What density distribution could be used for time delays?What is the correct inter-arrival time distribution in a Poisson process?Need help with P(D) in a Bayesian model2 User Queuing Model Probability ProblemWhat is the statistical steady state of this poisson process?Poisson bus arrival process with different arrival rateQueuing with Exponential impatience timesStaying time of last arrived traveler before given time in Poisson ProcessProbability: how to use hint for interarrival time questionSolving Exponential Distributions with Preemptive QueueingOn distribution of the time between two consecutive events (either arrival or departure) in an M/M/1 queueing system
CLI: Get information Ubuntu releases
Worshiping one God at a time?
Exit shell with shortcut (not typing exit) that closes session properly
When did hardware antialiasing start being available?
Imaginary part of expression too difficult to calculate
Print last inputted byte
Why does Surtur say that Thor is Asgard's doom?
How to understand 「僕は誰より彼女が好きなんだ。」
UK Tourist Visa- Enquiry
Output visual diagram of picture
Isn't the word "experience" wrongly used in this context?
What are the rules for concealing thieves' tools (or items in general)?
If I cast the Enlarge/Reduce spell on an arrow, what weapon could it count as?
Did Nintendo change its mind about 68000 SNES?
Does the Shadow Magic sorcerer's Eyes of the Dark feature work on all Darkness spells or just his/her own?
Weird lines in Microsoft Word
Does convergence of polynomials imply that of its coefficients?
Turning a hard to access nut?
How can an organ that provides biological immortality be unable to regenerate?
Single word to change groups
Do I need an EFI partition for each 18.04 ubuntu I have on my HD?
How are passwords stolen from companies if they only store hashes?
Print a physical multiplication table
Friend wants my recommendation but I don't want to
What density distribution could be used for time delays?
What is the correct inter-arrival time distribution in a Poisson process?Need help with P(D) in a Bayesian model2 User Queuing Model Probability ProblemWhat is the statistical steady state of this poisson process?Poisson bus arrival process with different arrival rateQueuing with Exponential impatience timesStaying time of last arrived traveler before given time in Poisson ProcessProbability: how to use hint for interarrival time questionSolving Exponential Distributions with Preemptive QueueingOn distribution of the time between two consecutive events (either arrival or departure) in an M/M/1 queueing system
$begingroup$
I am analyzing a process related with time delays produced in scheduled departures or arrivals of a distribution trucks. The process consists on:
The definition of the problem is:
- Each truck has associated some travels per day.
- Each travel has associated a departure time and a arrival time.
- Each travel could be delayed, so the actual departure/arrival time is later or
before the scheduled time. This delayed time is what I want to study. - The time is discretized in minutes.
- The source of delays could be whatever (There is not any kind of independence).
For example:
Truck 1 departs from point A to B. Truck 2 is waiting for truck 1 in point B. Normally there is a buffer time in order to avoid that the delay of A affects the departure of B. Once truck 1 arrives B, truck 2 need some minutes for receive the merchandise of truck 1. On other hand, truck 1 will return to A with more merchandise, so it will need time for be charged with merchandise. The buffer time is fixed but not the same for different scenarios, it could change a lot (sometimes too much, sometimes too few).
On other hand, a truck will try to start always at time (not before, not later). It means that if truck 1 arrives before, track 2 will not try to go ahead before scheduled (although it happens sometimes).
During the travel could happen whatever. A truck could go very fast and arrive sooner as expected, but it will try to arrive on time. A truck could be stuck in traffic jam and arrive too much as expected (if the delay exceeds too much it is considered an outlier). Other trucks travels could be affected for the interaction of truck 1 and track 2
I decided to separate positive values(delays) from the negative values(aheads). I fitted the histogram with an exponential distribution. The histogram and the exponential density distribution fitted of all trucks departures delay during a month looks like:
enter image description here
How could I face this problem? The exponential distribution is underestimatting the zero values. Is there any other distribution more left-side fitted?
