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# Class Note for EMSE 273 with Professor Dorp at GW

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Date Created: 02/07/15

SIMULATION MODELING AND ANALYSIS WITH ARENA T Altiok and B Melamed Chapter 2 Discrete Event Simulation The DES Paradigm The DES Discrete Event Simulation model possesses a state vector S consisting of system variables needed to describe the system evolution over time The system state at time 1 S09 is often a step function Whose jumps are triggered by events The DES model is driven by a clock and an event list Altiok Melam ed Simulation Modeling and Analysis with Arena Chapter 2 The structure of a DES event list Event List Head Event 4 Time 2211 Event List Tail Other elds A DES simulator executes the following algorithm 1 Set the simulation clock to an initial time usually 0 and then generate one or more initial events and schedule them 2 If the event list is empty then terminate the simulation run Otherwise nd the most imminent event and unlink it from the event list 3 Advance the simulation clock to the time of the most imminent event and execute it the event may stop the simulation 4 Loop back to Step 2 Altiok Melam ed Simulation Modeling and Analysis with Arena Chapter 2 Example A single FIFO machine FIFO JOb Buffer Machine JOb Arrivals Departures 0 Two event types arrival and process completion State St number of jobs in the system at time t State transitions arrival at time t St n gt St8 n 1 process completion at time t St n gt St8 n I Altiok Melam ed Simulation Modeling and Analysis with Arena 4 Chapter 2 Example Single machine With failures Failures FIFO Buffer Job Machine Job Arrivals Departures Repairs M NtVt Nt number of jobs in the buffer at time t 012 Vt process status at time t idle O busy 1 down 2 Altiok Melam ed Simulation Modeling and Analysis with Arena 5 Chapter 2 Single Machine With Failures Cont State transitions Job arrival nv p n v1 ifv 0 Job arrival nv gt n1 v ifv 1 Job arrival nv gt n1 v ifv 2 Service process completion n1gt 00 if n0 n1gt n11 if ngt0 Failure occurrence n1 p n2 Repair completion n2 gt n1 Altiok Melam ed Simulation Modeling and Analysis with Arena 6 Chapter 2 Monte Carlo Sampling and Histories Sampling random values from speci ed distributions uses random number generators RNGs RNGs generate pseudo random numbers uniform over 01 which are used to obtain values from any distribution to The term Monte Carlo is attributed to von Neumann amp Ulam for their work on random processes at Los Alamos Altiok Melam ed Simulation Modeling and Analysis with Arena 7 Chapter 2 Monte Carlo Sampling and Histories Cont Performance evaluation produces estimates of performance measures utilization average WIP levels etc from sample histories Performance measures are computed from multiple runs replications of a DES model Performance measure estimates from distinct replications form a statistical sample that is used to compute grand averages and confidence intervals for these performance measures Altiok Melam ed Simulation Modeling and Analysis with Arena 8 Chapter 2 Example Workstation Subject to Failures and Inventory Control Failures Warehouse Demand Machine Job Arrlvals Raw Departures Material lt Uns atis ed Repairs Demand A neverstarving machine produces product for a warehouse Inventory is managed by an Rr policy the machine stops producing when R 5 units are in the warehouse and resumes production when the inventory drops to r 2 The machine fails randomly Altiok Melam ed Simulation Modeling and Analysis with Arena 9 Chapter 2 Performance Measures and State Performance Measures probabilities of machine status machine throughput customer service level average nished inventory level System State SO VtKt Vt O Idle 1 Busy 2 Down Kt level of nished product in the warehouse Altiok Melam ed Simulation Modeling and Analysis with Arena 10 Chapter 2 Example Operational History of a Machine Subject to Failure Machine Status Idle Busy DOWH Busy Idle 5 5 I I c I a I I 3 4 I I c I I O 3 I I x I I 3 2 quotquot quot quot quotquotEquot quotquotquot IJ I I D 1 7 I I 0 l l i l l O O O O O O O O O O O O O O O O O N M V l0 0 N co o O N M V l0 0 Time Min F F F F F Here the reorder point is r 2 the target level is R 5 the machine has 3 states