advertisement

AN ABSTRACT OF THE THESIS OF Raghavendran D. Nagarj for the degree of Master of Science in Industrial Engineering presented on 8 December, 2003. Title: Linking Equations for the Analysis of a Serial Automated Workstation System. Abstract Redacted for privacy David S. Kim In this research, an analytical model for analyzing a production line consisting of a series of automated workstations with infinite buffers is developed. Automated workstations are assumed to have deterministic processing times, and independent exponentially distributed operating time between failures and repair times. The analytical model starts with existing results from a Markov chain model of two automated workstations in series, where analytical expressions are developed for the average number of jobs in the second workstation and its queue. This research focuses on the development of a set of linking equations that can be used to analyze larger systems using a two workstation decomposition approach. These linking equations utilize probabilities computed in the two-workstation Markov chain model to compute workstation parameters for a single workstation such that the first two moments of the inter-departure process from the twoworkstation system and the single workstation are the same. Simulations of a number of different 3-workstation and 10-workstation systems were carried out employing a range of workstation utilizations and processing time coefficients of variation. The results from these simulations were compared with those calculated with the analytical model and various two-parameter GJ/G/l approximations and linking equations present in the literature. The analytical model resulted in an average absolute percentage difference of less than 5% in the systems studied, and performed much better than general twoparameter GIG/i approximations. The analytical model was also robust in ranking the queues in the order of the average number ofjobs present in the queues. Linking Equations for the Analysis of a Serial Automated Workstation System by Raghavendran D. Nagarajan A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Science Presented December 8, 2003 Commencement June 2004 Master of Science thesis of Raghavendran D. Nagarajan presented on December 8, 2003. Redacted for privacy Major Professor, representing Industrial and Manufacturing Engineering Redacted for privacy Head of Industrial and Manufacturing Engineering Redacted for privacy Dean of the Graduate School I understand that my thesis will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my thesis to any reader upon request. Redacted for privacy Raghavendran D. Nagarajan, Author TABLE OF CONTENTS Pige Introduction 2 3 1 1.1 General objectives and motivation 1 1.2 System Description 1 1.3 General Approach 2 1.4 Organization of the document 3 Literature Review 2.1 GJIG/1 Approximations 4 2.2 Departure Process Approximations 8 2.3 Serial Automated Systems with Finite Buffers 9 Analytical Model 3.1 11 Finding Markov Chain Failure and Repair Probabilities 3.1.1 3.1.2 3.1.3 4 4 First moment of the processing time Second moment of the processing time Failure and repair probabilities for a WS 13 14 14 16 3.2 Distribution of inter-departure times from WS 2 18 3.3 Information from the Markov chain model 27 Evaluation of Methodology 30 4.1 Simulation of 3-workstation systems 31 4.2 Simulation of 10-workstation systems 35 TABLE OF CONTENTS (CONTiNUED...) 5 6 Results 5.1 Results for 3-workstation systems 39 5.2 Results for 10-workstation systems 46 5.3 Ranking workstations 49 Conclusion 50 Bibliography 51 Appendices 54 Appendix A Markov chain model for two automated workstations in series 55 Appendix B Input data for simulation of 3-workstation system 68 Appendix C Input data for simulation of 10-workstation system 80 Appendix D First two moments of the inter-departure time distribution 87 LIST OF FIGURES Figure Page 1 The two-workstation decomposition approach 11 2 Results for the 3_WSCV1L system 39 3 Results for the 3_WSCV2L system 40 4 Results for the 3_WSCV3_L system 41 S Results for the 3_WS_CV4_L system 41 6 Results for the 3_WS_CV5_L system 42 7 Results for the 3_WS_CV1H system 42 8 Results for the 3_WS_CV2_H system 43 9 Results for the 3WSCV3_H system 43 10 Results for the 3_WS_CV4H system 44 11 Results for the 3WS_CV5_H system 44 12 Results for the 3_WS_Dis system 45 13 Results for the 3WS_Sim system 45 14 Results for the 1OWS_CV1 system 46 15 Results for the 1 0_WS_CV2 system 46 16 Results for the 1 0_WSCV3 system 47 17 Results for the 10_WSCV4 system 47 18 Results for the 1 0_WS_CV5 system 48 19 Results for the 10_WS_CVO5 system 48 LIST OF TABLES Table The different factors and the levels for the simulations 31 2 Description of the different 3-workstation systems simulated 34 3 Additional 3-workstation systems simulated 35 4 Description of the different 10-workstation systems simulated 36 5 List of all the different theoretical models 37 6 Number of cases of incorrect workstation ranks for 10-workstation systems 49 LIST OF APPENDIX FIGURES Figure Al State transition diagram for the Markov chain model 58 A2 Macrostate Markov chain model 59 A3 Finding the microstate Markov chain transition probabilities 61 LIST OF APPENDIX TABLES Table Page Bi Input Data for the 3_WS CV1_L system 69 B2 Input Data for the 3_WS_CV2_L system 70 B3 Input Data for the 3_WS_CV3_L system 71 B4 Input Data for the 3_WS_CV4_L system 72 B5 Input Data for the 3_WS_CV5_L system 73 B6 Input Data for the 3_WS_CV1_H system 74 B7 Input Data for the 3WSCV2_H system 75 B8 Input Data for the 3_WS_CV3_H system 76 B9 Input Data for the 3_WS_CV4_H system 77 BlO Input Data for the 3_WS_CV5_H system 78 Bi 1 Input Data for the 3_WS_Dis system 79 B12 Input Data for the 3_WS_Sim system 79 Cl Input Data for the 1OWS_CV1 system 81 C2 Input Data for the 10_WS_CV2 system 82 C3 Input Data for the 10_WS_CV3 system 83 C4 Input Data for the 10_WS_CV4 system 84 C5 Input Data for the IQWS_CV5 system 85 C6 Input Data for the 1 0_WS_CVO5 system 86 Notation Expected value of a random variable Var[.] Variance of a random variable WS WorkStation f Failure probability of a WS r Repair probability of a WS f, Failure probability of WS i, = 1, 2, 3... Repair probability of WS i, i = 1, 2, 3... N Number of workstations in series in a system being considered MTBF Mean Time Between Failure for WS i MTTR Mean Time To Repair for WS i. P Total time spent by ajob in a WS (processing time) S, Processing rate of WS i when it is up u, Utilization of WS i u Utilization of any workstation Coefficient of Variation (CV) of processing times of workstation i Ca, Coefficient of Variation (CV) of inter-arrival times to workstation i Cd Coefficient of Variation (CV) of inter-departure times from workstation i 0a2 Variance of inter-arrival times to a workstation 0e2 Variance of service time ofajob on a workstation Arrival rate ofjobs to a workstation p Service rate ofjobs at a workstation W Expected waiting time of ajob in a queue Linking Equations for the Analysis of a Serial Automated Workstation System 1. INTRODUCTION 1.1 General Objectives and Motivation Queuing models have applications in the perfonnance evaluation of a variety of systems like communication networks, call centers and production lines. The focus of this research is directed towards a specific type of queuing system. This system is a series of workstations in which the work times of the individual servers (or workstations) can be assumed to be equal and deterministic. The significant randomness in the behavior of such machines comes from the random downtimes (time between workstation failures) and random repair times, both of which are normally assumed to be independent and exponentially distributed. In the past, researchers have used this queuing system to model the behavior of a series of automated workstations. The objective of this research is to develop an analytical model to analyze a system consisting of an arbitrary number of automated workstations in series. The use of two-moment GJIG/1 approximations, which seems to be a simple and a popular analytical approach for the analysis of such systems, results in potentially large errors for system performance measures (e.g., the average number of jobs in the system), especially when the coefficient of variation of processing times exceeds 1.5 (Hopp and Spearman, 1995). This provides the motivation for deriving an accurate, analytical method for analyzing such systems. A description of the exact system which has been studied in this research is given below. 1.2 System Description The system studied here is a production line consisting of a series of automated workstations, numbered from 1 through N. The buffer size between workstations is assumed to be infinite and the first workstation (WS 1) is assumed to have an unlimited supply of jobs. The workstations process a single job at a time and are assumed to have 2 deterministic, unit work times. The time between failure and the time to repair for each of the workstations are assumed to be independent and exponentially distributed. The performance measure of interest is the average number ofjobs in each workstation and its input queue. Knowing this and applying Little's Law the total time a job spends in each workstation and its input queue can be found. 1.3 General Approach The general approach used in this research is to model the continuous time, discrete event system described as a discrete time system. In prior research (Kim (2003)) a Markov chain model of a 2-workstation system has been developed, and analytical results for the average number of jobs in the second workstation and its queue have been derived (see Appendix A). This research uses the two-workstation Markov chain model as a building block to analyze systems containing more than two workstations. The general approach is to use a two-workstation decomposition approach, which involves analysis of the system workstation by workstation, and the approximate representation of the system feeding an input buffer as a single "aggregate" workstation. The main challenge in successfully using such an approach is estimating the departure process from a workstation which is occasionally starved. The analysis in this research begins by utilizing the stationary probabilities from the two-workstation Markov chain model to derive the probability distribution of the inter-departure times from the second workstation. The probability distribution of inter-departure times enables us to calculate the first two moments of the inter-departure times. This, in turn leads to a two-moment 2-workstation decomposition approach. The equations derived for the mean and variance of the workstation interdeparture times are referred to as "linking equations". The linking equations derived in this research enable the calculation of the parameters of the aggregate workstation described above. 1.4 Organization of document The rest of this document is organized in as follows. In chapter 2, a review of the literature describing approaches used in the analysis of systems similar to the one described here is given. In chapter 3, the linking equations that enable the calculation of parameters for an aggregate workstation are derived. In chapter 4, the methodology used in evaluating the performance of the analytical model using discrete event simulation is described. In chapter 5, the results of the simulations carried out on a number of automated workstation systems are presented. A comparison of the results obtained from the analytical model presented in this research with some of those given in the literature is also presented in this chapter. 