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 | | From: | Ritu | | Subject: | Queuing Theory Example | | Date: | 18 Jan 2005 18:11:09 -0800 |
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 | I have a problem releated to queuing theory in which i need to
estimate the number of servers required by the system
1. One server can server one user at a time
2. About 100 users in the system are there and will increase in future.
3. User arrival rate is 20 users at a time on an average.
4. Each user needs 30 minutes on an average with the server
5. The waiting time should not be more than 15 minutes for the user
6. There is only one 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!
Thanks
Ritu
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| | From: | Ritu | | Subject: | Re: Queuing Theory Example | | Date: | 18 Jan 2005 22:04:00 -0800 |
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| Thanks for your reply!!
I want to estimate the number of servers that are required 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 but i will try to answer
the questions you have mentioned.
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?
2.) Service time is the time they are logged on to server.And this also
may vary from user to user.Minimum time they are logged on is say 5
minutes. And Maximum is 60 minutes.But most people say spend about 25-
30 minutes on the server but not all.
Actually the above explanation is not at all technical I know.But I not
from this field. 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 as soo many different versions on net
that i almost got lost. So If you can help me explain the theory that i
can use for my problem or help me understand the formula for this.I
will be really grateful
Thanks again
Ritu
Karthik Thyagarajan wrote:
> If you need to solve your problem by queueing theory, there are some
> questions that need to be answered....
>
> 1) What is the probability distribution of arrivals? (for eg.,
Poisson,
> Exponential, Uniform etc.). If the process does not have poisson
> arrivals, this problem becomes a little difficult to solve using
> queueing theory concepts......("20 users at a time?"... Does this
mean
> arrivals occur in batches of 20?)
>
> 2) The service time info that you have mentioned is also a little
> confusing......what do you mean when you say "30 minutes on an
> average"......this again can be understood clearly only when the
> probability distribution of service times is known.....
> All the Best
> Karthik
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| | From: | aminer at generation.net | | Subject: | Re: Queuing Theory Example | | Date: | 19 Jan 2005 06:16:43 -0800 |
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| Hi Ritu,
I will try to answer your questions without getting into too much
mathematics, exemple: i will avoid to explain what is a poisson process
ect...
(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
>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 (1)
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 distribution..
You have to find empiricaly(by measurements...) what is the 'arrival
rate'.
For that, find the interval delta(t) that gives you 0 or1 (arrival..)
and you will obtain your 'lamda'(arrival rate).
Your model is M/M/1, and we know how to calculate the following:
Little formula:
average client in the 'system' = service rate x average waiting time in
the 'system'
average waiting time in the 'queue' =
average waiting time in the 'system' - (1/ service rate)
and
Utilization = arrival rate / service rate
R (response time) = 1 / (service rate - arrival rate)
> 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)at your are
not sa
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)
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 (1 - Utilization)) (1)
It's approximatly equal to:
Utilization^C/(service rate x (1 - Utilization^C)
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.
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| | From: | aminer at generation.net | | Subject: | Re: Queuing Theory Example | | Date: | 19 Jan 2005 18:36:21 -0800 |
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|
(adding 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 (1)
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)
> 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.
Ritu wrote:
> Thanks for your reply!!
>
> I want to estimate the number of servers that are required 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 but i will try to
answer
> the questions you have mentioned.
>
> 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?
>
> 2.) Service time is the time they are logged on to server.And this
also
> may vary from user to user.Minimum time they are logged on is say 5
> minutes. And Maximum is 60 minutes.But most people say spend about
25-
> 30 minutes on the server but not all.
>
> Actually the above explanation is not at all technical I know.But I
not
> from this field. 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 as soo many different versions on
net
> that i almost got lost. So If you can help me explain the theory that
i
> can use for my problem or help me understand the formula for this.I
> will be really grateful
>
> Thanks again
> Ritu
>
> Karthik Thyagarajan wrote:
> > If you need to solve your problem by queueing theory, there are
some
> > questions that need to be answered....
> >
> > 1) What is the probability distribution of arrivals? (for eg.,
> Poisson,
> > Exponential, Uniform etc.). If the process does not have poisson
> > arrivals, this problem becomes a little difficult to solve using
> > queueing theory concepts......("20 users at a time?"... Does this
> mean
> > arrivals occur in batches of 20?)
> >
> > 2) The service time info that you have mentioned is also a little
> > confusing......what do you mean when you say "30 minutes on an
> > average"......this again can be understood clearly only when the
> > probability distribution of service times is known.....
> > All the Best
> > Karthik
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| | From: | Karthik Thyagarajan | | Subject: | Re: Queuing Theory Example | | Date: | 18 Jan 2005 19:47:21 -0800 |
|
|
| If you need to solve your problem by queueing theory, there are some
questions that need to be answered....
1) What is the probability distribution of arrivals? (for eg., Poisson,
Exponential, Uniform etc.). If the process does not have poisson
arrivals, this problem becomes a little difficult to solve using
queueing theory concepts......("20 users at a time?"... Does this mean
arrivals occur in batches of 20?)
2) The service time info that you have mentioned is also a little
confusing......what do you mean when you say "30 minutes on an
average"......this again can be understood clearly only when the
probability distribution of service times is known.....
All the Best
Karthik
|
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| | From: | Ritu | | Subject: | Re: Queuing Theory Example | | Date: | 19 Jan 2005 18:42:20 -0800 |
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|
| wow!! Thanks so much to you all..you guys actually taught me this
thing. Thanks again!!
Regards
Ritu
ami...@generation.net wrote:
> (adding 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 (1)
>
>
> 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)
>
>
> > 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.
>
>
>
> Ritu wrote:
> > Thanks for your reply!!
> >
> > I want to estimate the number of servers that are required 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 but i will try to
> answer
> > the questions you have mentioned.
> >
> > 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?
> >
> > 2.) Service time is the time they are logged on to server.And this
> also
> > may vary from user to user.Minimum time they are logged on is say 5
> > minutes. And Maximum is 60 minutes.But most people say spend about
> 25-
> > 30 minutes on the server but not all.
> >
> > Actually the above explanation is not at all technical I know.But I
> not
> > from this field. 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 as soo many different versions on
> net
> > that i almost got lost. So If you can help me explain the theory
that
> i
> > can use for my problem or help me understand the formula for this.I
> > will be really grateful
> >
> > Thanks again
> > Ritu
> >
> > Karthik Thyagarajan wrote:
> > > If you need to solve your problem by queueing theory, there are
> some
> > > questions that need to be answered....
> > >
> > > 1) What is the probability distribution of arrivals? (for eg.,
> > Poisson,
> > > Exponential, Uniform etc.). If the process does not have poisson
> > > arrivals, this problem becomes a little difficult to solve using
> > > queueing theory concepts......("20 users at a time?"... Does this
> > mean
> > > arrivals occur in batches of 20?)
> > >
> > > 2) The service time info that you have mentioned is also a little
> > > confusing......what do you mean when you say "30 minutes on an
> > > average"......this again can be understood clearly only when the
> > > probability distribution of service times is known.....
> > > All the Best
> > > Karthik
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