A n ali sis K eluar an Si mu lasi St ead y-St at e

11.5 A n ali sis K eluar an Si mu lasi St ead y-St at e

² Consider a single run of a simulat ion model whose purpose is t o est i- mat e a st eady st at e, or long run, charact er ist ics of t he syst em.

² A ssume Y 1 ;Y 2 ; : : : are observat ions, which in general are samples of an aut ocorrelat ed t ime series.

² The st eady-st at e measure of per formance µ t o be est imat ed is de…ned by

² This result is independent of init ial condit ions, random number st reams, ...

11.5.1 I ni si al isasi B i as pada Si m ul asi St eady -St at e

There are several met hods of reducing t he point est imat or bias which is caused by using art i… cial and unrealist ic init ial condit ions in a st eady-st at e simulat ion.

1. Init ialize t he simulat ion in a st at e t hat is more represent at ive of long-

2. Divide t he simulat ion int o t wo phases, warm-up phaseand st eady st at e phase. Dat a collect ion doesn’t st art unt il t he simulat ion passes t he warm-up phase.

Consider t he example on page 452 (Example 12.13) ² A set of 10 independent runs, each run was divided int o 15 int ervals.

The dat a were list ed in Table 12.5 on page 453. ² Typicall we calculat e average wit hin a run. Since t he dat a collect ed

in each run is most likely aut ocor relat ed, a di¤erent met hod is used t o calculat e t he average across t he runs.

² Such averages are known as ensemble average. Several issues:

1. Ensemble average will reveal a smoot her and more precise t rend as t he number of replicat ions, R, is increased.

2. Ensemble average can be smoot her ed fur t her by plot t ing a moving average. In a moving average each plot t ed point is act ually t he average of several adjacent ensemble averages.

3. Cumulat ive averages become less variable as more dat a are aver- aged. Thus, it is expect ed t hat t he curve at left side (t he st art ing of t he simulat ion) of t he plot t ing is less smoot h t han t he right side.

4. Simulat ion dat a, especially from queueing models, usually exhibit s posit ive aut ocorrelat ion. T he mor e correlat ion present , t he longer it t akes for t he average t o approach st eady st at e.

5. I n most simulat ion st udies t he analyst is int erest ed in several mea- sures such as queuelengt h, wait ing t ime, ut ilizat ion, et c. Di¤erent performance measures may approach st ead st at e at di¤erent rat es. Thus it is import ant t o examine each performance measure indi- vididually for init ializat ion bias and use a delet ion point t hat is adequat e for all of t hem.

11.5.2 M et ode R epl i kasi Si mul asi St eady -St at e

² I f init ializat ion bias in t he point est imat or has been reduced t o a neg- ligible level, t he met hod of independent r eplicat ions can be used t o

² if signi…cant bias remains in t he point est imat or and a large number

of replicat ions are used t o reduce point est imat or variabilit y, t he result con… dence int erval can be misleading.

– The bias is not a¤ect ed by t he number of replicat ions R, but by delet ing more dat a (i.e. increasing ) or ext ending t he lengt h of each run (i.e. increasing .

– Increasing t he number of replicat ions R may produce short er con-

… dence int ervals around a “ wrong point ” , rat her t han ² I f d is t he number of observat ions t o delet e from a t ot al of n observa-

t ions, a rough rule is n-d should be at least 10d, or should be at least .

² Given t he run lengt h, t he number of replicat ions should be as many

as possible. Kelt on in 1986 est ablished t hat t here is lit t le value t o run more t han 25 replicat ions. So if t ime is available, make t he simulat ion longer, inst ead of making more replicat ions.

² See Example 12.14 on page 460 and Example 12.15 on page 461, where

Example12.15 demonst rat et hecases where few obsevat ions were delet ed.

A couple of not es as fewer observat ions were delet ed (d is smaller ):

1. The con…dence int erval shift s downwar d, re‡ect ing t he great er downward bias in as d decreases. This can be at t ribut ed as t he result of a “ cold st art ” .

2. The st andard error of , namely decreases as d decreases. As d de- cr eases, t he number of samples included in t he st at ist ics increases, reducing t he error range.

11.5.3 U k uran Sam pl e Si mul asi St eady-St at e

I n a st eady-st at e simulat ion, a speci…ed precision may be achieved eit her by increasing t he number of replicat ions R, or by increasing t he run lengt h .

Example 12.16 on page 462 shows an example of calculat ing R for a given precision.

Example 12.17 on page 463 shows an example of increasing for t he given precision (recall t he general rule: should be at least ).

11.5.4 B at ch M eans unt uk Est i m asi I nt er val Est i m a- t i on pada Si mul asi St eady-St at e

² One disadvant age of replicat ion is t hat dat a must be delet ed on each replicat ion.

² One disadvant ageof a single-replicat ion is it s dat a t end t o be aut ocor- relat ed.

² The met hod of bat ch mean divides t he out put dat a fr om onereplicat ion int o a few large bat ches.

² Treat t he means of t hese bat ches as if t hey were independent . See t he formula on page 463 and 464.

² The key isssue is t hat no commonly accept ed met hod for choosing an accet able bat ch size m. This is act ually one of t he r esearch areas in simulat ion.

– o Schmeiser found for a …xed t ot al samplesizet here is lit t le bene…t from dividing it int o more t han k = 30 bat ches.

– o Alt hough t here is typically aut ocorrelat ion bet ween bat ch means at all lags, t he lag-1 aut ocorrelat ion is usually st udied t o assess t he dependence bet ween bat ch means.

– o Thelag-1 aut ocorrelat ion bet ween bat ch means can beest imat ed using t he met hod described earlier. They should not be est imat ed from a small number of bat ch means, i.e. we need t o have large number of bat ches, t hough t he size of bat ches could be small.

– o If t he t ot al sample size is t o be chosen sequent ially (i.e. choose one for one experiment , choose anot her one for improvement et c.), t hen it is helpful t o allow t he bat ch size and number of bat ches t o grow as t he run lengt h increases.

² Example 12.18 on page 465.