4 2. LITERATURE REVIEW The most common analytical approach for the analysis of a series of automated workstations with infinite buffer capacities has been GJIGI1 queuing approximations used in conjunction with approximate models for workstation output departure processes (Whiff (1983), Hopp and Spearman (1995)). A GIIG/1 queue has a general distribution for independent times between arrivals to a workstation, generally distributed service times, and serves a single customer (job) at a time. This literature review is organized into three main sections. The first section is a review of GJIG/1 queuing approximations. The second section is a review of approximations of the departure process from an automated workstation. For the most part the focus in these first two sections is on twomoment approximations. The third section reviews research directed towards analyzing a series of automated workstations with finite buffers. 2.1 GIIG/1 Approximations A review of some of the methods present in the literature to analyze a GJJG/1 queue is given here. Because of the general nature and simplicity of the approximations that will be reviewed they are applicable to any single server queuing system, and are very easy to use. This simplicity and generality also leads to the expectation that they will not be good approximations for certain types of queuing systems, like the one being tested in this research. Lindley (1952), in one of the earliest works carried out in this area, derived a recursive integral equation for the distribution of the waiting times of jobs in a GIIG/1 system. As pointed out by Lindley (1952) and subsequent researchers, the integral equations yield analytically tractable solutions for very few cases. Since then, researchers have focused on developing approximate methods for finding the average number of jobs in a GJJG/l queue. There have been efforts to arrive at approximate methods by describing the number of jobs in the system as a diffusion process. A diffusion approximation refers to the concept of using the diffusion process (a continuous time, continuous state Markov process) to model the discrete queuing process. Thereupon, the probability distribution of the number of jobs in the system is usually arrived at by solving a system of differential equations. Heyman (1975) derived a diffusion approximation for the distribution of the waiting times by first deriving a diffusion approximation for the MIMI1 queue and using a similar heuristic for a GJJG/1 queue. Kobayashi (1974) derived a diffusion equation for the waiting time distribution, under the assumption that the difference in the number of jobs in the system has a normal distribution. Reiser and Kobayashi (1974) give a method to extend a diffusion approximation for the GIIG/1 queue to simple queuing networks. Through empirical investigation, these researchers have also shown that the accuracy of the diffusion approximation improves as the server utilization goes to unity. However, Shanthikumar and Buzacott (1980), as part of their research on the comparison of the various GIIG/l approximations, list some more diffusion approximations for the analysis of the GJJG/1 queue. They carried out numerical comparisons between these GJJG/1 approximations and conclude that "heuristic approximations" perform better than diffusion approximations. In this context a heuristic approximation is one where the researchers have used existing results for some special cases of the GTIG/1 system (e.g., MIGI1, MIM/1), combined with their own intuition, conjecture and empirical studies to arrive at bounds for the GJIG/1 case. A substantial amount of research has been directed at obtaining GJJG/1 approximations. A majority of these are two-moment approximations that express the average number of jobs in the system as a function of mean and variance of the interarrival and service time distributions. Kingman (1976) derived an upper-bound for the expected wait time for a job in the GJIGI1 queue. It has been shown that this upper bound can be used to obtain an approximation for the expected wait time by multiplying the bound by a factor, which LØJ makes the resulting expression exact for the MIG/1 queue (Marchal 1976). The expected waiting time in a GTIG/1 queue, given by Marchal (1976) is, w 1+A2o 2(1u) As shown by Marchal (1978) and also Medhi (1991), the upper bound for the average waiting time derived by Kingman (1976) can also be modified by adjusting it by a factor to arrive at, w" (u)(Ca2+Ce2')('1 1u) 2 )Li which is the same as the approximation used in Hopp and Spearman (1995). Daley (1977) derived an upper bound, which is also cited in Wolff (1988). Whitt (1984), derived a factor, which when multiplied with the upper bound derived by Daley (1977), resulted in an approximation for the expected wait time that is exact for the M/G/1 case. This approximation is given below. (1 + Ce2) * 4o + o (1 u)o] 2(1u) [(21u)l]+Ce2 Kramer and Lagenbach-Belz (1976) derive what is called the "refinement function" g() for different ranges of Ca2. The approximation for the average waiting time is of the following form, w " where, =1_122i-, 1u) )u) 2 (1_Ca2)21f 3u exp[-2(1u) Ca2+Ce2J C2 <1 exp[(1_u)(Ca2_1)lfC2l Ca2 +4Ce2 j Sakasegawa (1977) derived a heuristic 2-parameter approximation for the average queue length in a GJJGIs queue, given below: Ca2 +Ce2 Lq 1u 2 For s 1 the above expression is same as the GJJG/1 approximation given by Marchal (1978). In the Queueing Network Analyzer (QNA) described by Whitt (1983), a combination of GJJG/1 approximations given by Kramer and Lagenbach-Belz (1976) and Sakasegawa (1977) is used. The QNA uses the former for Ca2 <1 and the latter approximation for Ca2 1, resulting in, w" 1u) 2 )u) where, g= u) (1 Ca2 Iexpr_ 2(1 L L1forCa2 3u 1 2 Ca2+Ce2 ] for Ca2 <1 2.2 Departure Process Approximations Exact formulae for calculating the Cd2 value for a workstation (where Cd2 is the squared coefficient of variation of inter-departure times from a workstation) are available in only some special cases. For instance, in the case of the M/M/1 queue, the time between departures is exponentially distributed. For the cases where the arrival time and service time distributions follow a general distribution, the coefficient of variation of inter-departures, Cd2 can only be estimated. For a system consisting of a series of workstations the mean inter-arrival time and mean inter-departure time of all workstations is the same if the utilization of all workstations is strictly less than one (except for the first workstation which is assumed to have an infinite supply of jobs). Also, the arrival process to a workstation is the departure process from the prior workstation (with the exception of the first workstation). Therefore, the Ca2 value for a workstation other than workstation 1, is the Cd2 of the prior (upstream) workstation. There have been several different approximations developed for Cd2, for a workstation that does not have an infinite supply of work. These approximations are two- moment approximations that utilize the squared coefficient of variation, and average inter-departure times from prior workstations. Hopp and Spearman (1995) have proposed a simple method for calculating Cd2 as a weighted ratio of Ca2 and Ce2. The expression for calculating Cd2 being, Cd2 = u2Ce2 +(lu2)Ca2. Buzacott and Shanthikumar (1993) have derived approximations for Cd2 using the decreasing mean residual life (DMRL) property of the inter-arrival distribution. The approximations are as given below: Cd2 =(1u2) ICa2 +u2Ce2 Cd2 =1u2 +u2Ce2 +(Ca2 1+u2Ce2 }+u2Ce2 and l){(lu)(2u)+uCe(lu)} 2u + uCe2 These researchers point out that these approximations for Cd2 would work well only ifCa2 1. Similarly, according to Whitt (1993), there may be serious errors in the estimated values for the average number in system if Ca2 takes up values as high as 15 or more. Justification for the use of moment approximations in general is provided in Whitt (1982), where an examination of the approximation of the superposition of point processes by renewal processes is conducted. It is then suggested that distributions matching moments of the superposed processes can be used to represent the renewal distribution. These methods can, however, be used to fit a distribution only when the moments to be matched satisfy a set of conditions derived in the research. 2.3 Serial Automated Systems With Finite Buffers In contrast to a series of automated workstations with infinite buffer capacity, there has been much more research directed towards analyzing a series of workstations with finite buffers. In these systems the presence of finite buffers reduces throughput by causing blocking of workstations. A review of some of the studies that have focused on the analysis of serial production lines involving automated workstations is given below. Altiok (1982) presented a method to analyze a serial production line with blocking in which the workstations had Poisson distributed arrival and departure processes and exponential processing times. Buzacott (1966) analyzed an "automatic transfer line" with the aim of finding out the relationship 10 between capacity of buffers and the efficiency of the production line. Gershwin (1987) analyzed the automated production line system with finite buffers and presented an iterative method to evaluate approximate throughput and queue lengths. The method used in this study was to analyze a serial line as a sequence of 2-workstation lines. This process was iteratively applied to the whole line. Gershwin (1993) also presented a method to calculate the variance of the output of a tandem production system. Most of prior methods are numerical and thus require specially coded software to implement. In contrast the objective of this research is to build an analytical model for analyzing a production line consisting of a series of automated workstations with infinite buffers. 11 3. ANALYTICAL MODEL The approach taken in this research to develop an analytical method for analyzing an arbitrary number of automated workstations in series uses a two-workstation Markov chain model as a building block. The Markov Chain Model, given in Appendix A, provides an analytical expression for the average number of jobs in Workstation 2 (WS 2). This model is a discrete time approximation of the continuous time system of two workstations in series. Due to the discrete nature of the model, the workstation parameters required are the failure probability, f and repair probability, r in place of the mean time between failures (MTBF) and mean time to repair (MTTR). As found in Kim (2003), the failure probabilityf, is defined as the probability that a workstation will move into a down state at the end of the time cycle, if it had been up and processing ajob at the start of the time cycle. Similarly the repair probability, r is the probability that a workstation would be repaired at the end of the time cycle if it was in the down state at the start of the time cycle. Assume for now that we have a method to calculate the failure and repair probabilities from a workstation's MTBF and MTTR. The approach used for analytically calculating the average number of jobs in downstream workstations is called two-workstation decomposition and is outlined below. Figure 1. The two-workstation decomposition approach. 12 Two-workstation Decomposition Approach 1. First apply the Markov chain model to the first two workstations of a production system. Use a set of "linking equations" to calculate parameters of a single "aggregate" workstation (failure and repair probabilities) such that the first two moments of the departure process (inter-departure time process) from this aggregate workstation equals the first two moments of the departure process from the second workstation. In other words, using the linking equations, WS 1 and WS 2 can be now replaced by a single "aggregate workstation", that mimics the output process from WS 2 (Figure 1). 