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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.

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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