I think this process is very general, therefore should there be something similar in other fields (reliability, queue, etc.), but I don't find anything on internet. Do anyone know where could I learn about this kind of problem?
statistics probability-distributions bayesian queueing-theory reliability
asked Mar 13 at 10:20
Sergio Marrero MarreroSergio Marrero Marrero
104
$endgroup$
add a comment |
$begingroup$
I am analyzing a process related with time delays produced in scheduled departures or arrivals of a distribution trucks. The process consists on:
The definition of the problem is:
- Each truck has associated some travels per day.
- Each travel has associated a departure time and a arrival time.
- Each travel could be delayed, so the actual departure/arrival time is later or
before the scheduled time. This delayed time is what I want to study. - The time is discretized in minutes.
- The source of delays could be whatever (There is not any kind of independence).
For example:
Truck 1 departs from point A to B. Truck 2 is waiting for truck 1 in point B. Normally there is a buffer time in order to avoid that the delay of A affects the departure of B. Once truck 1 arrives B, truck 2 need some minutes for receive the merchandise of truck 1. On other hand, truck 1 will return to A with more merchandise, so it will need time for be charged with merchandise. The buffer time is fixed but not the same for different scenarios, it could change a lot (sometimes too much, sometimes too few).
On other hand, a truck will try to start always at time (not before, not later). It means that if truck 1 arrives before, track 2 will not try to go ahead before scheduled (although it happens sometimes).
During the travel could happen whatever. A truck could go very fast and arrive sooner as expected, but it will try to arrive on time. A truck could be stuck in traffic jam and arrive too much as expected (if the delay exceeds too much it is considered an outlier). Other trucks travels could be affected for the interaction of truck 1 and track 2
I decided to separate positive values(delays) from the negative values(aheads). I fitted the histogram with an exponential distribution. The histogram and the exponential density distribution fitted of all trucks departures delay during a month looks like:
enter image description here
How could I face this problem? The exponential distribution is underestimatting the zero values. Is there any other distribution more left-side fitted?
I think this process is very general, therefore should there be something similar in other fields (reliability, queue, etc.), but I don't find anything on internet. Do anyone know where could I learn about this kind of problem?
statistics probability-distributions bayesian queueing-theory reliability
asked Mar 13 at 10:20
Sergio Marrero MarreroSergio Marrero Marrero
104
$endgroup$
$begingroup$
"departure/arrivals" doesn't tell anything significant about the distribution to take as model; you shall specify more about the "physical" model underlying the data produced.
$endgroup$
– G Cab
Mar 13 at 10:38
$begingroup$
Okay. Thanks. I edited the question trying to give more information. Let me know if it is enough.
$endgroup$
– Sergio Marrero Marrero
Mar 13 at 10:57
$begingroup$
Well, that's better. However the most significant information would be about the dependence of the delays. If I understand properly you are recording the delays of the travels made by a number of trucks, where each truck makes multiple travels. If so, is it reasonable to assume that if a truck delays the first travel, then that will probably delay also the departure of the second, etc. ? and are the behaviour of the trucks independent, or there might be common causes that make them delay ? etc.
$endgroup$
– G Cab
Mar 13 at 14:15
$begingroup$
I redited the question. Thank you for your interest and help.
$endgroup$
– Sergio Marrero Marrero
Mar 13 at 15:42
add a comment |
$begingroup$
I am analyzing a process related with time delays produced in scheduled departures or arrivals of a distribution trucks. The process consists on:
The definition of the problem is:
- Each truck has associated some travels per day.
- Each travel has associated a departure time and a arrival time.
- Each travel could be delayed, so the actual departure/arrival time is later or
before the scheduled time. This delayed time is what I want to study. - The time is discretized in minutes.
- The source of delays could be whatever (There is not any kind of independence).
For example:
Truck 1 departs from point A to B. Truck 2 is waiting for truck 1 in point B. Normally there is a buffer time in order to avoid that the delay of A affects the departure of B. Once truck 1 arrives B, truck 2 need some minutes for receive the merchandise of truck 1. On other hand, truck 1 will return to A with more merchandise, so it will need time for be charged with merchandise. The buffer time is fixed but not the same for different scenarios, it could change a lot (sometimes too much, sometimes too few).
On other hand, a truck will try to start always at time (not before, not later). It means that if truck 1 arrives before, track 2 will not try to go ahead before scheduled (although it happens sometimes).
