APME_8 - Applied Probability Methods for Engineers Slide...

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Click to edit Master subtitle style Applied Probability Methods for Engineers Slide Set 8
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Click to edit Master subtitle style Chapter 20 of Winston’s Operations Research Book Queuing Theory
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Queuing Theory n A queue is simply a line n Many processes require that people or items wait for service or processing n We are interested in modeling such processes to answer questions about n How much time is spent in line? n What proportion of time is a server idle? n What is the average number of people/items waiting? n What is the probability distribution of the number of customers or of the waiting time?
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Queuing Terms n Arrival process: arrivals are called customers n We will assume customers arrive one at a time n Output/service process: we specify a service time distribution n We will assume service time is independent of # of customers in the system n We will consider both parallel and series processors
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Queuing Disciplines n Queue discipline gives rule(s) for order of service n First come, first served (FCFS) n Also called first in, first out (FIFO) n Last come, first served (LCFS) (aka LIFO) n Exiting from an elevator n Service in random order (SIRO) n Priority queuing n FCFS within each priority class, where higher priority classes served first
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Modeling Arrival Process n Let ti be time of the ith arrival n Ti = ti+1 – ti, the ith interarrival time n Assume Ti’s are independent, continuous and identically distributed and described by random variable A n A has density function a(t) n Let 1/λ denote mean interarrival time (measured in hours) n λ is the arrival rate (measured in arrivals per hour) n Under exponential interarrival assumption, E(A) = 1/λ and Var(A) = 1/λ2
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Exponential Assumption n Consider P(A > t + h| A ≥ t) = P(A > t + h, A ≥ t)/P(A ≥ t) = P(A > t + h)/P(A ≥ t) = e-λ(t + h)/e-λt = e-λh = P(A > h) n Interarrival times are exponential with parameter λ if and only if the number of arrivals during an interval of length t is Poisson with parameter λt n In general, if arrival rate is stationary, bulk arrivals cannot occur, and past arrivals do not affect future arrivals, then interarrival times are exponential and number of arrivals during time t is Poisson
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Erlang Distribution n Erlang is a distribution with rate parameter R, shape parameter k (k must be a positive integer), and density function n If T is Erlang distributed with parameters R and k, then E(T) = k/R and Var(T) = k/R2 n When k = 1, Erlang is exponential n Sum of k iid exponential random variables, each with parameter R gives Erlang with parameters k and R ( 29 ( 29 ( 29 1 1 ! k Rt R Rt e f t k - - = -
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Modeling Service Process n Service process is defined by a random variable S with density function s(t) and mean service time 1/µ (hours per customer) n Exponential assumption may be a poor assumption for service times n Sometimes use Erlang service times with shape parameter k and rate parameter kµ, which implies a mean service time of 1/µ
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APME_8 - Applied Probability Methods for Engineers Slide...

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