Heavy traffic approximation

In queueing theory, a discipline within the mathematical theory of probability, a heavy traffic approximation (sometimes heavy traffic limit theorem or diffusion approximation) is the matching of a queueing model with a diffusion process under some limiting conditions on the model's parameters. The first such result was published by John Kingman who showed that when the utilisation parameter of an M/M/1 queue is near 1 a scaled version of the queue length process can be accurately approximated by a reflected Brownian motion.

Heavy traffic approximations are typically stated for the process X(t) describing the number of customers in the system at time t. They are arrived at by considering the model under the limiting values of some model parameters and therefore for the result to be finite the model must be rescaled by a factor n, denoted

and the limit of this process is considered as n → ∞.

There are three classes of regime under which such approximations are generally considered.

Theorem 1. Consider a sequence of G/G/1 queues indexed by ${\displaystyle j}$.
For queue ${\displaystyle j}$
let ${\displaystyle T_{j}}$ denote the random inter-arrival time, ${\displaystyle S_{j}}$ denote the random service time; let ${\displaystyle \rho _{j}={\frac {\lambda _{j}}{\mu _{j}}}}$ denote the traffic intensity with ${\displaystyle {\frac {1}{\lambda _{j}}}=E(T_{j})}$ and ${\displaystyle {\frac {1}{\mu _{j}}}=E(S_{j})}$; let ${\displaystyle W_{q,j}}$ denote the waiting time in queue for a customer in steady state; Let ${\displaystyle \alpha _{j}=-E[S_{j}-T_{j}]}$ and ${\displaystyle \beta _{j}^{2}=\operatorname {var} [S_{j}-T_{j}];}$

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