During the travel could happen whatever. A truck could go very fast and arrive sooner as expected, but it will try to arrive on time. A truck could be stuck in traffic jam and arrive too much as expected (if the delay exceeds too much it is considered an outlier). Other trucks travels could be affected for the interaction of truck 1 and track 2
I decided to separate positive values(delays) from the negative values(aheads). I fitted the histogram with an exponential distribution. The histogram and the exponential density distribution fitted of all trucks departures delay during a month looks like:
enter image description here
How could I face this problem? The exponential distribution is underestimatting the zero values. Is there any other distribution more left-side fitted?
I think this process is very general, therefore should there be something similar in other fields (reliability, queue, etc.), but I don't find anything on internet. Do anyone know where could I learn about this kind of problem?
statistics probability-distributions bayesian queueing-theory reliability
asked Mar 13 at 10:20
Sergio Marrero MarreroSergio Marrero Marrero
104
$endgroup$
I am analyzing a process related with time delays produced in scheduled departures or arrivals of a distribution trucks. The process consists on:
The definition of the problem is:
- Each truck has associated some travels per day.
- Each travel has associated a departure time and a arrival time.
- Each travel could be delayed, so the actual departure/arrival time is later or
before the scheduled time. This delayed time is what I want to study. - The time is discretized in minutes.
- The source of delays could be whatever (There is not any kind of independence).
For example:
Truck 1 departs from point A to B. Truck 2 is waiting for truck 1 in point B. Normally there is a buffer time in order to avoid that the delay of A affects the departure of B. Once truck 1 arrives B, truck 2 need some minutes for receive the merchandise of truck 1. On other hand, truck 1 will return to A with more merchandise, so it will need time for be charged with merchandise. The buffer time is fixed but not the same for different scenarios, it could change a lot (sometimes too much, sometimes too few).
On other hand, a truck will try to start always at time (not before, not later). It means that if truck 1 arrives before, track 2 will not try to go ahead before scheduled (although it happens sometimes).
During the travel could happen whatever. A truck could go very fast and arrive sooner as expected, but it will try to arrive on time. A truck could be stuck in traffic jam and arrive too much as expected (if the delay exceeds too much it is considered an outlier). Other trucks travels could be affected for the interaction of truck 1 and track 2
I decided to separate positive values(delays) from the negative values(aheads). I fitted the histogram with an exponential distribution. The histogram and the exponential density distribution fitted of all trucks departures delay during a month looks like:
enter image description here
How could I face this problem? The exponential distribution is underestimatting the zero values. Is there any other distribution more left-side fitted?
I think this process is very general, therefore should there be something similar in other fields (reliability, queue, etc.), but I don't find anything on internet. Do anyone know where could I learn about this kind of problem?
statistics probability-distributions bayesian queueing-theory reliability
statistics probability-distributions bayesian queueing-theory reliability
asked Mar 13 at 10:20
Sergio Marrero MarreroSergio Marrero Marrero
104
asked Mar 13 at 10:20
Sergio Marrero MarreroSergio Marrero Marrero
104
edited Mar 13 at 15:41
Sergio Marrero Marrero
asked Mar 13 at 10:20
Sergio Marrero MarreroSergio Marrero Marrero
104
asked Mar 13 at 10:20
Sergio Marrero MarreroSergio Marrero Marrero
104
asked Mar 13 at 10:20
Sergio Marrero MarreroSergio Marrero Marrero
104
104
$begingroup$
"departure/arrivals" doesn't tell anything significant about the distribution to take as model; you shall specify more about the "physical" model underlying the data produced.
$endgroup$
– G Cab
Mar 13 at 10:38
$begingroup$
Okay. Thanks. I edited the question trying to give more information. Let me know if it is enough.
$endgroup$
– Sergio Marrero Marrero
Mar 13 at 10:57
$begingroup$
Well, that's better. However the most significant information would be about the dependence of the delays. If I understand properly you are recording the delays of the travels made by a number of trucks, where each truck makes multiple travels. If so, is it reasonable to assume that if a truck delays the first travel, then that will probably delay also the departure of the second, etc. ? and are the behaviour of the trucks independent, or there might be common causes that make them delay ? etc.
