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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

open allclose all

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

Simulate a renewal process:

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

Mean function:

In[1]:=1
✖
https://wolfram.com/xid/0fq7kxbq62jzm-dexx8w
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:

In[1]:=1
✖
https://wolfram.com/xid/0fq7kxbq62jzm-b727q5

Slice distribution functions:

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

Compute the probability of an event:

In[3]:=3
✖
https://wolfram.com/xid/0fq7kxbq62jzm-eije4n
Out[3]=3

Simulate the process:

In[7]:=7
✖
https://wolfram.com/xid/0fq7kxbq62jzm-cwx46h
Out[7]=7

Renewal process with interarrival times that follow a gamma distribution:

In[1]:=1
✖
https://wolfram.com/xid/0fq7kxbq62jzm-dt17zo

Slice distribution functions:

In[2]:=2
✖
https://wolfram.com/xid/0fq7kxbq62jzm-bx3mdt
Out[2]=2

Compute the probability of an event:

In[3]:=3
✖
https://wolfram.com/xid/0fq7kxbq62jzm-edu2v8
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0fq7kxbq62jzm-boqea0
Out[4]=4

Simulate the process:

In[5]:=5
✖
https://wolfram.com/xid/0fq7kxbq62jzm-g1cl2
Out[5]=5

Discrete Interarrival Distributions  (2)

Renewal process with interarrival times that follow a Pascal distribution:

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

Slice distribution functions:

In[37]:=37
✖
https://wolfram.com/xid/0fq7kxbq62jzm-b1x8wx
Out[37]=37

Compute the probability of an event:

In[5]:=5
✖
https://wolfram.com/xid/0fq7kxbq62jzm-h2gmnk
Out[5]=5

Simulate the process:

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

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

In[1]:=1
✖
https://wolfram.com/xid/0fq7kxbq62jzm-cmo9yx

Slice distribution functions:

In[2]:=2
✖
https://wolfram.com/xid/0fq7kxbq62jzm-bxs9jt
Out[2]=2

Compute the probability of an event:

In[3]:=3
✖
https://wolfram.com/xid/0fq7kxbq62jzm-owyr92
Out[3]=3

Simulate the process:

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

Parameter Estimation  (2)

Process parameter estimation:

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

Estimate the distribution parameters from sample data:

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

Approximate value of the mean:

In[1]:=1
✖
https://wolfram.com/xid/0fq7kxbq62jzm-ux8vdm
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0fq7kxbq62jzm-9reh83
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:

In[1]:=1
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https://wolfram.com/xid/0fq7kxbq62jzm-lxke66
In[2]:=2
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https://wolfram.com/xid/0fq7kxbq62jzm-ot7r9n
Out[2]=2

Mean number of arrivals during the first 10 milliseconds:

In[3]:=3
✖
https://wolfram.com/xid/0fq7kxbq62jzm-gdkb1w
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0fq7kxbq62jzm-bkp20x
Out[4]=4

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

RenewalProcess is a jump process:

In[6]:=6
✖
https://wolfram.com/xid/0fq7kxbq62jzm-fm6uww
In[9]:=9
✖
https://wolfram.com/xid/0fq7kxbq62jzm-t7cms7
Out[9]=9

RenewalProcess is not weakly stationary for any distribution:

In[1]:=1
✖
https://wolfram.com/xid/0fq7kxbq62jzm-dfnpbi
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0fq7kxbq62jzm-ey1jpy
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

Closed forms are not available for most properties:

In[1]:=1
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https://wolfram.com/xid/0fq7kxbq62jzm-dxx16r
In[2]:=2
✖
https://wolfram.com/xid/0fq7kxbq62jzm-k9rkcj
Out[2]=2

Obtain approximate values using inexact input or simulation:

In[3]:=3
✖
https://wolfram.com/xid/0fq7kxbq62jzm-ckrna
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0fq7kxbq62jzm-b4z7e
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.

✖
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.

✖
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

✖
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: 06-April-2025 ]}

✖
@misc{reference.wolfram_2025_renewalprocess, author="Wolfram Research", title="{RenewalProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/RenewalProcess.html}", note=[Accessed: 06-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: 06-April-2025 ]}

✖
@online{reference.wolfram_2025_renewalprocess, organization={Wolfram Research}, title={RenewalProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/RenewalProcess.html}, note=[Accessed: 06-April-2025 ]}