Asp Forum - Mathematical modeling correction

Ramine

3/19/2016 10:38:00 PM

Hello......

I have corrected something in my mathematical modeling,
i have substracted the Client node from the network, since
the Client users download the files in parallel...

Please reread my corrected mathematical calculations...

Here is the mathematical modeling of a queuing network that is
an ecommerce website and its calculations:

So, since the network of the those ecommerce websites
consist of queues interconnected in Tandem like this:

A -> M/G/c database servers queue -> M/M/1 Network queue -> A

The characteristic of the M/M/1 Network queue is:

Download speed of 60 Mbps
Upload speed up to 30 Mbps; or 12 Mbps in certain areas

I have choosen this, look here:

http://affaires.videotron.com/web/small-medium-companies/internet-services/internet-access/fibre...

Upload speed of 30 Mbps is equal to 3.75 Mbytes per second.

The mean size of the the files transfered is: 100 Kbytes

Adding the protocol overhead of 20% to the mean size of the files
gives 100 Kbytes * 120% = 120 Kbytes

And from the empirical tests on my harddisk , the service rates on each
server of
the M/M/c queue (c: is the number of servers) of the different database
transactions
of the ecommerce website such us read,write, delete is:

10266 read transactions per second

2053200 write transactions per second

3422 delete transactions per second.

and the percentage of the different transactions of the read,write and
delete,
is:

The read transactions are 70%

and the write transactions are 20%

and the delete transations are 10%

So the first moment that is the mean delay of the service of the M/G/c
queue of the database servers is:

M1 = 0.70 * (1/10266) + 0.20 * (1/2053200) + 0.10 * (1/3422)

so, M1 = 0. 0001

So the service rate of the each server in the M/G/c queue of the
database servers
queue is: 1/M1 = 10000 transactions per second.

And the service rate in the M/M/1 Network queue is: 32 transactions per
second

So now we have all the necessary numbers to do our calculation,
so since the Knee of the M/M/1 Network queue is at 50%, so this
will give 32 transactions per second * 50% = 16 transactions
per second.

So here is all the network:

A -> M/G/c database servers queue -> M/M/1 Network queue -> A

So the arrival rate A must not go beyond the Knee of the Network queue
that is 16 transactions per second , this is equal to 1382400
transactions per day,
, and the mean waiting time of the overall Network of queues that is in
tandem
is equal to:

The mean waiting time of the M/M/c database servers queue is:

D = Phi^c / Mu*(1 - Phi^c) [2]

Phi: is the utilization
and Mu: is the the service rate in server queue.

Phi = U(Density of circulation) / c (number of servers in the M/M/c queue)

This is equal then to:

Phi = (16 /10000) / c =0.0016 /c , so let's take that c, the number
of database servers is 2, so, this is equal to 0.0016 /2 = 0.0008

So, D = 0.0008^2 / (10000*(1- 0.0008^2)) = 0.000000000064 seconds

The mean response time R = D + 1/Mu = 0.000000000064 + 1/10000 = 0.0001
second

and the mean waiting time of the M/M/1 Network queue is:

D = (Phi/Mu) / (1- Phi)

and Phi = Lambda / Mu

Lambda is the arrival rate A

and Mu is the service rate of the M/M/1 queue.

this is equal to:

Phi = 16 / 32 = 0.5

D = (0.5/32) / (1-0.5) = 0.03 seconds.

The mean response time R = D + 1/Mu = 0.03 + 1/32 = 0.06 second

So the total mean response time of the Network is:

0.0001 second of the database servers queue + 0.06 second
of the Network queue, so the total mean response time of the network
of queues in Tandem is equal to: 0.06 seconds.

So the calculations are good.

-

I have updated my simulation of my queuing model simulation
of an M/M/c queue to version 1.03 , now it works with both
Delphi XE and Freepascal compilers , M means markovian and c
is the number of servers.

Here it is:

https://sites.google.com/site/aminer68/m-m-n-queuing-model-simulation-with-obj...

To do a simulation , just open the file called MMn.pas and you have
to change the following:

InterArrivals:=TExponentialDistribution.Create(420623,1.0/3.0);
ServiceTimes:=TExponentialDistribution.Create(220623,1.0/4.0);

The above 1.0/3.0 is the mean delay of the Exponential distribution
of the arrivals, you can change it for something else.

and 1.0/4.0 is the mean delay of the Exponential distribution
of the service of each server of the M/M/c queue, you can change it for
something else.

And with this simulation you can do easily the simulation
of an ecommerce website as i showed you before.

-

I have come to an interesting post...

Now i hope you have read my previous post titled:

"And here is how to do a simulation of the ecommerce websites"

Why i have done this mathematical calculations ?

To show you what permit exactly mathematical queuing theory ,
it permit to do a better QoS , and of course that's easy to
understand, but it permit also like in the Amdahl law of parallel
computing to model the ecommerce websites and to know how much the
ecommerce websites can handle of throughput and loads by
taking into account the Knee of the Network queue , and it
allows us to change theorically the characteristics of the M/G/c queue
of the database servers and the Network queue to be able to do
calculations before adding empirically more servers or more bandwidth ,
so that's optimization and that's good for your pocket.

