comp.programming

Ramine

3/16/2016 7:59:00 PM

Hello,

I correct:

Download speed of the client is:

10 Mbps is equal to 1.25 Mbytes.


Thank you,
Amine Moulay Ramdane.

On 3/16/2016 12:51 PM, Ramine wrote:
> Hello,
>
>
> I have made a mistake on my previous calculations, here is
> the correct 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 -> M/M/1 Client
> -> 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 characteristic of the Client queue is:
>
> Download speed of 10 Mbps
> Upload speed up to 1.6 Mbps
>
> Upload speed of 1.6 Mbps is equal to 200 Kbytes per second.
>
> Upload speed of 10 Mbps is equal to 1.25 Mps.
>
> 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 10000 transactions per second.
>
> And the service rate in the M/M/1 Network queue is: 32 transactions per
> second
>
> And the service rate in the M/M/1 Client queue is: 10.2 transactions per
> second.
>
> So now we have all the necessary numbers to do our calculation,
> so since the Knee of the M/M/1 Client queue is at 50% so this
> will give 10.2 transactions per second * 50% = 5.1 transactions
> per second.
>
> So here is all the network:
>
> A -> M/G/c database servers queue -> M/M/1 Network queue -> M/M/1 Client
> -> A
>
>
> So the arrival rate A must not go beyond the Knee of the Client queue
> that is 5.1 transactions per second , this is equal to 440640
> 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 = (5.1 /10000) / c =0.00051 /c , so let's take that c, the number
> of database servers is 2, so, this is equal to 0.00051 /2 = 0.000255
>
> So, D = 0.000255^2 / (10000*(1- 0.000255^2)) = 0.0000000000065025
> seconds
>
> The mean response time R = D + 1/Mu = 0.0000000000065025 + 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 = 5.1 / 32 = 0.159375
>
> D = (0.159375/32) / (1-0.159375) = 0.006 seconds.
>
> The mean response time R = D + 1/Mu = 0.006 + 1/32 = 0.04 second
>
> and the mean waiting time of the M/M/1 Client 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.
>
> Phi = 5.1 / 10.2 = 0.5
>
> D = (0.5/10.2) / (1-0.5) = 0.1 seconds.
>
> The mean response time R = D + 1/10.2 = 0.1 + 1/10.2 = 0.2 seconds
>
> So the total mean response time of the Network is:
>
> 0.0001 second of the database servers queue + 0.04 second
> of the Network queue + 0.2 second of the Client queue , so
> the total mean response time of the network of queues in Tandem
> is equal to: 0.24 seconds.
>
> So the calculations are good.
>
> And here is my explanation of my previous post that
> i have corrected more:
>
> And here is how to do a simulation of the ecommerce websites:
>
> You have to read and understand my previous post and
> after that use my following simulation program of
> an M/M/n queue:
>
> https://sites.google.com/site/aminer68/m-m-n-queuing-model-simulation-with-obj...
>
>
> use it to do your overall simulation of the ecommerce websites that
> have read-mostly workloads, since many ecommerce websites have
> read-mostly workloads, so the hyper-exponential service of the
> M/G/c queue of the database servers queue above can be approximated with
> an M/M/n queue when the writer and the delete transactions are
> less or equal to 30% of the total transactions and the ecommerce website
> has read-mostly workloads.
>
> So, since the network of the those ecommerce websites are
> interconnected in a serial manner like this:
>
> A -> M/G/c database servers queue -> M/M/1 Network queue -> M/M/1 Client
> -> A
>
> The M in M/G/c means markovian distribution of the arrivals to the
> M/G/c queue and the G in M/G/c is a general distribution of the service.
>
> A is the arrival rate to the network of queues.
>
> So you have to do your simulation using my above program
> for each queue in the network and after than the calculation
> for the waiting time and response time of the overall network of
> queues are easy , because your have to add them in a serial
> manner, since the network of queues are interconnected in
> a serial manner.
>
> And here is again my corrected mathematical modeling of
> the above case of the ecommerce websites:
>
