(2) INSTITUTO TECNOLÓGICO Y DE ESTUDIOS SUPERIORES DE MONTERREY CAMPUS MONTERREY DIVISIÓN DE INGENIERÍA Y ARQUITECTURA PROGRAMA DE GRADUADOS EN INGENIERÍA. Los miembros del comité de tesis recomendamos que el presente proyecto de tesis presentado por el Ing. Jesús Héctor Carranza Olivares sea aceptado como requisito parcial para obtener el grado académico de: Maestro en Ciencias con Especialidad en Sistemas de Manufactura. Comité de Tesis:. ____________________________ José Luis González Velarde Ph.D. Asesor. ________________________ Simme Douwe Flapper Ph.D. Co-asesor. _________________________ Jorge Limón Robles Ph.D. Sinodal. Aprobado:. _______________________ Dr. Francisco Ángel Bello Director del Programa de Graduados en Ingeniería Febrero, 2007.

(3) Dedicated to God Who has never let me down. My parents For their unconditional love. My brothers Alberto and Jorge, and my sister Nadia Who gave me the best childhood I could imagine. To my nephews Jorge and Bryan and my nieces Areli and Cristina. iii.

(4) Acknowledgements José Luis González Velarde Ph.D. For his help sharing with me some important references used in this thesis report. Simme Douwe Flapper Ph.D. For his role as a consultant and true guide. Jorge Limón Robles Ph.D. For his notable participation in two crucial classes in my Msc Program. Adenso Díaz Ph.D. For his important role as a coordinator of the Project “ALFA-LogInv: Logística Inversa y gestión respetuosa con el Medio Ambiente” under whose frame, this thesis report was developed. European Union Government For giving the funds that allow me to work abroad for three months in this thesis report. Daniel Sandoval For his exhaustive support in my company’s work issues when I was abroad. Joel Gloria and Reyes Cruz For their enthusiastic willingness to learn and to back my absence in my company. Jorge Torres and Juan de Dios Carrizales For their confidence in my personal project to study my Msc Program and to promote my scholarship from our company. Arnecom, S.A. de C.V. For being the adequate field to apply all what I learned during my Msc education period.. iv.

(5) Abstract Reverse Logistics has been stretching out worldwide, involving all the layers of supply chains in various industry sectors. While some actors in the chain have been forced to take products back, others have proactively done so, attracted by the value of used products. When customer demand is given and we have no other choice than accomplish it, we have to look for the most economical way to meet that demand, keeping in mind both quality and delivery within the customer expected time. In a framework of Reverse Logistics, under a scheme of production and rework, one key factor to deal with is to decide whether to carry out both production and rework as separated or combined. This thesis report examines behavior of operational cost over a mixture of production line’s configurations mainly differentiated by carrying out rework using an off-line repair station or doing it at on-line way, and also by sharing tooling or work force, and having a restriction where the average time an item spends in the system should be less than certain amount of time. Parameters such as percentage of good items, production rate (operators; crossfunctional workers), and rework/repair rate (repair men, cross-functional workers) were varied to evaluate system performance metrics. These parameters showed to be the most sensible ones to compare behavior of production/repair schemes carried out in a separated (off-line) or in a combined (on-line) configuration. Report presents five scenarios where three of them were solved using an analytical method based on a markovian open network with birth and death queues with infinite capacity and using an M/M/s queueing system to evaluate key performance metrics of the workstation, where the main difference among scenarios is how rework is carried out (off-line vs on-line). Other two remain scenarios were solved via simulation using Promodel where one scenario distinguishes from each other by what they share: tools or workforce. No matter what kind of method is used, system performance metrics to deal with are operational cost and the average time spent by a customer (or part) in the whole system, as main restriction.. Keywords: Reverse Logistics / production / rework, repair / operational cost / queueing theory. v.

(6) Table of contents. Dedicatory……………………………………………………………………………………………………….iii Acknowledgements…………………………………………………………………………………………..iv Abstract……………………………………………………………………………………………………………v Table of contents……………………………………………………………………………………..………vi List of figures………………………………………………………………………….….……..……………vii List of tables……………………………………………………………………………………………………..x I. Introduction ....................................................................................................1 Structure of the report ....................................................................................2 II. Literature Review ............................................................................................3 Definition and brief history of Reverse Logistics ................................................3 Delineation and scope .....................................................................................4 Rework/Repair ................................................................................................6 III. Analytical model description .............................................................................9 Preliminaries ...................................................................................................9 Analytical Model building ...............................................................................10 Terminology and notation I ...........................................................................11 IV. Analytical model procedure ............................................................................14 Scenario I Analysis ........................................................................................18 Scenario II Analysis.......................................................................................21 Scenario III-a Analysis...................................................................................26 V. Simulation model description..........................................................................32 Simulation of queueing systems .....................................................................32 Discrete-event simulation ..............................................................................33 VI. Simulation model procedure ...........................................................................38 Model Building ..............................................................................................38 Running parameters......................................................................................45 Terminology and notation II ..........................................................................48 Scenario III-b Analysis ..................................................................................49 Scenario IV Analysis ......................................................................................60 VII. Summary and results .....................................................................................71 VIII. Conclusions and future work ......................................................................92 References .........................................................................................................93 Appendix ............................................................................................................96. vi.

(7) List of Figures Figure 3.1. Flow diagram for scenario I……………………………………….…………………....18. Figure 3.2. Flow diagram to reach the optimal number of operators and tools in scenario I……………………………………………………………………………………...20. Figure 3.3. Flow diagram for scenario II…………..……………………………………………….21. Figure 3.4. Flow diagram to reach the optimal number of operators and tools in scenario II…………………………………………………………………………….……....25. Figure 3.5. Flow diagram for scenario III-a………………………………………………………..26. Figure 3.6. Flow diagram to reach the optimal number of operators and tools in scenario III-a……………………………………………………………………….…….....31. Figure 4.1. Single server queueing system………………………………………………………...32. Figure 4.2. Logic diagram of how discrete-event simulation works…………………….…35. Figure 4.3. Promodel ‘General Information’ window………………….…………….….………38. Figure 4.4. Promodel ‘Locations’ window………….……………………….………………….……39. Figure 4.5. Promodel ‘Entities’ window…………….……………………….……………….………39. Figure 4.6. Promodel ‘Path Networks’ window……….………………….………………….……40. Figure 4.7. Promodel ‘Resources’ window……….………………….………………………..……41. Figure 4.8. Promodel ‘Variables (global)’ window……….……….………………………..……42. Figure 4.9. Promodel ‘Process/Processing’ window……….……….…………………….….….43. Figure 4.10 Promodel ‘Arrivals’ window………………………………………………………………43 Figure 4.11 Promodel ‘Simulation Options’ window……….……….….…………………..……46 Figure 5.1. Flow diagram for scenario III-b………………………………………………………..49. Figure 5.2. Scenario III-b layout…………………………………………………….…………….49,51. Figure 5.3A Routing for Piece at ‘Stock_Raw’………………………………..…………………….52 Figure 5.3B Routing for Piece at ‘Worker’…………………………………..……………………….53 Figure 5.3C Routing for Piece at ‘Stock_FG’………………………………………………………..53 Figure 5.4. Plotted history of TH (TotProdRate) value…………………………………………56. Figure 5.5. Flow diagram to reach the optimal number of operators and tools in scenario III-b…………………………………………………………………………………59. Figure 5.6. Scenario IV layout………………………………………………………………………60,62. Figure 5.7A Routing for piece at ‘Stock_Raw’………………………………………………………62 Figure 5.7B Routing for piece at ‘Work_St’………………………………………………………....63 Figure 5.7C Routing for piece at ‘Stock_FG’…………………………………………………………64 vii.

(8) Figure 5.7D Routing for piece at ‘Stock_WIP’………………………………………………………64 Figure 5.7E Routing for piece at ‘Rep_St’……………………………………………………………65 Figure 5.7F Routing for piece at ‘Stock_FG (block one)’……………………………………….66 Figure 5.7G Routing for piece at ‘Stock_FG (block two)’……………………………………….66 Figure 5.8. Flow diagram to reach the optimal number of operators and tools in scenario IV…………………………………………………………………………………….70. Figure 6.1. Operational cost against arrival rate graphs along cost structures……….73. Figure 6.2A Labor cost behavior against ‘g’ under cost structure No 1……………….….74 Figure 6.2B Labor cost behavior against ‘Tp’ under cost structure No 1………………….74 Figure 6.2C Labor cost behavior against ‘g’ under cost structure No 2……………….….75 Figure 6.2D Labor cost behavior against ‘Tp’ under cost structure No 2………………….75 Figure 6.2E Labor cost behavior against ‘g’ under cost structure No 3……………….….76 Figure 6.2F Labor cost behavior against ‘Tp’ under cost structure No 3………………….76 Figure 6.3A Tooling cost behavior against ‘g’ under cost structure No 1…………….….78 Figure 6.3B Tooling cost behavior against ‘Tp’ under cost structure No 1……………….78 Figure 6.3C Tooling cost behavior against ‘g’ under cost structure No 2………………..79 Figure 6.3D Tooling cost behavior against ‘Tp’ under cost structure No 2……………….79 Figure 6.3E Tooling cost behavior against ‘g’ under cost structure No 3…………….….80 Figure 6.3F Tooling cost behavior against ‘Tp’ under cost structure No 3……………….80 Figure 6.4A Material cost behavior against ‘g’ under cost structure No 1…………….….82 Figure 6.4B Material cost behavior against ‘Tp’ under cost structure No 1……………...82 Figure 6.4C Material cost behavior against ‘g’ under cost structure No 2………………..83 Figure 6.4D Material cost behavior against ‘Tp’ under cost structure No 2…………..….83 Figure 6.4E Material cost behavior against ‘g’ under cost structure No 3…………….….84 Figure 6.4F Material cost behavior against ‘Tp’ under cost structure No 3…………..….84 Figure 6.5A Total operational cost behavior against ‘g’ under cost structure No 1 ……………………………………………………………………………….……………………86 Figure 6.5B Total operational cost behavior against ‘Tp’ under cost structure No 1 …………………………………………………………………………………………………….86 Figure 6.5C Total operational cost behavior against ‘g’ under cost structure No 2 …………………………………………………………………………………………………….87 Figure 6.5D Total operational cost behavior against ‘Tp’ under cost structure No 2 ………………………………………………………………………….………………………..87 viii.

(9) Figure 6.5E Total operational cost behavior against ‘g’ under cost structure No 3 …………………………………………………………………………….………………………88 Figure 6.4F Total operational cost behavior against ‘Tp’ under cost structure No 3 …………………………………………………………………………………………………….88. ix.

(10) List of Tables Table 6.1. Labor Cost Matrix……………………………………………………………….…………..71. Table 6.2. Tooling Cost Matrix…………………………………………………………….…………..71. Table 6.3. Material Cost Matrix……….………………………………………………….……………71. Table 6.4. Operational cost structure mixture table…..………………………….……………72. Table 6.5. Cost improvement percentage for every scenario across cost structures …………………………………………………………………………………………………….90. x.

(11) I. Introduction Twenty years ago, supply chains were busy fine-tuning the logistics of products from raw material to the end customer. Products are obviously still streaming in the direction of the end customer but an increasing flow of products is coming back. Besides this, distant sellers like e-tailers have to handle high return rates and many times at no cost for the customer. Reverse Logistics has been stretching out worldwide, involving all the layers of supply chains in various industry sectors. While some actors in the chain have been forced to take products back, others have proactively done so, attracted by the value of used products. One way or the other, Reverse Logistics has become a key competence in modern supply chains. (Dekker, de Brito, 2004). Inside Reverse Logistics Processes, the how viewpoint is meant to show how Reverse Logistics work in practice, how value is recovered from products. This case study focuses on rework/repair re-processing type as recovery option. (Appendix A). In a competitive field as manufacturing field it is, to carry out rework seems to be an additional load than a benefit itself; so companies are trying to reach this goal at the minimum operational cost. Knowing how operational cost is composed and how it reacts to changes across different external agents or internal variables, could help organizations to be aware of keeping under control such as variables. Five scenarios were studied; scenario I represents the initial situation where many companies are still working on, which means no rework is being done, and as a consequence of that, all defective parts should be scrapped and manufactured again. To have the current reference, we must calculate its parameters of interest and compare them with the same parameters of each scenario under analysis. These parameters of interest are: the average amount of time a part spends in the system (W), and the total operational cost (CT). Scenario II shows how companies face at first sight production and rework scheme, where the most used configuration is placing an off-line repair station with special repair men to carry out rework.. [1].

