### Abstract

Operator methods are used to systematically analyze the behavior of the Jackson network, considering rarely treated issues such as the transient behavior, and arbitrary subnetworks of the total system. By deriving the equations that govern an arbitrary subnetwork, one can see how the mean and variance for the queue length of one node as well as the covariance for two nodes vary in time. The transient behavior is estimated by deriving a stochastic upper bound for the joint distribution of the network in terms of a judicious choice of independent M/M/1 queue-length processes. The bound derived is one that cannot be derived by a sample-path ordering of the two processes. Moreover, one can stochastically bound from below the process for the total number of customers in the network by an M/M/1 system also. The network can then be approximated by the known transient distribution of the M/M/1 queue.

Original language | English (US) |
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Pages (from-to) | 379-393 |

Number of pages | 15 |

Journal | Journal of Applied Probability |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1984 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty