Curry G.L., Feldman R.M. Manufacturing Systems Modeling and Analysis
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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 vari-
ables (pp. 16 and 134).
probability density function (pdf) A function associated with a continuous ran-
dom 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 vari-
able (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).
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 con-
sistent 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).
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, 85
335
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
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, 216–218
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
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