LChrisman/Queuing theory application of MetaLog

This page contains brainstorming-level investigation into the application of the Keelin (MetaLog) distribution to queueing theory. Right now it is just a place for notes, may contain mistakes, etc.

Motivation

The paper

Baucells, Chrisman, Keelin and Xu (2025) , "On the properties of the Metalog distribution"

derives a closed-form formula for partial expectation.

Here I wonder whether this could be applied to a model of an M/G/1 queue, where service times have a MetaLog distribution, enabling a closed-form expression for partial expectations on the wait time distribution.

M/G/1 queue

Arrivals follow a Poisson process with an average arrival rate of [math]\displaystyle{ \lambda }[/math] (i.e., arrival rate is Exponential(λ)).

Service times are MetaLog distributed. And there is a single server.

Notation:

[math]\displaystyle{ \lambda }[/math]: Arrival rate. Reciprocal of expected time between successive arrivals.
[math]\displaystyle{ \mu }[/math]: Expected number of service completions per unit time. It is the reciprocal of mean service time.
[math]\displaystyle{ \rho = \lambda/\mu }[/math]: The utilization.
[math]\displaystyle{ W }[/math] = Mean wait time
[math]\displaystyle{ L }[/math] = Mean # of customers in the system ([math]\displaystyle{ L = \lambda W }[/math]).
[math]\displaystyle{ x }[/math] = Variable used for service-time.
[math]\displaystyle{ p_S(x) }[/math] = PDF of service-time distribution.
[math]\displaystyle{ F_S(x) }[/math] = CDF of the service-time distribution.
[math]\displaystyle{ M_S(y) }[/math] = Quantile function of service-time distribution. (A MetaLog)

For a functional system, [math]\displaystyle{ \rho \lt 1 }[/math]. If [math]\displaystyle{ \rho\ge 1 }[/math] then jobs arrive faster than they are serviced.

The Laplace-Stieltjes transform of [math]\displaystyle{ M_S(y) }[/math] is

[math]\displaystyle{ \mathcal{L}_p\{M_S\}(s) = \int_0^1 e^{- s M_S(y)} dy = \int_{-\infty}^\infty e^{-s x} dF_S(x) = E[e^{-s x}] }[/math]