But i have come to an interesting post...because
this modeling of an ecommerce website with mathematical
Queuing theory or simulation is not sufficient, because
it's necessary also to know how to limit the number of
connected internet users on the webserver to be able to
control the waiting time of the internet users to not go
beyond a not acceptable waiting time, and it's
important to give more priority to the write transactions
because that's critical for your pocket, because it makes
a company make more money, so my idea to realize
this requirement, is to use a FIFO synchonization semaphore
for each kind of database transactions such us read , write and delete ,
and you have to do a calculation of the time that
the internet users are waiting by computing it, if the time
go beyond an acceptable waiting time you will simply not process
those internet users and tell them to try to reexecute there
transactions again after a certain time... so, this requirement
and my solution that i have added is the right tool to make
a better QoS and to make your ecommerce website succeed.

-

Here is the book that i have read about capacity planning
with mathematical queuing theory, you have to read it:

Performance by Design: Computer Capacity Planning By Example

http://www.amazon.com/Performance-Design-Computer-Capacity-Planning/dp/...

But the researcher in this book is using Queuing with Multiple-Class
Models to model an ecommerce website, but i have not used his method
because i have modeled the M/G/c database servers with an
hyper-exponential service and i have approximate it with
an M/M/c queue , please take a look at my two previous posts that
have been corrected and that are titled:

"Here is the mathematical calculations of a queuing network"

and

" I have come to an interesting post"

To understand more my mathematical modeling of ecommerce websites etc.

-

The previous calculations are correct...

Now can we ask ourselves an important question..

What is the very important thing that this mathematical modeling
of a queuing network of an ecommerce website that i have done have
showed us ?

Here is my answer:

You have to know that this mathematical modeling that
i have done is very important, because it shows also
a very important thing, that the arrival rate is
is limited by the Network download bandwidth ,
since the arrival rate must not go beyond the Knee
of 50% of the M/M/1 queue of the Network queue , so
the Network of the ecommerce website that i have
modeled is limited by the Network download bandwidth,
since an operational law of queuing theory states that:
The rate of the jobs leaving any stable node must equal its
arrival rate.
-

An important mathematical deduction...

I will clear something on my mathematical calculations

Since i have mathematically modeled a queuing network
in Tandem of ecommerce websites..

I have to tell you something important..

Here is the network of the ecommerce websites:

A -> M/G/c database servers queue -> M/M/1 Network queue -> A

So as you have noticed the queues are interconnected in Tandem,
so this is good, but notice with me that the last queue
that is the M/M/1 Network queue is usually slower than the M/G/c
database servers queue that has an hyper-exponential service,
so my mathematical calculations have showed that the arrival
rate A to the network is limited by the slowest queue that is
in my example the M/M/1 Network queue, so since the
M/G/c data servers queue is the fastest, so no need to optimize
and to speed more the M/G/c data servers queue by adding more servers ,
and that is a very important thing to know, because this mathematical
deduction is infered from an operational law of queuing theory states
that: The rate of the jobs leaving any stable node must equal its
arrival rate and it is infered from the the Knee of the M/M/1 queue that
is equal to 50%, so the best way to optimize and to speed this network
is by adding more bandwidth on the slowest queue node like
the Network node, so be smart and don't forget this advise.

-

Here my other important mathematical deduction

If you read this book that i have read:

Performance by Design: Computer Capacity Planning By Example

http://www.amazon.com/Performance-Design-Computer-Capacity-Planning/dp/...

I have said the researcher in this book is using Queuing with
Multiple-Class Models to model an ecommerce website, but i have not used
his method because i have modeled the M/G/c database servers with
anhyper-exponential service and i have approximate it with
an M/M/c queue(c: is the number of servers), other than that this
book don't learn you as i have learned you that the network
of the ecommerce website is limited by the slowest node..

Other than that:

I have just read the following page, look at the the USL

(Universal Law of Computational Scalability) of Dr. Gunther,
he wrote this: (See: http://en.wikipedia.org/wiki/Neil_... )

--------------

The relative capacity C(N) of a computational platform is given by:

C(N) = N
------------------------------------------
1 + a (N - 1) + Ã? N (N - 1)

where N represents either the number of physical processors
in the hardware configuration or the number of users driving the
software application. The parameters a and Ã? represent respectively
the levels of contention (e.g., queueing for shared resources) and
coherency delay (i.e., latency for data to become consistent) in the
system. The Ã? parameter also quantifies the retrograde throughput
seen in many stress tests but not accounted for in either Amdahl's law
or event-based simulations.

----------

His website: http://www.perfdyn...

If you read carefully , you will see that Dr. Gunther is using this
model to predict scalability after he simulates a relatively small
number of vusers in LoadRunner ( because of licensing costs, it's
cost-effective) and after that he finds the coefficients of the
2nd-degree polynomial (quadratic equation) and then transform
those coefficients back to the USL parameters using the a = b - a
and Ã? = a.

And then he is extrapolating with the USL model to higher loads
to predict scalability.

Now my question follows:

Suppose we have obtained a small number of measured load-points
with Loadrunner or others tools, and we calculated the USL equation
to predict scalability of a webserver , how the USL model can predict
if the scalability/performance is limited by the network bandwidth
and not the server ? I think USL can not predict this.

So USL don't have the power of the Queuing theory that allows
us to deduce more facts that are important such us
the network is limited by the slowest node as i have
learned you on my previous posts.

Thank you,
Amine Moulay Ramdane.

comp.programming

Mathematical modeling correction

Ramine

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