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Lecture 11  Variability and Waiting Time Waiting timesA "queue" is made of a server and a queue in frontArrival time = average demand rate (# flow
units/timeCapacity (#flow units/time)We are interested in the waiting times in the queue and the queue length (inventory)Flow time = waiting time (in
queue) + activity timeTwo Causes of Waiting LinesSupply constrained process: capacity is lower than demand rate even in the absence of variability (peak
hours @ Subway)Waiting time is predicable
in this case, e.g., seasonal demandIncreasing capacityDemand constrained process: capacity is higher than
demand rate but there is variabilityIn the arrival process: the times between the arrival of two flow units is not constantIn the service process: the time to
process flow units differ from one to the otherIn the presence of variability
, queues can also arise if the implied utilization is bellow 100%Arrival:
Unpredicted volume swings Random arrivals (randomness is the rule, not the exceptionIncoming qualityProduct mixActivity time:Different service
requestInherent variationLack of standard operating proceduresQuality (scrap / network)Resources:Breakdowns/maintenanceOperator absenceSetup
timesRoutesVariable routingDedicated machines"No storage"! > more vulnerable to variabilityMeasures of variabilityAbsolute measure: standard
deviation of a random variable XStandard deviation is the most common measure of statistical dispersion
, measuring how spread out the sample values of a
random variable are. If the values are all close to the mean, then the standard deviation is close to zero. If many values are far from the mean, then the standard
deviation is far from zero. If all the values are equal, then the standard deviation is zeroRelative measure: coefficient of variation of a random x CV
x
= σ/μ
Random variables with CV < 1 are considered lowvariance, while those with CV > 1 are considered highvariance.Variability in Arrival ProcessInterarrival
TimesArrival time: the time at which a customer (service request) arrivesInterarrival time: the time between two consecutive arrivalsArrival rate: the
number of arrivals per unit of timeStationary arrival process: the expected number of customers arriving in a certain time interval is constant Depends on the
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 Fall '08
 JONNALAGEDDA

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