Mathematical modeling of ecommerce websites

[Ramine]

Hello............

Here is my ecommerce websites mathematical queuing theory
modeling that is good approximation:

A -> M/G/c database server 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 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 queue
of the M/G/c queue is:

M1 = p1/a + p2/b + p3/c [1]

and the second moment of each queue 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 each 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 queue 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
queue 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 / (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.


Thank you,
Amine Moulay Ramdane.