IE306Lec10 - IE306 SYSTEMS SIMULATION Ali Rıza Kaylan...

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Unformatted text preview: IE306 SYSTEMS SIMULATION Ali Rıza Kaylan kaylan@boun.edu.tr 1 LECTURE 10 OUTPUT DATA ANALYSIS OUTLINE OUTPUT PERFORMANCE MEASURES TYPES OF SIMULATION TERMINATING SIMULATIONS STEADY STATE SIMULATIONS WARM UP PERIODS TECHNIQUES FOR STEADY STATE SIMULATION COMPARISON OF ALTERNATIVE SYSTEM DESIGNS 2 OUTPUT PERFORMANCE MEASURES EXAMPLE: QUEUEING SYSTEM Total production of parts over the run (P) Average waiting time of parts in queue: N ∑ WQi i =1 N N = no. of parts completing queue wait WQi = waiting time in queue of ith part Maximum waiting time of parts in queue: max WQi i =1,...,N 3 OUTPUT PERFORMANCE MEASURES Simulation time : 20 Time-average number of parts in queue: 20 ∫0 Q (t ) dt 20 Q(t) = number of parts in queue at time t Maximum number of parts in queue: max Q (t ) 0≤t ≤20 Average and maximum total time in system (cycle time): P ∑TSi i =1 , max TSi P i =1,...,P 4 OUTPUT PERFORMANCE MEASURES Utilization of the machine (proportion of time busy) 20 ∫0 B(t ) dt 20 , 1 if the machine is busy at time t B(t ) = 0 if the machine is idle at time t 5 SIMULATION TYPES Terminating Known starting and stopping conditions Time frame is known (and finite) Steady-State Initial conditions are not always well defined No defined stopping condition (theoretically infinite) Interested in system response over the long run 6 WARM UP AND RUN LENGTH Warm-up period: Transient period Most models start empty and idle Empty: No entities present at time 0 Idle: All resources idle at time 0 In a terminating simulation: OK if realistic. In a steady-state simulation: Bias the output for a while after startup 7 WARM UP AND RUN LENGTH Remedies for initialization bias Better starting state, more typical of steady state Make the run so long that bias is overwhelmed Let model warm up, still starting empty and idle 8 WARM UP AND RUN LENGTH Warm-up and run length times? Most practical idea: preliminary runs, plots Simply “eyeball” them Be careful about variability — make multiple replications, superimpose plots Also, be careful to note “explosions” Possibility – different Warm-up Periods for different output processes To be conservative, take the max Must specify a single Warm-up Period for the whole model 9 TECHNIQUES FOR THE STEADY STATE SIMULATION • Method of Batch means • Method of Independent Replications • Regenerative Method 10 TECHNIQUES FOR THE STEADY STATE SIMULATION Method of Batch Means: One very long simulation run which is suitably subdivided into an initial transient period and n batches. Each batch: an independent run of the simulation experiment No observation are made during the transient period ( warm-up interval). How to choose batch interval size? Large / Small 11 TECHNIQUES FOR THE STEADY STATE SIMULATION Method of Batch Means: 12 TECHNIQUES FOR THE STEADY STATE SIMULATION Method of Independent Replications: Reasonable for systems with short transient periods. Independent runs of the simulation experiment (Different initial random seeds for the simulators' random number generator) Transient periods are removed 13 TECHNIQUES FOR THE STEADY STATE SIMULATION Method of Independent Replications: 14 COMPARISON OF ALTERNATIVE SYSTEM DESIGNS Example: Car Washing Center SYSTEM I. Parallel Stations ooo o o o oo o SYSTEM II. Service Stations in Series ooo o oo o o o 15 COMPARISON OF ALTERNATIVE SYSTEM DESIGNS Mean Performance Measure: E (Y ) = θ Simulation Results: { System II: { Y } } System I: Y 11,Y 21,...,Y n 1 1 12 ,Y 22 ,...,Y n 2 2 Hypothesis Testing Problem: H0 : θ 1 = θ 2 H 1: θ1 ≠ θ 2 16 COMPARISON OF ALTERNATIVE SYSTEM DESIGNS Test Statistics: Z= ( Y.1 − Y.2 ) − (θ 1 − θ 2 ) V (Y.1 − Y.2 ) ~ N (0,1) (Y.1 − Y.2 ) − (θ1 − θ 2 ) ~ t( ν ) T= ˆ V (Y − Y ) .1 .2 17 COMPARISON OF ALTERNATIVE SYSTEM DESIGNS Confidence Intervals: Confidence Level = 1 − α Three Possible Confidence Intervals: [ Y.1 − Y.2 0 [ 0 [ 0 Y.1 − Y.2 Y.1 − Y.2 18 COMPARISON OF ALTERNATIVE SYSTEM DESIGNS Case 1: Variances known: (Y .1 − Y. 2 ) ± z (1 − α / 2) σ 12 n1 + σ 22 n2 Case 2: Variances unknown: ˆ (Y.1 − Y. 2 ) ± t (ν ,1 − α / 2) V (Y.1 − Y.2 ) 19 COMPARISON OF ALTERNATIVE SYSTEM DESIGNS (σ 1 = σ 2 ) 2 Case 2a: Variances are equal 11 + n1 n2 (Y.1 − Y. 2 ) ± t (ν ,1 − α / 2)spooled s 2 2 ( n1 − 1)s1 + (n2 − 1) s2 = n1 + n2 − 2 2 pooled 2 Case 2b: Variances are unequal 2 2 (σ 1 ≠ σ 2 ) 2 s1 s2 (Y .1 − Y. 2 ) ± t (ν ,1 − α / 2) +2 n1 n2 ν= [s 2 1 [s 2 1 n1 + s n2 ] 2 2 2 n1 ] / (n1 − 1) + [s n2 ] / (n2 − 1) 2 2 2 2 20 COMPARISON OF ALTERNATIVE SYSTEM DESIGNS Comparison of Several Systems : θ 1 = θ 2 =... = θ k H0 : H1 : A t least one differs 21 ...
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