Glossary availability cellular manufacturing closed queueing network coefficient of variation (CV) conditional probability CONWIP

Transcription

1 Glossary availability The long-run average fraction of time that the processor is available for processing jobs, denoted by a (p. 113). cellular manufacturing The concept of organizing the factory into sub-factories with the capability to produce a technology group (p. 177). closed queueing network A network of queues in which no arrivals are possible from outside the network and no jobs within the network can leave (p. 242). coefficient of variation (CV) The standard deviation divided by the mean; usually restricted to positive random variables (p. 13). conditional probability The probability of event A given B is Pr(A B)/Pr(B) if Pr(B) 0(p. 2). Also used for random variables when information of one random variable is known and the distribution of the other random variable is desired (p. 27). CONWIP A production control strategy in which a constant level of work-inprocess is maintained within the facility and thus a form of pull-release control is used for jobs entering the system but not at each workstation (p. 241). correlation coefficient The covariance of two random variables divided by the product of the two standard deviations (p. 30). covariance The expected value of the product of the difference of one random variable and its mean multiplied by the difference of the second random variable and its mean (p. 29). cumulative distribution function (CDF) A function associated with a random variable giving the probability that the random variable is less than or equal to the specified value (p. 5). cycle time The time that a job spends within a system. The average cycle time is denoted by CT (p. 46). G.L. Curry, R.M. Feldman, Manufacturing Systems Modeling and Analysis, 2nd ed., DOI / , c Springer-Verlag Berlin Heidelberg

2 332 Glossary effective arrival rate The rate at which jobs enter the system, often denoted by λ e. Notice that λ often represents the rate that jobs come to the system and λ e represents the rate that jobs are allowed into the system (p. 73). effective processing time The time duration from when a job first has control of a processor or machine until the time at which the job releases the processor or machine so that it is available to begin work on another job; thus, it might include actual processing time plus a setup time or repair time in case of processor failure (p. 113). event A subset from the sample space, or a set of outcomes (p. 1). expected value The expected value of a discrete random variable is the sum over all possible values of the random variable times the probability that the value will occur; with continuous random variables, the integral replaces the sum (p. 10). group technology The analysis of processing operations with the goal of determining the similarity of the processing functions and, hence, the grouping of the associated parts for production purposes (p. 177). independence Random variables are independent if knowledge of the value of one random variable does not provide any information in predicting the value of the other random variables (p. 7). indicator function The indicator function for integers is a matrix with the value of 1 on the diagonal and 0 off the diagonal. If the matrix is square, it is an identity matrix (p. 170). job type Jobs with different routes or different processing characteristics are said to be of different job types (p. 48). joint distribution function The distribution function associated with two or more random variables (p. 24). kanban A production control strategy in which a maximum limit on work-inprocess at each workstation is maintained and thus a form of pull-release control is used at each workstation (p. 281). marginal distribution function The distribution function associated with one random variable, usually derived from a joint distribution function (p. 25). mean The mean of a random variable is its expected value (p. 11). memoryless property The lack of memory property is usually associated with a random variable that denotes the time at which an event occurs and the property implies that the probability of when the event will occur is the same as the conditional probability of when the event occurs given that the event has not yet occurred (p. 17). mixture of random variables The probabilistic selection of one random variable among a group of independent random variables (p. 35).