2. The parameters for the aggregate-workstation and the parameters for WS 3 can now be used in the Markov chain model to calculate the average number of jobs inWS3. 3. This process is then repeated for subsequent workstations assuming the aggregate workstation is now the first workstation in the system. The rest of this chapter is organized in the following manner. A method to convert the MTBF and MTTR values of a WS, into the failure and repair probability f and r is described in section 3.1. This then sets the stage for the application of the Markov chain model (Appendix A) for the first two workstations. Section 3.2 describes the derivation of the probability distribution for the inter-departure time from WS 2. One of the terms involved in this probability distribution isP, the probability that a job leaves WS 2 empty, which is the "information" calculated from the Markov chain model. The calculation of this term is described in section 3.3. 13 3.1 Finding Markov Chain Failure and Repair Probabilities In the two-workstation decomposition method described, a basic component of the method is to combine two adjacent workstations into one aggregate workstation. The end result of this is that it facilitates the use of the Markov chain model (Appendix A) on two new workstations, namely, the aggregate workstation and the workstation downstream of the aggregate workstation. However, before applying the Markov chain model and the two-workstation decomposition method, all of the continuous time workstation parameters (MTBF, MTTR) must be translated into Markov chain model parameters (failure and repair probabilities). This is described in this section, where the objective is to find expressions for Markov chain parameters for a workstation a function of the workstations MTBF and MTTR, such that the mean and variance of the time a job spends in a workstation is the same in continuous and discrete time. The approach used is to derive expressions for the mean and variance of time that a job spends in a workstation (processing time) in a discrete model with given failure and repair probabilities. These expressions can then be equated to known expressions for the mean and variance of time that a job spends in the workstation, which are functions of the MTBF and MTTR. The failure and repair probabilities can then be expressed as functions of workstation MTBF and MTTR. Let, P = the total time spent by a job in a workstation. It may be observed that P is the sum of the unit processing time and random downtime of the workstation, if any. The quantities E[PJ, E[P2] and Var[PJ for an automated workstation (in discrete time) having parameters f and r are derived as shown below. 14 3.1.1 First Moment of the processing time By conditioning on whether the workstation was up or down when the job enters the workstation we have, E[P] = E[P I =1] * P(I =1) + E[P 1 0] * P(I = 0) , where, I = 1 if a failure occurred when a job just moves into the workstation =0 otherwise. E[P] = f* + 1)(1 r)' r + 1 * (1f) where, k is the time to repair the workstation and hence k k (1) 1, 2. . co. We know that 0 when a failure has, indeed occurred. Evaluating (1) (2) 3.1.2 Second Moment of the processing time Conditioning once again on the indicator variable I, which was defined previously, E[P2] = E[P2 I 11* P(I = 1) + E{P2 I = 01* P(I = 0) I E[P2]=f*(k+1)2(1_r)lr+(1)2 *(1_f) Evaluating the right hand side of the above equation, r2 +f(2+r) r2 Var[P] E[P2] (E[P])2 (3) 15 Using (2) and (3) above, the following variance of the time spent by a job in a workstation having f and r as parameters, is calculated. Var[P] f(2 r f) (4) The above expression can also be arrived at using the method given below: Var[PJ = E1[Var(P I)] + Var1[E(P I)] (Ross, 1988), I I (5) where, I is as defined previously. Since the random variable Y I = 1 is a geometric random variable with the parameter r, Var(PII=1)= 1r r (6a) 2 If the workstation is up, the variance of the processing time is 0, because of deterministic unit processing times for the workstations. Var(PII=0)=O, (6b) Using (6a) and (6b), E1[Var(P Similarly, J)] f* (1 r) + O*(1_f) r f(1 r2 r) (7) 16 EJ[E(P I I)] (1+ + (1f) (8a) * f + (1f) * (1) EZ{E(P I 1)2] = (1 f+2fr+r2 (8b) r2 Using (8a) and (8b), f+2fr+r2 Var[E(PII)]= _1i+L2 (9) r) r2 Using (7) and (9), Var{P] j(2 ,which is the same as (4), given above. 3.1.3 Failure and repair probabilities for a WS The mean and variance of the time spent by a job in a workstation (in continuous time) can be expressed as a function of the workstation's MTBF and MTTR. For the system, being considered here, it is assumed that MTBF and MTTR for the workstations are independent and exponentially distributed. E[P] (MTBF + MTTR1 MTBF )S 17 In the automated workstation being considered in this study, the processing times all the workstations are assumed to be equal and deterministic, and hence without loss of generality, S = 1. Hence, E[P] MTBF + MTTR MTBF (10) The variance of the time spent by a job at a workstation has been derived by Kim and Alden (1997). Variance= (2MTTR2 "( 1 MTBF J1sJ (11) Similar to Equation (10), the value of S is equal to 1 in the above equation as well. The mean and variance of time spent by a job in a workstation can now be calculated. These two values can be matched with the expressions in f, r of the mean and variance of time spent in a workstation (Equations (2) and (4)). Solving the following equations, MTBF+MTTR MTBF r 2MTTR2f(2rf) MTBF ç gives 2MTTR MTBF+MTTR+2*MTBF*MTTR 18 (12) 2MTBF r= MTBF + MTTR +2 * MTBF * MTTR Thus, the parameters f and r of a workstation can be calculated from the MTBF and MTTR of the workstations. This is done so that the parametersf and r can then be used in the Markov chain model, described in Appendix A, to calculate the average number of jobs in the system. 3.2 Distribution of inter-departure times from WS 2 in the two-workstation Markov chain model Consider two workstations in series operating in discrete time with failure probability and repair probability r, for workstation i (i departure times from WS 2 1, 2). f, The mean and variance of the inter- can be calculated if the underlying probability distribution of the inter-departure times is known. Since the Markov chain model is discrete, this distribution will be a discrete probability distribution. Let, k = time between departure of jobs from WS 2. It can be seen that k is a positive- valued discrete random variable. Let, E: the event a job leaves WS F: the event ajob leaves WS 2 in a failed state; P(F) = 2 empty; P(E) = Since events E and F only occur when a job is leaving a workstation, the status of the workstations just before the occurrence of E is that WS 1 is down, and WS 2 is up, and 19 for F, WS 2 is up. Given that WS 2 is up, its probability of failure is independent of the contents of its input buffer. Hence the following holds true. P(EnF) P(E)*P(F) The probability of the event E, P is calculated using the Markov Chain model as shown in the next section. To derive the probability distribution of the inter-departures, we condition on specific job inter-departure times from WS 2. k=l The inter-departure time of jobs can be 1, only when the events Ec and Fc occurred at the departure of the th job. In other words, WS 2 should not be down and non-empty after the departure of the nth job. Hence, P(k =1)=(1_Pe)(1f2) k=2 Let I = 1, if WS 2 is empty but has not failed after the departure of the nth job, = 2, if WS 2 has failed but is not empty after the departure of the nt job, = 3, if WS 2 has failed and is also empty after the departure of the ntI job, =4, if WS 2 has not failed and is not empty. (13) 20 It can be seen that, P(I =:1)=Pe(1_f2) P(I=2)=(1P)f2 P(I )pef2 P(I =4) = (1 i )( f2) (14) It can be verified thatP(I =r)=1. We also haveP(k=q,q2I =4)=0, because if I = 4, then k = 1 is the only possibility. Using the law of total probability, P(k=2)=>P(k=2II =p)P(I =p) (15) For the case I = 1, for k to be equal to 2, WS 1, must have been repaired by the time the flth job departs from WS 2. Hence, the 2 time units are just times spent by the (n + 1)th job in getting processed in WS 1 and WS 2. Hence, P(k = 21 I = 1) = Using a similar argument, it can be seen that, P(k = 2 I I = 2) = P(k = 21 'n 3) = rr2 21 Hence, from (14), P(k 2) = J(1 (1 J)f2r2 + f2)ij + (15) k=3 We use the same indicator variable I,, as defined above. The next step would be to calculate the values of- P(k = 3 I I = m),m = 1,2,3. For the case I = 1, for k to be equal to 3, the (n + 1)th job must have spent 2 time units in WS 1 and its queue and 1 time unit in getting processed by WS 2. This means that, after the departure of the n job from WS 2, one time unit was used up in the repair of WS 1, and 2 time units were used up in the processing of the (n + l)t1 job on each of the two workstations. Hence, P(k=3JI =1)=(1i)r. For the case I = 2, for units in WS 2 (16) to be equal to k 3, the (n + l)t1I job must have spent two time without getting processed. This means that at the end of the first time unit after the departure of the job, WS 2 must still have been in a failed state. At the end of the second time unit after the departure of the nth job, WS 2 must have been repaired and at the end of the third time unit the (n + l)th job must have departed from WS 2. Hence, (17) P(k=3II =2)=(1r2)r2. For the case I =3, for k to be equal to 3, the (n l)th job could have spent 2 time units in WS 1 or in WS 2. We condition on the state of WS 1 after the departure of the ntl job. 22 Let, J, =0, if WS 1 = 1, is down when the nth job departs, and if WS 1 is up when the nth job departs. By definition, we have P(J =0) =1 P(J =1) = If J, = 1, then the (n + 1)t1 job must have spent 1 time unit in getting processed in WS 1, one time unit in WS 2 without getting processed and 1 time unit getting processed in WS 2. Hence we have the following, P(k=3II =3,J =1)=(1r2)r2. Consider the case when k (18) = 3, J, = 0 (andI = 3). WS 1 must necessarily get repaired at the end of the next time unit (the next time unit after the departure of the nh job) so that k = 3 is true. The probability of the occurrence time unit after the departure of the of this event is rj. At the end of the first nh job, WS 2 may or may not have been repaired. Let, K = 1, if, after the departure of the nth job from WS 2, the (n + 1) th job takes 1 time unit to arrive at WS 2. = 2, if, after the departure of the nth job from WS 2, the (n + l)th job takes 2 time units to arrive at WS 2. = 3, if, after the departure of the nthjob from WS 2, the (n + l)th job takes more than 2 time units to arrive at WS 2. The distribution for the indicator variable K is as follows. 23 P(K =1) =0 P(K =2) = P(K =3) =1 We therefore have, the following: P(k=3II=3,J=0,K=2)=r2+(1r2)r2 P(k=3II =3,J =O,K =3)=O (19) We now uncondition on the indicator variable, K,, to get the following. P(k=311,, =3,J,, =O)=r(r +r2(lr2)) (20) We now uncondition on the indicator variable, J,, (Equations 18 and 19 given above) to get the following. P(k=311,, =3)=(1ij){rj(r2 +r7(1r2))}+r1{(1r2)r2} (21) Finally, we uncondition on the indicator variable, I,, (Equations 14, 16, 17 and 21 above) to get the following. P(k = 3) = 1f2)[(1ij)ij]+(1P,,)f2[(1r2)rj +Jf2[(1r1){ij( +i(1r2))}+11{(1r2)r2}] k = c, C 4 We use the same indicator variable I,,, as defined above. 