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GUEs and queues. (English) Zbl 0980.60042
It is considered the stochastic process $$D_{.} = (D_k, k =1, 2, \ldots)$$ determined by the relations $D_k = \sup_{0=t_0<t_1<\ldots<t_{k-1}<t_k=1} \sum\limits_{i=1}^{k-1} \left[B_{i}(t_{k+1}) - B_{i}(t_{k})\right],$ where $$B_i$$ are independent Brownian motions. It is shown that the process $$D_{.}$$ is equidistributed with the process $$\sigma_{.} = (\sigma_k$$, $$k =1, 2, \ldots)$$, where $$\sigma_k$$ is the largest eigenvalue of the main $$k \times k$$ minor of a random infinite Hermitian matrix drawn from the Gaussian unitary ensemble (GUE). An application of this result in queueing theory is given. A series of interconnections of the main theorem with results in other mathematical subjects is described.

MSC:
 60G07 General theory of stochastic processes 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60K25 Queueing theory (aspects of probability theory)
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