(12) A second option to face a production and rework scheme is combining both operations at the same workstation carrying out also both production and rework by the same worker, that means as an “on-line” rework system, such as is presented in scenario III-a. When tooling cost appears as one important economic factor to deal with, trying to keep operational cost as minimum as possible, could guide companies to set a modified version of “on-line” rework scheme by sharing tools among workers to avoid some unnecessary costs. This configuration is carried out in scenario III-b. One of the issues to deal with in scenario II is the work load balance among operators and repair men, let say, because of the proportion of work load assigned to both kind of workers used to be quite different, it happened that for some customer demand levels, to add an additional repair man to meet customer demand resulted in a low utilization of the entire repair station becoming unprofitable the whole line. Scenario IV configuration tries to boost this circumstance.. Structure of the report An introductory paragraph to the paper was entered in this chapter I; in chapter II a literature review is presented; in chapter III, an overall description of the analytical model and its main purpose is described, including a general objective function description and the terminology and notation used; chapter IV is aim to depict the general procedure followed to solve scenarios I, II, and III-a where analytical models using queueing theory was applied and also additional detailed formulation for each scenario is presented; in chapter V, a general description of the simulation model including additional terminology is presented; chapter VI, describes a full procedure to solve scenarios III-b and IV using Promodel as simulation tool; chapter VII enclose a summary and analysis of results; finally chapter VIII hold recommendations and future work.. 2.

(13) II. Literature Review Definition and brief history of Reverse Logistics Though the conception of Reverse Logistics dates from long time ago, the denomination of the term is difficult to trace with precision. Terms like Reverse Channels or Reverse Flow already appear in the scientific literature of seventies, but consistently related with recycling (Guiltinan and Nwokoye, 1974; Ginter and Starling,1978). During the eighties, the definition was inspired by the movement of flows against the traditional flows in the supply chain, or as put by Lambert and Stock (1981), “going the wrong way” (see also Murphy, 1996, and Murphy and Poist, 1989). In the early nineties, a formal definition of Reverse Logistics was put together by the Council of Logistics Management, stressing the recovery aspects of reverse logistics (Stock, 1992): “… the term often used to refer to the role of logistics in recycling, waste disposal, and management of hazardous materials; a broader perspective includes all relating to logistics activities carried out in source reduction, recycling, substitution, reuse of materials and disposal.” In the same year, Pohlen and Farris (1992) define Reverse Logistics, guided by marketing principles and by giving it a direction insight, as follows: “… the movement of goods from a customer towards a producer in channel of distribution.” Kopicky et al. (1993) defines Reverse Logistics analogously to Stock (1992) but keeps, as previously introduced by Pohlen and Farris (1992), the sense of direction opposed to traditional distributions flows: “Reverse Logistics is a broad term referring to the logistics management and disposing of hazardous or non-hazardous waste from packaging and products. It includes reverse distribution…which causes goods and information to flow in the opposite direction of normal logistics activities.” 3.

(14) At the end of the nineties, Rogers and Tibben-Lembke (1999) describe Reverse Logistics stressing the goal and the processes (the logistics) involved: “The process of planning, implementing and controlling the efficient, cost-effective, flow of raw materials, in-process inventory, finished goods, and related purpose of recapturing value or proper disposal.” The European Working Group on Reverse Logistics, REVLOG (1998-), puts forward the following definition: “The process of planning, implementing and controlling the efficient, backward flows of raw materials, in-process inventory, packaging and finished goods, from a manufacturing, distribution or use point, to a point of recovery or point of proper disposal.” In summary, the definition of Reverse Logistics has change over time, starting with a sense of “wrong direction,” going through an overemphasis on environmental aspects, coming back to the original pillars of the concept, and coming finally to a widening of its scope. Delineation and scope Since Reverse Logistics is a relatively new research and empirical area, the reader may encounter in other literature terms like reversed logistics, return logistics, retro logistics, or reverse distribution, sometimes referring roughly to the same. Reverse Logistics is different from waste management as the latter mainly refers to collecting and processing waste efficiently and effectively. Reverse Logistics concentrates on those streams where there is some value to be recovered and the outcome enters a (new) supply chain. Reverse Logistics also differ from green logistics as that considers environmental aspects in all logistics activities and it has been focused specifically on forward logistics, i.e. from producer to customer (Rodrigue et al., 2001). The prominent environmental issues in logistics are consumption of nonrenewable natural resources, air emissions, congestion and road usage, noise pollution, and both hazardous and non-hazardous waste disposal (Camm, 2001). Finally, reverse logistics can be seen as part of sustainable development. The latter has been defined by Brundland (1998) in a report to the European Union as “to meet the 4.

(15) needs of the present without compromising the ability of future generations to meet their own needs.” In fact, one could regard reverse logistics as the implementation of that at the company level by making sure that society uses and reuses both efficiently and effectively all the value which has been put into the products. After having briefly introduced the topic or Reverse Logistics, we now go into the fundamentals or Reverse Logistics by analyzing the topic from four essential viewpoints: why,. how, what and who. •. Why are things returned.. •. How Reverse Logistics works in practice.. •. What is being returned.. •. Who is executing reverse logistics activities.. As the main topic of this thesis report is rework/repair as a recovery option, how viewpoint will be described in detail. The how viewpoint is meant to show how Reverse Logistics works in practice, how value is recovered from products. Recovery is actually only one of the activities involved in the whole reverse logistics process. First there is collection, next there is the combined inspection/selection/sorting process, thirdly there is recovery (which may be direct or may involve a form of processing), and finally there is redistribution (see Figure [A1] in Appendix A). Collection refers to bringing the products from the customer to a point of recovery. At this point the products are inspected, i.e. their quality is assessed and a decision is made on the type of recovery. Products can then be sorted and routed according to the recovery that follows. If the quality is (close to) as-good-as-new, products can be fed into the market almost immediately through reuse, resale, and redistribution. If not, another type of recovery may be involved that now demands more action, i.e. a form of processing. Reprocessing can occur at different levels: product level (repair), module level (refurbishing), component level (remanufacturing), selective part level (retrieval), material level (recycling), and energy level (incineration). Note that remanufacturing is “as-good-as-new recovery while refurbishing does not have to be so.. 5.

(16) Rework/Repair In many manufacturing settings, an item may require repair at one stage before it is passed on to the next manufacturing operation. A philosophy espoused by many manufacturers today is that of ‘quality at the source’. On production floor means not only to strive for better quality output, but for each stage of the process to be responsible for its own rework so that only good items are passed to the next stage. In addition to the philosophical reasons for online repair are economic or technological reasons. Identifying the precise nature of a defect may involve running additional diagnosis tests using the same test equipment that was used to perform the original functional results. (Jewkes, E., 1994). Jewkes states that when repair is carried out on-line, the processing and repair tasks compete for the same resource, so the issue of how the tasks should be scheduled arises. She also mention the importance of predict more accurately the manufacturing lead time in order to set customer delivery dates with greater confidence, and finally establishes what useful is information regarding variability of the flow time. With the preceding considerations, she looks at the distribution of flow times in a single-stage manufacturing system with repair when scheduling is carried out with the objective of minimizing mean customer flow time. Firms can handle defects that occur on an assembly line in a number of different ways. Traditionally, defective units were either scrapped or sent to a separate repair station. With the advent of Japanese production methodology in the US, line-stop buttons were introduced to assembly lines; when a defect is discovered, the assembly line was stopped and the defect was repaired immediately (on-line repair). There are many opportunities for errors on an assembly line. Each operator has a number of tasks to perform; an error on any task may lead to a defective product. Although a few errors may be inevitable, the outgoing quality level can be improved through inspection, with defective items either repaired or scrapped. But both the inspections and the repairs are costly, and may themselves be imperfect. In addition, the cost and accuracy of repair may depend on where in the process it is done. For example, one manufacturer told us that it is ten times as expensive to repair a defect in an automotive front-wheel-drive brake unit after it is fully assembled than it is to repair the same defect at the point at which it occurs. (Robinson, McClain and Thomas; 1990). 6.

