Lecture 11 Notes

# Lecture 11 Notes - Space IOE 202 lectures 11 and 12 outline...

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Space IOE 202: lectures 11 and 12 outline Announcements Last time... Queueing models — intro Performance characteristics of a queueing system Steady state analysis of an M / M / 1 queueing system Other queueing systems, including M / M / s and M / M / s / s Cost analysis of queueing systems IOE 202: Operations Modeling, Fall 2009 Page 1 Space Last time: characterizing behavior of uncertain continuous quantities Characterization through range of possible values and probability density function over that range Probability of taking on a particular value is 0 Probability of taking on a value in a given range equals to the integral of the density function over this range. Examples of 3 important distributions of continuous r.v.’s: Uniform ( U ( a , b )) – “everything between a and b equally likely” Normal ( N ( μ, σ )) – known quantilies in terms of standard deviations Exponential (Exponential with rate λ ) – memoryless! IOE 202: Operations Modeling, Fall 2009 Page 2

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Space Queues and queueing models Waiting in line: in a bank, a store, toll booth, tra c light Submitting jobs to a server, or to a printer, or to a repair shop One purpose of building queueing models is to analyze an existing system to quantify its performance characteristics (such as the average waiting time, average number of customers in line, what fraction of the servers are busy, etc.) Another purpose is to explore how the system can be made better Analysis of queueing systems sometimes can be performed analytically; sometimes, simulation analysis is needed. In the forthcoming lectures, we will mostly focus on the systems for which analytical results are available; however, they are too complicated to be derived from first principles, so a lot of formulas will be given , not derived... IOE 202: Operations Modeling, Fall 2009 Page 3 Space A very simple queueing system Jobs are submitted to the computer server every 15 seconds. It takes the server exactly 13 seconds to process each job; jobs are executed in order of submission, one at a time. 1. How long does a job wait to start? 2. At any given time, how many jobs are waiting in the queue? 3. What fraction of the time is the server busy? 4. The activity monitor displays IDs of the next 8 jobs to be processed (after the current job is completed). What fraction of the time is there room on this list? 5. Assuming the number of jobs submitted to the server is not going to change in the near future, is it worthwhile to replace it with a faster server? 6. What if the server is slower than usual, and so takes 14 seconds to complete each job? IOE 202: Operations Modeling, Fall 2009 Page 4
Space A more complicated queueing system Jobs are submitted to the computer server, on average, every 15 seconds ; the interarrival times are exponentially distributed . The time the server spends on each job is exponentially distributed, with the mean of 13 seconds .

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