 |
(I don't know why google is not formatting the text correctly?
anyway, here is my answer again to this post, and it's te last time
i am posting it.., i have added some clarifications and cleaning
some typos..)
Hi Ritu,
I will try to answer your questions without getting
into too much mathematics, example: i will avoid to
explain what is a poisson process etc...
(to not make it too long ...)
Ritu wrote:
>I have a problem releated to queuing theory in which i need to
>estimate the number of servers required by the system
>to make the waiting time as low as possible and also keeping
>the server number not to high.Actually I m not a mathematics person.
ok
>1. One server can server one user at a time
ok..
>2. About 100 users in the system are there and will increase
> in future
ok..
>4. Each user needs 30 minutes on an average with the server
So now you have the service rate.. right ?
>1.)The users don't arrive in batches as such. They might arrive
>one by one or sometimes 2-3 users can log on to server at one
>time.But still we can just assume that the queue is FIFO. So
>i guess this makes it poisson distribution . is this right?
You have to verify it...
But let's assume it's a poisson process..
You have to find empiricaly(by measurements...) what is
the 'arrival rate'.
For that, find the 'interval' h=delta(t) that gives you
0,1 (arrival..), so that the probability P{N(h) >= 2} = o(h)
(we say f(x) is in o(h) if: lim f(h)/h = 0 when h -> 0)
and you will obtain your 'lamda'(arrival rate).
Your model is M/M/1, and we know how to calculate
the following:
Utilization = arrival rate / service rate
Little formula:
average client in the 'system' = service rate x average waiting
time in the 'system'
also,
average waiting time in the 'queue' =
average waiting time in the 'system' - (1/ service rate)
and
T (average waiting time in the system) =
1 / (service rate - arrival rate)
Ritu wrote:
> 5. The waiting time should not be more than 15 minutes
> for the user
Waiting time in the system or in the queue ?
I will assume it's the waiting time in the queue..
>Now is it possible to estimate the optimized number of
>servers required to meet the above criteria. ? Can some
> explain with respect to the above example.
>It would be a great help!
ok...
>I am software person who is trying to make a
>simulation model with no knowledge of this thing.
>Though i m trying my best to do so but Queuing theory...
ok...
> So If you can help me explain the theory that i
>can use for my problem or help me understand...
ok...
So, after you have calculate the above formulas, to
answer the above question, you have to look at the result
of the following
formula:
average waiting time in the 'queue' =
average waiting time in the 'system' - (1/service rate)
If it's <= 15 minutes, it's ok.
But if it's not?
If not, try to add some servers in an M/M/C configuration..
Note: try to add servers with almost the same hardware
configuration...
after that you have to calculate the following formulas:
C(c, U) = Erlang formula = P(c) / (1 - Utilization)
note: c the number of servers..
P(c): means the probability that all the servers are busy
P(0): means the probability that there is no waiting time
in the queue, that means also: AT LEAST one server
among the C servers are not busy...
The average waiting time in the 'queue' =
C(c,U) / (service rate x c x (1 - Utilization)) (1)
It's approximatly equal to:
Utilization^C/(service rate x (1 - Utilization^C)
this approximation is exact for the M/M/1 and M/M/2 models,
but 'slightly' lower than the result in (1) if c > 2
and
Utilization = Density of circulation / C (number of servers)
Note: ^ means Power()
C means the number of servers
Response time = The average waiting time in the 'queue' +
(1 / service rate)
average numbers of users in the system = service rate x response time
average number of users in queue = service rate x average waiting time
in the 'queue'
Try with C = 2 , 3 ...
and find for wich C the result of (1) is <= 15 minutes ...
Hope this answer your questions.
Regards,
Amine Moulay Ramdane.
|
|
