Excerpt

## Table of Contents

List of Figures and Tables

List of Abbreviations

1 Introduction

1.1 Problem of the Thesis

1.2 Aims of the Thesis

1.3 Course of Research

2 Critical Perspective on Queuing Theory Approach

3 Theoretical Foundations

3.1 Queuing Theory

3.2 Probability Theory and Statistics

3.3 Production Logistics

3.4 Discrete Event System Simulation

4 Methodology

4.1 System Analysis

4.2 Raw Data Collection

4.3 Data Preparation and Modeling Procedure

4.4 Inputs for the Analysis

4.5 Output of the Analysis

4.6 Assumptions and Simplifications

5 Implementation

5.1 Exemplary Procedure Funnel 1

5.2 Table of Results

6 Final Consideration

6.1 Results and Critical Reflections

6.2 Implications for Further Research

6.3 Implications for Practice

Table of References

Affidavit

## List of Figures and Tables

Figure 1: Simple Queuing System

Figure 2 : The Funnel Model

Figure 3 : The Throughput Diagram

Figure 4: System Flowchart

Figure 5 : Distribution Arrivals Quarter 1

Figure 6 : Distribution with Best Fit for Arrival Data

Figure 7 : Exemplary Histogram Service Times Quarter 1

Figure 8: Histogram (excluding outliers)

Figure 9: Q-Q Plot Exponential Distribution-Histogram

Figure 10: Goodness of Fit Test for the Exponential Distribution

Table 1: System Components

Table 2: Timestamps Arrivals and Departures

## List of abbreviations

illustration not visible in this excerpt

## 1 Introduction

### 1.1 Problem of the thesis

Lufthansa Technik (LHT) is a provider of maintenance, repair and overhaul (MRO) services for aircraft. While LHT has traditionally performed the MRO services in-house, the subcontracted operations business is gaining momentum.

With about 90,000 outsourced MRO purchase orders (POs) per year, the process becomes increasingly complex. In the past, the company set the objective of achieving an average turnaround time (TAT) of five shop calendar days (SCD) for internal repair operations.

One of the main goals of LHT is to achieve an average TAT of fifteen SCD for the subcontracted MRO operations, between the monitoring points ‘Creation PO Requisition’ (i.e.: TS 29) and ‘Receive Unit Transaction’ (i.e.: TS 69). The current TAT is significantly higher, with substantial process variability.

The repair subcontracting value-chain consists of several processes, which fall under different areas of responsibility. Due to the stochastic nature of the series of processes and because many variables influence the overall performance of the system, a simulation study needs to be conducted. The present thesis aims to develop an analytical input model for a simulation study in technical procurement.

*Practical relevancy of the research problem*

The building of the analytical input model and subsequent simulation model is a substantial milestone towards the process optimization in the subcontracted MRO operations business.

The modeling of processes as an M/M/1 queuing system^{1} is the first step needed in order to

i) Simulate the processes and leverage optimization methods from the field of Production Logistics; Ileana Constantinescu: Process Mapping and Stochastic Input Modeling 2

ii) Identify whether the application of queuing theory elements for process description and analysis within technical procurement would aid in improving the processes.

*Theoretical relevancy of the research problem*

The thesis aims at applying already existing elements from queuing theory and statistics to a relatively new field - technical procurement, as to bring a contribution to the queuing science, an interface between queuing theory and practice.

### 1.2 Aims of the thesis

Since the end goal is to model each step where waiting time^{2} occurs as a queue, it is important to determine which probability distributions for the arrivals and service times are appropriate representations of the underlying processes. The previously mentioned distribution will serve as input models for a technical procurement simulation study.

A secondary goal of this paper is to critically review the queuing theory and acknowledge its inherent limitations. Chapter 5 will commence by showing that queuing models do, by no means, represent the universal tool of analysis, and continue by explaining its suitability for the analysis of the situation on hand.

### 1.3 Course of research

Departing from raw company data for the past year, for three quarters and approximately 22,000 records per quarter, appropriate probability distributions for the arrivals at each funnel^{3} in the process-sequence will be determined. Additionally, the processing time distribution shall be determined.

