Basic Queueing Theory CS 450 : Operating Systems Michael Saelee - PowerPoint PPT Presentation

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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