*(English)*Zbl 1041.15019

It was discovered recently by *J. Baik*, *P. Deift* and *K. Johansson*, [J. Am. Math. Soc. 12, No. 4, 1119–1178 (1999; Zbl 0932.05001)] that the asymptotic distribution of the length of the longest increasing subsequence in a permutation chosen uniformly at random from a process ${S}_{n}$, properly centered and normalised, is the same as the asymptotic distribution of the largest eigenvalue of an $n\times n$ GUE random matrix, properly centered and normalised, as $n\to \infty $. On the other hand, the eigenvalues of a GUE random matrix of dimension $n$ obey the same law as positions, after a unit length of time, of $n$ independent standard Brownian motions started from the origin and conditioned never to collide. These connections between random matrices, non-colliding processes and queues are studied.

The known theorem of classical queueing theory, which states that, in equilibrium, the output of a stable $M/M/1$ queue is Poisson, was firstly proved by *P. J. Burke* [Operations Research 4, No. 6, 699–704 (1956)]. Here, this output theorem for the $M/M/1$ queue is proved using reversibility and its extended version is used to obtain a representation for non-colliding Poisson processes and for non-colliding Brownian motions. Finally, output theorems are discussed generally and the analogue of the output theorem is given for Brownian motions with drift together with the first order asymptotics for directed percolation and directed polymers.

##### MSC:

15A52 | Random matrices (MSC2000) |

82D60 | Polymers (statistical mechanics) |

15A18 | Eigenvalues, singular values, and eigenvectors |

60K25 | Queueing theory |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60J65 | Brownian motion |