(17) Robinson, McClain and Thomas examined an automotive plant which used two different strategies for dealing with defects, and they considered a third one. The alternatives were: to stop the assembly line to repair a defect (line-stop strategy); to tag the defective unit and repair it in an off-line repair shop (repair-shop strategy); or to subdivide the assembly line and add buffer inventories to reduce the impact of repairing defects on-line (asynchronous strategy). They summarize the advantages of on-line repair inside three factors. The first factor is the length of the line, the second factor is the time to repair a defect and the third factor is the probability of producing a defect. They try to develop an intuitive feel for the relationships between these interacting factors and the relative advantages of the three repair strategies. Although ‘doing it right the first time’ is an important goal to pursue, defects and rework are common occurrences in a manufacturing process. The impact of this rework on the entire production process is, in general, not well understood. (Agnihothri and Kenett; 1995). Agnihothri and Kenett examined how the pattern of defects affects the performance of a process with 100% inspection followed by rework. Specifically they investigate the behavior of production lead time, work-in-process inventory, yield, and material handling costs for various defects levels in the manufacturing phase. They model the number of defects as a random variable having a general discrete distribution, and investigate the impact of the defect distribution on system performance metrics previously cited. They also provide management guidelines for short term control decisions such as identifying potential bottlenecks under increased workloads and allocating additional resources to release bottlenecks. In order to meet the long term goal of continuously decreasing defect levels, they propose a budget allocation method for process improvement projects. Proposal of this thesis report focuses on an analytical method based on a markovian open network with birth and death queues with infinite capacity and using an M/M/s queueing system to evaluate key performance metrics of the workstation, where the main difference among scenarios is how rework is carried out (off-line vs on-line). Other two remain scenarios 7.

(18) were solved via simulation using Promodel where one scenario distinguishes from each other by what they share: tools or workforce. By evaluating operational cost as the main performance metric and keeping in mind to accomplish a maximum preset amount of time an item could spend in the system, proposed model should tells us which scenario configuration behaves better when is run along selected range of parameters to be varied and across every cost structure defined.. 8.

(19) III. Analytical model description Preliminaries Queues (or waiting lines) are an unavoidable component of modern life. We are required to stand physically in queues in grocery stores, banks, department stores, amusement parks, movie theaters, etc. Although we don’t like standing in a queue, we appreciate the fairness that it imposes. Even when we use phones to conduct business, often we are put on hold, and served in a first-come-first-served fashion. Thus we face a queue even if we are in our own home. (Kulkarni, 1999). Queues are not just for humans, however. Modern communication systems transmit messages (like e-mails) from one computer to another by queuing them up inside the network in a complicated fashion. Modern manufacturing systems maintain queues (called inventories) of raw materials, partly finished goods, and finished goods throughout the manufacturing process. “Supply chain management” is nothing but the management of these queues! (Kulkarni, 1999). Typically, a queueing system consists of a stream of customers (humans, finished goods, messages) that arrive at a service facility, get served according a given service discipline, and then depart. The service facility may have one or more servers, and finite or infinite capacity. We start by introducing a standard nomenclature for queueing systems where customers form a single queue. Such queue is described as follows: 1. Arrival Process We assume that customers arrive one at time, and the successive inter-arrival times are iid. It is described by the distribution of the inter-arrival times, represented by special symbols as follows: •. M: Exponential;. •. G: General;. •. D: Deterministic;. •. Ek: Erlang with k phases, etc.. Note that the Poisson arrival process is represented by M (for memoryless, i.e., exponential inter-arrival times). 2. Service times 9.

(20) We assume that the service times of successive customers are iid. They are represented by the same letters as the inter-arrival times. 3. Number of servers Typically denoted by s (or k). All the servers are assumed to be identical, and that any customer can be served by any server. 4. Capacity Typically denoted by K. It includes the customers in service. If any arriving customer finds K customers in the system, he/she is permanently lost. If capacity is not mentioned, it is assumed to be infinite. Example 2.1 (Nomenclature) In an M/M/1 queue, customers arrive according to a Poisson process, request iid exponential service times and are served by a single server. The capacity is infinite. In an M/G/2/10 queue, the customers arrive according to a Poisson process and demand iid service times with general distribution. The service facility has two servers and a capacity to hold 10 customers.. Analytical Model building A markovian open network with birth and death queues with infinite capacity scheme has been considered to define formulation to be used in this model. In order to reduce complexity and do not lose certainty when calculating the output distribution after processing we used an M/M/s queueing system to evaluate the key performance parameters of the workstation. Parameters such as percentage of good items, production rate (operators; cross-functional workers), and rework/repair rate (repair men, cross-functional workers) were varied to evaluate selected performance metrics of each scenario. These parameters showed to be the most sensible ones to compare behavior of production/repair schemes carried out in a separated (off-line) or in a combined (on-line) configuration. At the end, model should calculate operational cost CT, which comes from the required number of employees/repair men, tools and material needed to fulfill customer demand. At the same time model should tell us which scenario achieves the goal at the minimum cost 10.

(21) under a restriction where the average time W of an item between its arrival and its leaving should be ‘y’ units of time; in others words, objective function is defined as follows: Min CT where: CT = CLAB + CTOL +CMAT Subject to: W < y (units of time);. Terminology and notation I Every formula used in this report is composed by three numbers separated each other for a period mark, where the number located at the left side of the period mark indicates the number of the chapter where formula is found, the number located at the center points out the number of scenario (or zero for general formulas), and the number located at the right side of the period mark refers to the order sequence of the formula inside each chapter. For all scenarios presented, following terminology was adopted: g:. percentage of produced items by the operator that are good (i.e., do not have to be reworked or scrapped);. λ:. average arrival rate to the work station, (parts per unit of time);. kp :. number of operators at the work station;. kr :. number of special repair men;. k:. number of cross-functional workers;. Tp :. exponential random variable for the operator’s service time;. E[P] : mean service time (processing) per operator or cross-functional worker at the work station; (unit of time per part); α:. proportion of time from the operator’s processing time, added to the mean processing time when parts are processed by a cross-functional worker at the work station;. Tr:. exponential random variable for the repair men’s service time;. E[R] : mean service time (repairing) per repair man or cross/functional worker at the repair station; (unit of time per part); δ:. proportion of time from the repair man’s repairing time added to mean repairing time when parts are repaired by a cross-functional worker at the work station; 11.