The steps in the input modeling will follow the conventional approach used and prescriptions from traditional literature: the data collection and preparation, the formulation of assumptions, the formulation of hypotheses for the expected probability distributions and distribution fitting.^{4}

## 2 Critical Perspective on the Queuing Theory Approach

This chapter will commence by describing the limitations of the queuing models, and continue by elaborating on their applicability to the analyzed system.

First, the queuing models are based on five key assumptions: the stationary process, stochastically independent arrivals, the memoryless property, exponentially distributed service times and steady state of the system. Buzen points out that the several assumptions, on which queuing models are based, are not always realistic and in any case difficult to verify.^{5}

As an additional limitation, queuing models cannot capture the transient behaviour, which is inherent to the manufacturing processes, whereas simulation can.

A third limitation results from the assumption of infinite buffer capacity. With the inventory reduction trends that aim at limiting or eliminating the buffers, this assumption is clearly unrealistic.

A fourth limitation stems from the first come-first serve assumption, which impedes the analysis of the system operational characteristics as well as the exploitation of the different scheduling policies.^{6}

Furthermore, as pointed out by van Dijk^{7}, while practitioners benefit from queuing insights, which aid in the conclusion of general practical rules for improving delays or process times, a hybrid analytic-simulation^{8} approach is often required to get these rules implemented.

Finally, the queuing concepts applied in practice have not evolved since last century, fact that might be due to the lack of communication between the theoretical researcher and the applied scientist.^{9} Likewise, as Kendall acknowledged, the developments in the theoretical research^{10} imply sophisticated mathematical procedures, which are usually too complicated to be applied in practice.

The above-mentioned considerations are arguments good enough to establish that the queuing models have their inherent limitations and are not the optimal tool for any type of analysis, which involves waiting time.

Particularly in standard processes, which are amenable to automation, it is beneficial to make use of customized tools. This motivated the development of the Funnel Model, of the Throughput Diagram and of the Logistics Operating Curves (LOCs) for the analysis of the production systems, tools which inherit elements from queuing theory, but that better describe measures of production systems in research and practice.^{11}

LOCs are a handy tool for analysis for the manufacturing industry.^{12} However, it should be pointed out that specifically because LOCs are designed to model manufacturing processes, they are not as well suited for the analysis of technical procurement. Allowing for the abovementioned considerations, it appears that queuing theory principles are more suitable and applicable to the situation on hand.

Firstly, the building of LOCs requires extensive data input which, due to the inherent differences in the processes, is either unavailable or irrelevant in technical procurement. Secondly, whereas the logistics operating curves are used for the analysis of a company’s internal production process, the production process dealt with in this thesis is split up in many sub-processes, each with its own level of documentation and process owners.

The existence of a large number of external process owners and very individualized flow of each order impedes the same level of detail for the documentation of processes.

Thirdly, the system analysed here is very fragmented and a simulation study needs to be conducted, in order to attempt to optimize the processes involved. To carry out a simulation using inputs such as inter-arrival times or arrival rates, we have to specify their probability distributions.^{13} Moreover, a simulation study requires inputs such as arrival rates or service time distribution, which can easily be computed via queuing theory principles.

While LOC can at most provide an approximation of the shape of the probability distribution, the use of statistics in conjunction with queuing theory enables the determination of the probability distributions for a data set.

Lastly, after the determination of the input models for the arrival rates and service durations, queuing theory is more flexible in that one can vary the arrival rates, number of servers, or processing sequence.

## 3 Theoretical Foundations

### 3.1 Queuing Theory

The research is contained within the general area of queuing theory, a theoretical discipline used to mathematically model real world processes that involve waiting and service times. Queuing theory has its origins in the early 1900s for the design of telephone systems.^{14} For analyzing production systems queuing theory plays an important role^{15}, since it provides a good conceptual model of waiting line conditions^{16}, providing one with an understanding of the influencing factors like arrival distribution, service distribution or number of servers.^{17}

By a *queuing process* will be meant a mathematically specified operation in which units arrive, wait and then leave.^{18} Units are arriving at the queue, and departing from the server, which, is conceptualized as a delay block.^{19}

illustration not visible in this excerpt

Figure 1: Simple Queuing System

In a queuing process, let 1/λ be the expected time between arrivals of two consecutive units, E [L] be the mean number of units in the system and E[S] is the mean time spent by a unit in the system. Little shows that if the three means are finite and the three stochastic processes strictly stationary^{20}, then

E[L] = λE[S]^{21}

, plausibility of which is proved by Morse, who shows that the relationship holds a number of particular situations.^{22} Furthermore, Little observes^{23} that the results are free of specific assumptions about arrival and service time distribution, independence of inter-arrival times, number of channels and queue discipline. Likewise, Morse maintains^{24} that the only situation when this relationship fails to hold is when arrivals come with rate λ but not all arrivals join the system.