24 For the case when I =1, WS 1 must have not been repaired for c 2 time units after the departure of the nh job. In addition to this, WS 2 must also have got repaired at the end of the c 1 time units, so that the (n + l)th job departs after c time units. Therefore, we have the following equation. P(k=c,c4JI =1)=(1-r1)2ij (22) Similarly, for the case when I,, =1, WS 2 must not have been repaired for c units and must have got repaired after c 2 time 1 time units, after the departure of the nth job from WS 2. Therefore we have, P(k=c,c4JI =2)=(1_r2)c_2r2 (23) For the case I =3, consider the indicator variable J as defined above. For the case J, =1 (that is, WS 1 is up when the n1 job departs WS 2), it can be seen that WS 2 must not have been repaired for c at the end of c 2 time units and subsequently, it should have got repaired 1 time units. Therefore we have, P(k=c,c4II =3,J =1)=(1_r2)c_2r2 (24) For the case I =3 and J =0, consider the indicator variable L, defined as given below. = 1, if, after the departure of the nth job from WS 2, the (n + l)th job takes c 2 time units or less to arrive at WS 2. = 2, if, after the departure of the nt job from WS 2, the (n + l)th job takes exactly c 1 time units to arrive at WS 2. 25 = 3, if, after the departure of the nth job from WS 2, the (n + l)th job takes more than c 1 time units to arrive at WS 2. The probability distribution for the indicator variable L is given below. P(L =l)=(1r)r, P(L =2)=(1,y31 P(L =3)=(1-iiY2 It can be verified that P(L = j) =1, holds true. It can be seen that ifL =1, then WS 2 must have taken c 1 time units to get repaired. Hence, we have, P(k=c,c4II =3,J =O,L =1)=(1_r2)c_2r2. Similarly, if L = 2, then WS 2 must have taken c 1 time units or less to get repaired. Hence, we have, P(k=c,c4II =3,J =O,L =2)=(lr2)'r2. It can be seen that ifL =3, then k cannot equal c. Hence we have, P(k=c,c4II =3,J =O,L =3)=O. 26 We now uncondition successively on the indicator variables used. We first uncondition on Ln , to get the following. cI C-4 j=I i=o "n =O)=(1_ij)c_3, *(1_r2)lr2 +(1r2)2r2 *(1_rj)1r1 P(k=c,c4II We uncondition on J, andI, successively and in that order, to get the following. P(k = c, c 4) =Pe(1f2)[(1 _rj)c_2ri] +(1 P)f2[(1 _r2)r2] +((lr2)2r2)(1i)} r1{(lr2)2r2}] The complete probability distribution for the inter-departure times is given below. P(k=1)=(1I)(1f2) P(k = 2) =](1f2)ij +(1P)f2r2 +1f2ijr P(k =3) =I(lJ)[(lij)i]+(lI)f2[(lr2)i] +]f2[(1ij){ij(r2 +r2(lr2))}+ij{(lr2)r2}] P(k = c, c 4) f2)[(1 r)2rJ + (1 Pe)f2[(1 r2) ci c-2 r2] c-4 [(1_r1){((1_r1)r1)(1r2yr2 +((1r2)2r2)(1r1)r1} + 1 j=i [1{(1r2) c-2 r2} ] The above distribution can be used to calculate the mean and the variance of the interdeparture times, if the value of P is known. The first two moments of the inter-departure times from WS 2 have been calculated and are as follows. 27 = Mean2 E{k] =1+ Ti E[k2]= r +f2(2+r)(2 +-+ f2r2( 1 1)f2(2+ij) (r+rrr)2 L'ii J The details pertaining to these calculations are given in Appendix D. 3.3 Information from the Markov Chain model Let X be the long term probability of the system of WS 1 and WS 2 being in a state which can lead to departure ofajob, given that a job, indeed departed. Let,I = 1, if there is exactly 1 job in WS 2 at the start of cycle, = 2, if there are 2 or more jobs in WS In the Markov chain model (Appendix A), 2 r0 at the start of cycle. and of finding zero and one job respectively in the WS WS 2 r1 2 refer to the long-term probabilities system (that is the number ofjobs in and its queue). It can be seen that, =1) = 2) = 1 The notation used in Appendix A for describing the long term probabilities involved with certain events is given below. ib, is the long term probability of finding WS 1 in state a, WS 2 in state b, given q jobs in the WS 2 system. The variable a takes a value of 1 or 0 depending on whether or not WS 1 is up. Similarly b takes values 1 or 0 depending on whether or not WS 2 is up. 1, then the long term probability of being in a state which can lead to a departure If I of ajob from WS 2, isz1 + ,r. Similarly, whenI = 2, this probability isa + ,r0. Using the law of total probability, X=(1 + + (if1 + 7t1)if1 By the definition ofI, ,r1,rI __=________________ X (1 ifX1TH2 + r) + (r1 + The distribution of inter-departures from WS 2 has now been completely defined, from which the mean and variance of inter-departures can be calculated. As discussed previously, the next step is to replace WS 1 and WS 2 by a single workstation whose output process mimics the output process of WS 2. This is done by matching the mean and variance of inter-departures from both cases. Thus, if Mean2 and Variance2 denote the mean and variance of inter-departure times from WS 2, the parameters of the aggregate-workstation, namely, f' and r' are obtained by solving the equations, Mean2 =1+ Variance2 which leads to, f'(2r'f') 29 f r= 2(1-2Mean2 +Mean) Mean2 Mean2 + 2(Mean2 Mean Variance2 1) Mean2 + Variance2 (16) 30 4. EVALUATION OF METHODOLOGY Discrete-event simulations of serial automated workstations were carried out to evaluate the performance of the analytical model. Arena 5.0 (Rockwell Software) was the simulation tool used to carry out these studies. The simulation results for the average number of jobs in the workstations and their input queues were compared with that predicted by the analytical model. The simulation of automated workstation systems was carried out until 1 million jobs had been processed by all the workstations present in a particular system. The output from these simulations was the average of the average number of jobs in system for each of the workstations over 30 replications. In addition, comparisons between the results given by the analytical model and those given by some of the GJIG/1 approximations discussed in the literature section are also given. The following factors were used to create a variety of serial workstation systems for analysis. 1. Number of automated workstations in the system. 2. Coefficient of variation of the processing time of the workstation. 3. Utilization of the workstations. Experiments were conducted at different "levels" of each of these factors as shown in the table below. 31 Table 1. The different factors and the levels for the simulations Factor Number of Levels Description of levels Number of WS, N 2 1. Number of workstations = 3 2. Number of workstations = 10 1. CV between 0 and 1 2. CV between 1 and 2 CV of processing 3. CV between 2 and 3 5 times, Ce 4. CV between 3 and 4 5. CV between 4 and 1. Utilization, u, 2 5 Utilization between 0.80 and 0.90 2. Utilization between 0.90 and 0.95 4.1 Simulation of 3-workstation systems The MTBF and MTTR values for these systems were generated randomly. The procedure is outlined below: 1. Let the CV of processing times (total time spent by a job in a WS) be between CV10 and CV/,jgh Let the utilization of the workstations be between and Uhj. 2. Effective speed Se of a WS may be defined as the product of the actual speed S of the workstation and the proportion of time the WS is up. Hence, we have, Se=SI MTBF (\MTBF + MTTR 32 It was ensured that all workstations had an effective speed of more than 0.5. In other words, the MTBF is always greater than the MTTR for any workstation. 3. Very large and very small MTBF and MTTR values (values greater than 10,000 and less than 0.1), if any, were discarded. 4. First, the effective speed of WS 1, Se1, was arrived at, by generating a random number between 0.5 and U high 5. Secondly, the CV of processing times on WS 1, Ce1, was arrived at, by generating a random number between and CVh,gh 6. The MTBF and MTTR values for WS 1 can be calculated using the following set of equations: cv2 MTBF1= 2(Se1 1) 2 (17) C2 MTTR1 2(Se1 1)Se1 The following is a description of the procedure used to derive the above set of equations. By definition, the following equations for Se1 and Ce12 hold true. MTBF MTBF + MTTR Se 2 Ce1 = 2(MTTRI2/MTBF) (1/Se)2 Solving for MTBF1 and MTTR1 in the above equations leads to equation set given in (17). 7. For subsequent workstations in the system, a similar procedure was adopted, except for the fact that the speeds of these workstations were random numbers generated between and 1. This was done to ensure that the utilization values U high 33 for the workstation was less than 1 and the speed of the workstation was more than speed of WS 1. The following table gives the list of all the experimental sets carried out on systems having 3 automated workstations. 34 Table 2. Description of the different 3-workstation systems simulated Label for Number of Experiment workstations systems simulated 3_WS_CV1_L 3 20 3_WS_CV2_L 3 20 Number of Description of the system CVs between 0 and 1 u, s between 0.8 and 0.9 CVs between 1 and 2 U s between 0.8 and 0.9 CVs between 2 and 3 3_WS_CV3_L 3 20 3_WS_CV4_L 3 20 u, s between 0.8 and 0.9 CVs between 3 and 4 CVs between 4 and 5 3 3_WS_CV5_L u, s between 0.8 and 0.9 20 u, s between 0.8 and 0.9 CVs between 0 and 1 3_WS_CV 1_H 3 20 3_WS_CV2_H 3 20 3_WS_CV3_H 3 s between 0.9 and 0.95 CVs between 1 and 2 u, s between 0.9 and 0.95 CVs between 2 and 3 20 U s between 0.9 and 0.95 CVs between 3 and 4 3_WS_CV4_H 3 20 U s between 0.9 and 0.95 CVs between 4 and 5 3_WS_CVS_H 3 20 s between 0.9 and 0.95 In addition to the experimental sets described in the above table, two other 3workstation systems were also simulated. In the first set of experiments, the second and the third Ji workstations were identical. In the second set of experiments, the second and the third workstations were ensured to have very different Ce values. In other words, the third workstation would have a Ce between 4 and 5, if the second workstation had a Ce between 0 and 1, and vice-versa. Five such systems were randomly generated and then the second and third workstations were inter-changed so that there were a total 10 different 3-workstation systems with the second and third workstations having dissimilar Ce values. The following table summarizes the two additional experimental sets of 3- workstation systems. Table 3. Additional 3-workstation systems simulated Label for Numberof Number of Experiment workstations systems simulated 3_WS_Dis 3 10 Descnption of the system Ce1 between 0 and 5. Ce3 between 0 and 1 if Ce2 between 4 and 5, and viceversa. u, s between 0.8 and 0.95 3WSSim 3 10 Ce1 between 0 and 5. Ce2 between 0 and 5. WS 2 and WS 3 identical. 4.2 Simulation of 10-workstation systems A total of 6 experimental sets having 10-workstations were simulated. In the first set, the Ce values of the workstations were allowed take values between 0 and 5. In the five subsequent sets of experiments, the Ce values of the workstations were allowed to take values within smaller intervals as summarized in the table below. Table 4. Description of the different 10-workstation systems simulated Label for Numberof Number of Experiment workstations systems simulated 10_WS_CVO5 10 10 Description of the system Ce values between 0 and 5. u.s between 0.8 and 0.95 Ce values between 0 and 1. 1 0_WS_CV 1 10 10 10_WS_CV2 10 10 1 0_WS_CV3 10 10 10_WSCV4 10 10 10_WSCV5 10 10 u, s between 0.8 and 0.95 Ce values between 1 and 2. u.s between 0.8 and 0.95 Ce values between 2 and 3. s between 0.8 and 0.95 Ce values between 3 and 4. s between 0.8 and 0.95 Ce values between 4 and 5. u,s between 0.8 and 0.95 Thus, there were 18 experimental sets in all 12 of which were systems having 3 workstations and the rest having 10 workstations. A total of 5 GI!G/1 approximations were compared with the analytical model. In the analytical model, the linking equations provided the mean and variance of inter-departure times from workstation i. For the GJJGI1 approximations, three different methods to calculate Cd,2 found in the literature were used. Thus, this resulted in a total 15 37 combinations against which the results from the analytical model were compared. The different combinations are listed in the table given below. Table 5. List of all the different theoretical models Label Calculation for average number in system Calculation of departure process Markov chain model Linking equations 2 Marchal (1976) Hopp and Spearman (1995) 3 Marchal (1976) Buzacott and Shantikumar (1993) 4 Marchal (1976) Buzacott and Shantikumar (1993) 1 Marchal (1978), 5 Hopp and Spearman (1995) Hopp and Spearman (1995) Marchal (1978), 6 Hopp and Speannan (1995) Buzacott and Shantikumar (1993) Marchal (1978), 7 Hopp and Spearman (1995) Buzacott and Shantikumar (1993) 8 Daley (1977) Hopp and Spearman (1995) 9 Daley (1977) Buzacott and Shantikumar (1993) 10 Daley (1977) Buzacott and Shantikumar (1993) 11 Kraemer Langenbach-Belz (1976) Hopp and Spearman (1995) 12 Kraemer Langenbach-Belz (1976) Buzacott and Shantikumar (1993) 13 Kraemer Langenbach-Belz (1976) Buzacott and Shantikumar (1993) 14 Whitt (1983) Hopp and Spearman (1995) 15 Whitt (1983) Buzacott and Shantikumar (1993) 16 Whitt (1983) Buzacott and Shantikumar (1993) The measure of performance used for the comparison of each of the 16 models listed above was the absolute percentage errors calculated as shown below. 