$endgroup$
– G Cab
Mar 13 at 14:15
$begingroup$
I redited the question. Thank you for your interest and help.
$endgroup$
– Sergio Marrero Marrero
Mar 13 at 15:42
add a comment |
$begingroup$
"departure/arrivals" doesn't tell anything significant about the distribution to take as model; you shall specify more about the "physical" model underlying the data produced.
$endgroup$
– G Cab
Mar 13 at 10:38
$begingroup$
Okay. Thanks. I edited the question trying to give more information. Let me know if it is enough.
$endgroup$
– Sergio Marrero Marrero
Mar 13 at 10:57
$begingroup$
Well, that's better. However the most significant information would be about the dependence of the delays. If I understand properly you are recording the delays of the travels made by a number of trucks, where each truck makes multiple travels. If so, is it reasonable to assume that if a truck delays the first travel, then that will probably delay also the departure of the second, etc. ? and are the behaviour of the trucks independent, or there might be common causes that make them delay ? etc.
$endgroup$
– G Cab
Mar 13 at 14:15
$begingroup$
I redited the question. Thank you for your interest and help.
$endgroup$
– Sergio Marrero Marrero
Mar 13 at 15:42
$begingroup$
"departure/arrivals" doesn't tell anything significant about the distribution to take as model; you shall specify more about the "physical" model underlying the data produced.
$endgroup$
– G Cab
Mar 13 at 10:38
$begingroup$
"departure/arrivals" doesn't tell anything significant about the distribution to take as model; you shall specify more about the "physical" model underlying the data produced.
$endgroup$
– G Cab
Mar 13 at 10:38
$begingroup$
Okay. Thanks. I edited the question trying to give more information. Let me know if it is enough.
$endgroup$
– Sergio Marrero Marrero
Mar 13 at 10:57
$begingroup$
Okay. Thanks. I edited the question trying to give more information. Let me know if it is enough.
$endgroup$
– Sergio Marrero Marrero
Mar 13 at 10:57
$begingroup$
Well, that's better. However the most significant information would be about the dependence of the delays. If I understand properly you are recording the delays of the travels made by a number of trucks, where each truck makes multiple travels. If so, is it reasonable to assume that if a truck delays the first travel, then that will probably delay also the departure of the second, etc. ? and are the behaviour of the trucks independent, or there might be common causes that make them delay ? etc.
$endgroup$
– G Cab
Mar 13 at 14:15
$begingroup$
Well, that's better. However the most significant information would be about the dependence of the delays. If I understand properly you are recording the delays of the travels made by a number of trucks, where each truck makes multiple travels. If so, is it reasonable to assume that if a truck delays the first travel, then that will probably delay also the departure of the second, etc. ? and are the behaviour of the trucks independent, or there might be common causes that make them delay ? etc.
$endgroup$
– G Cab
Mar 13 at 14:15
$begingroup$
I redited the question. Thank you for your interest and help.
$endgroup$
– Sergio Marrero Marrero
Mar 13 at 15:42
$begingroup$
I redited the question. Thank you for your interest and help.
$endgroup$
– Sergio Marrero Marrero
Mar 13 at 15:42
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
From what I could understand from your description (and from my engineering experience)
we can regard the time taken to travel a fixed distance approximately as a Poisson point process.
That is, they are normally distributed evenly according to the average speed, and deviate
from that with a Poisson distribution.
The arrival times of the first truck will then follows a Erlang distribution.
However, not considering early arrivals and given a sufficient buffer time, we can well approximate
the tail of the Erlang distribution, past the mean plus buffer time, as an exponential distribution.
The trucks following the first will add their delays approximately as the sum of exponential variables.
The delays accumulated by a chain of sequential trucks will then roughly follow a Gamma
distribution, which is the continuous analogous of a Erlang distribution.
This shows a parallel with the process of a telephone line, where the calls arrives as a Poisson process,
and the duration of the calls are exponentially distributed.
We can assimilate the trucks to calls arriving with a strict dispersion, and with a delay corresponding to the call
duration.
