Intermediate Production Storage Dimensioning Using Occupancy-dependent Key Performance Indicators

Transcription

1 Intermediate Production Storage Dimensioning Using Occupancy-dependent Key Performance Indicators Realize innovation. Key Performance Measures of a Storage Facility Contents The Full and Empty Portion as Performance Indicators Analyzing the Occupancy Performance Indicators of Queueing Systems and Little s Formula Applications of Little s Formula in Economics Energy-saving Measures for Queueing Systems Summary Page 2

2 The Full and Empty Portion as Performance Indicators We consider modeling an intermediate production storage with the basic object Buffer. Also the basic object ParallelProc can be used for this purpose. Since the parts can pass each other, much more applications are possible. The most frequent use is buffering to avoid unwanted waiting and blocking times of linked production resources. This regulating function can be analyzed using the following statistical methods of the Buffer which are shown on the tab Statistics. statrelativeemptyportion > 0: Maybe a waiting time of the successor could not have been avoided. statrelativefullportion > 0: Blockages of the predecessor can possibly occur. A full or empty Buffer is not able to effectively influence the material flow. Page 3 Analyzing the Occupancy Other key performance indicators can be derived from the relative occupancy, also known as utilization U (also on the tab Statistics). Let us consider a ParallelProc with a capacity C = 2 during the simulation time T = 10 min and 2 parts with processing times t 1 = 4 min and t 2 = 6 min. t 1 t 2 statrelativeoccupation: Interruptions such as Failures and Pauses are eliminated from the dwelling times. 2 min Failure statrelativeoccupationir: Processing times as well as interruptions are included. t 1 t 2 Page 4

3 Performance Indicators of Queueing Systems and Little s Formula A queueing system consists of a service station (SingleProc) and a waiting area (Buffer). We will determine the mean waiting time and the mean queueing length using the relative occupancy (utilization) of a resource without interruptions. We denote the sum of all waiting times by. Plant Simulation records the relative occupancy. The mean queueing length L is defined by. is the mean waiting time. We get and. Note that n is recorded by the statistics method statnumin. Page 5 Performance Indicators of Queueing Systems and Little s Formula If the simulation reached a steady state then the arrival rate and the leaving rate are equal. In other words, the system is in equilibrium. Such waiting lines are unpleasant experiences in all areas of life and therefore these queues were intensively investigated. The arrival rate and the inter-arrival time of the Source satisfy. Little s Formula: The ratio of the mean values of the processing time of the SingleProc and the inter-arrival time of the Source is the probability that the SingleProc is busy. Therefore the service time is at most equal to the inter-arrival time. Let L and L q be the mean number of parts in the entire system or in the buffer, resp., and W and W q are the corresponding mean dwelling times. Page 6

4 Performance Indicators of Queueing Systems and Little s Formula Little s Formula seems intuitively reasonable: = W Time W L inter-arrival time = total time W Actually: Note that it is not the mathematical proof, since L, and W are continuous random variables with arbitrary distributions. Using the equation we can verify Little s Formula by an experiment study in our simple queueing system. Page 7 Applications of Little s Formula in Economics Consider a general system with discrete flows which is in a stable state. An example can be a warehouse, a production facility or a financial system, such as banks or insurance companies. We can interpret L as Inventory Inv. The input and the output rate are frequently called Throughput Rate. The dwelling time W can be considered as Flow Time. Raw material Rate = 6000 k /year Production system Inv = 1200 k How long will it take until the product is processed? Time = 0,2 year Page 8

5 Energy Saving Measures for Queueing Systems Due to large inter-arrival times of orders, an operational Machine consumes an unnecessary amount of energy. Therefore the orders are to be collected in a Buffer. During this collection period the Machine is set to the energy-saving state StandBy. The energy state can be switched from Operational to StandBy only if both the Machine and the Buffer are empty. Therefore the essential system states are the total numbers of parts on the objects Buffer and Machine. The total numbers can be changed either by the entrance of a part into the Buffer or by the exit of a part out of the Machine (detected by the entrance control or the rear-triggered exit control) Page 9 Energy Saving Measures for Queueing Systems Trade off between energy-saving measure and storage costs Energy consumption Storage costs A reduced energy consumption leads to a significant increase of the storage costs. Page 10

6 Summary A full or empty production storage is not able to effectively influence the material flow. Corresponding statistical methods record such system states. The relative occupancy is determined by the sum of the dwelling times of all parts that are located in the considered system. From this sum we can derive the mean number of parts and the mean dwelling time in a system. Little s formula is of great importance for the discrete simulation and has many application in economics. Page 11 Thank you for your attention! Dr. rer. nat. Peter-Michael Schmidt Page 12