(22) µp :. mean service rate or capacity per operator or cross-functional worker at the work station; (parts per unit of time);. µr :. mean service rate or capacity per repair man or cross-functional worker; (parts per unit of time);. ρp :. average utilization rate for operators at the work station;. ρr :. average utilization rate for repair men;. P0p :. probability that work station is empty (station + stock room);. P0r :. probability that repair station is empty (station + stock room);. Pip :. probability that there were i customers (or parts) at the work station, (station + stock room, queue);. Pir :. probability that there were i customers (or parts) at the repair station, (station + stock room, queue);. Pkp :. probability that there were exactly kp (number of operators) customers (or parts) at the worker station, no queue;. Pkr :. probability that there were exactly kr (number of repair men) customers (or parts) at the repair station, no queue;. LQp :. average number of parts in queue at the work station;. LQr :. average number of parts in queue at the repair station;. Lp :. average number of parts at the work station, (in service + in queue);. Lr :. average number of parts at the repair station, (in service + in queue);. Wp :. average amount of time spent by any customer (or part) at the worker station (in service + in queue);. Wr :. average amount of time spent by any customer (or part) at the repair station; (in service + in queue);. WQp :. average amount of time spent in queue by any customer at the worker station;. WQr :. average amount of time spent in queue by any customer at the repair station;. W:. average amount of time spent by any customer in the entire system;. CL :. base payment fare per operator (monetary units per unit of time);. CTp :. acquisition cost for any processing tool (monetary units per tool);. CTr :. acquisition cost for any repairing tool (monetary units per tool);. M:. fixed material cost for any produced part at the workstation (monetary units per part);. d:. average percentage of the fixed material cost M added to any repaired part;. 12.

(23) Xp :. number of parts that can be processed by a processing tool before it becomes unusable (parts per tool);. Xr :. number of parts that can be repaired by a repairing tool before it becomes unusable (parts per tool);. a:. percentage points over the base payment fare applied to a repair man, a<1;. b:. percentage points over the base payment fare applied to a cross-functional worker, b<1, b>a;. c:. percentage points over repairing tool’s unit cost based on the unit cost of a processing tool;. e:. proportion of parts that a repairing tool can process based on the number of parts that a processing tool can process;. H:. proportion of the tooling cost associated to its holding cost;. CT :. total operational cost per produced part;. CLAB : labor cost per produced part; CTOL : tooling cost per produced part; CMAT : material cost per produced part. N:. Upper limit value for λ. 13.

(24) IV. Analytical model procedure Model is formulated to set a frame to analyze the impact to carry out production and rework combined in a single stage and one (type of) product production line against carrying out in a separated way (off line rework). This model assumes that all rework is successful, no transport and set up times are consider and machines and operator never fails; parts arrive at a rate of λ according to a Poisson process to a stock area with an infinite capacity room. Service time at the workstation is an exponential random variable with k identical servers. Parts form a single line and get served by the next available server on a first come, first serve (FCFS) basis. After processing, a fraction ‘g’ of parts goes directly to the finish good stock area with an infinite capacity room, while a defective faction of parts ‘1-g’, could follow any of the next ways depending which scenario we are referring to: 1. goes to scrap and it’s necessary to manufacture that portion of parts, newly. 2. goes to special repair station to be reworked. 3. is repaired at the same workstation immediately after processing. Five scenarios have been analyzed, but only three of them were chosen to be solved using an analytical method. For each scenario the main objective function is to minimize the total operational cost CT for given preliminary parameters and under one main restriction that is to accomplish certain number of hours (or any unit of time) as the maximum length of time an item spends in the system. For scenarios I, II and III-a, exact formulation based on queueing theory was used to compute first, the total amount of time an item spends in the system (W); and second, to calculate the related operational cost (CT), composed by labor, tooling and material cost. A primary condition to evaluate performance of queueing systems is to classify them according the type of arrival and processing distribution, number of parallel servers and its capacity. To start analyzing, we have to ensure that queueing system is stable. To settle this condition down we referred to Theorem 8.9 (Limiting Distribution of an M/M/s Queue) from Kulkarni, V.G., 1999, p268; which establishes that an M/M/s (M/M/k in our case) with an arrival rate of λ and ‘s’ (k) identical servers each with service rate µ is stable if and only if: 14.

(25) ρ = λ/sµ < 1. (3.0.1). Any part entering to service will spend an average E[P] amount of time, so its service rate is defined by: µ = 1/E[P]. (3.0.2). Once we confirm that queue is stable, its limiting distribution is given by: Pi= P0ρi where 1/i! (λ/µ)i. if 0 ≤ i ≤ s-1. ρi =. (3.0.3) ss/s!(ρ)i. if i ≥ s. and s-1. P0 = [ Σ (λ/µ)i/ i! + (ss/s!) (ρs/1-ρ) ]-1; see appendix B an algorithm to calculate P0 for stable i=0. M/M/s queueing systems with an infinite capacity room. After limiting distribution is calculated, now we are able to compute the average expected number of parts in the system, which is given by:. ∞. L = Σ iPi i=0. but, for systems with infinite capacity room this equation can be simplified to: L = λ/µ + Psρ/(1-ρ)2. (3.0.4). And using Little’s Law (Appendix C, [Little, J.D.C., 1961]), we get: 15.

(26) W = L / λ = 1/µ + Ps/sµ(1-ρ)2. (3.0.5). To compute CT, we have to refer as the sum of the labor, tooling and material cost: CT = CLAB + CTOL +CMAT. (3.0.6). Due to every component of the total cost depends on each scenario configuration, formulas to compute labor, tooling and material cost should be addressed at each scenario analysis section. Anyway some general premises can be settled down. Labor Cost Labor cost is composed by a base payment fare per unit of time multiplied by the number of workers (operators and repair men, or cross functional workers) needed to meet customer demand and divided by the arrival rate from the customer. Payment fare has set in CL monetary units per unit of time for a worker that only processes parts, which we call. operator. Payment fare is increased a quantity of ‘a’ percentage points for a worker that only repairs, which we call repair men. Finally, payment fare is increased a quantity of ‘b’. percentage points for a worker that can process and also can repair, which we call crossfunctional worker, where b>a; a,b,<1. In order to measure any component of operational cost in the same units, we have set every formula in monetary units per produced part. Tooling Cost Tooling cost is basically integrated by the number of tools needed to perform each particular operation and its associated acquisition cost plus the holding cost, which represents a fixed fraction ‘H’ of the tooling cost over an annual basis. We are preliminary assuming that any operator needs only one tool to do his job and they have their own tool. Unit cost for a repairing tool is a ‘c’ percentage higher than a unit cost for a processing tool, and is capable to process only an ‘e’ percentage of the parts than a processing tool can process.. 16.

