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# IE 429, Parisay, January 2003

Review of Probability and Statistics: Experiment outcome: constant, random variable Random variable: discrete, continuous Sampling: size, randomness, replication Data summary: mean, variance (standard deviation), median, mode Histogram: how to draw, effect of cell size

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## IE 429, Parisay, January 2003

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1. Review of Probability and Statistics: • Experiment outcome: constant, random variable • Random variable: discrete, continuous • Sampling: size, randomness, replication • Data summary: mean, variance (standard deviation), • median, mode • Histogram: how to draw, effect of cell size • Probability distribution: how to draw, mass function, • density function IE 429, Parisay, January 2003

2. Review of Probability and Statistics (cont): • Relationship of histogram and probability distribution • Cumulative probability function: discrete and continuous • Standard distributions: parameters, other specifications • Read Appendix C and D IE 429, Parisay, January 2003

3. Review of Queuing Theory: • Basic queuing system M/M/1 • Simulation of the M/M/1 system • Comparison of output from four situations • Performance measures for queuing system: server utilization, • waiting time in line, waiting time in system, number in line, • number in system, max number in line, probability that a • customer waits in line more than x unit of time, probability • that a customer has to wait, probability that system is empty IE 429, Parisay, January 2003

4. Analysis Options • Educated guessing • Queuing theory • Requires additional assumptions about the model • Popular, simple model: M/M/1 queue • Interarrival times ~ exponential • Service times ~ exponential, indep. of interarrivals • E(service) < E(interarrival) • Steady-state (long-run, forever) • Problems: validity, estimating means, time frame • Often useful as first-cut approximation Source: Systems Modeling Co.

5. Lq = average number of customers in line Ls = average number of customers in server L = average number of customers in system Wq = average waiting time in line Ws = average waiting time in server, service time W = average waiting time in system = number of customers being served per unit of time, service rate   = number of customers arriving to the system per unit of time, arrival rate = server utilization (traffic intensity)

6. Analysis of Basic Queuing System 2. Based on the theoretical M/M/1 IE 429, March 99

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