> Here is my ecommerce websites mathematical queuing theory
> modeling that is good approximation..
>
> Here is the ecommerce website network organized as inter-connected
> queues in serial manner:
>
> A -> M/G/c database servers queue -> M/M/1 Network queue -> M/M/1 Client
> -> A
>
> The M in M/G/c means markovian distribution of the arrivals to the
> M/G/c queue and the G in M/G/c is a general distribution of the service.
>
> A is the arrival rate to the network of queues.
>
> M/G/c database server has an hyper-exponential service, i think
> that to not get into simulation, since many ecommerce websites
> have read-mostly workloads, the hyper-exponential service of the
> M/G/c queue of the database servers can be approximated with an
> M/M/n queue when the writer and the delete transactions are less or
> equal to 30% of the total transactions and the ecommerce website has
> read-mostly workloads.
>
> And now i will give you the mathematical equations of the M/G/c
> queue that has an hyper-exponential service that permit us to
> model a database server, here they are:
>
> The mean that is the the mean time of the service of each server
> of the M/G/c queue is:
>
> M1 = p1/a + p2/b + p3/c [1]
>
> and the second moment of each server is:
>
> M2 = 2*p1/a^2 + 2*p2/b^2 + 2*p3/c^2
>
> and the variance of each queue is:
>
> variance = M2 - M1^2
>
> a , b and c are the service rates of the different transactions
> such us read,write, delete.
>
> and p1 , p2 and p3 are the percentage of the transactions.
>
> And when you calculate and get M1 you will then calculate the
> mean service time that is 1/M1 and you will plug it on your M/M/c
> queue that is an approximation of the M/G/n queue of the database
> server that has an hyper-exponential service on read-mostly workloads,
> using the arrival rate.
>
> So for the M/M/c queue we have that:
>
> A good approximation of waiting time of the M/M/c 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)
>
> U = Lambda / Mu
>
> Lambda is the arrival rate A to the M/M/c queue
>
> and Mu is the service rate of each server of the M/M/C queue.
>
> And the response time of the M/M/c queue is:
>
> R = D + 1/Mu
>
> and the perceived throughput of the M/M/c queue that is Pt = 1/R
>
> And the mean number of transactions on the system is:
>
> Ns = Lamda*R
>
> Lambda is the arrival rate to the M/M/c queue.
>
> And the mean number of transactions on the M/M/c waiting queue is:
>
> Nq = Lambda*D
>
> So from the above equation [1] we get the service rate of each
> server of the M/M/c queue of the database server that is: 1/M1
>
> so we plug that in equation [2] of the M/M/c queue, so we get:
>
> D = Phi^c / ((1/M1)*(1 - Phi^c))
>
> So we get the response time of the M/M/c that is:
>
> R = D + 1/(1/M1) => R = D + M1.
>
> For the other M/M/1 queues of the Network queue and the client queue
> we have the following equation:
>
> The waiting time of M/M/1 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.
>
> And you have to not forget that in the M/M/1 Network queue
> you have a protocol overhead that is approximatly equal to 20%
> so you have to multiply the mean size of the files to be transferred on
> the M/M/1 Network queue by 120% and calculate after that the service
> rate of the M/M/1 Network queue. And you have to not forget about the
> Knee of the M/M/c queue of the database server that is equal to 74% and
> the Knees of the other M/M/1 queues that is equal to 50%.
>
> So since the queues of the ecommerce website to be modeled are organized
> in a serial manner, so the calculations are easy now, so i will let you
> do the calculations easily now.
>
> -
>
> 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 the Client
> queue, and it allows us to change theorically the characteristics
> of the M/G/c queue of the database servers and the Network queue
> and the Client 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
> Queing 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 Queing 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.
>
>
>
>
> Thank you,
> Amine Moulay Ramdane.
>
>
>