Random Incidence Under Erlang Arrivals

What this lecture covers

This worked problem extends random incidence, normally studied for Poisson processes, to a process whose inter-arrival times are Erlang of order 2 instead of exponential. It asks for the distribution of the length of the inter-arrival interval that contains a fixed time T*, and shows the answer is an Erlang random variable of order 3.

You should already know Poisson processes, the exponential distribution, and the random-incidence paradox for Poisson processes before watching. Afterward, you'll be able to decompose a non-Poisson renewal process into an underlying Poisson process to reuse known Poisson results.

Key ideas

• Random incidence paradox: the inter-arrival interval containing a fixed time T* is generally longer, in distribution, than a typical inter-arrival interval.
• Erlang random variable: an Erlang variable of order k is the sum of k i.i.d. exponential random variables, with mean k/lambda.
• Reinterpreting Erlang-2 arrivals: since each Erlang-2 inter-arrival gap has the same distribution as two exponential gaps, you can insert a fictitious arrival in the middle of each gap.
• Recovering a Poisson process: after inserting the fictitious arrivals, the resulting finer-grained process is an ordinary Poisson process with rate lambda, with the original Erlang arrivals as every other arrival.
• Random incidence for Poisson gives Erlang-2: within the underlying Poisson process, the standard random-incidence result says the interval containing T* is Erlang of order 2.
• Adding the extra piece: because T* can land in either half of an Erlang-2 gap, an additional exponential piece is added, giving a total of three independent exponentials, i.e. an Erlang of order 3.

Before you watch

• Review the definition of an Erlang random variable as a sum of exponentials.
• Know the random-incidence result for Poisson processes (the interval containing a fixed point is Erlang-2, not exponential).
• Be comfortable with the idea of splitting one random variable into a sum of independent pieces.

Check your understanding

1. Why is inserting a fictitious arrival in the middle of each Erlang-2 gap valid, and why does it produce a Poisson process?
2. What is the random-incidence result for a plain Poisson process, and how is it reused here?
3. Why does the interval around T* end up as a sum of three exponentials rather than two?
4. How would the answer change if the underlying process had Erlang-3 inter-arrival times instead of Erlang-2?