<p>I was <a href="https://editorialexpress.com/cgi-bin/conference/download.cgi?db_name=SSCWatBC&amp;paper_id=150" rel="noreferrer">reading a paper</a> by <em>Platz and Østerdal</em> (2014) titled "<strong>The curse of the first-in-first-out queue discipline</strong>". I was wondering about a particular section of it, particularly the part about last-in-first-out mechanisms. <span class="math-container">$R(t)$</span> is defined as the cumulative arrival distribution, and they try and show:</p>

<blockquote>
  <p><em>Let <span class="math-container">$R(t)$</span> be a Nash Equilibrium for the FIFO (first-in-first-out) queue discipline. Then <span class="math-container">$R(t)$</span> is Lipschitz continuous.</em></p>
</blockquote>

<p>This is the second Lemma on page 2.</p>

<p>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 <span class="math-container">$R(t)$</span> is a unique Nash equilibrium for FIFO? </p>

CC BY-SA 4.0
2019-04-16T11:57:42.683Z
I couldn't open the paper.
Marcelo Gelati
CC BY-SA 4.0
2019-04-16T19:12:48.033Z
@MarceloGelati That's what older me gets for not putting in the author and title. I updated the link. Should work now.
Kitsune Cavalry
CC BY-SA 4.0
2019-04-16T20:50:56.800Z
Are you still interested in the question?
Marcelo Gelati
CC BY-SA 4.0
2019-04-16T21:42:20.037Z
Hell yeah buddy
Kitsune Cavalry
CC BY-SA 4.0
2019-04-17T16:22:25.713Z
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
CC BY-SA 4.0
2019-04-18T14:22:56.367Z
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
CC BY-SA 4.0
2022-07-10T20:32:58.387Z
Lipschitz continuity is used to proff that $R(t)$ is absolutely continuous an then the following formula holds: $R(t) = R(0) + \int_0^t r_+(t)\text{d}t$
Davius
<microeconomics>
Queuing Discipline and Lipschitz Continuity
Kitsune Cavalry
# Queuing Discipline and Lipschitz Continuity

(Asked by: username_0 on 2015-10-12)

I was [reading a paper](https://editorialexpress.com/cgi-bin/conference/download.cgi?db_name=SSCWatBC&paper_id=150) 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?

**Tags:** microeconomics

> **Comments:**
> **username_1**: I couldn't open the paper.  
> **username_0**: @username_1 That's what older me gets for not putting in the author and title. I updated the link. Should work now.  
> **username_1**: Are you still interested in the question?  
> **username_0**: Hell yeah buddy  
> **username_2**: 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.  
> **username_1**: 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.  
> **username_3**: Lipschitz continuity is used to proff that $R(t)$ is absolutely continuous an then the following formula holds: $R(t) = R(0) + \int_0^t r_+(t)\text{d}t$  

---

## 0 answers