There are three basic elements that comprise a queuing system: the entities that do the waiting in quest for their resources^{25}, the resources for which the waiting is done^{26}, generically called servers and the space where the waiting is done, called a queue.^{27}

Authors such as Cassandras and Lafortune segment the components of queuing system models into the specification of stochastic models (e.g. for arrival and service processes), structural parameters (e.g. the capacity of the queue and number of servers) and operating policies used (e.g. prioritization).^{28}

Kendall^{29} denoted a queuing system by an arrival distribution, a service distribution, the number of servers, the buffer capacity, population size and service discipline.^{30}

The M/M/1 model to be used in this thesis is characterized by a memoryless^{31} distribution for the arrival and service, by a single server, infinite buffers and FIFO service discipline.

### 3.2 Probability Theory and Statistics

In this section several concepts from probability theory and statistics will be defined, which shall serve as framework of understanding the steps made in the current input modeling process.

For a discrete random variable, a *probability distribution* is a mathematical expression representing the variable of interest.^{32} Likewise, a distribution of values indicates that there is a probability associated with the occurrence of any particular value.^{33}

A *histogram* is a plot of distribution of frequency of occurrence^{34}, which is useful to identify the shape of a distribution.

The *mean* of a probability distribution is the expected value of its variable.^{35} In probability theory, the *expected value* (E[X]) or mean of a random variable equals the weighted average of all possible values the respective random variable can take on.^{36} The weights used in computing this average correspond to the probabilities in case of a discrete random variable, or densities in case of a continuous random variable.^{37}

For a *stationary process*^{38}, statistical properties such as the mean values and probability density^{39} do not vary with time.

The *variance* ( Var[X] ) is a measure of the dispersion of a random variable about its mean.^{40} Consequently, the larger Var[X] the more likely the random variable is to take on values far from its mean E[X].

Based on the underlying nature of the processes and system under consideration, the null hypotheses to be tested will be that the arrivals at each funnel are Poisson distributed, while their associated service durations are exponentially distributed.

The *Poisson distribution* is a discrete distribution often used to count the number of occurrences of some event in a given time interval.^{41} In this thesis, the mean E[X] and variance Var[X] are of concern, which in the case of the Poisson distribution both equal the arrival rate^{42}:

E[X] = Var[X] = λ.^{43}

A useful indicator whether data is Poisson distributed is to calculate the mean and standard deviation of the number of events counted in a particular time interval.^{44}

Additionally, if the condition that the standard deviation approximately equal radical of the mean fails to hold, the data is not Poisson distributed.^{45}

The second distribution of concern is the *exponential distribution* is a continuous distribution used to measure the time interval needed to perform some activity.^{46} One of the most important properties of the exponential distribution is that it possesses the *memoryless property*^{47}, also called the Markov property.^{48}

The latter implies that the past history of a random variable that is exponentially distributed plays no role in predicting its future.^{49} In stochastic modeling, random variables such as interarrival times and service times are frequently modeled as exponentials.^{50}

Reasons for the use of the exponential distribution in this thesis include its analytical tractability and relationship to the Poisson process.

Furthermore, models with exponentially distributed inter-arrival and service times are known to provide conservative estimates of the system behavior.^{51}

In this thesis, the mean and the variance are of concern^{52}, which in the case of the exponential distributions are given by the formulas:

E[X] = and Var[X] =

### 3.3 Production Logistics

This section elaborates on several key logistics concepts, which are conductive to the very understanding of waiting line conditions.

Essentially, both the Funnel formula and Little’s Law formulate the relationship between and the manufacturing inventory, the performance and the throughput time.^{53} The *work content*, key parameter for the Funnel Model^{54} is a measure of the planned time for an operation at a workstation stated in planned hours.^{55}

Furthermore, Nyhuis defines the formula for the WC^{56}: WC = where WC is the work content expressed in hours (hrs), LS is the lot size expressed in units, top is the processing time per piece expressed in min/piece and ts is the setup time per lot expressed in min.