And there is the similar problem that if the delay (call duration) overpass a given threshold, it will block the following
truck (call).
Therefore a Gamma / Exponential distribution looks to be a suitable model to fit your data.
answered Mar 13 at 23:43
G CabG Cab
20.3k31341
$endgroup$
$begingroup$
Thank you very much for such elaborated answer. I will study in deep along this week. Another question have arosen to my mind, and it is related with the discretization of the time. I assumed that the process was defined over continuos time, but actually I have minute-by-minute records, so I don't know if I should assume continuous or discrete time. If I assume continous I should think on pmf (probability mass function) and for discret on pdf (probabilty density function). Thanks on advanced!
$endgroup$
– Sergio Marrero Marrero
21 hours ago
$begingroup$
@SergioMarreroMarrero: hope I caught the core of your problem and structure of data. As for continuous/discrete, well in any case you have a discrete frequency table from your data to match with the model, and the exponential pdf treats the time as continuous. Concerning the parameter for the number of variables in series, if the number of "chained" trucks is low and almost constant, you will opt for matching that, otherwise you can approx with a continuous one.
$endgroup$
– G Cab
19 hours ago
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3146378%2fwhat-density-distribution-could-be-used-for-time-delays%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
From what I could understand from your description (and from my engineering experience)
we can regard the time taken to travel a fixed distance approximately as a Poisson point process.
That is, they are normally distributed evenly according to the average speed, and deviate
from that with a Poisson distribution.
The arrival times of the first truck will then follows a Erlang distribution.
However, not considering early arrivals and given a sufficient buffer time, we can well approximate
the tail of the Erlang distribution, past the mean plus buffer time, as an exponential distribution.
The trucks following the first will add their delays approximately as the sum of exponential variables.
The delays accumulated by a chain of sequential trucks will then roughly follow a Gamma
distribution, which is the continuous analogous of a Erlang distribution.
This shows a parallel with the process of a telephone line, where the calls arrives as a Poisson process,
and the duration of the calls are exponentially distributed.
We can assimilate the trucks to calls arriving with a strict dispersion, and with a delay corresponding to the call
duration.
And there is the similar problem that if the delay (call duration) overpass a given threshold, it will block the following
truck (call).
Therefore a Gamma / Exponential distribution looks to be a suitable model to fit your data.
answered Mar 13 at 23:43
G CabG Cab
20.3k31341
$endgroup$
$begingroup$
Thank you very much for such elaborated answer. I will study in deep along this week. Another question have arosen to my mind, and it is related with the discretization of the time. I assumed that the process was defined over continuos time, but actually I have minute-by-minute records, so I don't know if I should assume continuous or discrete time. If I assume continous I should think on pmf (probability mass function) and for discret on pdf (probabilty density function). Thanks on advanced!
$endgroup$
– Sergio Marrero Marrero
21 hours ago
$begingroup$
@SergioMarreroMarrero: hope I caught the core of your problem and structure of data. As for continuous/discrete, well in any case you have a discrete frequency table from your data to match with the model, and the exponential pdf treats the time as continuous. Concerning the parameter for the number of variables in series, if the number of "chained" trucks is low and almost constant, you will opt for matching that, otherwise you can approx with a continuous one.
$endgroup$
– G Cab
19 hours ago
add a comment |
$begingroup$
From what I could understand from your description (and from my engineering experience)
we can regard the time taken to travel a fixed distance approximately as a Poisson point process.
That is, they are normally distributed evenly according to the average speed, and deviate
from that with a Poisson distribution.
The arrival times of the first truck will then follows a Erlang distribution.
However, not considering early arrivals and given a sufficient buffer time, we can well approximate
the tail of the Erlang distribution, past the mean plus buffer time, as an exponential distribution.
The trucks following the first will add their delays approximately as the sum of exponential variables.
The delays accumulated by a chain of sequential trucks will then roughly follow a Gamma
distribution, which is the continuous analogous of a Erlang distribution.
This shows a parallel with the process of a telephone line, where the calls arrives as a Poisson process,
and the duration of the calls are exponentially distributed.
We can assimilate the trucks to calls arriving with a strict dispersion, and with a delay corresponding to the call
duration.
