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In various stochastic models the random equation of implicit renewal theory appears where the real random variable S and the stochastic process Ψ with index space and state space R are independent. By use of stochastic approximation the distribution function of S is recursively estimated on the basis of independent or ergodic copies of Ψ. Under integrability assumptions almost sure L1-convergence is proved. The choice of gains in the recursion is discussed. Applications are given to insurance mathematics (perpetuities) and queueing theory (stationary waiting and queueing times).

For a GI/G/1 queue we derive new lower and uper bounds on the expected stationary waiting time which are a function of more than the first two moments of the inter-arrival and service times. When the probability for a delay is low, the bounds tend to be better than other bounds. This is illustrated in an example of a Uniform/Uniform/1 queue.

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Recursive estimation of distributional fix-points

Tightness of the stationary waiting time in heavy traffic

Bounds on the expected waiting time in a GI/G/1 queue: upgrading for low traffic intensity

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3 results

Recursive estimation of distributional fix-points

Tightness of the stationary waiting time in heavy traffic

Bounds on the expected waiting time in a GI/G/1 queue: upgrading for low traffic intensity