3 Glossary 333 outcome An element of the sample space (p. 1). offered workload See workload. Poisson process A renewal process formed by the sum of exponential random variables (pp. 16 and 134). probability density function (pdf) A function associated with a continuous random variable such that a probability that the random variable is between to values equals the integral of the function between those values (p. 6). probability mass function (pmf) A function associated with a discrete random variable giving the probability that the random variable equals the independent variable (p. 6). probability space A three-tuple (Ω,F,Pr) where Ω is a sample space, F is a collection of events from the sample space, and Pr is a probability measure that assigns a number to each event contained in F (p. 1). pull A general control strategy applied to a system that has a limit applied to its work-in-process. After the maximum number of jobs are within the system, further jobs are allowed into the system only when they are pulled into the system by other jobs departing from the system (pp. 241 and 267). push The standard operating assumption for open queueing networks in which jobs enter the system whenever they arrive to the system or according to a schedule independent of the system status (pp. 241 and 267). random variable A function that assigns a real number to each outcome in the sample space (p. 4). renewal process A process formed by the sum of nonnegative random variables that are independent and identically distributed (p. 134). routes The sequence of processing steps for a job (p. 48). routing matrix A matrix of probabilities, P =(p ij ), where p i, j is the probability that an arbitrary job leaving Workstation i will be routed directly to Workstation j (p. 139). sample space A set consisting of all possible outcomes (p. 1). squared coefficient of variation (SCV) The variance divided by the square of the mean value (usually restricted to positive random variables) (p. 13). standard deviation The square root of the variance (p. 11). step-wise routing matrix A routing matrix indicating the probability of moving from processing step to processing step instead of from workstation to workstation (p. 169). switching rule The probabilities that indicate the probabilistic branching for jobs as they depart from one workstation and get routed to another (p. 139).

4 334 Glossary throughput rate The number of completed jobs leaving the system per unit of time. The throughput rate averaged over many jobs is denoted by th (p. 47). variance The variance of a random variable is the expected value of the squared difference between the random variable and its mean. Equivalently, it is the second moment minus the square of the mean (p. 11). work-in-process The number of jobs within a system that are either undergoing processing or waiting in a queue for processing. The average work-in-process is denoted by WIP (p. 46). workload The total amount of work that is required of a workstation per unit of time and is determined by the sum of the total arrival rate (per time unit) for each product type multiplied by its associated mean processing time (in time units consistent with the arrival rate) (p. 159). workstation A collection of one or more identical machines or resources (p. 47). workstation mapping function Gives the workstation assigned to each step of the production plan (p. 168).

5 Index algorithm marginal distribution analysis exponential, 249 non-exponential, 267 mean value analysis Excel, 272 exponential, 247 multi-product, 257, 260, 263 non-exponential, 253 arrival process closed networks, 246, 256 merging streams, 134 multiple product, 160 random branching, 219 SCV for merging streams, 141 total arrival rate, 140 asymptotic approximation, 134 availability, 113, 328 balance equations, 73 batch models batch move, 198 batch network example, 222 batch type service, 209 departure SCV, 220 setup reduction, 206 workstations after batch service, 213 Bernoulli, 14 Bernoulli decomposition, 136 binomial, 14 Bortkiewicz, L.V., 16 breakdowns, 113 cellular manufacturing, 177 central limit theorem, 22 Chebyshev, P.L., 10 closed queueing network, 241, 255 coefficient of variation, 13 conditional expectation, 33 conditional probability definition, 2 probability density function, 27 probability mass function, 26 confidence interval, 100 convolution, 8, 9 CONWIP, 241 correlation coefficient, 29 covariance, 29 Coxian distribution, 89, 282, 327 cumulative distribution function definition, 5 joint, 24 properties, 5 cycle time, 46 de Moivre, A., 21 decomposition, 128, 282 departure process, 125 batch moves, 204 batch service, 211, 220 batch setups, 209 deterministic routings, 175 splitting streams, 135, 214 deterministic routing, 174 diagrams, 73 distributions Bernoulli, 14 binomial, 14 continuous uniform, 16 Coxian, 89 discrete uniform, 13 Erlang, 18 exponential, 17 memoryless property, 17,