And there is the similar problem that if the delay (call duration) overpass a given threshold, it will block the following
truck (call).
Therefore a Gamma / Exponential distribution looks to be a suitable model to fit your data.
answered Mar 13 at 23:43
G CabG Cab
20.3k31341
$endgroup$
$begingroup$
Thank you very much for such elaborated answer. I will study in deep along this week. Another question have arosen to my mind, and it is related with the discretization of the time. I assumed that the process was defined over continuos time, but actually I have minute-by-minute records, so I don't know if I should assume continuous or discrete time. If I assume continous I should think on pmf (probability mass function) and for discret on pdf (probabilty density function). Thanks on advanced!
$endgroup$
– Sergio Marrero Marrero
21 hours ago
$begingroup$
@SergioMarreroMarrero: hope I caught the core of your problem and structure of data. As for continuous/discrete, well in any case you have a discrete frequency table from your data to match with the model, and the exponential pdf treats the time as continuous. Concerning the parameter for the number of variables in series, if the number of "chained" trucks is low and almost constant, you will opt for matching that, otherwise you can approx with a continuous one.
$endgroup$
– G Cab
19 hours ago
add a comment |
$begingroup$
From what I could understand from your description (and from my engineering experience)
we can regard the time taken to travel a fixed distance approximately as a Poisson point process.
That is, they are normally distributed evenly according to the average speed, and deviate
from that with a Poisson distribution.
The arrival times of the first truck will then follows a Erlang distribution.
However, not considering early arrivals and given a sufficient buffer time, we can well approximate
the tail of the Erlang distribution, past the mean plus buffer time, as an exponential distribution.
The trucks following the first will add their delays approximately as the sum of exponential variables.
The delays accumulated by a chain of sequential trucks will then roughly follow a Gamma
distribution, which is the continuous analogous of a Erlang distribution.
This shows a parallel with the process of a telephone line, where the calls arrives as a Poisson process,
and the duration of the calls are exponentially distributed.
We can assimilate the trucks to calls arriving with a strict dispersion, and with a delay corresponding to the call
duration.
And there is the similar problem that if the delay (call duration) overpass a given threshold, it will block the following
truck (call).
Therefore a Gamma / Exponential distribution looks to be a suitable model to fit your data.
answered Mar 13 at 23:43
G CabG Cab
20.3k31341
$endgroup$
From what I could understand from your description (and from my engineering experience)
we can regard the time taken to travel a fixed distance approximately as a Poisson point process.
That is, they are normally distributed evenly according to the average speed, and deviate
from that with a Poisson distribution.
The arrival times of the first truck will then follows a Erlang distribution.
However, not considering early arrivals and given a sufficient buffer time, we can well approximate
the tail of the Erlang distribution, past the mean plus buffer time, as an exponential distribution.
The trucks following the first will add their delays approximately as the sum of exponential variables.
The delays accumulated by a chain of sequential trucks will then roughly follow a Gamma
distribution, which is the continuous analogous of a Erlang distribution.
This shows a parallel with the process of a telephone line, where the calls arrives as a Poisson process,
and the duration of the calls are exponentially distributed.
We can assimilate the trucks to calls arriving with a strict dispersion, and with a delay corresponding to the call
duration.
And there is the similar problem that if the delay (call duration) overpass a given threshold, it will block the following
truck (call).
Therefore a Gamma / Exponential distribution looks to be a suitable model to fit your data.
answered Mar 13 at 23:43
G CabG Cab
20.3k31341
answered Mar 13 at 23:43
G CabG Cab
20.3k31341
answered Mar 13 at 23:43
G CabG Cab
20.3k31341
answered Mar 13 at 23:43
G CabG Cab
20.3k31341
20.3k31341
$begingroup$
Thank you very much for such elaborated answer. I will study in deep along this week. Another question have arosen to my mind, and it is related with the discretization of the time. I assumed that the process was defined over continuos time, but actually I have minute-by-minute records, so I don't know if I should assume continuous or discrete time. If I assume continous I should think on pmf (probability mass function) and for discret on pdf (probabilty density function). Thanks on advanced!