38 Absolute Percentage Error = IQsim 'sim I S where, Qsjm 'sim x 100%, = the average number ofjobs in system from simulation, = the average number ofjobs in system predicted by a model. The graphs for the average of the absolute percentage errors in each of the 18 experimental sets are given in the next chapter. 39 5. RESULTS 5.1 Results for 3-workstation systems For each of the 3-workstation system categories mentioned in Table 2 there were 20 different systems simulated on Arena. As shown in Table 2, there were 20 different 3 workstation systems simulated. From each of those simulations, the average number of jobs in WS 3 was obtained and the Absolute Percentage Error (APE) was calculated as shown previously. The following graph shows the average of the 20 APE values obtained for the average number of jobs in system for WS 3. In the graphs that follow, the x-axis represents the different models for instance, model 1 represents the analytical model that was derived in this research. 3 WS CVI L Average Errors for average number of Jobs in WS 3 and its queue 0.45 0.4 0.35 0 0.3 0.25 o 0.2 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Index for the model Figure 2. Results for the 3_WS_CV1_L system. 16 17 16 19 40 In the case of the 3-workstation systems mentioned in Table 3 and the 10-workstation systems mentioned in Table 4, the number of systems simulated on Arena was 10. 3 WS CV2 L Average Errors for average number of jobs in WS 3 and its queue 0.45 0.4 0.35 0 0.3 C 0.25 0.2 a 0.15 a 0.1 0.05 1 2 3 4 5 6 7 8 9 10 11 12 13 Index for model Figure 3. Results for the 3_WS_CV2_L system 14 15 16 41 3 WS CV3 L Average Errors for average number of jobs in WS 3 and Its queue 0.45 0.4 0.35 0 0.3 C 0.25 0.2 .0 4 a 0.15 a 0.1 0.05 0 2 4 3 5 7 6 8 9 10 11 12 13 14 15 16 Index of model Figure 4. Results for the 3_WS_CV3_L system 3 WS CV4 L Average Errors for average number of jobs in WS 3 and Its queue 045 0.4 0.35 0 0.3 0.25 0.2 .0 4 a 0.15 4 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Index for model Figure 5. Results for the 3_WS_CV4_L system 14 15 16 42 3 WS CV5 L Average Errors for average number of jobs in WS 3 and its queue 0.45 04 0.35 0 0.3 025 a. a 0.2 .0 a 0.15 0.1 0.05 1 2 4 3 5 6 8 7 9 10 11 12 13 14 15 16 Index for model Figure 6. Results for the 3_WS_CV5_L system 3 WS CVI H Average Errors for average number of jobs In WS 3 and Its queue 045 0.4 0.35 0 0.3 C 0.25 02 a 0.15 a 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Index of model Figure 7. Results for the 3_WS_CV1_H 14 15 1 43 3 WS CV2 H Average Errors for average number of Jobs in WS 3 and its queue 0.45 0.4 035 0 w 0.3 C S S 0.25 a- C 0.2 .0 a e 0.15 0.1 005 0 I 2 3 4 5 6 7 8 9 10 11 13 12 14 15 16 Index for model Figure 8. Results for the 3_WS_CV2_H 3 WS CV3 H Average Errors for average number of Jobs in WS 3 and its queue 0.45 0.4 0.35 0 Ui & 0.3 C S a 0.25 a. 0.2 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 index for model Figure 9. Results for the 3_WS_CV3_H 14 15 1 44 3 WS CV4 H Average Errors for average number of jobs In WS 3 and its queue 0.45 0.4 0.35 0 0.3 0.25 0.2 .0 0 a 0.15 0.1 0.05 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 Index for model Figure 10. Results for the 3_WS_CV4_H 3 WS CV5 H Average Errors for average number of jobs In WS 3 and Its queue 0.45 04 0.35 0 0.3 C 0.25 a 0.2 .0 0 0.15 a 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 Index for model Figure 11. Results for the 3_WS_CV 5_H 13 14 15 1 3_WS_Dis Average Errors for average number of jobs In WS 3 and its queue o 45 0.4 0.35 2 0.3 0.25 0.2 0.15 0.1 0.05 0 2 3 4 5 6 7 8 9 10 12 11 13 14 15 16 13 14 15 16 Index for model Figure 12. Results for the 3_WS_Dis 3_WS_Sim Average Errors for average number of jobs in WS 3 and its queue 0.45 0.4 0.35 0 0.3 025 6 02 a a 0.15 0.1 0.05 0 2 3 4 5 6 7 8 9 10 11 12 Index for model Figure 13. Results for the 3_WS_Sim 5.2 Results for 10-workstation systems lowscv1 Average Errors for Average number of jobs In a WS and Its queue 0.350 0.300 4-1 5-2 3 0.250 w a *-5 4-6 8 C 0.200 4-- 7 e a. 0.150 .0 10 11 5 12 13 0.100 14 * 15 16 0.050 0000 2 3 4 5 8 7 6 9 10 WorkStation Figure 14. Results for the 1O_WSCV1 1O_WS_CV2 Average Errors for Average number of Jobs In a WS and its queue 0.350 0.300 .-1 4-2 w S 3 0.250 4 *5 - 0.200 4-- 7 0 a. S .0 6 8 C -.9 10 0.150 11 0 12 -- 0.100 13 --.-. 14 15 16 0.050 0.000 1 2 3 4 5 6 7 8 WorkStation Figure 15. Results for the 1O_WS_CV2 9 10 47 1 OWS_CV3 Average Errors for Average number of jobs in a WS and its queue 0.350 0 300 U-2 w C 3 0250 4 -a-- 5 --6 C 8 9 0200 4--i a a- a a 0.150 10 .0 11 C ,'__\\S*.k / 0 100 0 050 12 13 -*-- 14 5-15 ____- 16 0.000 2 3 4 5 6 7 8 9 10 WorkStation Figure 16. Results for the 1O_WS_CV3 system 1O_WS..CV4 Average Errors for Average numb.r of Jobs In a WS and Its queue 0.350 0300 .-1 U w C * 5 C 8 0200 4-7 C a- C a a 2 3 _r. 4 0.250 0.150 10 11 C 12 .--..---.,......---, 13 0100 14 -*-- 15 16 0050 0 000 1 2 4 3 5 6 7 8 9 WorkStation Figure 17. Results for the 1 OWS_CV4 system iC 1O_WS_CV5 Average Errors for Average number of jobs in a WS and Its queue 0.350 0.300 .-1 5-2 3 0.250 4 Mi 5-5 a C 8 0.200 '--.7 a a. a 0 0.150 10 ii .0 a 12 I! 13 0.100 14 15 16 0 050 A AflA 12345678950 WorkStation Figure 18. Results for the 1 O_WS_CV5 system 1O_WS_CVO5 Average Errors for Average number of jobs in a WS and Its queue 0.700 0.600 .-1 5 2 3 p0.500 4 UI 5-5 a --6 8 C +7 0.400 a aa a --9 0.300 10 .0 11 a 12 13 0.200 14 - 15 16 0100 0.000 2 3 4 5 6 7 8 WorkStation Figure 19. Results for the 1O_WS_CVO5 system 9 10 49 5.3 Ranking workstations For 10-workstation systems, the workstations were ranked on the basis of the average number of jobs in the workstations and its queue, as given by the simulation. In other words, the workstation with the highest value for the average number of jobs was given a rank of 1. This ranking of workstations was carried out for workstations 3 through 10. Such a ranking was also done for the predicted values given by the theoretical model and each of the 15 GJJGI1 departure process combinations. The number of times, a workstation was wrongly ranked by a model was counted. The results for this count are given in Table 5. For example, it can be seen from the table that, in case of the 1 OWSCV 1 system, the theoretical model gave a wrong workstation rank in 18 cases out of a total of 80 cases, whereas the K-L-Belz 1 combination gave a wrong workstation rank in 45 cases. Table 6. Number of cases of incorrect workstation ranks for 10-workstation systems Model Index 10 WS CV1 10 WS CV2 10 WS CV3 10 WS CV4 10 WS CV5 10- WS- CVO5 1 18 7 8 6 0 14 2 41 31 18 24 21 27 3 44 39 28 36 28 32 4 44 39 28 33 28 32 5 44 27 14 18 15 23 6 43 31 25 28 24 26 7 43 30 25 28 24 26 8 41 31 18 24 19 27 9 44 39 28 35 26 32 10 44 39 28 33 26 32 11 45 26 12 18 12 20 12 44 29 25 26 24 21 13 44 29 23 26 24 21 14 45 27 14 18 15 23 15 44 31 25 28 24 26 16 44 30 25 28 24 26 6. CONCLUSION The analytical model developed in this research results in a model for analyzing a series of automated workstations that clearly outperforms other existing analytical approaches (Two-moment GIL/Gil approximations). In most cases tested, the various GJIG/l approximations examined in this research resulted in higher average absolute percentage differences (compared to simulation results) for the average number ofjobs in the system. For the analytical model, the typical average difference was less than 5%. In addition, the model seems to perform well for systems where the coefficient of variation of processing times was greater than 4.0. Among the GTIGI1 approximations used, the ones given by Kraemer and Lagenbach-Belz (1976) and Whitt (1973) seem to give better results than the rest. The analytical model presented here, also gave fewer errors in the ranking of queues than the GIIG/1 approximations. The implications of this study are two-fold. First, it has been shown that there exists an accurate analytical approximation for calculating the average number of jobs in the workstation and its queue for systems discussed in the study. Secondly, the analytical model does give consistently better results than other approximate methods studied. 51 Bibliography Altiok, T., 1982, "Approximate analysis of exponential tandem queues with blocking", European Journal of Operations Research, Volume 11, 390 398. Buzacott, J. A. and Shanthikumar, J. G., 1993, "Stochastic models of manufacturing systems", Prentice-Hall International Series in Industrial Engineering. Buzacott, J. A., 1967, "Automatic transfer lines with buffer stocks", International Journal of Production Research, Volume 5, Number 3, 183 200. Feinberg, B.N. and Chiu, S. 5., 1987, "A Method to Calculate Steady State Distributions of Large Markov Chains," Operations Research, Vol. 35, No. 2, pp. 282-290. Gershwin B. S., 1992, "Variance of Output of a Tandem Production System", Queueing Networks with Finite Capacity, Proceedings of the Second International Workshop held in Research Triangle Park, North Carolina, May 28-29, 1992, 291 302. Gershwin, B. S., 1987, "An efficient decomposition method for the approximate evaluation of tandem queues with finite storage space and blocking", Operations Research, Volume 35, Number 2, March April 1987, 291 305. Heyman, D. P., "A diffusion model approximation for the GIIG/1 queue in heavy traffic", Bell System Technical Journal, Volume 54, Number 9, 1637 1646. Hopp W. J. and Spearman M. L., 1995, "Factory Physics: foundations of manufacturing management", 2 edition, McGraw-Hill Higher Education. Kim, D.S., 2003, "The "A/All" Queue: A Queuing Model for an Automated Workstation Receiving Arrivals from an Automated Workstation", Submitted to TIE Transactions. 52 Kim, D.S. and Alden, J. M., 1997, "Estimating the distribution and variance of time to produce a fixed lot size given deterministic processing times and random downtimes", International Journal of Production Research, Volume 35, Number 12, 3405 3414. Kim, D.S., and Smith, R. L., 1995, "An Exact AggregationlDisaggregation Algorithm for Large Scale Markov Chains," Naval Research Logistics, Vol. 42, pp. 1115-1128. Kingman, J. F. C., 1961, "The single server queue in heavy traffic", Proceedings of the Cambridge Philosophical Society, Mathematical and Physical Sciences, Volume 57, 902 904. Klincewicz, J. G. and Whitt W., 1984, "On approximations for Queues, II: Shape constraints", AT&T Bell Laboratories Technical Journal, Volume 63, Number 1, January 1984, 139-162. Kobayashi, H., 1972, "Applications of the diffusion approximations to queueing networks: Part I Equilibrium queue distributions", Journal of the Association for Computing Machinery, Volume 21, 316-329. Kramer, W., and Lagenbach-Belz, 1976, "Approximate Formulae for the Delay in the Queueing system GJJGI1 ", Proceedings of the Eighth International Teletraffic Congress, Melbourne, 10 17 November 1976, 235.1 235.8. Lindley, D. V., 1951, "The Theory of Queues with a single server", Proceedings of the Cambridge Philosophical Society, Mathematical and Physical Sciences, Volume 48, 277 289. Marchal G. W., 1976, "An approximate formula for waiting time in single server queues", AIIE Transactions, Volume 8, Number 4, 473 474. Medhi J., 1991, "Stochastic Models in queueing theory", 2uid edition, Academic Press. 