Modeling time dependent arrival rates using gamma distribution
Jiggy
I am modeling the patient arrival rates during a mass casualty event
and as we all know, they are time dependent and not constant. I am
trying to use gamma distribution to model the arrival process. I am
aware that gamma distribution is characterized by two parameters
namely the shape and the scale parameter. Now during a disaster
situation, the patients arriving to the hospital usually arrive in two
waves, the first wave arrives within 5-30 minutes from the time the
event occurs and the second wave after 30 mins may be. Following are
my questions:
1. Relationship between the gamma parameters keeping in mind the
arrival rate. How will the two parameters impact the arrival rate and
pattern?
2. How can I find the gamma parameters that would give the above said
delay times?
3. I want to model the arrival as gamma distribution over a time
horizon with the time horizon divided into several small time
intervals and within each of the small time intervals the arrival rate
follows an exponential distribution. How do I model this in the
simulation model designed by me. I need what way should I specify the
same in the create block to which represents arrivals in my model.
Thanks a lot.
Amita.
sbm...@gmail.com
You are giving yourself a headache by using a Gamma distribution
because you'll have a tough time explaining what the parameters mean
to your customer. Better to select a 1 parameter distribution that
has the approximate shape of the Gamma you have in mind. That should
be good enough since you are only approximating arrivals.
For each arrival slug you have 2 random variables, the total number of
patients and their arrival pattern. You could generate your patient
count at sim start from 1 distribution and dump them into a queue.
E.g. the number of total patients over the 25 minute period is N(1000,
250) or something.
Then you will need realizations of an arrival pdf as an order
statistic to ping the patient queue and inject the patients into the
system. So say the time dependent arrival pattern over 25 minutes is
lognormally distributed (14, 11) or whatever. I'd have to think about
this. But after you generate the n patients, you would generate n
arrival times and then sort those. Then use the arrival times to ping
the the patient queue.
Replay the same process for slug 2 with different parameters. This
assumes of course that the # of patients and their arrival patterns
are not correlated.
Without knowing what sim package you use, I can't tell you the actual
mechanics of the model design.
SteveM
Jiggy
Thank you so much for your reply....I am using ARENA 10.0 to model
this....it would be great if you can explain me to do this giving an
example... :) . Shall wait for your nxt reply.
sbm...@gmail.com
Sorry. Can't help you there. I use Extend not Arena.
Good Luck,
SteveM
Jiggy
Hey no problem. Thanks a lot for your reply though :)
DB Fuller
I agree with sbma...@gmail.com; do you really have a strong reason to
use the gamma?...
I think it is the a similar idea to sbma...@gmail.com's, but I believe
you should separate the number of patients from their arrival times. I
also understand the two arrival waves are independent, so you can just
do the same thing twice starting at different times.
My fisrt idea would look something like building a histogram
backwards. First, generate the total number of patients that will
arrive in, say, the first wave (repeat for the second) and how long
this wave will last. Then divide the lenght of the wave in as many
intervals as you see fit (you can randomise this amount if you like).
The next step is to take the area under the distribution you are using
in each interval (sorry I can't help you with choosing the gamma
parametres... – but see below) and multiply it by the total number of
arrivals; this will give you the number of arrivals in each interval.
I have absolutely no idea how to do this in Arena and even if it can
be done, you'll have to mind the rounding problems (unless you can
have fractional patients arriving – pardon the dark humour ¦¬) ). With
the number of arrivals of each interval, you can use the Exponential
distribution (as you propoused) to find the intervals; this should be
straight foward now.
HTH and tell me if you need clarification, I'm in a bit of a hurry
right now,
DB Fuller
PS: Just checked a software I have here; if you can tell me explicitly
three things: mean, std. deviation and skewness, I can tell you alpha
and beta for the gamma distribution.