(27) Material Cost Material cost is composed by the acquisition cost of all needed components to manufacture a part and it is a fixed number for every part at the worker’s station. When any produced part at the worker’s station need any kind of reparation, an average percentage ‘d’ of its original cost is added, because the use of replacement parts, basically.. 17.

(28) Scenario I Analysis. No rework is considered, all defective part should be processed newly.. ∞. 1 λ. ∞. Scrap. 2. 1-g. . . . .. g. k E[P]. ∞ FG. Figure [3.1] Flow diagram for scenario I In this scenario after processing a faction of parts 1-g goes to scrap because they were found them as defective. In order to meet customer demand, that fraction of parts 1-g should be reprocessed newly, but after each reprocessing operation a portion of parts equal to 1-g goes to scrap again. So, at the end, to deliver a λ rate of good parts, workstation must receive and additional rate of parts to reach demand rate from the customer. First we start by calculating the necessary elements to compute W, a key performance metric of the system. Due to only a proportion of parts equal to ‘g’ will be good after the initial processing a true rate of gµ parts will come out per each server, so then, from equation (3.0.1) we can calculate the minimum number of operators to meet customer demand, by: ρp=λ/kpgµp , ρp < 1, kp > 0, integer; In our model, from equations (3.0.3) we can compute P0, ρi and Pi, for i=kp: Kp-1. P0p = [ Σ (λ/gµp)i/ i! + (kpKp/kp!) (ρpKp/1-ρp) ]-1 i=0. ρkp = kpkp/kp!(ρp)kp and finally 18.

(29) Pkp = (P0p)(kpKp/kp!)(ρp)Kp Applying equation (3.0.4) to our terminology, we get: Lp = λ/gµp + Pkpρp/(1-ρp)2 Then, from equation (3.0.5) we have: Wp= Lp / λ, or Wp =1/gµp + (1/kpgµp) [Pkp/(1-ρp)2]. (3.1.1). Once W was computed, CT should be calculated, so next formulas to get every component of the operational cost are depicted. Labor cost for scenario I is composed by the product of the base payment fare per operator (CL) and the number of operator required (kp), being divided by the arrival rate from the customer demand. So this equation could be written down as follows: CLAB = CL kp / λ. (3.1.2). For scenario I, tooling cost is only composed by the operator’s activity, so we can obtain it by dividing the cost for one processing tool (CTp) by the number of produced parts that a processing tool is useful before it becomes unusable (Xp), and adding it its holding cost which is computed by multiplying CTp by the number of operators (or tools, according rule: number of tools = number of operators) and dividing it by the product of the true arrival rate or parts (λ/g) and the number of working hours in one year due to parameter H is on an annual basis. So, tooling cost per unit of time, can be written down as follows: CTOL = CTp / Xp + CTp kp H g / λ (#wrk.hrs/year). 19. (3.1.3).

(30) Due to no rework is carry out in scenario I, the fraction of parts that are defective then are scrapped and consequently, this amount of parts need to be manufactured newly, incurring in additional full base material cost. Equation is written as follows: CMAT = M / g. (3.1.4). Finally from equation (3.0.6), total operational cost of the workstation, is calculated as follows: CT = CLAB + CTOL + CMAT Figure [3.2] shows the flow diagram to reach the optimal number of workers and tools for scenario I, needed to compute W and CT; which later in the form of graphs are presented in results section. Start. λ=1 kp=1. Run/Calculate. W≤3. No. kp = kp+1. Yes λ = λ+1. No. Yes λ>N. End. Figure [3.2] Flow diagram to reach the optimal number of operators and tools in scenario I. 20.

(31) Scenario II Analysis. All defective parts should be reworked by special repair men ("off-line rework").. 1. ∞. 1. ∞. λ. 2. . . . .. 1-g. kp E[P]. 2. . . .. kr. E[R]. g. ∞. Figure [3.3] Flow diagram for scenario II After processing, a faction of parts 1-g go to a common stock area with an infinite capacity room prior to be reworked at a repair station because they were found them as defective. In this repair station, service time is also an exponential random variable with an expected value E[R] and using k identical servers. Parts form a single line and get served by the next available server on a (FCFS) basis. All rework is successful, so after rework, all parts can go to the finish good stock area. After confirm queue stability with equation (3.0.1) we can also use this equation to calculate the minimum number of operators and repair men necessary to meet customer demand: ρp=λ/kpµp , ρp < 1, Kp > 0, integer; for the worker’s sation, and ρr=λ(1-g)/krµr , ρr < 1, Kr > 0, integer; for the repair man’s station. where µp and µr can be calculated from (3.0.2): µp = 1/E[P], and µr = 1/E[R] Now, from equations (3.0.3) we can compute P0, ρi and Pi, for i=k, for worker’s station and repair man’s station, as follows: 21.

(32) Kp-1. P0p = [ Σ (λ/µp)i/ i! + (kpKp/kp!) (ρpKp/1-ρp) ]-1, for worker’s station, and; i=0. Kr-1. P0r = [ Σ (λ(1-g)/µr)i/ i! + (krKr/kr!) (ρrKr/1-ρr) ]-1, for repair man’s station. i=0. Then, Pkp = (P0p)(kpKp/kp!)(ρp)Kp, for worker’s station, and; Pkr = (P0r)(krKr/kr!)(ρr)Kr, for repair man’s station. Now from equation (4.4) we can compute the expected number of parts in the system by: Lp = λ/µp + Pkpρp/(1-ρp)2, being served and in queue at the worker’s station, and; Lr = λ(1-g)/µr + Pkrρr/(1-ρr)2, being served and in queue at the repair man’s station. And from equation (3.0.5) we will be able to calculate the average amount of time a part spends in the system: Wp= Lp / λ, or Wp =1/µp + (1/kpµp) [Pkp/(1-ρp)2], at the worker’s station, and; Wr= Lr / λ(1-g), or Wr =1/µr + (1/krµr) [Pkr/(1-ρr)2], at the repair man’s station. Due to not all parts go through the same routing, the average amount of time an incoming part spends in the system is given by: W = gWp + (1-g)(Wr+Wp) this can be simplified as: W = Wp + (1-g)Wr. (3.2.1). Knowing W can allow us to determine if the production line is capable to satisfy a certain lead time required by the customer. Now is turn to compute every component of operational cost. 22.

