represents the distribution of times for the Markov process mproc to pass from the initial state to final states f for the first time.
- FirstPassageTimeDistribution is also known as first hitting time.
- The Markov process mproc can be a DiscreteMarkovProcess or ContinuousMarkovProcess.
- The probability for time t in a FirstPassageTimeDistribution is equivalent to Probability[x[t]∈f∧∀τ,0<τ<tx[τ]∉fxi,xmproc], where i is the initial state.
- If mproc is already in the target state, FirstPassageTimeDistribution gives a distribution for the mean recurrence time.
- If the chain is absorbing and the target states are non-absorbing, FirstPassageTimeDistribution gives a distribution conditional on reaching the target states.
- FirstPassageTimeDistribution represents a discrete phase-type distribution for discrete-time Markov processes and a continuous phase-type distribution for continuous-time Markov processes.
- FirstPassageTimeDistribution can be used with such functions as Mean, Quantile, PDF, and RandomVariate.
Examplesopen allclose all
Basic Examples (1)
A taxi is located either at the airport or in the city. From the city, the next trip is to the airport with probability 1/4, or to somewhere else in the city with probability 3/4. From the airport, the next trip is always to the city. Model the taxi, using a discrete Markov process, with state 1 representing the city and state 2 representing the airport, starting at the airport:
A gambler, starting with 3 units, places a 1-unit bet at each step, with a winning probability of 0.4 and a goal of winning 7 units before stopping. Find the expected playing time until the gambler achieves his goal or goes broke. The gambling process can be modeled as a discrete Markov process, where state represents the gambler having units:
The Hubble space telescope carries six gyroscopes, with a minimum of three required for full accuracy. The operating times of the gyroscopes are independent and exponentially distributed with failure rate . If a fourth gyroscope fails, the telescope goes into sleep mode, in which further observations are suspended. It requires an exponential time with mean to put the telescope into sleep mode, after which the base station on Earth receives a sleep signal and a shuttle mission is prepared. It takes an exponential time with mean before the repair crew arrives at the telescope and has repaired the stabilizing unit with the gyroscopes. In the meantime, the other two gyroscopes may fail. If the last gyroscope fails, the telescope will crash. Suppose , , and , all with units of inverse years:
Properties & Relations (4)
Autosimplification to ExponentialDistribution or other phase-type distributions:
Wolfram Research (2012), FirstPassageTimeDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/FirstPassageTimeDistribution.html.
Wolfram Language. 2012. "FirstPassageTimeDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FirstPassageTimeDistribution.html.
Wolfram Language. (2012). FirstPassageTimeDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FirstPassageTimeDistribution.html