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Unformatted text preview: Waiting Line Management Queuing systems: basic framework & key metrics Customer Population Arrival Queue Service Exit 1. Average utilization (% time server busy) 2. Average queuing time 3. Average queue length (# of customers in line) 4. Average system time (queuing + service) 5. Average # of customers in the system (in line + being served) Life without variability Customer Population Arrival Queue Service Exit Arrival stream: 1 every 5 minutes Service time: Exactly 5 minutes Average utilization Average queuing time Average queue length Average system time Average no. in system 50% 5 0.5 100% 5 1 1 every 10 minutes Exactly 5 minutes Queuing Models… Arrival rate λ Service rate μ Single queue, single server (M/M/1) Assume • Time between arrivals is Exp(λ) • Time between services is Exp( ) Queuing Models: Arrivals T = time between arrivals Assume T is an exponential random variable with rate λ λ 1 ] [ = T E Expected time between arrivals: , ) ( ≥ = t e t f t T λ λ Probability density function for the exponential distribution: Queuing Models: Arrivals If we want to know how many customers arrive in a given time period, we can use the Poisson distribution. , ! ) ( ) ( ) ( ≥ = n n e T n P T n T N λ λ PN(T)(n) is the probability that the number of arriving customers in any period of length T is exactly n Time between arrivals is Exp( ) Poisson arrivals at rate λ N(T) = number of arrivals in T time units Examples:...
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 Spring '10
 CHANG
 Management, Poisson Distribution, Probability theory, Exponential distribution, Poisson process, λ Wq

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