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Several posts on capacity planning tells me that the only way for me to determine what hardware I need is to test it myself and benchmark it.

Benchmark your code on hardware similar to what you'll be using in production, identify any bottlenecks, then determine how much of a workload your current hardware can handle, and/or how much hardware horsepower you need to handle your target workload

Suppose however, that I cannot afford to do so for various reasons. I can use benchmarks already published on other websites, but I cannot benchmark anything myself.

Is there an easy way to estimate what impact any given upgrade to a server will make, knowing only the specifications of my current server and the upgrade? (other than simply - it will be faster)

Hello World
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    You can always make a guesstimate based on other people's tests, but it will remain just that. – Jenny D Aug 11 '15 at 14:36
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    If you can't adequately afford to test your upgrade, ask for a larger budget. If you can't test, performance is the least of your problems. – Reaces Aug 11 '15 at 14:40

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If you buy rather than build your software you can always contact the vendor for a sizing estimate. Some have even have specific sizing tool kits.

A number of vendors have, for specific applications/workloads reference architectures, such as the Microsoft SQL Server Data Warehouse Fast Track Reference Architecture which is implemented/benchmarked with both HP and Dell hardware (and possibly others).

If you don't buy of the shelf but you don't know what your application stack is actually doing under the hood either (because you don't understand the impact of your own code and you neither do any profiling, benchmarking nor testing) you simply can't know where your money will be spent best and should keep it in your pocket.

For instance: a single threaded application won't go any faster regardless of how many additional cores you throw at it. When your application is CPU-bound adding expensive flash storage won't shave off a single micro-second.

HBruijn
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