Stochastic Process → Poisson Process → Definition → Examples → Arrival, Inter-Arrival Times → Memoryless Property

**Poisson Process: Memoryless Property**

Memoryless property of the Poisson process can take any of the following three shapes

It simply says, time and results should not be overlapped. For example, if the results of the first hour is given then the first hour and its result should be subtracted in the subsequent counting. The following example will make the matter more clear.

**Example 1.** An office having opening timings from 9:00 to 5:00 receive fax messages according to a Poisson process at a mean rate of 10 per hour.

**(a)** Find the probability that the first message receive after 9:10.

**(b)** Given that no message arrived until 9:10. What is the probability that the first message receive after 9:15?

**(c)** Given that the second message received at 9:05. What is the probability that the third message arrive after 9:20?

**(d)** Given that the 50^{th} message received at 12:00. What is the probability that the 51^{st} message arrive after 1:00?

**Solution**

**(a)** Here λ = 10 and *t* = 10/60 = 1/6 hour

**(b)** Here the first 10 minutes should be subtracted from the second 15 minutes (according to the memoryless property). Hence, λ = 10 and *t* = 15 – 10 = 5 minutes = 1/12 hour

**(c)** The second message arrived at 9:05. So, *X*_{1}+*X*_{2} = 5/60 = 1/12. The time for the third message is 9:20. So, according to the memoryless property, the first five minutes should be subtracted from these 20 minutes 20/60 – 5/60 = 1/3 – 1/12 = 1/4.

**(d)** The 50^{th} message arrived at 12:00. So, *X*_{1}+*X*_{2}+….+*X*_{50} = 3 hour. The time for the 51^{st} message is 1:00 (after 4 hours). So, according to the memoryless property, the first three hours should be subtracted from these 4 hours 4 – 3 = 1.

**Example 2.** Cars pass through a certain location of a road at a rate of one every two minutes.

**(a)** What is the probability that exactly four cars pass the location in ten minutes?

**(b)** What is the probability that exactly ten cars pass the location in twenty minutes, given that in the first eight minutes three cars pass the location minutes?

**Solution**

**(a)** Here λ = 1 every two minutes = 0.5 per minute, *t* = 10 minutes and *n* = 4.

**(b)** Here λ = 1 every two minutes = 0.5 per minute, *t* = 20 minutes and *n* = 10. Now, according to the memoryless property of the Poisson process the first time *t* and *n* must be subtracted from the second *t* and *n*. So,

*t* = 20 – 8 = 12 and *n* = 10 – 3 = 7