Since it frequently is necessary to express how long the workstation is occupied in shop calendar days (SCD), Nyhuis defines the parameter *operation time* (TOP [SCD]), calculated by dividing the work content (WC [hrs]) of a workstation by its maximum output rate (ROUTmax [hrs/SCD]).^{57}

**[...]**

^{1} Cf. ( Kendall, D.G 1953), pp. 338-354.

^{2} Essentially, a waiting time before being serviced, as described in section 3.1, 3. 3 and 4.1 and illustrated by figures 1, 4.

^{3} Essentially, a waiting time before being serviced, as described in section 3.1, 3. 3 and 4.1 and illustrated by figures 1, 4.

^{4} Cf. (Banks 2007), pp. 375-384.

^{5} Cf. (Buzen 1978), p.175-194.

^{6} Cf. (Jackman 1993), p. 805.

^{7} Cf. (van Dijk 1997), p. 463-476.

^{8} As it is the case for the present analysis where the calculated arrival rates and service times will feed a simulation study

^{9} Cf. (Howard 1968), cited by (Bhat 1969), p. 280.

^{10} Cf. (Kendall 1964), pp. 1-13

^{11} Cf. (Nyhuis 2005), pp.417-422

^{12} Cf. (Nyhuis 2009), p. 35.

^{13} Cf. (Law 2007), p. 275.

^{14} Cf. (Koenigsberg 1991), p.38.

^{15} Cf. (Reiner 2010), pp. 63, 92.

^{16} Cf. (Vazsonyi 1979), p.64.

^{17} Cf. (Vazsonyi 1979), p.64.

^{18} Cf. (Little 1961), p. 384.

^{19} Cf. (Cassandras 2008), p.35.

^{20} Cf. (Bhat 2002), p. 13.

^{21} Cf. (Little 1961), p 383.

^{22} Cf. (Morse 1958), p. 22.

^{23} Cf. (Little 1961), p 387.

^{24} Cf. (Morse 1958), p. 75.

^{25} To be further referred to as ‘units’.

^{26} Cf. (Cassandras 2008), p. 35.

^{27} Cf. (Cassandras 2008), p.35.

^{28} Cf. (Cassandras 2008), p. 430.

^{29} Cf. (Kendall 1953), p. 338-354.

^{30} Cf. (Kendall, 1953), p. 338-354.

^{31} For a concise explanation of the memoryless property, refer to section 2.2.

^{32} Cf. (Levine 2005), p. 197.

^{33} Cf. (Kirkup 2002), p.85.

^{34} Cf. (Banks 2010), p. 310.

^{35} Cf. (Levine 2005), p. 189.

^{36} Cf. (Hamming 1991), p. 38.

^{37} Cf. (Sheldon 2007), p.64.

^{38} Cf. (Bhat 2002), p. 13.

^{39} Cf. (Gardiner 1996), p. 28-30.

^{40} Cf. (Law 2007), p. 223.

^{41} Cf. (Barlow 2005), p 31.

^{42} Mean rate of arrivals per unit of time.

^{43} For further information regarding exponential distribution, refer to (Law 2007, p. 308-310).

^{44} Cf. (Kirkup 2002), p. 152.

^{45} Cf. (Kirkup 2002), p. 153.

^{46} Cf. (Barlow 2005), p 31.

^{47} Cf. (Stewart 2008), p. 136.

^{48} Cf. (Steward 2008), p. 138.

^{49} Cf (Stewart 2008), p. 137.

^{50} Cf (Stewart 2008), p. 388.

^{51} Cf. (Banks 2010), p. 285.

^{52} For further information regarding exponential distribution, refer to (Law 2007, p. 283-285).

^{53} Cf. (Loedding 2008), p. 3.

^{54} Cf. (Nyhuis 2009), p. 17.

^{55} Cf. (Nyhuis 2009), p. 19.

^{56} Cf. (Nyhuis 2009), p. 18.

^{57} Cf. (Nyhuis 2009), p.20.

- Quote paper
- Ileana Constantinescu (Author), 2011, Process Mapping and Stochastic Input Modeling, Munich, GRIN Verlag, https://www.grin.com/document/181775

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