Idle Busy Down and Kt stock on hand At time t 0 V 0 0 machine is idle and K0 4 warehouse has 4 units Altiok Melam ed Simulation Modeling and Analysis with Arena 1 1 Chapter 2 Sample System Behavior Machine Status Idle Busy Down Busy Idle Stock On Hand A A 100 110 120 130 A 140 150 160 l l l l l l O O O O D l l l l l l O O O O N M N 00 O C C Vl In Time Min At time t 35 a customer arrives demanding 3 units and K35 1 The reorder point r is down crossed and the machine becomes busy replenishing the inventory to the target level R Thereafter the inventory level gradually increases with no demand arrivals of until time t 65 Where K 65 4 Altiok Melam ed Simulation Modeling and Analysis with Arena 12 Chapter 2 Sample System Behavior Cont Machine B Idl Status Idle Busy Down 1153 e 6 I 5 5 7 g I f 4 i 5 3 i x l l 8 2 quot quotl a 1 o i i 5 i I O O O O O O O O O O O O O O O O 0 Time Min At time t 69 a customer arrives and places a demand that depletes the stock onhand so K69 0 unsatisfied demand is unful lled At time t 75 the unit that started processing at time t 65 is completed so K 75 0 At time t 87 the machine fails and repair begins Down state Altiok Melam ed Simulation Modeling and Analysis with Arena 13 Chapter 2 Sample System Behavior Cont Machine Bus Idle Status Idle Busy Down y 6 I I 5 5 I I f 4 i I I O 3 I I x g 2 I a 1 i o I I I I IiI I I O O O O O O O O O O O O O O O O C Time Min Repair is completed at time t 119 and the machine enters the Busy resuming processing the unit that was interrupted during failure At time t 101 a customer arrives with demand 1 so K101 1 Thereafter the inventory level gradually increases with no demand arrivals between times t 127 and t 157 Then the target level R 5 is reached and the machine production is suspended Altiok Melam ed Simulation Modeling and Analysis with Arena 14 Chapter 2 Distribution of Machine Status Consider the machine status over the time interval 0T T1 is the total idle time over 0T T B is the total busy time over 0T T D is the total down time over 0 T The probability distribution of machine status is estimated as the ratios of time spent in a state to total simulation time T 358 39 Prmachine idle I 0261 T 165 T 8735157119 39 Prmachine busy 2 B 0545 T 165 1 11987 39 Prmachine down 2 D 0194 11 165 Altiok Melam ed Simulation Modeling and Analysis with Arena 15 Chapter 2 Machine Throughput Throughput is the effective processing rate namely the number of job completions departures per unit time CT be the number of job completions in the machine over the time interval 0T Throughput is estimated as Altiok Melam ed Simulation Modeling and Analysis with Arena 16 Chapter 2 Customer Service Level Consider customers arriving at the warehouse over the time interval 0T with product demands N S is the total number of arriving customers Whose demand is totally satis ed over 0T N T is the is the total number of arriving customers over 0T J k is the unmet quantity of the demand of customer k M is the total number of customers with unmet demand over 0 T The customer service level over 07 is estimated as N 2 52 32 206667 NT 3 The customer average of unmet demand over 07 is estimated as N N T 2T J k AVeraged ov r Z J k Averaged over J k 1 CUStomerS Wlth j k 1 Total Number of T unmet demand NT Customers Altiok Melam ed Simulation Modeling and Analysis with Arena 17 Chapter 2 Distribution of Inventory Levels Consider the longterm probability that the number of nished units in the warehouse K is at some given level k estimated as total time spent with k units in stock Prk units in stock 2 t tal ti 0 me In particular for k 0 Prstockout 0036 The corresponding estimated distribution is given below PrKk 0036 0279 0218 0121 0297 0048 Altiok Melam ed Simulation Modeling and Analysis with Arena 18 Chapter 2 Average Inventory onhand The average inventory on hand in the warehouse is estimated as 5 K Z kPrKk 2506 k0 This is a consequence of the general time average formula T fKtdt 0 T K Altiok Melam ed Simulation Modeling and Analysis with Arena 19 Chapter 2 DES Modeling Languages and Tools 0 ArenaSIMAN Promodel Extend G2 GPSS SLAM MODSIM General purpose programming languages Altiok Melam ed Simulation Modeling and Analysis with Arena 20 Chapter 2

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