Basic Queueing Theory CS 450 : Operating Systems Michael Saelee <lee@iit.edu>
Agenda - Queueing theory? Huh? - Probability refresher / Crash course - Queueing theory & Kendall’s notation - Mean value analysis of basic queues
§ Queueing Theory? Huh?
Remember, we started our discussion of scheduling at a high level — “policy” - mostly described heuristics -based (i.e., hand-wavy) approaches
to obtain empirical data, we can examine a “live” operating system’s scheduler
to exercise rigor , should leverage some branch of mathematics well-suited to analyzing scheduling systems … queueing theory!
queueing theory models wait queues using probability theory - e.g., arrival/service rate distributions - supports mathematical analysis & rigor
wide application: - checkout lines - telecom switch - traffic light system - network quality of service
we’ll barely scratch the surface — queueing theory is an area ripe for research — but you’ll see some basic applications - will also help explain underpinnings of simulators used for scheduling!
§ Probability refresher / Crash course
Probability theory = quantitative analysis of random phenomena - assign a weighted probability to every event in a sample space ( Ω ) - use these probability distributions to better understand the behavior of the phenomena
Core abstraction: random variable - a R.V . is a function that maps the sample space onto numeric values (e.g., X: Ω → ℝ ) - discrete R.V .s map to a countable set - continuous R.V .s map events onto an uncountable set (e.g., real-valued)
E.g., double coin toss (discrete event space) Ω = { TT, TH, HT, HH } 0 , if ω = TT 1 , if ω = TH X ( ω ) = 2 , if ω = HT 3 , if ω = HH
Typically interested in a variety of statistics of random variables (and corresponding events): - probability of event n : P ( X = n ) (or p ( n ) ) - expected value (mean): E ( X ) - variance: σ 2 ( X ) ; standard deviation: σ ( X ) - coefficient of variance: C X = σ ( X )/ E ( X ) (unitless measure)
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sha1_base64="3hdbwTSVxBON+FiA5gMGQhY0lE=">ACD3icbZDLSgMxFIYz9VbrbdSlm2BRKgxlptbqplB047KCvUA7lEyaUMzmSHJCGXoG7jxVdy4UMStW3e+jWk7C209kPDx/+eQnN+LGJXKtr+NzMrq2vpGdjO3tb2zu2fuHzRlGAtMGjhkoWh7SBJGOWkoqhpR4KgwGOk5Y1upn7rgQhJQ36vxhFxAzTg1KcYKS31zN6oV3lZ9WoK+uLxBOnElSmViQVx2rZJ1bZevCqvTMvF20ZwWXwUkhD9Kq98yvbj/EcUC4wgxJ2XHsSLkJEopiRia5bixJhPAIDUhHI0cBkW4y2cCT7TSh34o9OEKztTfEwkKpBwHnu4MkBrKRW8q/ud1YuVfuQnlUawIx/OH/JhBFcJpOLBPBcGKjTUgLKj+K8RDpENROsKcDsFZXHkZmqWio/munK9dp3FkwRE4BgXgEtQA7egDhoAg0fwDF7Bm/FkvBjvxse8NWOkM4fgTxmfP/s6mMs=</latexit> e.g., 6-sided dice roll P ( X = n ) = p ( n ) = 1 6 , n = 1 , 2 , 3 , 4 , 5 , 6 — probability mass function X Note: P ( X = n ) = 1 n X E ( X ) = n · p ( n ) = 3 . 5 n
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