ssumtions onening these sooun times:

3 (-LRE) The sooun times (G () ) fo stte e suosed to hve identil geometil distibution G () given by obbility 0. (3-LRE) The sooun times ( ) fo stte 0 e suosed to hve identil geometil distibution given by obbility 0. (4-LRE) The ndom sequenes (G () i ) nd ( i ) e suosed to be indeendent. Fom the oeties of the geometi distibution nd (- LRE) we get nd := E[G () ] = = 0 () := V (G() ) = = 0 (3) The sme holds fo stte 0, whee (3-LRE) imlies nd 0 0 := E[ ] = = 0 (4) := V (G(0) ) = 00 = 0 (5) nd 0 e the men sooun times, so we hve E[Y ] = + 0 (6) Now suose tht we efom simultion extly until the fist smles of sooun times g () in stte nd the fist sooun times g (0) in stte 0 e omleted. We still mesue the sme thee vlues n; ; s bove. So in ontst to the evious hte, whee n ws fixed nd nd wee deending on n, now is fixed nd n = n() nd = () e deending on! We still use v = n. In ft we hve Now onside nd R () = v = X = X = R () := = R (0) := = g () (7) g (0) (8) X = X = G () (9) (0) is the men vlue of geometilly distibuted ndom vibles, tht e indeendent beuse of (4-LRE). If is lge enough, we my ly the entl limit theoem stting tht R () hs oximtely noml distibution with men nd vine =. Ou mesued vlue = is indeed ust smle = = = P = g() of R (), so we my dedue onfidene intevls fo P ( = < z = ) (z) () using the well known funtion (z) = nom(z) nom( z) fo onfidene levels. A simil esult holds fo R (0) nd 0 : P ( 0 v= < z 0 = ) (z) () Using (4-LRE), we see tht R () nd R (0) e indeendent. But the ftion R () R () + R (0) (3) of two noml distibuted ndom vibles is etinly not noml distibuted! Howeve, this does not ontdit the esult of [Sheibe 999] stting the noml distibution of, s we will show in the following: R () R () + R (0) + 0 = R() ( + 0 ) (R () + R (0) ) (R () + R (0) ) ( + 0 ) = R() 0 R (0) (R () + R (0) ) ( + 0 ) = (R() ) 0 (R (0) 0 ) (R () + R (0) ) ( + 0 ) (R() ) 0 (R (0) 0 ) = R () ( + 0 ) ( + 0 ) 0 ( + 0 ) R(0) ( + 0 ) (4) The eltive(!) eo of this oximtion tends to 0, if R () ohes 0. So the eision ohes nd R (0) of the oximtion will inese with, s the vines = nd 0 = onvege to zeo. In onsequene, fo lge the distibutions must be vey simil, s soon s thei vines e smll omed to nd 0. As the lss of noml distibuted ndom vibles is losed unde finite sums nd multilition with onstnt vlue, (4) is obviously noml distibuted. Its men is 0, nd its vine tuns out to be := ( + 0 ) 4 (5) Substituting the vlues fom () to (5), we see tht (5) nd () e identil w..t. the oximtions (4) to (7): v n v 3 ( n ) Putting ll things togethe, we get the following summy:

