Probability Distributions PDF

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Probability Distributions...

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Probability Distributions 1. Arrival distribution a. Poisson distribution which specifies that n customers will arrive in T time periods  ( ) for n= 0,1,2,… i. The exponential distribution describes the probability that the next customer will arrive in the next T time periods and the service times b. Interarrival times  the time between customer arrivals 2. Service time distribution a. Exponential distribution i. Probability = ( ) ii. Assumption: 1. Each service time is independent of those that preceded it 2. Very small and very large service times are possible Using Waiting-Line Models to Analyze Operations Operations Managers can use waiting-line models to balance the gains that might be made by increasing the efficiency of the service system against the costs of doing so or not making improvements Operating characteristics of a system:

1. Line length 2. Number of customers in system 3. Waiting time in line 4. Total time in system 5. Service facility utilization The best way to analyze a waiting-line problem is to relate the characteristics and their alternatives to dollars Single-Server Model Assumptions: 1. The customer population is infinite and all customers are patient 2. The customers arrive according to a Poisson distribution with a mean arrival rate of λ 3. The service distribution is exponential with a mean service rate of μ 4. The mean service rate exceeds the mean arrival rate 5. Customers are served on a first-come first served basis 6. The length of the waiting line is unlimited Formulas: ( ) Multiple Server Model Asssumptions: 1. The customer population is infinite and all customers are patient 2. The customers arrive according to a Poisson distribution with a mean arrival rate of λ 3. The service distribution is exponential with a mean service rate of μ 4. The mean service rate exceeds the mean arrival rate 5. Customers are served on a first-come first served basis 6. The length of the waiting line is unlimited 7. There are s identical servers 8. The service distribution for each server is exponential with a mean service time of 1/ μ 9. It should always be the case that sμ exceeds λ Little’s Law Little’s law = a fundamental law that related the number of customers in a waiting-line system to the arrival rate and waiting time of customers

 You only need to know two of the parameters to estimate the thirdDIm Can also be used for manufacturing processes to estimate the average work-in-process (L) using Little’s Law  Basis for measuring the effects of process improvements on the work-in-process at the facility  Not applicable in situations where the customer population is finite Finite-Source Model If N is greater than 30 customers the single-server model with the assumption of an infinite customer population is adequate  otherwise finite-source model...