53 Reiser, M. and Kobayashi, H., 1974, "Accuracy of the diffusion approximation for some queueing systems", IBM Journal of Research and Development, Volume 18, 110 124. Sakasegawa, H., 1977, "An approximate formula Lq a p'° 1(1 p) ", Annals of the Institute of Statistical Mathematics, Volume 29, Part A, 67 75. Shanthikumar, J. G. and Buzacott, J. A., 1980, "On the approximations to the single server queue", International Journal of Production Research, Volume 18, Number 6, 761 773. Whitt W., 1982, "Approximating a Point Process by a Renewal Process, I: Two basic methods", Operations Research, Volume3O, 125-477. Whitt, W., 1983, "The Queueing Network Analyzer", The Bell System Technical Journal, Volume 62, Number 9, 2779 2815. Whitt W., 1984, "On approximations for Queues, I: Extremal Distributions", AT&T Bell Laboratories Technical Journal, Volume 63, Number 1, January 1984, 115 138. Whitt W., 1984, "On approximations for Queues, III: Mixtures of Exponential Distributions", AT&T Bell Laboratories Technical Journal, Volume 63, Number 1, January 1984, 163 175. Whitt, W., 1993, "Approximations to the GIJG/1 queue", Production and Operations management, Volume 2, Number 2, 114 161. Wolff R. W., 1988, "Stochastic modeling and the theory of queues", Prentice-Hall International Series in Industrial and Systems Engineering. 54 APPENDICES 55 APPENDIX A 56 A.1 Markov Chain Model of Two Automated Workstations in Series (Kim (2003)) We develop a Markov chain model of two automated workstations in series assuming there is an infinite supply of unprocessed jobs before the first workstation and infinite storage space between the workstations. This Markov chain model is a discrete time model where the fixed processing time t, serves as the discrete time unit (it is assumed that both workstations produce at the same speed when up). By the discrete nature of the model, the operating times between failures, and repair times will follow geometric distributions. In most automated workstations, this type of discrete approximation is sufficiently accurate since the fixed processing times are normally much smaller than the time between failures and repair times. The mechanics of the discrete time Markov chain are as follows: State transitions occur at the end of each time step. Any workstation that is down at the beginning of a time step may be repaired even if the workstation is empty at the beginning of the time step. A workstation that is up and not empty at the beginning of a time step will complete its job even if it moves to a down state at the end of the time step. Any jobs completed at the end of a time step are moved out of the workstations and new jobs are moved into the workstations even if a workstation moves to a down state. Note that ajob may be moved out of both workstations, and ajob moved into both workstations at the end of a time step. Workstations that are up at the beginning of a time step but idle because they are starved, cannot change to a down state at the end of a time step. Since the first workstation always has jobs to process, its output represents the input to the second workstation (from an automated workstation). We assume that the average processing capacity of the first workstation is strictly less than that of the second workstation. This ensures that the queue size (feeding the second workstation) will not 57 steadily increase over time. The objective of the model is to analyze the behavior of the second workstation and its input queue, which we will refer to as the "system". We let the state of the Markov chain at time unit n, X, = workstation i and x, E {O,1}. (x1 ,x2, N), where = status of x, = 0 if the workstation is down a the beginning of a time step, and x, = 1 if the workstation is up a the beginning of a time step. N is the number of jobs in workstation 2 plus the number in the buffer between workstation 1 and 2. p is used to denote the transition probability matrix of this Markov chain. If workstation i is up and operating (an unprocessed job is in the workstation) at the beginning of the time step, it remains up during the time step and may transition to a down state with probability f, at the end of the time step. The job being processed in this cycle will be completed and moved out of the workstation. If workstation i is down and under repair at the beginning of the time step, it remains down during the time step and is repaired with probability r, at the end of the time step. If an unprocessed job was present in this workstation at the beginning of the time step, the job remains unprocessed and must remain in the workstation until the workstation is repaired before being processed. Since the actual workstations are assumed to have exponentially distributed operating times between failures, and repair times, the probabilities function of the fixed processing time t, MTBF , f, and r, are computed as a and MTTR. MTBF, is the mean operating time between failures for workstation i, and MTTR1 is the mean repair time for workstation i. The probabilities J and r, are computed such that the mean time to process a job, and the variance of the time to complete a job in the discrete model match that for the actual workstation. Without loss of generality let t = 1, and let P = the total time spent by ajob in workstation i. In Kim and Alden (1997), and Hopp and Spearman (1995) it is shown that E[P. I I MTBF + MTTR and Var[F] MTBF 2* MTTR2 MTBF The formulas for J and ,are derived in Section 3.1.3 of this document. For the discrete model, E[1J=1+1- and Var[]= r 2,fromwhich J and i arefoundas: r 2MTTR1 (1) MTBF + MTTR +2 * MTBF * MTTR r.= 2MTBF MTBF, + MTTR +2 * MTBF * MTTR (2) A state transition diagram of the Markov chain model is shown in Figure Al. Number In System o 1 2 3 (1,1) (1,0) Ii) (0,1) (0,0) Figure Al. State transition diagram for the Markov chain model I1, The transition probabilities between any two states in Figure Al are functions of the workstation status at time n and n+1. For example the transition probability from state (1,1,2) to (1,0,2) equals (1 f1 )f2. When there are no customers in the system (i.e., workstation 2 is starved), the transition probabilities reflect the assumption that workstation 2 cannot fail if it is starved. A.2 Derivations of the and Average Number in System and Distribution of the Number in System To derive analytical expressions for the average number in system and the distribution of the number of jobs in the system, we take advantage of the transition structure of the Markov chain, and Markov chain aggregation/disaggregation results (Kim and Smith (1995), Feinberg and Chiu (1987)). Figure 1 was constructed in such a way that the number in system defines a natural partitioning of the system states. Following the terminology defined in Kim and Smith (1995), a set of four states in the Markov chain that represent the same number in system will form what is termed a macrostate. The transitions from macrostate to macrostate also constitutes a Markov chain (Kim and Smith (1995)). The solution to this macrostate Markov chain represents the solution to the queuing model since the macrostates represent the number in system, and the macrostate Markov chain steady state probabilities will equal the sum of the steady state probabilities of all states contained in the macrostate (Kim and Smith (1995)). A diagram of the macrostate Markov chain is shown in Figure A2. ... Figure A2. Macrostate Markov chain model A.3 Microstate Markov Chain Transition Probabilities To compute the transition probabilities of the macrostate Markov chain (denoted by P), the states within the macrostate must be examined in isolation (all transitions after a macrostate is left are ignored). The states within a macrostate are referred to as microstates, and the process realized by viewing the microstates in isolation constitutes a Markov chain (Kim and Smith (1995)). These Markov chains are referred to as microstate Markov chains. If the steady state probabilities of the microstate Markov chain are known, then they can be used to calculate the macrostate Markov chain transition probabilities (Kim and Smith (1995)). To find the transition probabilities of the microstate Markov chains (denoted by N for the microstate chain associated with N customers in system) we take advantage of the transition structure. Consider the microstate chain that corresponds to zero customers/jobs in the system. All transitions leaving the set of states contained in this chain must eventually return to these states (due to ergodicity), and re-enter the set of states from only a single state. Thus the transition probabilities for those transitions leaving the microstate Markov chain states are known. This is shown in Figure A3. 61 Number In System 0 1 r(1f)___ (1,1) Transitions leaving the microstate chain -b- (1,0) 4-, r )(1 f2) ------1 (0,1) - I. (1ij)A (0,0) Figure A3. Finding the microstate Markov chain transition probabilities The transition probability matrix for the microstate Markov chain for zero customers/jobs in the system is, rjf (1rj)(1f2) (1r)f2 ('-f) 'if (1ii)(1f2) (1r)f2 0 (1ii) 0 ij(1r2) (1rj)r2 (1j)(1rf2) r1(1f2) p rjr2 We next address the microstate Markov chain transition probabilities when there are two or more customers in the system. In Figure 1 it can be seen that all transitions that increase the number in system occur when workstation 1 is up and workstation 2 is down (with the exception of 0 jobs in the system). Similarly all transitions that decrease the number in system occur when workstation I is down and workstation 2 is up. Furthermore the Markov chain structure (transition into, within, leaving) of the states within a macrostate is the same when the number in system is two or greater. Therefore p' p for I, J p 2, and using the same reasoning as ised to find p° we get 1 (lf1)(1f2) i(1f2) (1-f1)f2 (1f1)r2 rr2 f1(1f2) fJ2 (1rj)(1f2) (1)f2 (1fl(1r2) Jr2 J(1r2) ii(lr2) (li)r2 1 I for I2 (1 , )(lrL )J To find p' requires knowledge of the steady state probabilities of p° since transitions leaving the microstate chain (for one in the system) to the "left" in Figure 1 may return to the microstate chain via two different states (as shown in Figures 1 and 3). ' = [rj r Let r] represent the steady state probabilities of the microstate Markov chain for I in the system, where is the steady state probability for the microstate representing workstation 1 in state i, and workstation 2 in statej. Let a + Then, (lfl(lf2) p (1J)f2 f1(1f2) fj r(1f2) (1r1)(lf2) (lr1)f2 r1f2 a(1f1)+(1a)(1J)r2 (1a)(1f1)(1r2) af1 +(la)fr2 (1a)f(1r2) r1(1r2) (1i)r2 (1i)(1rf2) A.4 Microstate Markov Chain Solutions Analytical solutions for the microstate Markov chains can be derived as functions of the workstation failure and repair probabilities. These are presented next. Microstate Markov chain for N = 0. ri(r2 +i(-1+f2 +r2)) 71.0ii ir2 -,j r2 f2i2 71.01,0 ijr2 -,j r2 =1 (-1+i)(r2 +r2)) 'V2 71.000 f2(-1+) 1V.2 r2 Microstate Markov chain for N = 1. _[ri+a(-1+f1+r)(-1+r2)+r2f1r2iir2][r2+r(-1+f2+r2)1 1,1 C 7r1,0_ (r1(-1+r2)+(-1+f1)r2)(f2 +f1(1af2 r2 +ar2)) C [(f1(-1+f2)+f2(-1+r3)(r2 +(-1+f2 +r2))] C 7r0,0_ [(f1(-1+f2)+f2(-1+r3)(f2 +f1(-1+a+f2 +r2 ar2))] C where, i(r2+i(-1+f2+r2)) ij(r2 +r2))f2r +i(-1+f2 and 2 // +r) 0 0 64 C f22(1 + ij)2 + r2)(1 + a + f2(-1--li)(2li ar1 +2r2 2ijr2 +arjr2) f12(-1 + + ar2) +(ij r2 +rjr2)(a+ Wi +r2 ar2 r1r2 +aijr2) +f1[-2f22(l+j)+2(l+a)(l+r2)(ij r2 +rjr2) ar, 2aij + 4r2 + f2(-2 + a + 4r1r2 + 2aijr2)] Microstate Markov chain for N 2. 