4 Fo lge, (3) is oximtely(!) noml distibuted with men E[Y ] nd vine (5), nd we hve the smle =n of (3): =n = = n= = = = + v= Ou nlysis leds to the sme esults s [Sheibe 999], but now =n tuns out to be smle fom noml distibution, while in the oiginl e the vine of ws onstuted in fily omlited wy with n -osteioi gument fom n; ;. It is now le to see whee lge smle onditions e neessy: should llow lition of the the entl limit theoem: > 0 (6) (whih should be suffiient, s hee we hve geometi distibutions), nd the oximtion (4) should be suffiiently eise: = << ; 0 = << 0 (7) whih is equivlent to ( )n >> 3 Confidene intevls fo LRE (v )n ; >> 3 v (8) Although the -osteioi nlysis fom [Sheibe 999] esults in noml distibution, the utho of [Sheibe 999] did not intodue onfidene intevls, ehs beuse the oof did not ove =n to be smle! But with ou oh, it is vey ntul to onstut onfidene intevls. In ddition, we e now ble to ome the esults of the LRE with the bth mens method. Using the well known eltion (z) = nom(z) nom( z) fo onfidene levels we get (z) = P ( z (E[Y ] =n)= z) = P ( E[Y ] =n z ) The vlues (:64) 0:9, (:96) 0:95 nd (:58) 0:99 e fequently used to get 90%, 95% o 99% of onfidene. As () 0:68, using s eo mesue oesonds to onfidene level of only 68%! So oding to the suggestions in [Sheibe 999] ited bove, simultion should ledy be stoed (s being eise enough), s soon s the 68% onfidene intevl hs dius of 5% of the men vlue. At tht time, the moe usul 95% onfidene intevl hs dius of bout :96 5%= 9:75% of the men vlue! We do not think tht this eision is high enough fo litions nd suggest tht simultions should be efomed without this stoing iteium. Disussion nd exeimentl esults Obviously, ondition (-LRE) will not be tue in most el simultion senios. On the othe hnd, the one ste tnsition obbilities led to oet vlues fo the LAG-- utooeltion of the seies (Y i ). Unless dditionl dt is mesued fom the simultion, the ssumtion of Mkovin behviou imlies estimtes on the highe tye utooeltions tht n hdly be imoved. Ou sooun time nlysis (5) shows tht thee is big influene of the sooun time vine on the esulting onfidene intevls. If the el vines e smlle thn es. 0, the LRE-lgoithm will esult in unneessily lge intevls. On the othe hnd, the intevls e too smll if the el sooun time vines e lge thn es. 0. In the following we esent the esults of few tests to illustte this behviou. Using thee diffeent tyes of time seies (Y i ) nd fo nge of smles fom 0 5 to 0 7 we mesued 99% onfidene intevls fo the LRE lgoithm nd fo n imlementtion of the Lw-Cson lgoithm fo the bth mens method (see e.g. [Btley et l. 987]). The exmles wee onstuted fom sevel disete Mkov hins (X i ) with Y i =, X i G, whee the sets G wee hosen subsets of the stte ses. The intevls due to the bth men method do not ely on the LRE hyotheses, so they vy oetly with the diffeent vines of the sooun times. We fist stt with setting tht fits to (-LRE), i.e. the time seies (Y i ) indeed me fom -Node Mkov hin with Y i = X i. We hose = 0: nd 0 = 0:06. Plese note tht the othe tnsition obbilities e uniquely defined by nd 0. As exeted, the intevls e lmost identil: Confidene intevls, -Node-Chin, _=0. ext esult LRE/sooun, mesued ob. bth mens, mesued ob. LRE/sooun, 99% intevl bth mens, 99% intevl e+06 e+06 5e+06 e+07 smle size n The seond test used seies with sooun times of signifint smlle vine. It is bsed on the following k-node Mkov hin with G = f; :::; kg, i.e. Y i =, X i k.

5 q q q q q q q k k+ k stges k+ k Confidene intevls, high vine hin, _=0. ext esult LRE/sooun, mesued ob. bth mens, mesued ob. LRE/sooun, 99% intevl bth mens, 99% intevl e+06 e+06 5e+06 e+07 smle size n Plese note tht in ll exmles, the ssumtion (4-LRE) bout the indeendeny of the sooun times is vlid! This is tue lso fo ny Mkovin bith-deth oess, so (-LRE) nd (3-LRE) seem to be moe itil thn (4-LRE). We tied this exmle with k = 0 nd = 0:99. As exeted, the onfidene intevls e muh lge thn fo the bth mens method: Confidene intevls, Elng-tye hin, _=0. ext esult LRE/sooun, mesued ob. bth mens, mesued ob. LRE/sooun, 99% intevl bth mens, 99% intevl e+06 e+06 5e+06 e+07 smle size n The thid test used 4-Node hin with highly vint sooun times, whee G = f; g: q kq d b d b = d b b d b d b d b b b d = d d k We tied n exmle with k = 00, so the sooun times e build by mixtues of two geometi distibutions diffeing by fto of 00. As exeted, the onfidene intevls e wy too smll in this se: Conlusion Fom the exmles, the disdvntges of the -Node oh n be lely seen. On the othe hnd, this oh n be esily inooted into e event simultions, see e.g. [Gög nd Sheibe 996]. Hee the bth mens method is not lible t ll. So egdless of these bove oblems, the LRE lgoithm is vey useful in this field. Ou new nlysis shows whee imovements e ossible: At the time, we e woking on n lgoithm mesuing nd 0 dietly fom the times seies insted of using the onditions (-LRE) nd (3-LRE). Initil esults on the gined onfidene intevls fo non-geometi sooun times e vey omising. REFERENCES [Btley et l. 987] Btley, Pul; Fox, Bennett L.; nd Shge, Linus E A Guide to Simultion. Singe, New Yok [Gög nd Sheibe 996] Gög, C.; Sheibe, F. "The RESTART/LRE Method fo Re Event Simultion." In Poeedings of the 996 Winte Simultion Confeene. Coondo, Clifoni USA, [Sheibe 984] Sheibe, F. "Time Effiient Simultion: The LRE-lgoithm fo oduing emiil distibution funtions with limited eltive eo." In AEÜ, 38, 994, [Sheibe 999] Sheibe, F. "Relible Evlution of Simultion Outut Dt: A simlified Fomul Bsis fo the LRE-Algoithm" In Poeedings of the MMB 99, Tie. VDE Velg Belin, ISBN , Mny dditionl efeenes n be found t the following web ge: