Find the number of dentists so less than a specific number of patients are waiting at specific time t
up vote
3
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Historical data:
Am I right to think about this question from the perspective of Poisson Distribution? I think I have to find something like $P(Xle 3)$, but I don't know how to find $lambda$, the mean number of occurrences in this case.
In this case, we need to make sure there are less than $3$ patients at a specific time $t$.
How do you find the number of dentists so less than a specific number of patients are waiting at specific time t?
probability probability-theory statistics probability-distributions
asked Nov 22 at 10:44
Yolanda Hui
15810
add a comment |
up vote
3
down vote
favorite
Historical data:
Am I right to think about this question from the perspective of Poisson Distribution? I think I have to find something like $P(Xle 3)$, but I don't know how to find $lambda$, the mean number of occurrences in this case.
In this case, we need to make sure there are less than $3$ patients at a specific time $t$.
How do you find the number of dentists so less than a specific number of patients are waiting at specific time t?
probability probability-theory statistics probability-distributions
asked Nov 22 at 10:44
Yolanda Hui
15810
Firstly, you can never make sure of this $-$ however many dentists you have, there will always be a chance, however small, that more than three people will have to wait in line. Secondly, we are missing too much data anyway. For instance, how many patients are expected in a typical hour?
– TonyK
Nov 22 at 11:04
Added the historical data, they don't allow more than 3 people to wait in the line. The case already specified that. @TonyK
– Yolanda Hui
Nov 22 at 11:07
What on earth are those arrival times? Are you measuring them to the ten-thousandth of a minute? Anyway, now you have to decide what is an acceptable probability threshold for the chance of having to turn away a patient in, say, a given day. This should be some small number, maybe 0.001 (but its exact value is a business decision, not a mathematical one).
– TonyK
Nov 22 at 11:24
Yes. We are only concerned about the mathematical value though, that's how it was thrown to us by the professor.
– Yolanda Hui
Nov 22 at 12:17
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Historical data:
Am I right to think about this question from the perspective of Poisson Distribution? I think I have to find something like $P(Xle 3)$, but I don't know how to find $lambda$, the mean number of occurrences in this case.
In this case, we need to make sure there are less than $3$ patients at a specific time $t$.
How do you find the number of dentists so less than a specific number of patients are waiting at specific time t?
probability probability-theory statistics probability-distributions
asked Nov 22 at 10:44
Yolanda Hui
15810
Historical data:
Am I right to think about this question from the perspective of Poisson Distribution? I think I have to find something like $P(Xle 3)$, but I don't know how to find $lambda$, the mean number of occurrences in this case.
In this case, we need to make sure there are less than $3$ patients at a specific time $t$.
How do you find the number of dentists so less than a specific number of patients are waiting at specific time t?
probability probability-theory statistics probability-distributions
probability probability-theory statistics probability-distributions
asked Nov 22 at 10:44
Yolanda Hui
15810
asked Nov 22 at 10:44
Yolanda Hui
15810
edited Nov 23 at 18:53
asked Nov 22 at 10:44
Yolanda Hui
15810
asked Nov 22 at 10:44
Yolanda Hui
15810
asked Nov 22 at 10:44
Yolanda Hui
15810
15810
Firstly, you can never make sure of this $-$ however many dentists you have, there will always be a chance, however small, that more than three people will have to wait in line. Secondly, we are missing too much data anyway. For instance, how many patients are expected in a typical hour?
– TonyK
Nov 22 at 11:04
Added the historical data, they don't allow more than 3 people to wait in the line. The case already specified that. @TonyK
– Yolanda Hui
Nov 22 at 11:07
What on earth are those arrival times? Are you measuring them to the ten-thousandth of a minute? Anyway, now you have to decide what is an acceptable probability threshold for the chance of having to turn away a patient in, say, a given day. This should be some small number, maybe 0.001 (but its exact value is a business decision, not a mathematical one).
– TonyK
Nov 22 at 11:24
Yes. We are only concerned about the mathematical value though, that's how it was thrown to us by the professor.
– Yolanda Hui
Nov 22 at 12:17
add a comment |
Firstly, you can never make sure of this $-$ however many dentists you have, there will always be a chance, however small, that more than three people will have to wait in line. Secondly, we are missing too much data anyway. For instance, how many patients are expected in a typical hour?
– TonyK
Nov 22 at 11:04
Added the historical data, they don't allow more than 3 people to wait in the line. The case already specified that. @TonyK
– Yolanda Hui
Nov 22 at 11:07
What on earth are those arrival times? Are you measuring them to the ten-thousandth of a minute? Anyway, now you have to decide what is an acceptable probability threshold for the chance of having to turn away a patient in, say, a given day. This should be some small number, maybe 0.001 (but its exact value is a business decision, not a mathematical one).
