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Birthday Problem in Continuous time

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I encountered a practical problem and want a help (my Google skill fail T^T). Basically, this is a birthday problem in continuous space, is it hard to solve?

Problem Statement

A computer has N cores and there are M tasks within a day. Each core can only process one job at a time and each take T seconds to complete. And tasks can arrive at anytime (continuous time) with uniform probability.


Q1. Given N,M,T. What is the probability that the event of all cores being occupied (processing a task) happen during a day?

Q2. Given M,T. Find N such that P(not all cores being occupied) > 0.9, 0.99, 0.999, 0.9999, 0.99999?

If there are approximation methods, the background information is that T is small e.g. 0.1 second, and M is much larger 10^7.

probability birthday

share|cite|improve this question

edited Mar 15 at 5:48

Martin Sleziak

44.9k10122275

asked Jul 21 '18 at 3:35

Jack TangJack Tang

101

$endgroup$

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  These are results in queueing theory If you search on that you can find lots of information.
  $endgroup$
  – Ross Millikan
  Jul 21 '18 at 4:07
  

  
  
  
  


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  Tasks cannot "arrive at any time with uniform probability" because there is no way to assign probabilities to the set of all non-negative real numbers with uniform probability. So you need to consider some other probability distribution, such as an Exponential distribution.
  $endgroup$
  – awkward
  Jul 21 '18 at 12:17
  

  
  
  
  


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  @awkward I should clarify that I assume all M tasks must arrive within a day, so it is not unbounded.
  $endgroup$
  – Jack Tang
  Jul 22 '18 at 2:42
  

  
  
  
  


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  @RossMillikan Thanks! I borrowed a book from libbrary today. Queueing theory is exactly what I am looking for.
  $endgroup$
  – Jack Tang
  Jul 22 '18 at 7:02
  

  
  
  
  


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  This is a $G/D/1$ queue with a nontrivial interarrival distribution, so I suggest either using standard approximations from queueing theory or creating a simulation to estimate the results.
  $endgroup$
  – Math1000
  Jul 24 '18 at 1:43
  

  
  
  
  

add a comment | 

0

$begingroup$

I encountered a practical problem and want a help (my Google skill fail T^T). Basically, this is a birthday problem in continuous space, is it hard to solve?

Problem Statement

A computer has N cores and there are M tasks within a day. Each core can only process one job at a time and each take T seconds to complete. And tasks can arrive at anytime (continuous time) with uniform probability.


Q1. Given N,M,T. What is the probability that the event of all cores being occupied (processing a task) happen during a day?

Q2. Given M,T. Find N such that P(not all cores being occupied) > 0.9, 0.99, 0.999, 0.9999, 0.99999?

If there are approximation methods, the background information is that T is small e.g. 0.1 second, and M is much larger 10^7.

probability birthday

share|cite|improve this question

edited Mar 15 at 5:48

Martin Sleziak

44.9k10122275

asked Jul 21 '18 at 3:35

Jack TangJack Tang

101

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  These are results in queueing theory If you search on that you can find lots of information.
  $endgroup$
  – Ross Millikan
  Jul 21 '18 at 4:07
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Tasks cannot "arrive at any time with uniform probability" because there is no way to assign probabilities to the set of all non-negative real numbers with uniform probability. So you need to consider some other probability distribution, such as an Exponential distribution.
  $endgroup$
  – awkward
  Jul 21 '18 at 12:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @awkward I should clarify that I assume all M tasks must arrive within a day, so it is not unbounded.
  $endgroup$
  – Jack Tang
  Jul 22 '18 at 2:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @RossMillikan Thanks! I borrowed a book from libbrary today. Queueing theory is exactly what I am looking for.
  $endgroup$
  – Jack Tang
  Jul 22 '18 at 7:02
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This is a $G/D/1$ queue with a nontrivial interarrival distribution, so I suggest either using standard approximations from queueing theory or creating a simulation to estimate the results.
  $endgroup$
  – Math1000
  Jul 24 '18 at 1:43
  

  
  
  
  

add a comment | 

$begingroup$

I encountered a practical problem and want a help (my Google skill fail T^T). Basically, this is a birthday problem in continuous space, is it hard to solve?

Problem Statement

A computer has N cores and there are M tasks within a day. Each core can only process one job at a time and each take T seconds to complete. And tasks can arrive at anytime (continuous time) with uniform probability.


Q1. Given N,M,T. What is the probability that the event of all cores being occupied (processing a task) happen during a day?

Q2. Given M,T. Find N such that P(not all cores being occupied) > 0.9, 0.99, 0.999, 0.9999, 0.99999?

If there are approximation methods, the background information is that T is small e.g. 0.1 second, and M is much larger 10^7.

probability birthday

share|cite|improve this question

edited Mar 15 at 5:48

Martin Sleziak

44.9k10122275

asked Jul 21 '18 at 3:35

Jack TangJack Tang

101

$endgroup$

share|cite|improve this question

edited Mar 15 at 5:48

Martin Sleziak

44.9k10122275

asked Jul 21 '18 at 3:35

Jack TangJack Tang

101

share|cite|improve this question

edited Mar 15 at 5:48

Martin Sleziak

44.9k10122275

asked Jul 21 '18 at 3:35

Jack TangJack Tang

101

edited Mar 15 at 5:48

Martin Sleziak

44.9k10122275

edited Mar 15 at 5:48

Martin Sleziak

44.9k10122275

asked Jul 21 '18 at 3:35

Jack TangJack Tang

101

asked Jul 21 '18 at 3:35

Jack TangJack Tang

101