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Probability · Lecture 55 of 76 · 9:43

Random Incidence Under Erlang Arrivals

Study guide

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

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

Vocabulary

random incidence (phrase): The fact that arriving at a random time tends to land you in a longer-than-average interval.
Random incidence also applies to non-Poisson arrival processes.
renewal process (phrase): A process where the gaps between arrivals are independent and follow the same distribution, but not necessarily exponential.
This problem studies random incidence in a general renewal process.
fictitious (adjective): Invented or imagined rather than actually observed, used as a trick in a calculation.
We insert a fictitious arrival in the middle of each gap.
finer-grained (adjective): Divided into smaller, more detailed pieces.
The finer-grained process is now an ordinary Poisson process.
underlying (adjective): Forming the hidden basic structure beneath something else.
We decompose the process into an underlying Poisson process.
Erlang random variable (phrase): A random variable equal to the sum of several independent exponential random variables.
An Erlang random variable of order 3 appears in the final answer.
exponential distribution (phrase): A probability pattern often used to model waiting times, with a constant rate.
Each inter-arrival gap is built from the exponential distribution.
inter-arrival time (phrase): The gap in time between one arrival and the next.
Here the inter-arrival time follows an Erlang-2 distribution.
Poisson process (phrase): A process where arrivals happen at random times, at a constant average rate.
Inserting a fictitious arrival recovers a plain Poisson process.
decompose (verb): To break something down into simpler parts.
You decompose a non-Poisson process into an underlying Poisson process.
paradox (noun): A result that seems strange or contradictory at first, but is actually true.
Random incidence is a well-known probability paradox.
sub-process (noun): A smaller process contained within a larger one.
The Poisson sub-process trick simplifies the calculation.
reinterpret (verb): To understand something in a new way.
We reinterpret each Erlang-2 gap as two exponential gaps.
arrival (noun): An event that happens at a particular point in time in a random process.
Each arrival marks the end of one inter-arrival interval.
rate (noun): How often something happens on average per unit of time.
The underlying process has rate lambda.
gap (noun): The empty space or time between two events.
T* can land in either half of an Erlang-2 gap.
order (of a random variable) (noun): The number of pieces summed together to form an Erlang random variable.
The answer is an Erlang random variable of order 3.
reuse (verb): To use something again for a new purpose.
We reuse the known Poisson random-incidence result.
typical (adjective): Usual or average, representing the normal case.
The interval containing T* is longer than a typical inter-arrival interval.
i.i.d. (phrase): Short for independent and identically distributed: separate variables that follow the same pattern and don't affect each other.
An Erlang variable is a sum of i.i.d. exponential variables.
insert (verb): To add something into an existing sequence.
We insert a fictitious arrival in the middle of each gap.
distribution (noun): A description of how likely different values of a random variable are.
We find the distribution of the interval containing T*.
extend (verb): To make an idea apply to a larger or new situation.
This worked problem extends random incidence beyond Poisson processes.
trick (noun): A clever method that makes a hard problem easier to solve.
The Poisson sub-process trick reuses known results.
afterward (adverb): At a later point, after something else has happened.
Afterward, you'll be able to decompose non-Poisson processes into Poisson ones.

From the YouTube description

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
View the complete course: http://ocw.mit.edu/6-041SCF13
Instructor: Jimmy Li

License: Creative Commons BY-NC-SA
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