Assignment 5. Simulation for Logistics. Monti, N.E. Yunita, T.
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1 Assgnment 5 Smulaton for Logstcs Mont, N.E. Yunta, T. November 26, 2007
2 1. Smulaton Desgn The frst objectve of ths assgnment s to derve a 90% two-sded Confdence Interval (CI) for the average watng tme from an M/M/1 queung smulaton by usng the bootstrap procedure. The second objectve s to asses f ths nterval covers the analytcal smulaton response. The methodology used s to apply the bootstrap procedure to four dfferent smulaton experments, each of them wth the same traffc rate but wth dfferent number of replcatons (m) and wth dfferent run lengths (c). In ths way, the nfluence of these smulaton settngs can be assessed when determnng the bootstrap confdence nterval. We used the Arena software to perform the M/M/1 smulaton experments. The arrval tmes and the servce tmes were smulated wth a sngle PRN stream, but each replcaton had non-overlappng PRN streams. The smulaton settngs for each experment are as follows: Smulaton 1 Traffc Rate: ρ = 0.3 Number of Replcatons: m = 10 Run Length: c = 1,000,000 [customers] Smulaton 2 Traffc Rate: ρ = 0.3 Number of Replcatons: m = 50 Run Length: c = 1,000,000 [customers] Smulaton 3 Traffc Rate: ρ = 0.3 Number of Replcatons: m = 10 Run Length: c = 100 [customers] Smulaton 4 Traffc Rate: ρ = 0.3 Number of Replcatons: m = 50 Run Length: c = 100 [customers] - 1 -
3 The smulatons outputs for each experment are shown n Table 1. Table 1 Smulaton outputs m = 10 m = 50 m = 10 m = 50 c = 1,000,000 [customers] c = 1,000,000 [customers] c = 100 [customers] c = 100 [customers] Replcaton Nr. W Replcaton Nr. W Replcaton Nr. W Replcaton Nr. W Bootstrap Procedure After obtanng the smulaton results, we performed the bootstrap procedure to calculate the confdence nterval for each experment. Standard bootstrappng assumes that the orgnal observatons are IID. In general, a smulaton experment can have m replcatons for each combnaton. The smulated - 2 -
4 observatons wthn combnaton are denoted as Smulaton for Logstcs Assgnment 5 w, gve the average smulaton output for combnaton, namely, w, 1 2 K wm. These observatons Accordng to Klejnen (2008), the steps to perform the bootstrappng are the followng: 1. Obtan the bootstrapped observatons w r by re-samplng the orgnal observatons and replacng each w r n combnaton. In ths case, the sample sze s kept constant. Thus, the bootstrap observatons w 1, w 2, K, wm may have frequences f 1,, f m such that f f m = m. 2. Compute the average of the bootstrapped smulaton outputs per combnaton: w. 3. Repeat steps 1 and 2 n each combnaton to obtan a bootstrap sample sze B = 1,000 values. So, for each combnaton, we have a set of bootstrap averages w w, K w. { } 1, 2, Sort the bootstrap averages w b (b = 1, 2,..., 1000) n step 3 from the smallest to the largest value and defne the CI for combnaton. A bootstrap (1-α ) CI s [ w, ]. For example, for α = 0.1, the lower lmt s the 50 th ordered ( Bα / 2) w( B [1 α / 2]) value of the bootstrapped averages and the upper lmt s the 950 th ordered value (see Deflandre and Klejnen, 2002). Further, we denote the average of the bootstrapped dstrbuton for combnaton as w. After determnng the 90% two-sded CI wth the bootstrap procedure, we calculated the analytcal smulaton response for each experment to verfy f these values are ncluded n the bootstrap ntervals. To do ths, two dfferent approaches were used accordng to the smulaton run length of the experments. When consderng the smulaton experments wth a long run length, we can assume that n these cases the steady state s reached. Therefore, to determne the analytcal smulaton response we used the followng expresson: W analytcal = (1) (1 ρ) µ µ where ρ s the traffc rate and µ s the average servce rate. The W analytcal calculated n Equaton 1 can only be appled n a long run-length smulaton, as the system reaches a steady state. However, when consderng the smulaton experments wth a short run-length, we cannot guarantee that a steady state has been reached. Deflandre and Klejnen (2002) suggest that the true mean for a short run-length smulaton can be estmated from the average output of a bg number of replcatons. Therefore, n ths case we decded to consder the average smulaton output from 100,000 replcatons as the W. analytcal The bootstrappng procedure was then performed n the four smulaton settngs. The results of the bootstrap dstrbuton, the CI, and the analytcal smulaton responses are shown n Table 2 and Table 3. Further, the dstrbutons of the bootstrapped w
5 smulaton outputs for each smulaton experment are shown through Fgure 1 to Fgure 4. Table 2 Bootstrappng Dstrbuton Summary Smulaton Run-length (c) [customers] Replcatons (m) Average of bootstrap dstrbuton ( w ) Varance 1 1,000, E ,000, E E E-06 Table 3 CI Summary Smulaton Run-length Replcatons Lower Upper Interval Analytcal Response (c) (m) lmt lmt length ( W analytcal ) [customers] 1 1,000, E ,000, E E E Fgure 1 Bootstrap dstrbuton for Smulaton 1-4 -
6 Fgure 2 Bootstrap dstrbuton for Smulaton 2 Fgure 3 Bootstrap dstrbuton for Smulaton 3-5 -
7 Fgure 4 Bootstrap dstrbuton for Smulaton 4 In all the smulaton experments the analytcal smulaton response les nsde the bootstrap CI when consderng a level of sgnfcance α = 0.1, except for Smulaton 1. However, f we consder α = 0.05 for Smulaton 1, then the CI s ncreased and t covers the W, although t s stll very close to the nterval's lower lmt. analytcal The smulatons outputs ndcate that the bootstrap dstrbuton varance s bgger for a short run-length compared to a long run-length. These results are expected, consderng that as long as the smulaton approaches the steady state then the smulaton outputs varance decreases. Therefore, the CI ncreases when the smulaton run-length becomes shorter (see Table 2 and Table 3). Furthermore, we found out that the CI for hgher replcatons has a smaller length than for lower replcatons. As each bootstrapped value n the dstrbuton s generated as an average of the m re-sampled orgnal values, a hgher number of replcatons produce bootstrapped averages { w 1, w 2, K, w 1000} that are closer to each other. In ths case, we can see that the varance of the bootstrapped dstrbuton s smaller as the number of replcatons ncreases (see Table 2). All n all, we can conclude that the bootstrap procedure helps to derve a CI for the smulaton outputs n dfferent smulaton settngs. However, t should be noted that the CI s more precse (havng a smaller nterval) when the re-samplng s performed usng the smulaton outputs from a steady state smulaton (long run-length). Also, we should be careful of usng the bootstrap procedure n expensve smulatons where only a few number of replcates are avalable (e.g. m = 2). In these stuatons, the resamplng may gve the same result many tmes. Therefore, when usng a few number of replcatons, the bootstrappng procedure may not generate a relable CI, and therefore, other procedures must be used to derve a CI
8 References Deflandre, D. and Klejnen, J.P.C. (2002). Statstcal analyss of random smulatons: Bootstrap Tutoral. Department of Informaton Management/Center for Economc Research (CentER), Tlburg Unversty (KUB), The Netherlands. No. 58. ISSN Klejnen, J.P.C. (2008). DASE: Desgn and Analyss of Smulaton Experments. Sprnger Scence + Busness Meda, New York, USA
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