## Abstract

We consider a push pull queueing network with two servers and two types of job which are processed by the two servers in opposite order, with stochastic generally distributed processing times. This push pull network was introduced by Kopzon and Weiss, who assumed exponential processing times. It is similar to the Kumar-Seidman Rybko-Stolyar (KSRS) multi-class queueing network, with the distinction that instead of random arrivals, there is an infinite supply of jobs of both types. Unlike the KSRS network, we can find policies under which our push pull network works at full utilization, with both servers busy at all times, and without being congested. We perform fluid and diffusion scale analysis of this network under such policies, to show fluid stability, positive Harris recurrence, and to obtain a diffusion limit for the network. On the diffusion scale the network is empty, and the departures of the two types of job are highly negatively correlated Brownian motions. Using similar methods we also derive a diffusion limit of a re-entrant line with an infinite supply of work.

Original language | English |
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Pages (from-to) | 201-217 |

Number of pages | 17 |

Journal | Performance Evaluation |

Volume | 67 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2010 |

### Bibliographical note

Funding Information:We would like to thank Serguei Foss for useful discussions on stability of Markov chains, and fluid and diffusion approximations of queueing networks. The author’s research was supported in part by Israel Science Foundation Grant 249/02 and 454/05 and by European Network of Excellence Euro-NGI.

## Keywords

- Diffusion limits
- Fluid models
- Infinite virtual queues
- Petite bounded sets
- Positive Harris recurrence
- Push pull
- Queueing networks

## ASJC Scopus subject areas

- Software
- Modeling and Simulation
- Hardware and Architecture
- Computer Networks and Communications