Re: Gamma Distribution application for modeling arrivals

On Aug 5, 7:08 pm, Jiggy <joshiamit...@xxxxxxxxx> wrote:
On Aug 5, 2:33 pm, Peter <peterp...@xxxxxxxxxxx> wrote:



I want to model
the arrival rate as follows:

Say the duration for which the victims are arriving
is 4 hrs (Time
horizon for the event). I want to divide this time
horizon into many
small subintervals say of 5 minutes each. The arrival
within these
subintervals follow an exponential distribution with
arrival rate
lamda, But the overall distribution of the arrival
shall follow gamma
distribution. How do I estimate different shape and
scale parameters
in this case and how do I use this logic to represent
an expression?

Unless I misunderstood the intention of your model, I think that you can't get a gamma distribution out of this. If the waiting time to the next event is "locally" exponentially distributed, the "global" distribution will also be exponential.
Something about the link between the exponential and gamma distributions... If you have 10 poisson processes going on simultaneously, with an average waiting time AWT between events, the waiting time for events coming from one of those processes will be exponentially distributed with mean waiting time AWT. The average waiting time for an event coming from _any_ of those independent processes will be 10*AWT. If, on the other hand, you "start waiting" at some point in a poisson process with average time between events AWT, and wait until the 10th event, the time for that 10 event to occur will be distributed as gamma(k=10,scale=AWT).
Watch out for the parameter confusion! You can express the parameter of both exponential and gamma distributions either as their scale or as the event rate. One is the opposite of the other. If you have, for example, an average of 2 events (e.g. people entering the hospital) per minute, the average waiting time up to the first event is 0.5 minute - that makes sense, right? So, whatever you are using for software, you either feed it the number of events per minute or the number of minutes per event. The first is a rate parameter, often written as lambda, and the second a scale parameter, often written as theta. The rate parameter is the inverse of the scale parameter and vice versa.
If you would at some point have to model the number of events in a unit of time, then you would use the Poisson distribution, and that one usually takes the event rate lambda as parameter. Note that it scales with your unit of time: 3/minute, for example, is equivalent to 15/5 minutes, 180/hour, etc.

Given that you have a finite number of victims of a bombing, you might think of a model that has a certain minimum delay (for ambulances to arrive at the spot), with maximum event rate (maybe proportional to distance and the number of ambulances available) dropping down as victims enter the hospital. I am thinking of some Poisson processes running in parallel, but with decreasing event rates as you are bringing in more and more people from the bomb site. Well, just thinking out loud.

For example, when we want to mention that the arrival
follows
exponential distribution with a mean of 3 then we
represent it as expo
(3) right?? How do I represent in the case I
mentioned above.

I also confused with the scale parameter and the
arrival rate...is it
that in my case arrival rate value is equal to the
scale parameter
value??

Thanks.
Amita

If I understood correctly, suppose I want to simulate the arrivals for
a period of 240 minutes (4hrs), and I divide this time horizon into 8
30 minutes intervals 0-30, 30-60, 60-90, 90-120, 120-150 and so on
till 210-240. does this mean K =8? and I assume that within each of
these subintervals, patients arrive at a rate say 5/min = lamda the
rate parameter and the equivalent scale parameter will be 0.2 right??
then I express the arrivals as EXPO (GAMMA (8,0.2))...right??? Now as
I know, the arrivals have two important parameters : one is the number
of total people coming and the other is the arrival pattern. How do I
take care of the number of people generated using the distribution
which I just explained above. Say I want to generate a total of 400
people coming into the simulated system. Just for your information, I
am using ARENA simulation software to model the arrivals.

Thanks a lot for your help
Amita.

Hello!!

Can you please help me :)
.

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