2l >2 i,i = (,j(_1+T2)+(_1+f1)r2)*(_r2 +ij(-1+f2 +r2)) (f2 r >2 1,0 = (,j(_l+r2)+(_1+fi)r2)*(_f2 +j(-1+f2 +r2)) (f2 +, ,r->2 0,1 = >2 0,0 ))2 +i f2ij +r2 ir2 f1(-1+f2 f2i +r2 i f1(-1+f2 +r2))2 (fl(_l+f2)+(_1+li)f2)*(_r2 +i(-1+f2 +r2)) (f2 +l f2rj +r2 ljr2 f1(-1+f2 +r2))2 i (J(_l+f2)+(_1+li)f2)*(_f2 +f1(-1+f2 +r2)) f1(-1+f2 +r2))2 (f2 +i fij +r2 A.4 Macrostate Markov Chain The macrostate Markov chain depicted in Figure 2 has the following transition probability matrix structure. p00 p01 0 0 0 0 0 Pio 0 p11 p12 0 0 0 0 q r s 0 0 0 0 0 q r s 0 0 0 q [o rs 0 0 0 q 0 r s 65 The transition probabilities will be computed from the results of the microstate Markov chain analysis. Let r = [r0 , , ,r2,...] represent the steady state probabilities of P Because of the simple structure of the macrostate Markov chain, it is straightforward to show that, (q s)p10 p10 + p01 '0= (q s) + (q (3) p01 p12 p10 + p01 s)p01 p10 + p01 (qs)+ (1 (q s / q) s) + (1s/q) (4) 0112 p10 +p01 Poi Pu p10 + p01 (5) p01p12 p10 + p01 p01 p12 + = (q s) + for I (6) 3. P01P12 plo +pol An expression for the average number in system can then be obtained +2K + 3KJ + 4K[-iJ + 5K[J Average Number in System 2 + p01 p12 (1s / q) p10 + p01 K where (q Simplifying we get, s) p10 + p01 +2+ Average Number in System = P12 2s qs + sq (q_)2) (7) Expressions for the transition probabilities in P are found using the results from the microstate Markov chains and are as follows. i(r2 +i(-1+f2 +r2)) pOl r1r2ijr2 -f2 + Fir2 r2 [(f(1 +f2)+f2(-1+r))(r2 +r(-1+f2 +r2))] D P12 s= (r1(-1+r2)+(-1+f1)r2)(f2 +J(1af2 r +ar2)) D (i(_1+r2)+(_1+J)r2)*(_f2 +f (-1+f2 +r2)) (f2 +ij f2i +r2 ir2 f1(-1+f2 +r2))2 (fi(_1+f2)+(_1+,i)f2)*(_r2 +ri(-1+f2 (f2 +ij f2r1 +r2 -ir2 f1(-1+f2 +r2)) q where, D = f2(l--) f2(---l+f2 +r2)(-1+b+f2 +r2 br2) +f2(-1+r)(2ij br1 +2r2 2rjr2 +br1r2) +(r r2 +rr2)(b+r bi +r2 br2 ijr2 +br1r2) +J[-2f22(l+i)+2(l+b)(-1+r2)(r r2 +r1r2) +f2(-2+b+4r1 2bij +4r, br2 4ijr2 +2br1r2)] 67 b= i(r2 +(-1+f2 +r2))f2i2 APPENDIX B Table Bi Input Data for the 3WSCV1L system WS1 WS2 MTBF 2.339 2.192 17.238 3.685 MTTR 1.122 0.317 2.201 1.798 1.049 0.171 2.576 4.221 0.912 3.939 0.732 3.336 2.461 4.534 9.596 0.11 5.809 0.549 0.972 2.936 0.484 0.088 1.281 1.42 0.624 1.306 0.397 0.78 0.688 1.705 2.701 0.024 1.03 0.212 0.541 0.431 WS3 MTBF MTTR MTBF MTTR 6.44 2.017 3.809 1.08 930.331 3.47 264.035 2.762 1784.16 26.381 1319.243 3.528 1.06 7.552 1.761 0.33 2.527 0.634 7.108 2.234 1 4.915 1.224 0.362 7.128 2.054 10.59 2.208 3.387 0.273 15.309 1.755 2.266 1.008 4.44 2.019 5.256 0.734 35.028 2.975 4.72 3.392 1.004 1.436 19.524 0.599 1307.683 25.621 4.914 0.648 1.85 0.117 27.026 2.727 13.278 3.046 101.211 22.619 1.552 5.636 8253.806 61.188 42.433 2.967 254.153 8.536 169.651 9.342 6.202 1.485 13.544 2.673 2.499 1.698 0.524 0.952 22.279 0.559 174.958 2.551 70 Table B2 Input Data for the 3_WS_CV2_L system WS2 WS1 MTBF 7.987 10.397 9.719 25.992 24.726 13.136 5.601 6.476 52.651 6.724 4.555 58.15 11.785 9.709 10.214 4.359 21.53 14.755 8.792 8.353 MTTR 5.626 6.301 7.438 7.112 9.023 8.208 5.129 6.4 6.682 6.438 2.744 8.612 5.249 3.456 3.227 2.967 8.441 8.185 6.144 2.706 WS3 MTBF MTTR MTBF MTTR 5.107 2.509 13.346 6.36 21.677 6.454 2.731 9.15 6.453 3.524 6.224 2.957 224.452 22.64 2377.372 60.256 70.513 12.184 51.885 10.581 21.613 9.116 28.802 10.017 5.934 4.029 4.07 2.88 2.199 5.992 3.842 3.59 7295.83 70.923 6530.329 59.355 11.088 7.359 5.394 8.18 10.093 3.776 7.89 3.157 1337.268 40.255 711.966 19.813 16.312 4.747 17.503 4.748 25.982 4.226 35.329 6.562 275.299 21.705 551.4 31.022 10.186 23.872 9.572 3.959 97.059 13.227 88.922 13.894 24.457 7.695 14.401 5.523 2.914 6.584 3.284 7.995 51.244 7.788 23.944 4.257 71 Table B3 Input Data for the 3_WS_CV3_L system WS1 WS2 MTBF MTTR 31.817 15.313 85.017 25.182 12.904 8.853 22.818 13.303 45.801 14.673 15.958 11.917 16.107 9.839 20.009 9.557 49.427 16.766 56.536 20.03 75.634 24.356 36.346 12.805 16.017 15.867 95.468 24.47 96.103 16.939 53.234 17.318 24.788 12.77 103.486 22.224 30.602 12.041 50.974 20.741 MTBF MTTR 46.202 13.029 957.23 55.539 31.523 13.104 39.021 14.819 139.997 19.29 41.638 19.077 WS3 MTBF MTTR 24.824 45.865 11.596 26.507 33.937 9.526 15.762 27.751 12.465 24.202 72.159 23.641 126.851 33.765 344.139 34.089 212.144 246.019 24.831 379.822 40.955 238.855 30.843 184.498 31.866 46.424 132.35 22.887 524.543 25.968 15.474 11.129 8.698 1793.832 70.365 232.578 27.716 5861.3 135.346 1599.747 81.468 294.437 29.483 338.892 31.933 59.122 21.282 45.442 15.387 47.582 1358.611 68.012 557.165 29.39 85.524 17.962 152.453 20.532 130.668 22.549 134.231 124.7 831.664 31.427 97.72 233.13 19.187 46.806 72 Table B4 Input Data for the 3_WS_CV4_L system WS1 WS2 WS3 MTBF MTTR MTBF MTTR MTBF MTTR 57.845 23.733 432.714 55.809 139.163 30.558 148.453 39.87 56.147 511.733 70.27 437.02 33.732 19.485 28.316 84.496 35.104 68.413 105.304 31.334 656.929 71.548 265.13 42.017 102.531 34.804 307.13 45.153 433.25 62.405 37.324 26.546 101.628 38.075 94.849 36.63 153.614 36.537 565.438 56.768 3092.181 162.955 33.899 24.096 32.725 68.328 27.121 63.815 32.394 20.078 77.302 28.938 85.873 36.759 148.186 44.156 1471.101 87.351 3796.458 145.844 33.306 21.817 78.966 34.564 78.348 30.712 32.989 21.893 71.585 31.614 63.902 29.27 63.094 28.624 42.695 144.15 40.003 247.453 29.664 20.284 23.795 70.321 28.411 53.75 109.742 31.768 2387.386 143.708 421.039 55.899 43.067 22.175 71.927 24.686 130.904 30.153 64.323 32.304 41.921 139.393 41.435 145.388 34.747 25.271 24.08 77.717 36.434 53.575 183.594 40.07 5086.872 168.888 2267.432 119.343 40.954 25.019 38.347 96.181 34.872 91.913 73 Table B5 Input Data for the 3WSCV5L system WS1 MTBF 222.809 105.238 141.624 65.924 37.749 260.014 299.854 47.207 177.082 68.217 51.672 115.875 48.394 41.919 193.519 47.785 327.195 102.226 84.181 62.719 MTTR WS2 MTBF 3441.336 207.254 1475.599 247.771 66.136 1553.229 1405.169 97.646 1070.197 111.609 MTTR WS3 MTBF 634.494 67.389 212.339 51.965 195.1 63.3 48.125 149.29 1899.934 36.396 255.874 70.823 36.999 50.241 64.081 70.402 791.781 126.114 73.023 124.473 8991.207 35.25 130.867 53.286 53.169 129.112 2348.507 48.456 95.291 51.91 40.947 45.689 94.399 88.461 48.004 93.477 243 563.441 43.122 50.41 66.821 81.884 40.659 64.624 82.245 50.338 52.767 979.244 118.758 1521.225 42.408 74.851 38.997 65.159 66.115 2261.993 170.311 6986.513 47.143 244.606 61.166 384.463 40.635 173.554 47.648 233.692 39.487 144.339 112.131 47.806 MTTR 95.02 59.815 148.98 65.587 43.795 111.271 311.8 58.512 146.32 43.182 41.95 64.116 45.917 47.21 125.578 39.486 292.51 72.28 67.384 59.543 74 Table B6 Input Data for the 3_WS_CV 1_H system WS1 MTBF 1.147 9.705 1.925 1.304 5.293 0.35 10.721 5.493 2.693 0.776 2.998 1.884 0.475 3.637 4.775 1.229 0.512 2.212 2.761 5.519 WS2 MTTR 1.116 1.313 0.462 0.992 2.023 0.185 1.499 1.267 1.967 0.284 1.062 0.779 0.216 2.01 2.066 0.931 0.411 0.275 2.025 0.477 WS3 MTBF MTTR 0.514 0.402 14.055 0.559 21.362 3.115 0.513 0.306 7.568 2.026 0.282 0.686 41.933 1.768 32.091 3.534 0.251 0.156 1.939 0.546 0.513 0.133 4.934 1.67 1.912 5.006 2.809 1.242 1.224 3.545 1.723 1.707 1.041 1.08 1203.164 20.071 0.665 0.41 819.02 3.257 MTBF MTTR 0.527 0.438 68.07 5.268 14.723 2.214 2.49 1.543 7.547 2.013 2.94 1.259 51.842 2.941 12.907 1.48 2.849 1.651 4.334 1.063 8.21 1.967 2.835 0.961 5.209 1.626 2.848 1.291 6.133 1.933 0.782 0.484 0.224 0.154 10.252 0.553 0.341 0.209 232.444 2.593 75 Table B7 Input Data for the 3_WS_CV2_H system WS1 MTBF 10.503 56.489 30.561 100.124 13.251 9.738 9.298 10.885 12.61 71.604 9.292 132.904 6.055 11.325 8.312 8.197 19.8 5.687 90.951 42.159 MTTR 5.359 12.79 4.919 11.99 4.44 5.006 5.629 5.027 3.794 11.132 6.875 16.18 3.85 3.289 5.299 4.341 7.459 5.539 15.707 6.134 WS2 WS3 MTBF MTTR 10.162 3.95 MTBF MTTR 2.725 6.977 82.524 10.351 197.862 17.33 609.41 36.275 32.252 8.532 14.484 6.218 5.301 2.664 12.985 4.581 35.438 8.287 207.243 15.596 12.583 8.046 2387.262 69.389 8.592 4.426 28.314 5.239 11.35 5.428 3.962 9.391 22.409 6.159 3.222 4.007 271.567 18.031 545.724 29.938 135.303 16.113 850.684 42.109 1045.355 41.172 44.549 10.918 12.844 5.07 10.087 5.247 27.79 9.565 76.575 13.284 203.544 18.746 11.718 7.176 1218.341 36.718 6.139 3.149 35.164 6.535 9.777 5.05 4.103 9.755 16.036 4.807 7.073 5.986 329.87 20.073 327.331 18.86 76 Table B8 Input Data for the 3_WS_CV3_H system WS1 MTBF MTTR 458.566 16.65 95 18.517 19.862 39.242 49.349 13.022 22.778 16.342 11.957 12.647 24.041 21.287 13.519 9.39 74.257 78.909 46.208 11.318 15.196 11.581 100.915 20.626 45.455 12.19 18.572 16.001 71.345 20.457 47.217 17.301 13.042 10.306 48.338 20.27 30.678 10.567 17.191 16.717 WS2 MTBF 3494.887 23.661 101.422 12.286 21.016 93.927 99.338 163.038 89.209 17.102 16.939 279.746 227.8 13.596 150.541 116.382 25.803 39.833 69.081 19.748 WS3 MTTR 100.417 15.529 17.882 9.647 9.463 21.93 24.536 29.568 15.728 11.966 11.155 31.586 34.917 9.868 26.733 28.094 16.51 13.684 18.203 15.481 MTBF MTTR 1712.149 75.491 17.385 11.588 120.949 20.586 20.77 15.301 36.893 18.473 128.662 25.56 104.107 25.433 134.694 23.538 102.001 21.785 13.532 8.819 27.311 17.603 232.855 26.023 164.669 28.754 11.224 8.291 140.976 24.779 64.236 16.512 13.082 8.494 64.628 17.999 107.418 25.222 18.375 15.343 77 Table B9 Input Data for the 3_WS_CV4_H system WS1 MTBF 31.962 43.304 260.64 27.075 331.817 28.668 59.854 47.409 137.661 73.829 59.083 490.255 50.148 43.857 36.766 112.026 46.008 275.942 226.944 27.899 MTTR 27.824 24.591 55.276 26.707 54.632 25.812 33.738 30.471 36.318 29.495 27.177 71.194 29.281 22.114 30.574 37.811 31.691 44.929 46.491 25.112 WS3 WS2 MTBF 40.82 66.116 609.744 38.088 MTTR 1042.911 30.089 84.581 61.547 316.483 121.406 131.643 6098.93 55.231 59.953 78.011 42.826 226.121 51.638 1022.251 562.552 27.836 29.159 31.45 64.967 31.419 23.676 37.305 33.82 58.462 37.883 42.419 187.851 27.491 22.777 29.356 46.239 29.134 90.974 75.947 20.791 MTBF MTTR 40.088 29.697 33.924 75.073 683.994 74.883 32.393 28.725 81.01 1098.878 23.832 19.015 87.816 37.808 55.917 29.928 290.954 52.043 154.155 41.399 96.108 34.967 912.823 79.945 56.997 24.348 78.04 33.315 28.433 43.655 160.928 32.94 35.699 20.147 2012.268 125.105 684.507 64.879 21.961 27.591 fk Table B 10 Input Data for the 3_WS_CV5_H system WS1 MTBF 210.87 55.665 65.023 672.385 90.203 206.276 212.762 278.89 167.555 46.893 416.26 225.067 94.266 554.873 136.045 175.168 101.385 185.658 41.602 93.498 MTTR 56.213 40.053 37.233 100.195 38.869 59.016 64.203 74.823 50.216 46.401 83.464 61.148 44.364 87.627 46.608 62.467 51.18 50.076 32.829 45.554 WS2 MTBF 348.525 88.145 84.984 2783.834 177.074 444.566 348.06 448.663 310.381 56.299 1148.937 443.755 MTTR WS3 MTBF 467.596 58.752 MTTR 85.854 62.716 35.116 51.014 46.403 105.491 41.958 164.477 7441.415 286.652 202.528 59.196 63.453 417.672 83.768 76.228 83.55 62.814 454.244 84.776 79.305 430.087 85.334 66.545 492.895 37.556 33.249 44.608 649.908 87.929 112.087 75.12 464.837 73.005 148.187 58.594 125.661 48.451 2134.361 157.317 1413.894 134.734 73.206 220.647 56.292 296.435 56.55 248.243 60.722 240.768 99.03 41.384 128.929 47.784 65.06 549.829 95.989 346.064 73.835 46.996 60.704 37.844 150.034 51.237 180.531 63.893 79 Table Bi 1 Input Data for the 3_WS_Dis system WS1 MTBF MTTR 109.513 109.513 51.576 51.576 8.467 8.467 4.42 4.42 22.073 22.073 51.229 51.229 38.033 38.033 8.197 8.197 1.484 1.484 19.086 19.086 WS2 MTBF MTTR 133.804 51.561 0.894 3.818 81.523 51.429 1.448 2.823 53.704 41.65 1.665 1.226 358.421 74.069 14.668 1.393 64.391 49.229 0.74 1.171 WS3 MTBF 3.818 133.804 2.823 81.523 1.665 53.704 14.668 358.421 1.171 64.391 MTTR 0.894 51.561 1.448 51.429 1.226 41.65 1.393 