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Quantitative Finance · Glossaire

Qu'est-ce que « M/M/1 queue » ?

Definition 8.8 Quantitative Methods · Chapitre 8 — Markov Chains and Queues

The M/M/1 queue has Poisson arrivals at rate λ\lambda, exponential service times with rate μ\mu and one server; the number in the system is a birth–death process with λn=λ\lambda_n = \lambda and μn=μ\mu_n = \mu. (Kendall’s notation: Markov arrivals, Markov service, one server.)

A birth–death chain: from state n the only moves are up at rate _n and down at rate _n. A best queue counted in orders is one, with limit orders as births and market orders and cancellations as deaths.
Figure 8.1. A birth–death chain: from state nn the only moves are up at rate λn\lambda_n and down at rate μn\mu_n. A best queue counted in orders is one, with limit orders as births and market orders and cancellations as deaths.
The M/M/1 queue: mean number in the system against utilisation, from the stationary law and from 200 000 simulated customers per point, whose time-averaged count agrees with W (Little’s law). Near saturation the simulation converges slowly. Data: the chapter’s tutorial, seeded.
Figure 8.2. The M/M/1 queue: mean number in the system against utilisation, from the stationary law and from 200 000 simulated customers per point, whose time-averaged count agrees with λW\lambda W (Little’s law). Near saturation the simulation converges slowly. Data: the chapter’s tutorial, seeded.
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