Queueing Theory, I/O arrays

Computer Organization and Design: The Hardware/Software Interface

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CS152 Computer Architecture and Engineering Lecture 24 I/O and Storage Systems April 29, 2004 John Kubiatowicz (www.cs.berkeley.edu/~kubitron) lecture slides: http://inst.eecs.berkeley.edu/~cs152/ 4/28/04 ©UCB Spring 2004 CS152 / Kubiatowicz Lec24.2 Recap: Nano-layered Disk Heads ° Special sensitivity of Disk head comes from “Giant Magneto-Resistive effect” or (GMR) ° IBM is leader in this technology Same technology as TMJ-RAM breakthrough we described in earlier class. Coil for writing 4/28/04 ©UCB Spring 2004 CS152 / Kubiatowicz Lec24.3 Disk Latency = Queueing Time + Controller time + Seek Time + Rotation Time + Xfer Time Order of magnitude times for 4K byte transfers: Average Seek: 8 ms or less Rotate: 4.2 ms @ 7200 rpm Xfer: 1 ms @ 7200 rpm Recap: Disk Device Terminology 4/28/04 ©UCB Spring 2004 CS152 / Kubiatowicz Lec24.4 Disk I/O Performance Response time = Queue + Device Service time 100% Response Time (ms) Throughput (Utilization) (% total BW) 0 100 200 300 0% Proc Queue IOC Device Metrics: Response Time Throughput latency goes as T ser ×u/(1-u) u = utilization
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4/28/04 ©UCB Spring 2004 CS152 / Kubiatowicz Lec24.5 ° Queueing Theory applies to long term, steady state behavior Arrival rate = Departure rate ° Little’s Law : Mean number tasks in system = arrival rate x mean reponse time Observed by many, Little was first to prove Simple interpretation: you should see the same number of tasks in queue when entering as when leaving. ° Applies to any system in equilibrium, as long as nothing in black box is creating or destroying tasks “Black Box” Queueing System Arrivals Departures Introduction to Queueing Theory 4/28/04 ©UCB Spring 2004 CS152 / Kubiatowicz Lec24.6 ° Server spends a variable amount of time with customers Weighted mean m1 = (f1 x T1 + f2 x T2 +. ..+ fn x Tn)/F = Σ p(T)xT σ 2 = (f1 x T1 2 + f2 x T2 2 +...+ fn x Tn 2 )/F – m1 2 = Σ p(T)xT 2 -m1 2 Squared coefficient of variance : C = 2 /m1 2 - Unitless measure (100 ms 2 vs. 0.1 s 2 ) ° Exponential distribution C = 1 : most short relative to average, few others long; 90% < 2.3 x average, 63% < average Hypoexponential distribution C < 1 : most close to average, C=0.5 => 90% < 2.0 x average, only 57% < average Hyperexponential distribution C > 1 : further from average C=2.0 => 90% < 2.8 x average, 69% < average Avg. A Little Queuing Theory: Use of random distributions Avg. 0 Proc IOC Device Queue server System 4/28/04 ©UCB Spring 2004 CS152 / Kubiatowicz Lec24.7 ° Disk response times C 1.5 (majority seeks < average) ° Yet usually pick C = 1.0 for simplicity Memoryless, exponential dist Many complex systems well described by memoryless distribution! ° Another useful value is average time must wait for server to complete current task: m1(z) Called “Average Residual Wait Time” Not just 1/2 x m1 because doesn’t capture variance Can derive m1(z) = 1/2 x m1 x (1 + C) No variance C= 0 => m1(z) = 1/2 x m1 Exponential C= 1 => m1(z) = m1 A Little Queuing Theory: Variable Service Time Proc IOC Device Queue server System Avg. 0
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Queueing Theory, I/O arrays - Recap: Nano-layered Disk...

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