# Queuing Discipline and Lipschitz Continuity

I was reading a paper by Platz and Østerdal (2014) titled "The curse of the first-in-first-out queue discipline". I was wondering about a particular section of it, particularly the part about last-in-first-out mechanisms. $$R(t)$$ is defined as the cumulative arrival distribution, and they try and show:

Let $$R(t)$$ be a Nash Equilibrium for the FIFO (first-in-first-out) queue discipline. Then $$R(t)$$ is Lipschitz continuous.

This is the second Lemma on page 2.

I am not quite understanding why we are trying to show this property to lead into the third Lemma. What is the importance of Lipschitz continuity here? A basic intuitive answer should be okay, but some math would help me I think. Are they trying to ultimately show that $$R(t)$$ is a unique Nash equilibrium for FIFO?

• I couldn't open the paper. – Marcelo Gelati Apr 16 '19 at 11:57
• @MarceloGelati That's what older me gets for not putting in the author and title. I updated the link. Should work now. – Kitsune Cavalry Apr 16 '19 at 19:12
• Are you still interested in the question? – Marcelo Gelati Apr 16 '19 at 20:50
• It appears that Lipschitz continuity is used later mostly related to the LIFO queue results. Moreover, Lipschitz continuity is used in fixed point theorems, and Nash Equilibrium is essentially a definition of a fixed point. – Alecos Papadopoulos Apr 17 '19 at 16:22
• I understand it in another way; perhaps these two sentences might help: "For an agent arriving at a time $t$ where $R(t)$ has no jump, the time of service is deterministic" and "note that [...] Lemma 3 implies that $R$ has no jumps for $t > 0$". I mean, if the function is Lipschitz continuous, you get a deterministic time of service and because of that you can guarantee lemma 4; otherwise, you couldn't. – Marcelo Gelati Apr 18 '19 at 14:22