## Queues with time-dependent arrival rates: The transition through saturation |

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a0 dt actual expected queue arrival and departure arrival and service arrival rate builds arrival rate increases average queue length boundary condition bution cause the queue coefficients db(t decrease the variance departure processes dependent arrival deterministic approximation deterministic model Deterministic queueing deterministic theory differential equation diffusion equation discrete distribution functions equation like 23 equi equilibrium distri equilibrium distribution equilibrium queue length equilibrium solution 29 estimate the queue expected queue length large compared length becomes infinite length becomes large length in units librium distribution mean queue length nearly constant original integer units parameters postulate queue behavior queue distribution queue length becomes queue length distribution queue to vanish queue vanishes queueing process Queues with time-dependent random variables random walk range real queue reflecting barrier rescaled rush hour scale service rate solution of 25 stochastic arrival time-dependent arrival rates traffic intensity approaches transition through saturation variance per step zero