74.069 0.74 49.229 Table B12 Input Data for the 3WS_Sim system WS1 MTBF MTTR 24.74 153.972 180.243 7.484 97.3 49.437 4.991 21.441 31.894 152.509 WS2 WS3 MTBF MTTR MTBF MTTR 17.604 8.006 16.067 16.067 8.006 57.645 29.433 3.003 29.433 3.003 37.698 6549.276 191.394 6549.276 191.394 3.819 188.466 46.973 188.466 46.973 293.861 31.807 28.091 293.861 31.807 63.804 7.979 777.055 777.055 63.804 2.109 1.013 0.352 1.013 0.352 4.099 8.404 13.192 13.192 4.099 17.262 71.371 31.901 71.371 31.901 154.55 23.354 3964.678 154.55 3964.678 APPENDIX C Table Cl Input Data for the 10_WS_CV1 system WS1 MTBF 0.78 0.429 26.544 0.615 WS2 MTTR 0.503 0.125 2.103 0.265 4.447 39.868 5.293 15.497 WS5 1.764 3.584 410.376 0.881 1693.265 1.816 143.23 15.459 2.212 664.869 1.281 14.901 MTTR 0.461 3.332 1430.595 24.766 1.386 0.264 0.234 326.113 6.108 0.534 1.942 0.951 1.115 4.347 0.691 5.006 5233.805 39.03 2.023 15.75 2.701 1.609 994.893 15.143 1.122 0.744 MTBF MTBF MTTR 0.861 0.42 3.831 0.21 16.497 0.681 8.463 0.593 0.428 18.34 WS6 WS4 WS3 MTBF MTTR 1.315 2.652 3.652 0.71 1138.665 17.563 4.892 0.751 75.969 5.509 1.478 6.207 200.641 16.088 1491.297 0.791 0.582 3.762 2.711 26.483 MTBF 1.826 39.299 766.908 17.697 16.58 1.875 87.96 219.361 1.505 392.507 MTTR 0.909 2.962 18.015 2.896 1.041 1.118 3.469 3.034 0.369 6.521 WS8 WS7 MTBF MTTR MTBF MTTR MTBF MTTR 0.531 0.286 6.794 2.499 1.91 0.625 5.247 8.219 1.785 1.966 0.368 79.822 30.71 5506.876 35.929 4828.942 46.284 3398.558 0.415 3.036 1.88 1.244 0.7 0.395 24.044 2.762 4.558 0.456 942.259 13.486 3.043 1.226 0.454 1.472 0.222 2.7 1.646 61.741 0.293 0.19 5.748 1.692 242.534 61.26 1.394 8.807 0.287 7.128 7.001 2.651 3.595 0.576 1.689 14.196 5.267 811.049 17.469 954.298 21.802 177.581 WS9 MTBF 5.853 10.455 45.028 1.456 5.121 2.597 17.137 52.506 4.88 583.576 MTTR 2.245 1.619 0.822 0.313 0.179 1.051 3.172 1.442 1.035 14.265 WS1O MTBF MTTR 1.003 2.965 81.121 3.897 22.783 1526.834 2.799 0.797 33.547 3.441 2.133 5.505 28.465 4.287 184.401 2.614 10.755 4.274 1.894 0.177 Table C2 Input Data for the 10 WSCV2 system WS1 WS2 MTBF 4.074 34.446 23.595 MTTR 17.925 8.687 19.839 10.086 5.302 9.838 8.608 5.591 3.959 WS5 3.67 4.824 9.02 4.729 2.948 4.26 4.259 6.718 MTBF 8.292 624.261 77.421 52.868 23.246 545.518 21.115 14.499 18.669 10.636 MTBF MTTR 3.587 2.654 31.187 1154.282 45.751 79.468 13.391 13.278 7.501 9.139 38.342 74.021 12.584 8.813 32.167 45.44 6.134 8.658 3.915 40.358 5.37 3.411 6.781 28.704 6.467 5.115 4.78 13.605 6.78 MTTR 6.655 WS6 MTBF MTTR MTBF MTTR 4.467 3.31 3.791 2.214 220.406 14.603 1199.355 29.955 34.179 6.618 39.501 9.527 114.122 12.858 37.036 6.053 23.608 6.252 20.479 4.794 231.234 15.739 51.211 8.574 232.605 20.126 43.644 5.729 16.215 7.524 4.985 3.189 50.218 8.554 35.691 10.597 4.445 2.813 10.268 4.698 WS9 MTBF 11.106 8244.169 39.072 45.193 26.288 76.498 261.41 10.769 64.058 7.982 WS4 WS3 MTTR 7.152 89.457 8.285 10.513 8.329 12.049 20.986 7.099 13.473 4.55 MTBF MTTR 4.883 3.039 761.878 34.893 23.177 5.69 180.136 16.354 68.538 14.036 419.042 24.631 118.095 11.388 9.266 5.945 11.535 7.309 3.258 3.807 WS7 WS8 MTBF MTTR 4.593 8.547 25.025 687.037 68.249 12.745 MTBF MTTR 6.623 4.467 1234.089 29.266 49.212 7.887 84.793 10.716 30.499 8.656 6.116 60.073 98.393 8.816 8.608 5.631 24.622 5.274 4.642 8.321 175.471 40.45 49.633 60.132 3.314 31.959 10.958 18.461 6.958 5.705 9.103 2.282 5.426 5.117 WS1O MTBF MTTR 3.081 5.102 1768.759 56.898 64.016 11.276 35.561 8.345 8.114 45.84 158.552 9.787 107.797 12.069 7.247 3.218 9.002 25.098 7.485 10.853 Table C3 Input Data for the 1 OWSCV3 system WS1 MTBF 12.323 21.252 54.972 51.956 27.102 66.054 39.722 15.129 30.001 20.675 WS2 MTTR 11.102 11.392 21.014 16.205 12.352 15.747 11.609 12.157 12.986 16.136 WS5 MTBF 14.25 87.857 104.149 312.081 67.834 691.8 582.292 39.161 142.68 15.465 MTBF 19.961 MTTR 16.02 22.82 9.903 185.755 31.856 145.655 25.178 54.667 20.725 1299.979 76.654 83.108 15.408 36.324 16.987 57.89 15.862 30.709 13.803 WS6 MTTR 10.559 22.757 20.166 35.162 17.963 47.088 50.383 18.17 24303 9.788 MTBF 14.936 101.008 98.486 123.635 68.115 146.087 161.299 16.362 82.136 29.779 MTTR 8.724 23.998 23.752 20.207 15.945 24.459 21.173 10.468 24.387 13.624 WS9 MTBF MTTR 26.395 16.999 29.218 12.947 105.935 22.464 109.609 25.497 65.062 20.613 194.401 30.619 659.631 52.955 25.616 16.887 145.752 30.654 22.778 12.985 WS3 MTBF 12.651 66.018 190.937 110.531 181.438 160.946 107.248 17.12 84.695 33.809 MTTR 9.359 17.521 32.173 21.625 30.846 21.725 23.007 10.875 19.083 16.848 WS7 MTBF 24.47 73.348 162.684 427.436 MTTR 13.148 20.606 30.381 44.971 124.95 21.492 193.313 22.221 172.469 26.108 12.634 8.699 110.784 18.808 30.597 14.289 WS1O MTBF MTTR 17.115 10.334 100.675 23.857 161.865 28.511 93.774 22.005 131.62 23.297 612.746 37.825 302.949 33.918 24.912 11.064 59.11 21.202 25.268 17.427 WS4 MTBF 16.429 42.377 70.588 480.361 154.724 1102.723 353.947 26.868 33.207 18.059 MTTR 10.223 19.2 17.329 43.611 31.685 64.818 34.573 17.244 11.674 11.564 WS8 MTBF 18.741 67.98 147.237 MTTR 12.64 17.113 23.598 240.579 30.404 76.057 21.587 237.25 24.155 342.913 30.726 22.297 14.585 80.47 17.236 23.612 13.172 84 Table C4 Input Data for the 1 0_WS_CV4 system WS1 MTBF 31.646 108.297 31.686 59.129 66.563 24.596 31.962 59.083 27.227 26.177 MTTR 23.817 35.223 26.654 22.599 24.35 22.761 27.824 27.177 26.513 23.995 WS5 MTBF 65.872 141.917 58.19 205.676 878.509 32.207 39.041 262.311 49.611 46.601 MTTR 3 1.876 35.722 32.661 36.995 84.667 23.285 28.211 44.156 29.135 29.383 WS4 WS2 WS3 MTBF MTTR 39.36 21.075 145.134 32.319 33.413 24.312 310.859 47.685 133.471 39.558 53.407 29.698 52.383 30.479 305.976 58.216 32.442 28.087 40.737 29.619 MTBF MTTR 59.297 32.678 389.895 54.301 35.301 23.689 218.586 48.399 78.45 783.352 24.54 31.226 45.885 30.306 121.036 37.745 33.789 20.479 37.958 27.952 WS6 MTBF 83.097 428.205 49.525 190.135 248.992 39.812 48.293 76.503 41.654 42.043 MTTR 36.572 59.789 30.568 46.508 41.74 29.164 30.624 29.481 26.517 32.382 WS9 MTBF MTTR 65.383 28.316 130.441 32.28 35.727 22.702 126.727 38.579 228.803 43.253 56.083 31.949 49.893 28.723 236.807 40.76 36.429 21.9 33.618 21.408 WS7 MTBF 55.802 137.96 45.355 111.627 212.471 34.517 62.747 157.509 48.167 38.213 MTBF 40.937 675.827 59.166 140.947 377.839 47.788 MTTR 22.976 65.95 33.398 35.789 53.282 26.458 34.481 22.809 335.669 58.213 42.462 29.317 48.607 29.365 WS8 MTTR MTBF MTTR 27.294 32.928 32.575 33.709 50.388 21.379 31.224 35.866 28.027 40.85 1 20.889 55.584 26.697 34.528 52.129 22.081 24.532 31.115 19.972 22.1 WS1O MTBF MTTR 61.998 32.831 420.577 59.72 34.325 20.149 282.555 56.812 157.87 42.236 29.18 48.464 33.357 19.442 83.711 29.436 27.57 38 27.817 21.98 277.099 45.496 174.522 327.407 32.257 43.588 85.108 34.277 29.925 19.565 Table C5 Input Data for the 10_WSCV5 system WS1 MTBF MTTR 47.996 46.339 71.832 48.839 76.121 43.12 57.729 36.266 173.36 50.637 129.56 49.002 137.915 126.548 126.737 128.123 53.045 42.598 48.068 43.286 WS5 MTBF 73.928 104.435 114.858 143.74 834.173 652.449 410.159 522.103 891.721 734.772 MTTR 51.499 47.681 55.62 53.26 97.656 99.26 79.748 87.004 94.155 102.535 WS2 MTBF 66.174 105.96 138.488 134.577 373.871 974.763 213.849 204.052 913.32 375.834 MTTR 50.561 148.841 189.696 374.533 247.171 230.22 389.923 443.262 1293.381 MTBF 63.252 91.817 51.206 56.622 109.243 48.429 98.413 83.613 372.48 118.878 312.866 57.362 1219.868 52.368 278.174 244.118 101.391 68.602 388.178 WS6 MTBF 41.974 128.648 WS4 WS3 MTTR 34.517 46.039 45.717 61.452 70.052 70.139 56.212 75.108 78.503 129.138 WS9 MTBF MTTR 76.051 78.756 128.85 97.953 591.545 701.297 923.254 444.304 268.788 1467.337 48.167 45.605 49.025 52.73 91.629 86.771 115.294 71.265 58.543 130.369 MTTR MTTR 40.325 39.808 39.785 42.75 57.384 51.083 52.43 39.716 140.101 72.357 2173.361 163.829 68.877 317.227 77.212 359.506 80.822 134.191 302.796 71.826 70.38 311.793 74.777 61.702 67.371 1875.114 132.842 WS7 MTBF 76.466 100.004 241.968 207.687 3002.533 523.953 MTBF 47.796 86.389 155.986 MTTR 49.574 46.94 64.601 66.922 197.349 93.208 44.265 140.851 273.379 57.865 1084.378 116.454 560.964 97.952 WS1O MTTR MTBF 58.34 48.527 92.912 39.576 198.524 56.068 83.819 41.419 5000.063 240.205 719.029 104.531 48.597 166.361 58.299 239.253 56.295 243.849 110.95 1070.278 WS8 MTBF 85.367 138.325 143.417 158.236 350.464 197.028 808.265 278.627 452.356 491.249 MTTR 52.884 55.73 51.284 48.578 76.102 49.909 114.47 66.085 74.377 89.842 Table C6 Input Data for the 1 0_WS_CVO5 system WS1 MTBF MTTR 40.496 29.978 192.325 64.653 9.674 6.215 14.045 3.731 17.875 8.316 10.42 7.168 56.263 42.386 78.454 42.125 110.784 29.811 111.991 33.791 WS5 MTBF 16.216 108.405 102.639 1027.865 130.014 19.837 6.465 9.06 239.084 2635.436 MTTR 9.157 15.368 WS2 MTBF 0.412 1212.072 24.63 22.371 188.134 9.675 MTTR 0.227 96.131 12.751 3.52 43.876 3.96 55.631 33.321 124.401 38.384 1106.189 68.197 25.087 3.462 WS4 MTBF MTTR 6.491 10.837 6.969 0.96 6.266 2.067 305.611 56.381 7.676 1.786 0.986 0.54 0.533 0.93 75.93 31.945 228.865 26.761 5.805 70.512 MTBF MTTR 92.705 44.484 120.096 13.044 2.497 0.986 80.938 13.499 118.596 43.401 WS7 WS6 MTBF 4.26 41.697 34.524 222.089 157.009 143.234 78.793 WS3 MTTR 2.779 3.904 15.228 44.836 52.98 57.907 45.606 0.296 36.437 114.368 49.595 7.946 4.121 2.158 1.064 19.322 3327.876 183.731 133.264 95.605 20.478 WS9 MTBF MTTR 86.012 45.306 557.849 74.282 45.524 19.12 693.976 74.68 30.86 6.456 53.216 25.563 43.142 20.374 142.776 42.442 374.641 75.982 3837.041 183.516 MTBF 64.863 1866.338 2.275 428.508 75.592 47.066 0.712 58.95 766.676 121.489 MTTR 129.131 55.577 123.181 404.715 66.601 53.771 30.534 51.058 62.307 12.273 WS8 MTBF 2.583 71.868 MTTR 1.287 17.876 3.337 28.037 154.917 0.946 6.961 35.244 84.807 1987.914 20.86 125.043 26.578 36.31 20.111 74.339 0.455 39.42 25.808 25.836 178.859 45.652 2093.03 141.033 99.1 296.08 60.943 23.811 WS1O MTTR MTBF 52.802 34.273 203.025 35.456 23.13 9.099 1668.385 152.507 48.736 198.583 17.813 33.625 17.432 33.382 72.886 277.213 386.524 70.732 2.468 14.911 E:A APPENDIX D The exact expressions for the first two moments of the inter-departure times from WS 2 were evaluated from the probability distribution derived. First moment of the inter-departure time distribution: First, the quantity cP(k c) is evaluated. cP(k = c) P(1_f2)[(1__ri)c_2ri} +(1_P)f2[(1_r2)c_2r21 =c +ff2[(1r3{((1iy3i) c=4 ci c-4 (lr2)'r2 +((1r2)'2r2)(lr1)r1} +r1{(1_r2)c_2r2}] 1P(-1 + ,)2(1 +3i)(i(-1+ r2) r2)r2 1 _f2ri(_1+r2)2fri (l-2i Next, the first moment of the inter-departure times is evaluated as shown below. cP(k = c) (1 P )(1 f2) +2(Pe(1f2)T +(1P)f2r2 +Pf2ir2) e +3)IP(lf2)(lr1)r +(1Pe)f2(1r2)r2 + r2(1 r2))) rl(1_r2)r2)} IP(_l+r1)2(l+3,)(r1(_1+r2)_r2)r2 + 1 r1(ri(-1+r2)r2)r2 (1) 1 f2r1(-1+r2)2(r (--1-2r1 +Pe(-1+ri)2(1--3ri))r2 I L+3(-1+1(-1+)2)(-1 +)r22)) I j As shown previously, __=________________ X (1,r0 r1)(rj +,r1)+(,r1 +,r,1)r1 >2 >2 The exact expressions (in terms off1 , , f2 , r) for each of the terms in the right hand side of the above equation has been given in the Markov chain model (Appendix A). Substituting these expressions in equation (1) above we get, E[kJ =1+ Second moment of the inter-departure time distribution: First, the quantity c2P(k = c) is evaluated. c2P(k = c) P(1_f2)[(l_,)c2r1] +(1_Pe)f2[(1_r2)c_2r2] =c2 +Pf2[(lr1){((lr1y3r1) c=4 c-I c-4 (1 r2)r2 +((l r22r2)(l +ri{(l_r2)c_2r2}] 2 Ti 2 + r2 (5 + 9r2) 2 f(_l+r32(_2_9,i2(_1+r2)2 r2(5+9r2)+i(-1+r2)(5+18r2)) (ij +r2 l2) Next, the complete expression for the second moment of inter-departure time distribution is evaluated as shown below. c2P(k = c) =12(1_ P )(1 f2) +22(Pe(1f2)1i +(1-P)f2r2 +Pf2r1r2) 32JP(1_f2)(1_i)ii +(1-P)f2(1-r2)r2 +r2(1_r2)))+(1_r2)r2)} + (_1+)2 (2+(S+91))f(1 Ti )2 * 2 2+ r2 (5 + 9r2) 2 12 r2(5+9r2)+,(-1+r2)(5+18r2)) (ij+r2ir2)2 Simplifying equation (2) above we get, E[k2}= r22+f2(2+r2)p12 r2 ++ f2r2(ii-1)f2(2+ii) (r+rrr)2 1 (2)