# On The Waiting Time for A M/M/1 Queue with Impatience

## TL;DR

This paper analyzes the waiting time distribution in an M/M/1 queue with impatience, showing convergence to a limiting distribution with power-law or exponential tails depending on the arrival and service rates.

## Contribution

It introduces a model for queue impatience with last-in-first-out answering and proves the convergence of waiting times to a specific distribution with tail behavior.

## Key findings

- Waiting time converges to a limiting distribution W_T.
- W_T has a power-law tail when λ=μ.
- W_T has an exponential tail when λ≠μ.

## Abstract

This paper focuses on the problem of modeling the correspondence pattern for ordinary people. Suppose that letters arrive at a rate $\lambda$ and are answered at a rate $\mu$. Furthermore, we assume that, for a constant $T$, a letter is disregarded when its waiting time exceeds $T$, and the remains are answered in {\it last in first out} order. Let $W_n$ be the waiting time of the $n$-th {\it answered} letter. It is proved that $W_n$ converges weekly to $W_T$, a non-negative random variable which possesses a density with {\it power-law} tail when $\lambda=\mu$ and with exponential tail otherwise. Note that this may provide a reasonable explanation to the phenomenons reported by Oliveira and Barab\'asi in \cite{OB}.