(33) Labor Cost In this scenario, labor cost is integrated by operator and repair men labor cost, in general, both are composed by the product of payment fare CL for an operator and the number of operators (kp), plus the product of (1+a)CL payment fare, that is the new payment fare applied to a repair man, and the number of repair men required (kr); then both divided by the arrival rate from the customer demand. This equation could be written down as follows: CLAB = [CL kp + (1+a)CLkr] / λ. (3.2.2). Tooling Cost For scenario II, tooling cost is composed by operators and by repair men’s activity, so we can obtain tooling cost from the operator’s activity by dividing the cost for one processing tool (CTp) by the number of produced parts that a processing tool can process before it becomes unusable (Xp); and adding its correspondent holding cost. To obtain tooling cost from repair men’s activity we should divide the cost for one repairing tool (CTr) by the number of produced parts that a repairing tool can process before it becomes unusable (Xr); and adding its correspondent holding cost. By summarizing both, we can get tooling cost for scenario II, as follows: CTOL = CTp/Xp + CTpkpH/λ(#wrk.hrs/year) + CTr/Xr + CTrkrH/λ(1-g)(#wrk.hrs/year) where: CTr= (1+c)CTp and, Xr= e Xp so then, CTOL = CTp/Xp +CTpkpH/λ(#wrk.hrs/year) + CTp(1+c)/eXp + CTp(1+c)krH/λ(1-g)(#wrk.hrs/year). 23.

(34) Simplifying: CTOL = CTp/Xp[1+(1+c)/e] + CTpH/λ(#wrk.hrs/year)[kp+(1+c)kr/(1-g)]. (3.2.3). Material Cost In scenario II, due to rework instead of manufacture newly, only a percentage (d) of the fixed material cost per part (M), is added to the material cost of the amount of parts (1-g) that were found as defective in workstation. So then, formulas to compute material cost from operator and from repair men’s activity, CMAT = M + M d;. (3.2.4). or CMAT = M(1+d); Finally from (3.0.6) we get the operational cost of the workstation. CT = CLAB + CTOL + CMAT Figure [3.4] shows the flow diagram to reach the optimal number of workers and tools for scenario II, needed to compute W and CT; which later in the form of graphs are presented in results section.. 24.

(35) Start. λ=1 kp=1 kr=1. Run/Calculate. W≤3. No. kp = kp+1. Yes Run/Calculate. λ = λ+1. No. λ>N. Yes W≤3. No. kr = kr+1. Yes End. Figure [3.4] Flow diagram to reach optimal number of operators and tools in scenario II. 25.

(36) Scenario III-a Analysis. Operators do all rework themselves ("on-line rework") where each operator has his own tool. λ(1-g) 1. ∞. λ. Tp. 2. . . . .. Tr. λ. ∞. s Ts. Figure [3.5] Flow diagram for scenario III-a In this scenario operator (cross-functional worker) will spend a time Tp to produce an item that he found as good, and a time Tp + Tr when after processing he found a defective item, and it need to be repaired. Tp and Tr are both exponential random variables independent one from each other, with mean E[P] and E[R], respectively. Due to cross-functional usage in this scenario, standard processing time Tp used in scenarios I and II are affected by an ‘α’ (alpha) factor as described below: TpIII-a = (1+α)TpI where suffix III-a and I, refer to the scenario under analysis. At this time is appropriate to formulate an additional terminology, specific for this scenario: Ts:. general distributed random variable for the cross functional worker’s service time, including processing and repair.. E[S]:. mean weighted service time (processing or processing plus repairing) at the repair station; unit of time per part;. µs :. mean service rate or capacity per cross-functional worker; parts per unit of time;. In order to save terminology, we will keep Tp as originally stated in section III for every scenario. Knowing λ, E[P], E[R] and g, we must calculate first the mean weighted processing time E[S] as follows: E[S] = g E[P] + (1-g) (E[P]+E[R]). 26.

(37) or E[S] = E[P] + E[R](1-g). (3.3.1). As stated earlier, processing mean times E[P] and E[R] comes from exponential random variables Tp and Tr, respectively; but the mean weighted service time E[S] now is related to the processing random variable Ts which won’t be an exponential distributed variable anymore, but it will be a general (G) distributed random variable. So for this scenario, our model becomes an M/G/k queueing system. Knowing that no exact method known to calculate W in this type of models is available, we start setting down some preliminaries using the system M/G/1, according Ross, Sheldon M., 2000; p.458 and 459. For an arbitrary queueing system, let us define the work in the system at any time t to be the sum of the remaining service times of all customers in the system at time t. Imagine that entering customers are forced to pay money (according to some rule) to the system. We would then have the following basic cost identity: average rate at which the system earns = λ x average amount an entering customer pays (3.3.2) and considering the following cost rule: each customer pays at a rate of y/unit time when his. remaining service time is y, whether he is in queue or in service. Thus, the rate at which the system earns is just the work in the system; so the basic identity yields that: V= λ E[amount paid by a customer] Now, let S and WQ denote respectively the service time and the time a given customer spends waiting in queue. Then, since the customer pays at a rate of S per unit of time while he waits in queue and at a rate of S-x after spending an amount of time x in service, we have S. E[amount paid by a customer] = E [ SWQ + ∫ (S-x)dx ] 0. and thus 27.

(38) V = λ E[SWQ] + λ E[S2]/2. (3.3.3). It should be noted that the preceding is a basic queueing identity and as such is valid in almost all models. In addition, if a customer’s service time is independent of his wait in queue (as is usually, but not always the case), then we have from equation (3.3.3) that V = λ E[S]WQ + λ E[S2]/2. (3.3.4). Now, for an arbitrary customer in an M/G/1 system: Customer’s wait in queue = work as seen by an arrival But, due to Poisson arrivals, the average work as seen by an arrival will be equal to V, the time average work in the system. Hence for the model M/G/1, WQ = V The preceding in conjunction with the identity V = λ E[S]WQ + λ E[S2]/2 yields the so-called Pollaczek-Khintchine formula, WQ = λE[S2]/2(1- λE[S]). (3.3.5). where E[S] and E[S2] are the first two moments of the service distribution. Now let’s consider the M/G/k system, for an arbitrary arrival, we will have the following identity: work in system when customer arrives = k x time customer spends in queue + R. (3.3.6). where R is the sum of the remaining service times of all other customers in service at moment when a new arrival enters service. 28.

