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RenewalProcess[rdist]

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

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Continuous Interarrival Distributions  
Discrete Interarrival Distributions  
Parameter Estimation  
Applications  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
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RenewalProcess

RenewalProcess[rdist]

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)

Simulate a renewal process:

Mean function:

Scope  (6)

Continuous Interarrival Distributions  (2)

Renewal process with interarrival times that follow an Erlang distribution:

Slice distribution functions:

Compute the probability of an event:

Simulate the process:

Renewal process with interarrival times that follow a gamma distribution:

Slice distribution functions:

Compute the probability of an event:

Simulate the process:

Discrete Interarrival Distributions  (2)

Renewal process with interarrival times that follow a Pascal distribution:

Slice distribution functions:

Compute the probability of an event:

Simulate the process:

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

Slice distribution functions:

Compute the probability of an event:

Simulate the process:

Parameter Estimation  (2)

Process parameter estimation:

Estimate the distribution parameters from sample data:

Approximate value of the mean:

Applications  (1)

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:

Mean number of arrivals during the first 10 milliseconds:

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

Properties & Relations  (2)

RenewalProcess is a jump process:

RenewalProcess is not weakly stationary for any distribution:

Possible Issues  (1)

Closed forms are not available for most properties:

Obtain approximate values using inexact input or simulation:

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.

CMS

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

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: 01-May-2026]}

BibLaTeX

@online{reference.wolfram_2025_renewalprocess, organization={Wolfram Research}, title={RenewalProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/RenewalProcess.html}, note=[Accessed: 01-May-2026]}