# How Long Might We Wait at Random?

## TL;DR

This paper investigates the distribution of maximum queue length in a random customer-server model, highlighting the complexity of extending known results from simpler systems and discussing potential generalizations.

## Contribution

It provides algebraic expressions for maximum queue length in a two-server system and explores the challenges of extending these to three servers, reviewing related queue wait time distributions.

## Key findings

- Algebraic expressions derived for two-server maximum queue length.
- Complexity increases significantly with three servers, making algebraic solutions difficult.
- Discussion on generalizing maximum wait time distributions from M/M/1 queues.

## Abstract

In discrete time, customers arrive at random. Each waits until one of three servers is available; each thereafter departs at random. We seek the distribution of maximum line length of idle customers. Algebraic expressions obtained for the two-server scenario do not appear feasible here. We also review well-known distributional results for maximum wait time associated with an M/M/1 queue and speculate about their generalization.