## On the Saturation Rule for the Stability of Queues (1998)

Venue: | J. Appl. Prob |

Citations: | 28 - 9 self |

### BibTeX

@ARTICLE{Baccelli98onthe,

author = {Francois Baccelli and Serguei Foss},

title = {On the Saturation Rule for the Stability of Queues},

journal = {J. Appl. Prob},

year = {1998},

volume = {32},

pages = {494--507}

}

### OpenURL

### Abstract

This paper focuses on the stability of open queueing systems under stationary ergodic assumptions. It defines a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule: `saturate' the queues which are fed by the external arrival stream; look at the `intensity' of the departure stream in this saturated system; then stability holds whenever the intensity of the arrival process, say satisfies the condition ! , whereas the network is unstable if ? .

### Citations

78 |
Instability of FIFO queueing networks
- Bramson
- 1994
(Show Context)
Citation Context ...ernal monotonicity property is not satisfied. Note that the conditions stating that the traffic intensity be less than one at each queue is not sufficient for monotonicity, even in the FIFO case (see =-=[9]-=-). 12 Kelly-type Networks with Synchronization Constraints These networks were defined in [2]. The routing is mechanism is the same as above, but the service discipline is such that, on a given queue,... |

70 | Subadditive ergodic theory - Kingman - 1973 |

68 | Asymptotic Methods in Queueing Theory - BOROVKOV - 1984 |

41 | Stochastically recursive sequences and their generalizations
- Borovkov, Foss
- 1992
(Show Context)
Citation Context ... queues) tend to 1 in probability. The proposed method provides an alternative to the methods based on regeneration (Harris-recurrence [15] in the Markov-case, Renovating Events in the nonMarkov case =-=[8]-=-). Regeneration is replaced by separability. An interesting point when comparing these methods is that regeneration has to be effective (for instance, in the Harris-recurrent case, it is necessary to ... |

21 |
Ergodicity of Jackson-type queueing networks
- Baccelli, Foss
- 1994
(Show Context)
Citation Context ...e j n , and is routed to stationsj n ), then the network falls in the refined monotone separable framework, provided the routing sequences satisfy the two stationarity properties described below (see =-=[3]-=-). Let K be the number of stations. ffl We assume that there exist K non-decreasing, integer-valued sequences ' k l , k = 1; : : : ; K, such that when denoting F k n = ' k 1 + \Delta \Delta \Delta + '... |

7 |
The throughput of a series of buffers
- Kelly
- 1982
(Show Context)
Citation Context ...the examples of the final section). So subadditive ergodic theory is central for a much wider class than the class of models based (directly or indirectly) on random graph structures (see for example =-=[11]-=-, [2], [1], [5], [6]). 2 The Monotone Separable Framework for Maximal Daters Arrival Point Process Let N be a marked point process with points fT n g n2ZZ and marks f n g n2ZZ , wheresn 2 (K; K). This... |

3 |
Stability and maximum departure rate of certain open queueing networks having finite capacity constraints
- Lavenberg
- 1978
(Show Context)
Citation Context ...y designed for Markovian queueing systems, wherescould be obtained by computing the steady state of a Markov chain of smaller dimension than that of the initial non-saturated system (see for instance =-=[13]-=-). The aim of the present paper is to set a natural framework in which this rule can be rigorously proved when dropping the Markovian assumptions and replacing them by standard stationary ergodic assu... |

2 |
An improved version of Kingmanâ€™s subadditive ergodic theorem
- Liggett
- 1985
(Show Context)
Citation Context ...ve ergodic theorem, see [16]. ffl Homogeneity can also be weakened and be replaced by some homogeneity in distribution. Concerning the relevant extension of Kingman's subadditive ergodic theorem, see =-=[14]-=-. Several other extensions are possible, like for instance replacing the arrival point process by a random measure. 3 Proof of the Saturation Rule Stationarity Assumptions Assume the variables f n ;sn... |

1 |
On a Class of Stochastic Evolution Equations arising in Queueing Theory
- Baccelli, Liu
- 1992
(Show Context)
Citation Context ...amples of the final section). So subadditive ergodic theory is central for a much wider class than the class of models based (directly or indirectly) on random graph structures (see for example [11], =-=[2]-=-, [1], [5], [6]). 2 The Monotone Separable Framework for Maximal Daters Arrival Point Process Let N be a marked point process with points fT n g n2ZZ and marks f n g n2ZZ , wheresn 2 (K; K). This poin... |

1 |
Structural, Timed and Stochastic Properties of Unbounded Free Choice Petri Nets
- Baccelli, Foss, et al.
- 1994
(Show Context)
Citation Context ...st firing time of whole net for the [m; n] restriction of the arrival point process. Conditions 1-3 of the framework are always satisfied. The validity of condition 4 of the framework is discussed in =-=[4]-=-. This class contains generalized Jackson networks, although it is much more general. 13 Polling Systems For a wide class of polling models, it is possible to prove stability under general assumptions... |

1 |
Scheduling and Stability of a Class of Parallel Processing Systems
- Bambos, Walrand
- 1993
(Show Context)
Citation Context ...the final section). So subadditive ergodic theory is central for a much wider class than the class of models based (directly or indirectly) on random graph structures (see for example [11], [2], [1], =-=[5]-=-, [6]). 2 The Monotone Separable Framework for Maximal Daters Arrival Point Process Let N be a marked point process with points fT n g n2ZZ and marks f n g n2ZZ , wheresn 2 (K; K). This point process ... |

1 |
On Stationary Tandem Queuing networks with Job Feedback
- Bambos, Wasserman
- 1994
(Show Context)
Citation Context ...inal section). So subadditive ergodic theory is central for a much wider class than the class of models based (directly or indirectly) on random graph structures (see for example [11], [2], [1], [5], =-=[6]-=-). 2 The Monotone Separable Framework for Maximal Daters Arrival Point Process Let N be a marked point process with points fT n g n2ZZ and marks f n g n2ZZ , wheresn 2 (K; K). This point process is no... |

1 |
On the Boundedness of Queue-Length Process in Polling Systems
- Chernova
(Show Context)
Citation Context ...uting and walking mechanism with regenerative structure). The properties (1)-(4) are still valid, but (3)-(4) only hold in distribution. The constant fl(0) coincides with the well-known constant (see =-=[10]-=-). A Non-separable Network for which the Rule Does Not Hold Consider an assembly queue with two independent Poisson arrival streams with the same intensity =2. The system starts empty. Whenever there ... |

1 |
Stability of Generalized Jackson Networks. A nn
- Meyn, Down
- 1993
(Show Context)
Citation Context ...(either the time to inactivity or the number of customers in some queues) tend to 1 in probability. The proposed method provides an alternative to the methods based on regeneration (Harris-recurrence =-=[15]-=- in the Markov-case, Renovating Events in the nonMarkov case [8]). Regeneration is replaced by separability. An interesting point when comparing these methods is that regeneration has to be effective ... |

1 |
Almost Subadditive Extensions of Kingman's Ergodic Theorem
- Schruger
- 1991
(Show Context)
Citation Context ...;l\Gamma1] (N) ffi ` (see x3); -- fU [\Gamman;0] (N)g is uniformly integrable and U [\Gamman;0] (N)=n ! 0 a.s. as n !1. Concerning the relevant extension of Kingman's subadditive ergodic theorem, see =-=[16]-=-. ffl Homogeneity can also be weakened and be replaced by some homogeneity in distribution. Concerning the relevant extension of Kingman's subadditive ergodic theorem, see [14]. Several other extensio... |