Hello,

More of my philosophy about the knee of an M/M/n queue and more..

I am a white arab from Morocco, and i think i am smart since i have also
invented many scalable algorithms and algorithms..

Here is the mathematical equation of the knee of an M/M/n queue in
queuing theory in operational research:

1/(n+1)^1/n

So then an M/M/1 has a knee of 50% of the utilization, and the one of
an M/M/2 is 0,578, so i correct below:

More of my philosophy about the network topology in multicores CPUs..

I invite you to look at the following video:

Ring or Mesh, or other? AMD's Future on CPU Connectivity

https://www.youtube.com/watch?v=8teWvMXK99I&t=904s

And i invite you to read the following article:

Does an AMD Chiplet Have a Core Count Limit?

Read more here:

https://www.anandtech.com/show/16930/does-an-amd-chiplet-have-a-core-count-limit

I think i am smart and i say that the above video and the above article
are not so smart, so i will talk about a very important thing, and it is
the following, read the following:

Performance Scalability of a Multi-core Web Server

https://www.researchgate.net/publication/221046211_Performance_scalability_of_a_multi-core_web_server

So notice carefully that it is saying the following:

"..we determined that performance scaling was limited by the capacity of
the address bus, which became saturated on all eight cores. If this key
obstacle is addressed, commercial web server and systems software are
well-positioned to scale to a large number of cores."

So as you notice they were using an Intel Xeon of 8 cores, and the
application was scalable to 8x but the hardware was not scalable to 8x,
since it was scalable only to 4.8x, and this was caused by the bus
saturation, since the Address bus saturation causes poor scaling, and
the Address Bus carries requests and responses for data, called snoops,
and more caches mean more sources and more destinations for snoops that
is causing the poor scaling, so as you notice that a network topology of
a Ring bus or a bus was not sufficient so that to scale to 8x on an
Intel Xeon with 8 cores, so i think that the new architectures like Epyc
CPU and Threadripper CPU can use a faster bus or/and a different network
topology that permits to both ensure a full scalability locally in the
same node and globally between the nodes, so then we can notice that a
sophisticated mesh network topology not only permits to reduce the
number of hops inside the CPU for good latency, but it is also good for
reliability by using its sophisticated redundancy and it is faster than
previous topologies like the ring bus or the bus since
for example the search on address bus becomes parallelized, and it looks
like the internet network that uses mesh topology using routers, so it
parallelizes, and i also think that using a more sophisticated topology
like a mesh network topology is related to queuing theory since we can
notice that in operational research the mathematics says that we can
make the queue like M/M/1 more efficient by making the server more
powerful, but we can notice that
the knee of a M/M/1 queue is around 50% , so we can notice that
by using in a mesh topology like internet or inside a CPU you can
by parallelizing more you can in operational research both enhance the
knee of the queue and the speed of executing the transactions and it is
like using many servers in queuing theory and it permits to scale better
inside a CPU or in internet.

Thank you,
Amine Moulay Ramdane.