By Y.C. Tay
This e-book is an creation to analytical functionality modeling for desktops, i.e., writing equations to explain their functionality habit. it truly is available to readers who've taken college-level classes in calculus and likelihood, networking, and working platforms. this isn't a coaching handbook for turning into knowledgeable functionality analyst. particularly, the target is to aid the reader build easy versions for examining and knowing the platforms within which they're . Describing a sophisticated method abstractly with mathematical equations calls for a cautious number of assumptions and approximations. those assumptions and approximations make the version tractable, yet they need to now not get rid of crucial features of the method, nor introduce spurious homes. to assist the reader comprehend the alternatives and their implications, this e-book discusses the analytical types in 20 examine papers. those papers hide a extensive variety of issues: processors and disks, databases and multimedia, worms and instant, and so forth. An Appendix offers a few questions for readers to workout their realizing of the types in those papers. desk of Contents: Preliminaries / thoughts and Little's legislations / unmarried Queues / Open structures / Markov Chains / Closed structures / Bottlenecks and circulation Equivalence / Deterministic Approximations / temporary research / Experimental Validation and research / research with an Analytical version
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Applied Discrete constructions through Alan Doerr & Kenneth Levasseur is approved lower than an inventive Commons Attribution-NonCommercial-ShareAlike three. zero usa License.
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Extra resources for Analytical Performance Modeling for Computer Systems
1). This shows graphically how M/G/1 becomes unstable as λ approaches μ, so steady state is impossible for λ = μ. 1 μ Why does an M/G/1 queue blow up this way for λ < μ? After all, the server still has some idle time (ρ = 0). This behavior is caused by the randomness in the service and arrival times. For example, if service times are random, then as ρ increases (the server becomes busier), it becomes more likely that the queue will contain a long job, whose service time will be added to every job that has to wait for it.
1 T1 T2 T3 T4 T5 . . 1: Events occur at times T1 , T2 , . , and an arrival at time t observes the next event at Tn . Tn − t is a residual life. What is E(Tn − t)? Assume exponential inter-event times, Ti+1 − Ti ∼ Exponential(μ). One might argue that, by the memoryless property, 1 . 2) E(Tn − t) = On the other hand, since t is arbitrary, we expect E(Tn − t) = which contradicts Eq. 1). 3) 22 3. OPEN SYSTEMS since EX 2 = V arX + (EX)2 . 2, we get 1 2 E(Tn − t) = 1 1 μ + 1 = , 2μ 2 μ μ so Eq. 1) is correct.
This chapter considers two such techniques: Average Value Approximation and fluid approximation. 1 AVERAGE VALUE APPROXIMATION (AVA) Very often in performance modeling, a derivation would make an approximation by replacing a random variable X by its mean EX. I call this Average Value Approximation (AVA). To illustrate, although EXY = (EX)(EY ) in general, AVA gives E(XY ) ≈ E((EX)Y ) = (EX)EY, since EX is a constant. , V arX is small. 1 Let X and Y be random variables, EX = μX , EY = μY , and ρXY = E(X − μX )(Y − μY ) .