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}\{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]

From the Pollaczek-Khinchine transform equation:

[math]\displaystyle{ \mathcal{L}\{W\} = {{(1-p) s \mathcal{L}\{p\}(s)} \over {s - \lambda (1-\mathcal{L}\{p\}(s))}} = {{(1-p)s} \over {s - \lambda + \lambda \mathcal{L}\{M_S \}(s)}} }[/math]

Laplace transforms of MetaLog

Two-term

[math]\displaystyle{ M(p) = a_0 + a_1 \ell(p) }[/math]

[math]\displaystyle{ \mathcal{L}\{M\}(s) = a_0 + {a_1 \over 2}\left( \phi\left({s\over 2}\right) - \phi\left({{s+1}\over 2}\right)\right) }[/math]

where [math]\displaystyle{ \phi(z) = {{\Gamma'(z)}\over{\Gamma(z)}} }[/math] is the digamma function.

k-terms

[math]\displaystyle{ M(y) = \sum_{i=1}^k a_i B_i(y) }[/math]

Note: Paper uses [math]\displaystyle{ Y_i(y) }[/math] where I use [math]\displaystyle{ B_i(y) }[/math] for the basis functions.

[math]\displaystyle{ \mathcal{L}\{M\}(s) = \sum_{i=1}^k a_i \mathcal{L}\{B_i\}(s) }[/math]

The poly terms ([math]\displaystyle{ i \in \{1,4,5,8,9,...\} }[/math]

[math]\displaystyle{ B_i(y) = a_i (y - 0.5)^d }[/math] where [math]\displaystyle{ d = \left\lfloor{{i-1}\over 2}\right\rfloor }[/math].

Then

[math]\displaystyle{ \mathcal{L}\{B_i(y)\}(s) = \sum_{j=0}^d \left( \begin{array}{c}d \\ j\end{array} \right) \left( -{1\over 2}\right)^{m-j} {{\gamma( j+1,s)} \over {s^{j+1}}} }[/math]

where [math]\displaystyle{ \gamma(s,x) }[/math] is the incomplete gamma function.

Apply standard rules:

[math]\displaystyle{ \mathcal{L}\{x^j} = {{j!}\over{s^{j+1}} }[/math]
Time shifting: [math]\displaystyle{ \mathcal{L}\{f(x-c) u(x-c)\}(s) = e^{-c s} \mathcal{L}\{f(x)\}(s) }[/math]

where [math]\displaystyle{ u(x) = (x\ge 0) }[/math] is the unit function.

Since [math]\displaystyle{ -0.5\le y-0.5 \le 0.5 }[/math], [math]\displaystyle{ \mathcal{L}\{ \sum_{i=0}^n c_i x^i \} = \sum_{i=0}^n c_i {{i!}\over{s^{i+1}}} }[/math]