– TonyK
Nov 22 at 11:24
Yes. We are only concerned about the mathematical value though, that's how it was thrown to us by the professor.
– Yolanda Hui
Nov 22 at 12:17
Firstly, you can never make sure of this $-$ however many dentists you have, there will always be a chance, however small, that more than three people will have to wait in line. Secondly, we are missing too much data anyway. For instance, how many patients are expected in a typical hour?
– TonyK
Nov 22 at 11:04
Firstly, you can never make sure of this $-$ however many dentists you have, there will always be a chance, however small, that more than three people will have to wait in line. Secondly, we are missing too much data anyway. For instance, how many patients are expected in a typical hour?
– TonyK
Nov 22 at 11:04
Added the historical data, they don't allow more than 3 people to wait in the line. The case already specified that. @TonyK
– Yolanda Hui
Nov 22 at 11:07
Added the historical data, they don't allow more than 3 people to wait in the line. The case already specified that. @TonyK
– Yolanda Hui
Nov 22 at 11:07
What on earth are those arrival times? Are you measuring them to the ten-thousandth of a minute? Anyway, now you have to decide what is an acceptable probability threshold for the chance of having to turn away a patient in, say, a given day. This should be some small number, maybe 0.001 (but its exact value is a business decision, not a mathematical one).
– TonyK
Nov 22 at 11:24
What on earth are those arrival times? Are you measuring them to the ten-thousandth of a minute? Anyway, now you have to decide what is an acceptable probability threshold for the chance of having to turn away a patient in, say, a given day. This should be some small number, maybe 0.001 (but its exact value is a business decision, not a mathematical one).
– TonyK
Nov 22 at 11:24
Yes. We are only concerned about the mathematical value though, that's how it was thrown to us by the professor.
– Yolanda Hui
Nov 22 at 12:17
Yes. We are only concerned about the mathematical value though, that's how it was thrown to us by the professor.
– Yolanda Hui
Nov 22 at 12:17
add a comment |
1 Answer
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First of all get an estimate of the inter-arrival rate from the dataset. Calculate interarrival times as $I_i=a_i-a_{i-1}$ where $a_i$'s are arrival times and $a_0=0$. Then $hatlambda=frac{n}{sum I_i}.$ Now let's say that only $m$ many patients are allowed to wait and you want $n$ many dentists. So, at a time $t$ you want inter-arrival less than or equal to $m+n$ between time $(t-10,t]$, otherwise, the clinic has to return some patients. So, you want $N_t-N_{t-10}left(stackrel{d}{=}N_{10}right)leq m+n.$ Now $N_t-N_{t-10}sim textbf{Poi}(10hatlambda)$ and as @TonyK mentioned in the comment that there is no way you can make sure that with probability one. However, you can do it for expectation or quantiles.
For an example, if you want on an average how many dentists you need then you answer should be $E(N_t-N_{t-10}leq m+n)Rightarrow 10hatlambdaleq m+nRightarrow ngeq 10hatlambda-m.$
Again if you want how many doctors you need such that with probability at most $p$ the clinic don't have to return a patient, then find the smallest non-negative integer $k_0$ such that $P(N_{10}>k_0)leq p.$ Then the answer should be $ngeq k_0-m.$
answered Nov 23 at 13:26
John_Wick
1,199111
$p$ is the probability the clinic doesn't have to turn away a client over which time period? (And less importantly, is $n$ the number of arrival times, or the number of dentists?)
– TonyK
Nov 23 at 13:36
$n$ is the number of dentists, and apparently, if you fix any time $t$ then by memoryless property probability that the clinic has to turn away a patient doesn't depend on time $t$. And I am looking at the marginals.
– John_Wick
Nov 23 at 13:42
@John_Wick I have come up with the result using your method, we need to hire exactly 2 dentists. Can you confirm if my result is correct?
– Yolanda Hui
Nov 23 at 16:44
add a comment |
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1 Answer
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1 Answer
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up vote
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First of all get an estimate of the inter-arrival rate from the dataset. Calculate interarrival times as $I_i=a_i-a_{i-1}$ where $a_i$'s are arrival times and $a_0=0$. Then $hatlambda=frac{n}{sum I_i}.$ Now let's say that only $m$ many patients are allowed to wait and you want $n$ many dentists. So, at a time $t$ you want inter-arrival less than or equal to $m+n$ between time $(t-10,t]$, otherwise, the clinic has to return some patients. So, you want $N_t-N_{t-10}left(stackrel{d}{=}N_{10}right)leq m+n.$ Now $N_t-N_{t-10}sim textbf{Poi}(10hatlambda)$ and as @TonyK mentioned in the comment that there is no way you can make sure that with probability one. However, you can do it for expectation or quantiles.
