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RenewalProcess
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RenewalProcess

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represents a renewal process with interarrival times distributed according to rdist.

Details

  • RenewalProcess is a continuous- or discrete-time and discrete-state process.
  • RenewalProcess is a discrete-state and continuous-time or discrete-time process depending on rdist.
  • The state is the number of events in the interval 0 to and .
  • The distribution rdist can be any continuous or discrete distribution with positive domain.
  • RenewalProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.

Examples

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Basic Examples  (2)Summary of the most common use cases

Simulate a renewal process:

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In[1]:=1
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https://wolfram.com/xid/0fq7kxbq62jzm-eop4dp
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Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0fq7kxbq62jzm-viw31c
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Out[2]=2

Mean function:

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In[1]:=1
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https://wolfram.com/xid/0fq7kxbq62jzm-dexx8w
Direct link to example
Out[1]=1

Scope  (6)Survey of the scope of standard use cases

Continuous Interarrival Distributions  (2)

Renewal process with interarrival times that follow an Erlang distribution:

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In[1]:=1
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https://wolfram.com/xid/0fq7kxbq62jzm-b727q5
Direct link to example

Slice distribution functions:

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In[2]:=2
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https://wolfram.com/xid/0fq7kxbq62jzm-ozs4nd
Direct link to example
Out[2]=2

Compute the probability of an event:

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In[3]:=3
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https://wolfram.com/xid/0fq7kxbq62jzm-eije4n
Direct link to example
Out[3]=3

Simulate the process:

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In[7]:=7
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https://wolfram.com/xid/0fq7kxbq62jzm-cwx46h
Direct link to example
Out[7]=7

Renewal process with interarrival times that follow a gamma distribution:

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In[1]:=1
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https://wolfram.com/xid/0fq7kxbq62jzm-dt17zo
Direct link to example

Slice distribution functions:

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In[2]:=2
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https://wolfram.com/xid/0fq7kxbq62jzm-bx3mdt
Direct link to example
Out[2]=2

Compute the probability of an event:

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In[3]:=3
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https://wolfram.com/xid/0fq7kxbq62jzm-edu2v8
Direct link to example
Out[3]=3
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In[4]:=4
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https://wolfram.com/xid/0fq7kxbq62jzm-boqea0
Direct link to example
Out[4]=4

Simulate the process:

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In[5]:=5
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https://wolfram.com/xid/0fq7kxbq62jzm-g1cl2
Direct link to example
Out[5]=5

Discrete Interarrival Distributions  (2)

Renewal process with interarrival times that follow a Pascal distribution:

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In[2]:=2
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https://wolfram.com/xid/0fq7kxbq62jzm-8b5vy
Direct link to example

Slice distribution functions:

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In[37]:=37
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https://wolfram.com/xid/0fq7kxbq62jzm-b1x8wx
Direct link to example
Out[37]=37

Compute the probability of an event:

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In[5]:=5
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https://wolfram.com/xid/0fq7kxbq62jzm-h2gmnk
Direct link to example
Out[5]=5

Simulate the process:

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In[6]:=6
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https://wolfram.com/xid/0fq7kxbq62jzm-2nmik
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Out[6]=6

Renewal process with interarrival times that follow a Borel–Tanner distribution:

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In[1]:=1
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https://wolfram.com/xid/0fq7kxbq62jzm-cmo9yx
Direct link to example

Slice distribution functions:

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In[2]:=2
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https://wolfram.com/xid/0fq7kxbq62jzm-bxs9jt
Direct link to example
Out[2]=2

Compute the probability of an event:

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In[3]:=3
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https://wolfram.com/xid/0fq7kxbq62jzm-owyr92
Direct link to example
Out[3]=3

Simulate the process:

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In[4]:=4
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https://wolfram.com/xid/0fq7kxbq62jzm-ggneeu
Direct link to example
Out[4]=4

Parameter Estimation  (2)

Process parameter estimation:

