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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 Probability distribution: how to draw, mass function, density function

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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

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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

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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.

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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)

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

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M / M / 1 / GD / / (Section 4) Little’s queuing formula This is independent of number of servers, queue discipline, interarrival time dist., service time.

M / M / 1 / GD / / (Section 4) Little’s queuing formula This is independent of number of servers, queue discipline, interarrival time dist., service time.

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