For an example, if you want on an average how many dentists you need then you answer should be $E(N_t-N_{t-10}leq m+n)Rightarrow 10hatlambdaleq m+nRightarrow ngeq 10hatlambda-m.$
Again if you want how many doctors you need such that with probability at most $p$ the clinic don't have to return a patient, then find the smallest non-negative integer $k_0$ such that $P(N_{10}>k_0)leq p.$ Then the answer should be $ngeq k_0-m.$
answered Nov 23 at 13:26
John_Wick
1,199111
$p$ is the probability the clinic doesn't have to turn away a client over which time period? (And less importantly, is $n$ the number of arrival times, or the number of dentists?)
– TonyK
Nov 23 at 13:36
$n$ is the number of dentists, and apparently, if you fix any time $t$ then by memoryless property probability that the clinic has to turn away a patient doesn't depend on time $t$. And I am looking at the marginals.
– John_Wick
Nov 23 at 13:42
@John_Wick I have come up with the result using your method, we need to hire exactly 2 dentists. Can you confirm if my result is correct?
– Yolanda Hui
Nov 23 at 16:44
add a comment |
up vote
0
down vote
First of all get an estimate of the inter-arrival rate from the dataset. Calculate interarrival times as $I_i=a_i-a_{i-1}$ where $a_i$'s are arrival times and $a_0=0$. Then $hatlambda=frac{n}{sum I_i}.$ Now let's say that only $m$ many patients are allowed to wait and you want $n$ many dentists. So, at a time $t$ you want inter-arrival less than or equal to $m+n$ between time $(t-10,t]$, otherwise, the clinic has to return some patients. So, you want $N_t-N_{t-10}left(stackrel{d}{=}N_{10}right)leq m+n.$ Now $N_t-N_{t-10}sim textbf{Poi}(10hatlambda)$ and as @TonyK mentioned in the comment that there is no way you can make sure that with probability one. However, you can do it for expectation or quantiles.
For an example, if you want on an average how many dentists you need then you answer should be $E(N_t-N_{t-10}leq m+n)Rightarrow 10hatlambdaleq m+nRightarrow ngeq 10hatlambda-m.$
Again if you want how many doctors you need such that with probability at most $p$ the clinic don't have to return a patient, then find the smallest non-negative integer $k_0$ such that $P(N_{10}>k_0)leq p.$ Then the answer should be $ngeq k_0-m.$
answered Nov 23 at 13:26
John_Wick
1,199111
$p$ is the probability the clinic doesn't have to turn away a client over which time period? (And less importantly, is $n$ the number of arrival times, or the number of dentists?)
– TonyK
Nov 23 at 13:36
$n$ is the number of dentists, and apparently, if you fix any time $t$ then by memoryless property probability that the clinic has to turn away a patient doesn't depend on time $t$. And I am looking at the marginals.
– John_Wick
Nov 23 at 13:42
@John_Wick I have come up with the result using your method, we need to hire exactly 2 dentists. Can you confirm if my result is correct?
– Yolanda Hui
Nov 23 at 16:44
add a comment |
up vote
0
down vote
up vote
0
down vote
First of all get an estimate of the inter-arrival rate from the dataset. Calculate interarrival times as $I_i=a_i-a_{i-1}$ where $a_i$'s are arrival times and $a_0=0$. Then $hatlambda=frac{n}{sum I_i}.$ Now let's say that only $m$ many patients are allowed to wait and you want $n$ many dentists. So, at a time $t$ you want inter-arrival less than or equal to $m+n$ between time $(t-10,t]$, otherwise, the clinic has to return some patients. So, you want $N_t-N_{t-10}left(stackrel{d}{=}N_{10}right)leq m+n.$ Now $N_t-N_{t-10}sim textbf{Poi}(10hatlambda)$ and as @TonyK mentioned in the comment that there is no way you can make sure that with probability one. However, you can do it for expectation or quantiles.