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In[1]:=1
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https://wolfram.com/xid/0fq7kxbq62jzm-45b7g2
Direct link to example

Estimate the distribution parameters from sample data:

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In[2]:=2
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https://wolfram.com/xid/0fq7kxbq62jzm-epi747
Direct link to example
Out[2]=2

Approximate value of the mean:

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In[1]:=1
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https://wolfram.com/xid/0fq7kxbq62jzm-ux8vdm
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0fq7kxbq62jzm-9reh83
Direct link to example
Out[2]=2

Applications  (1)Sample problems that can be solved with this function

Messages arrive at a communication line according to a two-phase hyperexponential distribution with phase probabilities 0.4 and 0.6. The average arrival times for the two phases are 4.8 milliseconds and 0.8 milliseconds, respectively. Simulate the process for 100 milliseconds:

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In[1]:=1
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https://wolfram.com/xid/0fq7kxbq62jzm-lxke66
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0fq7kxbq62jzm-ot7r9n
Direct link to example
Out[2]=2

Mean number of arrivals during the first 10 milliseconds:

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In[3]:=3
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https://wolfram.com/xid/0fq7kxbq62jzm-gdkb1w
Direct link to example
Out[3]=3

Compare with the value obtained from simulation of the time slice distribution:

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In[4]:=4
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https://wolfram.com/xid/0fq7kxbq62jzm-bkp20x
Direct link to example
Out[4]=4

Properties & Relations  (2)Properties of the function, and connections to other functions

RenewalProcess is a jump process:

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In[6]:=6
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https://wolfram.com/xid/0fq7kxbq62jzm-fm6uww
Direct link to example
Copy to clipboard.
In[9]:=9
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https://wolfram.com/xid/0fq7kxbq62jzm-t7cms7
Direct link to example
Out[9]=9

RenewalProcess is not weakly stationary for any distribution:

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In[1]:=1
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https://wolfram.com/xid/0fq7kxbq62jzm-dfnpbi
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0fq7kxbq62jzm-ey1jpy
Direct link to example
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

Closed forms are not available for most properties:

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In[1]:=1
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https://wolfram.com/xid/0fq7kxbq62jzm-dxx16r
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0fq7kxbq62jzm-k9rkcj
Direct link to example
Out[2]=2

Obtain approximate values using inexact input or simulation:

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In[3]:=3
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https://wolfram.com/xid/0fq7kxbq62jzm-ckrna
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
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https://wolfram.com/xid/0fq7kxbq62jzm-b4z7e
Direct link to example
Out[4]=4
Wolfram Research (2012), RenewalProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/RenewalProcess.html.
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Wolfram Research (2012), RenewalProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/RenewalProcess.html.

Text

Wolfram Research (2012), RenewalProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/RenewalProcess.html.

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Wolfram Research (2012), RenewalProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/RenewalProcess.html.

CMS

Wolfram Language. 2012. "RenewalProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RenewalProcess.html.

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Wolfram Language. 2012. "RenewalProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RenewalProcess.html.

APA

Wolfram Language. (2012). RenewalProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RenewalProcess.html

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Wolfram Language. (2012). RenewalProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RenewalProcess.html

BibTeX

@misc{reference.wolfram_2025_renewalprocess, author="Wolfram Research", title="{RenewalProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/RenewalProcess.html}", note=[Accessed: 04-April-2025 ]}

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@misc{reference.wolfram_2025_renewalprocess, author="Wolfram Research", title="{RenewalProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/RenewalProcess.html}", note=[Accessed: 04-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_renewalprocess, organization={Wolfram Research}, title={RenewalProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/RenewalProcess.html}, note=[Accessed: 04-April-2025 ]}

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@online{reference.wolfram_2025_renewalprocess, organization={Wolfram Research}, title={RenewalProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/RenewalProcess.html}, note=[Accessed: 04-April-2025 ]}