Jiggy
Thanks for the reply, I am using Gamma distribution as it gives
flexibility for choosing the shape of the distribution and also it is
found to be used commonly to model the non homogeneous poisson
process. The arrivals are going to be exponential in nature with a
mean lamda. but lamda follows a gamma distribution with parameters
alpha and beta...so my experiment is going to be that I will allow 400
victims in the system, and try to perform an analysis by changing the
arrival pattern and see which of the arrivals are sensitive on the
system performance..so in this case, the number of victims is fixed,
the rate at which I inject them (as steve mentioned) is what I need to
analyze. It would be great if you can make me understand this by
giving an hypothetical example. Also, you said that if I give you
mean , std deviation and Skewness, you can tell me gamma parameters,
but unfortunately I do not have data to obtain teh same.
Thanks.
Amita.
Jiggy
I need to know how the arrival rate, shape parameter and scale
parameter are related and how do i specify it as an expression in
create block. I was thinking that it should be specified as EXPO
(GAMMA (shape, scale)).I want to know, depending upon the arrival rate
that I decided, can I get the values of shape and scale???
sbm...@gmail.com
Take my word for it, modeling probability distributions of pdf
parameters in the real world, (especially when it's not a purely
physical process) is nuts. Remember rule #1 about sim, "All models
are wrong. Some are useful." Unless this is an academic exercise
that compels you to load up on complexity, stick with something
simpler that gives you something that is "useful". And if you forget
rule #1, then remember rule #2, if an analytically pristine model is
rejected by the client because the analysis confuses him, the model is
NOT useful.
But what the heck to do I know? OR has shot itself in the foot for
decades building analytically dense models that were quickly consigned
to the trash heap because they were obtuse. So go ahead. Pile it
on. Generate pdfs of the pdfs of the parameters even. A little more
mathematical detritus won't hurt.
SteveM
DB Fuller
I agree with SteveM. What I proposed is too complicated, but I only
tried to help Amita do what I understood he wanted to do.
I think I understand better what is needed. I believe the proposed
experiment could be done with Exponentially distributed arrivals for
different chosen lambda (as many scenarios as you like, but each with
a fixed lambda for its duration). If you really want the lambda to be
gamma distributed, you actually need a number generator to create the
lambdas, then just create the scenarios.
So, it's not that different from what I said earlier, but much
simpler: generate many values for lambda with a number generator
(sorry I don't have one for the gamma) then run many separate
scenarios with these lambdas. Won't that give the same results?
If you want to use different gammas, just repeat everything.
HTH,
DB Fuller
Jiggy
Thanks..looks like you hv understood what I am trying to do....Can you
explain me how to do this? As in, stepwise procedure so that I can go
ahead and do it... please it will be very helpful to me Fuller :)
Jiggy
Hello,
I have figured out the way to get the time intervals which I am
looking for the patients to arrive at the hospital. The run length of
my simulation model is 1440 mins (24hrs). Can you explain me what is
the relationship between the number of victims generated and the
expression EXPO(GAMMA(alpha,beta))?? As, I know the mean and variance
of gamma both are dependent upon the parameters and hence changing
either of the value will change my mean as well as my variance...can u
explain me how do I take care of this..suggestions on possible
analysis with arrival pattern and rate is also welcomed..
thanks.
Amita
DB Fuller
I thought you were going to tell the software how many victims you
wanted. Isn't that expression the expression
("EXPO(GAMMA(alpha,beta))") for the time interval? I think I don't
understand what you need...
Take a look at this applet: http://www.fortunecity.co.uk/meltingpot/back/340/product/java/cdfdemomain.html,
which gives the graph for your parameters. It might help clarify your
thoughts.
I don't know this software, I've just googled for something like it:
http://www.mathwave.com/articles/gamma_distribution.html. The trial
version might also help you understand the distributions' behaviour if
it does what I think it does (i.e., dinamically graphing the
distribution).
I'm sorry I'm not really following your doubts anymore, but you can
try to clarify things.
DB Fuller