6 336 Index gamma, 19 generalized Erlang (GE), 89, 285 geometric, 15 log-normal, 22 mixture of generalized Erlangs (MGE), 286 normal, 21 Poisson, 15 Weibull, 20 Weibull parameters, 36 effective arrival rate, 73 effective processing time, 113 entity, 323 Erlang, 18 Erlang models, 85, 87 event-driven, 323 Excel equation generation, 150 gamma function, 36 goal seek, 37 inverse distributions, 322 matrix inverse, 97 mean value analysis, 272 simulation, 62, 98, 150 t-statistic, 100 Weibull parameters, 36 expected value definition, 10 property, 11 exponential, 17 exponential random variate, 322 factory models deterministic, 54 deterministic routing, 174 multiple product networks, 159 processing step paradigm, 167 serial workstations, 125 single product networks, 138 single workstation, 69 various forms of batching, 197 factory performance general networks, 138 failures, 114, 328 finite queues, 285 flow shop, 48 future event, 323 gamma distribution, 19 gamma function, 19, 36 gamma random variate, 322 Gauss, K., 21 general distribution models, 93, 95 general service models, 91, 253 generalized Erlang (GE), 89, 285 generator, 286, 290 geometric, 15 glossary, 331 Gosset, W.S., 16 group technology, 177 i.i.d., 33 independence, 7, 28 job shop, 48 job type, 48 joint cumulative distribution function, 24 probability density function, 25 probability mass function, 24 just-in-time, 241 kanban, 241, 267, 281 Kendall notation, 76 log-normal, 22, 322 marginal probability density function, 25 probability mass function, 25 marginal distribution analysis exponential, 249 non-exponential, 267 Markovian routing, 136 matrix inverse, 97 mean, 11 mean value analysis, 242 Excel, 272 exponential, 245, 247 multi-product, 257, 260, 263, 267 multi-servers, 249, 267 non-exponential, 253 memoryless property, 17, 85 merging streams, 133 mixture of generalized Erlangs (MGE), 286 mixtures of random variables, 35 multiple product networks, 159 multiple servers, 249, 267 multiple streams, 139 multivariate distributions, 24 network, 125, 222 network approximations, 138 non-identical servers, 81 nonserial network models, 133, 139 normal, 21 normal random variate, 322 offered workload, 159

7 Index 337 open systems multiple streams, 139 single product, 145 operator-machine interactions, 116 performance measures cycle time, 46 throughput rate, 47 work-in-process, 46 phase-type models, 89 Poisson, 15 Pollaczek and Khintchine formula, 91 probability, 1 conditional, 2 measure, 1 properties, 1 space, 1 probability density function conditional, 27 definition, 6 joint pdf, 25 marginal pdf, 25 probability mass function conditional, 26 definition, 6 joint, 24 marginal, 25 processing step, 48 processing step paradigm, 167 processing time variability, 111 pull, 241, 267 push, 241, 267 queueing models Erlang-2/M/1/3, 87 G/G/1 approximation, 93 G/G/c approximation, 95 GE-2/Erlang-2/1/3, 89 limited buffer, 285 M/Erlang-2/1/3, 86 M/G/1, 91 M/M/1, 77, 78 cycle time, 80 M/M/1/n, 69 non-identical servers, 81 Pollaczek and Khintchine formula, 91 queueing network models closed, 241 open, 125, 133 queueing notation, 76 random numbers, 321 random sized batches, random variables convolution, 8 correlation coefficient, 29 definition, 4 fixed sum, 32 independent, 7, 28 mixture, 35 nonnegative, 9 random sum, 34 random variate, 322 exponential, 322 gamma, 322 log-normal, 322 normal, 322 Weibull, 322 re-entrant flow, 48 relative arrival rates, 245 reliability, 114, 328 renewal process, 134 repairs, 113, 114, 328 routing, 48 routing matrix, 139 sample space, 1 scale parameter, 19, 20 serial network model, 128, 213, 293 setups, 206 shape parameter, 19, 20 simulation, 62, 98, 150 single server, 90 skewness, 23 solutions to linear systems, 97 splitting streams, 135 squared coefficient of variation, 13 departure SCV, 127 service SCV, 163 standard deviation, 10 standard normal, 322 steady-state, 69, 73 sums of random variables fixed, 32 random, 34 switching probabilities, 164 switching rule, 139 throughput rate, 47 two-node systems, 284 uniform, continuous, 16 uniform, discrete, 13 utilization, 84 multiple products, 162 single product, 92 variance coefficient of variation, 13

8 338 Index definition, 11 property, 12 Venn diagrams, 2 Weibull distribution, 20, 36 Weibull random variate, 322 Weibull, W., 20 WIP formula, 52 limits constant, 241 kanban, 281 production control pull, 241 push, 241 work-in-process, 46 workload, 159, 162 workstations, 46, 47