(39) The foregoing follows because while the arrival is waiting in queue, work is being processed at a rate of k per unit of time (since all servers are busy). Thus, an amount of work k x time in queue is processed while he waits in queue. Now, all of this work was present when he arrived and in addition the remaining work on those still being served when he enters service was also present when he arrived ― so we obtain equation (3.3.6). For an illustration, suppose that there are three servers all of whom are busy when the customer arrives. Suppose in addition, that there are no other customers in the system and also that the remaining service times of the three people in service are 3, 6 and 7. Hence, the work seen by the arrival is 3 + 6 + 7 = 16. Now the arrival will spend 3 time units in queue, and at the moment he enters service, remaining times of the other two customers are 6 - 3 = 3 and 7 – 3 = 4. Hence R = 3 + 4 = 7 and as a check of equation (3.3.6) we see that 16 = 3 x 3 + 7. Taking expectations of equation (3.3.5) and using the fact that Poisson arrivals see time averages, we obtain V = kWQ + E[Rs] which, along with equation (3.3.5), would enable us to solve for WQ if we could compute E[Rs]. However there is no known method for computing E[Rs] and in fact, there is no known formula for WQ. The following approximation for WQ was obtained from Nozaki and Ross (1978) by using the foregoing approach:. k. WQ. ≈. 2. λ E[S ](E[S]) 2(k-1)!(k-λE[S]). 2. k-1. ∑ n=0. (λE[S]) n!. k-1. n. +. (λE[S]). k. (k-1)!(k-λE[S]). (3.3.7). E[S2] was calculated from the general formula of variance: V[S] = E[S2] – E[S]2, from Scheaffer and McClave, 1995. Appendix D describes an algorithm to calculate WQ for stable M/G/s queueing systems with an infinite capacity room. Finally from (3.0.5) we can compute W: W = WQ + E[S]. (3.3.8) 29.

(40) Labor Cost In this scenario, labor cost is integrated only by a cross-functional worker’s activity. Payment fare now is set to (1+b)CL that corresponds to the payment fare applied to a cross-functional worker, and it must be multiplied by the number of workers required (k), as it is shown below: CLAB = (1+b)CLk/λ. (3.3.9). Tooling Cost In scenario III-a, two different types of operations are made by the same worker using a different tool for each operation, so then, it is convenient to set formulation according this scheme. By dividing the acquisition cost for a processing tool CTp by the number of parts a processing tool can process Xp, and adding it its related holding cost, we obtain the tooling cost per produced part from the worker’s activity for exclusive processing operations. Then, to calculate the tooling cost per produced part from the worker’s activity derived only from repairing operations, we divide the acquisition cost for a repairing tool CTr by the number of parts a repairing tool can process Xr, and adding its correspondent holding cost. Finally, by summarizing both components, we can get the formula to compute the tooling cost per produced part for this scenario, as it is written down here: CTp/Xp + CTpkH/λ(#wrk.hr/year) + CTr/Xr + CTrkH/λ(1-g)(#wrk.hr/year) or, CTp/Xp + CTpkH/λ(#wrk.hr/year) + CTp(1+c)/eXp + CTp(1+c)kH/λ(1-g)(#wrk.hr/year) (3.3.10) this can be simplified as, (CTp/Xp)[1+(1+c)/e] + [CTpkH/λ(#wrk.hr/year)][1+(1+c)/(1-g)]. 30.

(41) Material Cost Due to material cost is fixed no matter who processed the part, for scenario III-a material cost is calculated as same as for scenario II, the only difference is that in scenario II processing and repairing operations are made by different workers while in scenario III-a both operations are made by the same worker. At the end, this condition is meaningless when computing material cost. So then, from equation (3.2.4) we calculate material cost per unit of time for this scenario: CMAT = M + M d; or CMAT = M(1+d);. Figure [3.6] shows the flow diagram to reach the optimal number of workers and tools for scenario III-a, needed to compute W and CT; which later in the form of graphs are presented in results section. Start. λ=1 k=1. Run/Calculate. W≤3. No. k = k+1. Yes λ = λ+1. No. Yes λ>N. End. Figure [3.6] Flow diagram to reach optimal number of operators and tools in scenario III-a 31.

(42) V. Simulation model description. Simulation of queueing systems A queueing system is described by its calling population, the nature of the arrivals, the service mechanism, the system capacity, and the queueing discipline. In the single channel queue depicted in figure [4.1], the calling population is infinite; that is, if a unit leaves the calling population and joins the waiting line or enters service, there is no change in the arrival rate of other units that could need service. Arrivals for service occur one at a time in a random fashion; once they join the waiting line, they are eventually served. In addition, service times are of some random length according to a probability distribution which does not change over time. The system capacity has no limit, meaning that any number of units can wait in line. Finally, units are served in the order of their arrival by a single server or channel.. Waiting line. Calling population. Server. Figure [4.1] Single server queueing system Arrivals and services are defined by the distribution of the time between arrivals and the distribution of service times, respectively. For any single or multichannel queue, the overall effective arrival rate must be less than the total service rate, or the waiting line will grow without bound. When queues grow without bound, they are termed “explosive” or unstable. The state of the system is the number of units in the system and the status of the server, busy or idle. An event is a set of circumstances that causes an instantaneous change in the state of the system. In a single channel queueing system, there are only two possible events that can affect the state of the system. They are the entry of a unit into the system (the arrival event) and the completion of service on a unit (the departure event). The queueing 32.

(43) system includes the server, the unit being serviced (if one is being serviced), and the units in the queue (if any are waiting). The simulation clock is used to track simulated time. Now how the events just described occur in simulated time? Simulations of queueing systems generally require the maintenance of an event list for determining what happens next. The event list tracks the future times at which the different types of events occur. In simulation, events usually occur at random times, the randomness imitating uncertainty in real life. For example, it is not known with certainty when the next customer will arrive at a grocery store, or how long the bank teller will take to complete a transaction. In these cases, a statistical model of the data is developed either from data collected and analyzed or from subjective estimates and assumptions. (Banks, Carson II, Nelson, Nicol,2005).. Discrete-event simulation A discrete-event simulation is one in which the state of a model changes at only discrete, but possibly random, set of time points, known as event times. Two or more traffic units often have to be manipulated at one and the same time point. Such “simultaneous” movement traffic at a time point is achieved by manipulating units of traffic serially at that time point. The “transaction-flow world view” often provides the basis for discrete-event simulation. In this view, a system consists of discrete units of traffic that move (“flow”) from point to point in the system while competing with each other for the use of scarce resources. (Schriber, Brunner, 1996). Most simulation software, including Promodel, presents a process-oriented world view to the user for defining models. This is the way most humans tend to think about systems that process entities. When describing a system it is natural to do so in terms of the process flow. Entities begin processing at activity A then move onto activity B and so on. In discrete-event simulation, these process flow definitions are translated into a sequence of events for running the simulation: first event 1 happens (an entity begins processing at activity A), then event 2 occurs (it completes processing at activity A), and so on. Events in simulation are of two types: scheduled and conditional, both of which create the activity delays in the simulation to replicate the passage of time.. 33.