$endgroup$
– Sergio Marrero Marrero
21 hours ago
$begingroup$
@SergioMarreroMarrero: hope I caught the core of your problem and structure of data. As for continuous/discrete, well in any case you have a discrete frequency table from your data to match with the model, and the exponential pdf treats the time as continuous. Concerning the parameter for the number of variables in series, if the number of "chained" trucks is low and almost constant, you will opt for matching that, otherwise you can approx with a continuous one.
$endgroup$
– G Cab
19 hours ago
add a comment |
$begingroup$
Thank you very much for such elaborated answer. I will study in deep along this week. Another question have arosen to my mind, and it is related with the discretization of the time. I assumed that the process was defined over continuos time, but actually I have minute-by-minute records, so I don't know if I should assume continuous or discrete time. If I assume continous I should think on pmf (probability mass function) and for discret on pdf (probabilty density function). Thanks on advanced!
$endgroup$
– Sergio Marrero Marrero
21 hours ago
$begingroup$
@SergioMarreroMarrero: hope I caught the core of your problem and structure of data. As for continuous/discrete, well in any case you have a discrete frequency table from your data to match with the model, and the exponential pdf treats the time as continuous. Concerning the parameter for the number of variables in series, if the number of "chained" trucks is low and almost constant, you will opt for matching that, otherwise you can approx with a continuous one.
$endgroup$
– G Cab
19 hours ago
$begingroup$
Thank you very much for such elaborated answer. I will study in deep along this week. Another question have arosen to my mind, and it is related with the discretization of the time. I assumed that the process was defined over continuos time, but actually I have minute-by-minute records, so I don't know if I should assume continuous or discrete time. If I assume continous I should think on pmf (probability mass function) and for discret on pdf (probabilty density function). Thanks on advanced!
$endgroup$
– Sergio Marrero Marrero
21 hours ago
$begingroup$
Thank you very much for such elaborated answer. I will study in deep along this week. Another question have arosen to my mind, and it is related with the discretization of the time. I assumed that the process was defined over continuos time, but actually I have minute-by-minute records, so I don't know if I should assume continuous or discrete time. If I assume continous I should think on pmf (probability mass function) and for discret on pdf (probabilty density function). Thanks on advanced!
$endgroup$
– Sergio Marrero Marrero
21 hours ago
$begingroup$
@SergioMarreroMarrero: hope I caught the core of your problem and structure of data. As for continuous/discrete, well in any case you have a discrete frequency table from your data to match with the model, and the exponential pdf treats the time as continuous. Concerning the parameter for the number of variables in series, if the number of "chained" trucks is low and almost constant, you will opt for matching that, otherwise you can approx with a continuous one.
$endgroup$
– G Cab
19 hours ago
$begingroup$
@SergioMarreroMarrero: hope I caught the core of your problem and structure of data. As for continuous/discrete, well in any case you have a discrete frequency table from your data to match with the model, and the exponential pdf treats the time as continuous. Concerning the parameter for the number of variables in series, if the number of "chained" trucks is low and almost constant, you will opt for matching that, otherwise you can approx with a continuous one.
$endgroup$
– G Cab
19 hours ago
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3146378%2fwhat-density-distribution-could-be-used-for-time-delays%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
"departure/arrivals" doesn't tell anything significant about the distribution to take as model; you shall specify more about the "physical" model underlying the data produced.
$endgroup$
– G Cab
Mar 13 at 10:38
$begingroup$
Okay. Thanks. I edited the question trying to give more information. Let me know if it is enough.
$endgroup$
– Sergio Marrero Marrero
Mar 13 at 10:57
$begingroup$
Well, that's better. However the most significant information would be about the dependence of the delays. If I understand properly you are recording the delays of the travels made by a number of trucks, where each truck makes multiple travels. If so, is it reasonable to assume that if a truck delays the first travel, then that will probably delay also the departure of the second, etc. ? and are the behaviour of the trucks independent, or there might be common causes that make them delay ? etc.
$endgroup$
– G Cab
Mar 13 at 14:15
$begingroup$
I redited the question. Thank you for your interest and help.
$endgroup$
– Sergio Marrero Marrero
Mar 13 at 15:42