For an example, if you want on an average how many dentists you need then you answer should be $E(N_t-N_{t-10}leq m+n)Rightarrow 10hatlambdaleq m+nRightarrow ngeq 10hatlambda-m.$
Again if you want how many doctors you need such that with probability at most $p$ the clinic don't have to return a patient, then find the smallest non-negative integer $k_0$ such that $P(N_{10}>k_0)leq p.$ Then the answer should be $ngeq k_0-m.$
answered Nov 23 at 13:26
John_Wick
1,199111
First of all get an estimate of the inter-arrival rate from the dataset. Calculate interarrival times as $I_i=a_i-a_{i-1}$ where $a_i$'s are arrival times and $a_0=0$. Then $hatlambda=frac{n}{sum I_i}.$ Now let's say that only $m$ many patients are allowed to wait and you want $n$ many dentists. So, at a time $t$ you want inter-arrival less than or equal to $m+n$ between time $(t-10,t]$, otherwise, the clinic has to return some patients. So, you want $N_t-N_{t-10}left(stackrel{d}{=}N_{10}right)leq m+n.$ Now $N_t-N_{t-10}sim textbf{Poi}(10hatlambda)$ and as @TonyK mentioned in the comment that there is no way you can make sure that with probability one. However, you can do it for expectation or quantiles.
For an example, if you want on an average how many dentists you need then you answer should be $E(N_t-N_{t-10}leq m+n)Rightarrow 10hatlambdaleq m+nRightarrow ngeq 10hatlambda-m.$
Again if you want how many doctors you need such that with probability at most $p$ the clinic don't have to return a patient, then find the smallest non-negative integer $k_0$ such that $P(N_{10}>k_0)leq p.$ Then the answer should be $ngeq k_0-m.$
answered Nov 23 at 13:26
John_Wick
1,199111
answered Nov 23 at 13:26
John_Wick
1,199111
answered Nov 23 at 13:26
John_Wick
1,199111
answered Nov 23 at 13:26
John_Wick
1,199111
1,199111
$p$ is the probability the clinic doesn't have to turn away a client over which time period? (And less importantly, is $n$ the number of arrival times, or the number of dentists?)
– TonyK
Nov 23 at 13:36
$n$ is the number of dentists, and apparently, if you fix any time $t$ then by memoryless property probability that the clinic has to turn away a patient doesn't depend on time $t$. And I am looking at the marginals.
– John_Wick
Nov 23 at 13:42
@John_Wick I have come up with the result using your method, we need to hire exactly 2 dentists. Can you confirm if my result is correct?
– Yolanda Hui
Nov 23 at 16:44
add a comment |
$p$ is the probability the clinic doesn't have to turn away a client over which time period? (And less importantly, is $n$ the number of arrival times, or the number of dentists?)
– TonyK
Nov 23 at 13:36
$n$ is the number of dentists, and apparently, if you fix any time $t$ then by memoryless property probability that the clinic has to turn away a patient doesn't depend on time $t$. And I am looking at the marginals.
– John_Wick
Nov 23 at 13:42
@John_Wick I have come up with the result using your method, we need to hire exactly 2 dentists. Can you confirm if my result is correct?
– Yolanda Hui
Nov 23 at 16:44
$p$ is the probability the clinic doesn't have to turn away a client over which time period? (And less importantly, is $n$ the number of arrival times, or the number of dentists?)
– TonyK
Nov 23 at 13:36
$p$ is the probability the clinic doesn't have to turn away a client over which time period? (And less importantly, is $n$ the number of arrival times, or the number of dentists?)
– TonyK
Nov 23 at 13:36
$n$ is the number of dentists, and apparently, if you fix any time $t$ then by memoryless property probability that the clinic has to turn away a patient doesn't depend on time $t$. And I am looking at the marginals.
– John_Wick
Nov 23 at 13:42
$n$ is the number of dentists, and apparently, if you fix any time $t$ then by memoryless property probability that the clinic has to turn away a patient doesn't depend on time $t$. And I am looking at the marginals.
– John_Wick
Nov 23 at 13:42
@John_Wick I have come up with the result using your method, we need to hire exactly 2 dentists. Can you confirm if my result is correct?
– Yolanda Hui
Nov 23 at 16:44
@John_Wick I have come up with the result using your method, we need to hire exactly 2 dentists. Can you confirm if my result is correct?
– Yolanda Hui
Nov 23 at 16:44
add a comment |
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Firstly, you can never make sure of this $-$ however many dentists you have, there will always be a chance, however small, that more than three people will have to wait in line. Secondly, we are missing too much data anyway. For instance, how many patients are expected in a typical hour?
– TonyK
Nov 22 at 11:04
Added the historical data, they don't allow more than 3 people to wait in the line. The case already specified that. @TonyK
– Yolanda Hui
Nov 22 at 11:07
What on earth are those arrival times? Are you measuring them to the ten-thousandth of a minute? Anyway, now you have to decide what is an acceptable probability threshold for the chance of having to turn away a patient in, say, a given day. This should be some small number, maybe 0.001 (but its exact value is a business decision, not a mathematical one).
– TonyK
Nov 22 at 11:24
Yes. We are only concerned about the mathematical value though, that's how it was thrown to us by the professor.
– Yolanda Hui
Nov 22 at 12:17