## On Deciding Stability of Constrained Homogeneous Random Walks and Queueing Systems (2000)

Venue: | Mathematics of Operations Research |

Citations: | 5 - 4 self |

### BibTeX

@INPROCEEDINGS{Gamarnik00ondeciding,

author = {David Gamarnik},

title = {On Deciding Stability of Constrained Homogeneous Random Walks and Queueing Systems},

booktitle = {Mathematics of Operations Research},

year = {2000},

pages = {2002}

}

### OpenURL

### Abstract

We investigate stability of some scheduling policies in queueing systems. To the day no algorithmic characterization exists for checking stability of a given policy in a given queueing system. In this paper we propose a certain generalized priority policy and prove that the stability of this policy is algorithmically undecidable. We also prove that stability of a homogeneous random walk in Z d + is undecidable. To the best of our knowledge this is the first undecidability result in the area of stability of queueing systems and random walks in Z d + . We conjecture that stability of other common policies like First-In-First-Out and priority policy is also an undecidable problem.

### Citations

522 |
Introduction to the theory of computation
- Sipser
- 1997
(Show Context)
Citation Context ...exists which given "Turing machine+input word" will tell whether the Turing machine halts on the input word. For a definition of a Turing machine and discussion of a Turing halting problem s=-=ee Sipser [26]-=-. Recently Blondel, et al. [3] demonstrated that global stability of a piece-wise affine dynamical system is undecidable. This was established via reduction from so called counter machines (see [16],[... |

284 |
Formal Languages and Their Relation to Automata
- Hopcroft, Ullman
- 1969
(Show Context)
Citation Context ...]. Recently Blondel, et al. [3] demonstrated that global stability of a piece-wise affine dynamical system is undecidable. This was established via reduction from so called counter machines (see [16],=-=[17]-=-), which is a variation of a Turing machine. We will also use counter machines as a reduction tool. We will show how a counter machine can be embedded into a single station queueing system, and how ha... |

266 | What’s decidable about hybrid automata
- Henzinger, Kopke, et al.
- 1998
(Show Context)
Citation Context ...tivated by similar results in the area of control theory. It is known that certain dynamical systems like hybrid systems or piecewise affine systems can simulate a Turing machine, see Henzinger et al.=-=[15]-=-, Koiran et al. [19]. The stability of these systems is then reduced to the halting problem of the Turing machine which is a 2 classical example of an undecidable problem: no algorithm exists which gi... |

225 | On positive Harris Recurrence of Multiclass Queueing Networks: A Unified approach via Fluid Limit Models
- Dai
- 1995
(Show Context)
Citation Context ...ation model) is not bigger than one. Many results have demonstrated that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks ([5],=-=[6]-=-,[10],[20],[24]), deterministic fluid networks ([2],[6],[7],[8],[9]), deterministic adversarial networks ([1],[4],[12],[14]). One of the earliest result in the area were obtained by Rybko and Stolyar ... |

109 | D.P.: Adversarial queueing theory
- Borodin, Kleinberg, et al.
(Show Context)
Citation Context ...bility. The results were obtained primarily in the context of stochastic networks ([5],[6],[10],[20],[24]), deterministic fluid networks ([2],[6],[7],[8],[9]), deterministic adversarial networks ([1],=-=[4]-=-,[12],[14]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simple priority policy can lead to instability in some queueing ne... |

107 | Universal Stability Results for Greedy Contention-resolution Protocols
- Andrews, Awerbuch, et al.
- 1996
(Show Context)
Citation Context ... stability. The results were obtained primarily in the context of stochastic networks ([5],[6],[10],[20],[24]), deterministic fluid networks ([2],[6],[7],[8],[9]), deterministic adversarial networks (=-=[1]-=-,[4],[12],[14]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simple priority policy can lead to instability in some queuein... |

100 | Distributed scheduling based on due dates and buffer priorities
- Kumar, Lu
- 1991
(Show Context)
Citation Context ...el) is not bigger than one. Many results have demonstrated that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks ([5],[6],[10],=-=[20]-=-,[24]), deterministic fluid networks ([2],[6],[7],[8],[9]), deterministic adversarial networks ([1],[4],[12],[14]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and L... |

96 |
Ergodicity of stochastic processes describing the operation of open queueing networks
- Rybko, Stolyar
- 1992
(Show Context)
Citation Context ...s not bigger than one. Many results have demonstrated that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks ([5],[6],[10],[20],=-=[24]-=-), deterministic fluid networks ([2],[6],[7],[8],[9]), deterministic adversarial networks ([1],[4],[12],[14]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and... |

78 |
Instability of FIFO queueing networks
- Bramson
- 1994
(Show Context)
Citation Context ...unication model) is not bigger than one. Many results have demonstrated that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks (=-=[5]-=-,[6],[10],[20],[24]), deterministic fluid networks ([2],[6],[7],[8],[9]), deterministic adversarial networks ([1],[4],[12],[14]). One of the earliest result in the area were obtained by Rybko and Stol... |

54 |
On the stability of multiclass queueing networks: a relaxed sufficient condition via limiting fluid processes
- Stolyar
- 1995
(Show Context)
Citation Context ...dman [25] showed that even FIFO policy can be unstable in stochastic networks. Instability of FIFO was later demonstrated in an adversarial queueing setting by Andrews et al. [1]. Dai [6] and Stolyar =-=[27]-=- established that stability of a fluid deterministic queueing network implies stability of a stochastic queueing network. A similar result was established by Gamarnik [12], which connects stability of... |

44 | Stability of adaptive and non-adaptive packet routing policies in adversarial queueing networks
- Gamarnik
- 1999
(Show Context)
Citation Context ... [2] and Dai and Vande Vate [7]. Goel [14] constructed a complete characterization of adversarial queueing networks which are stable under the usual load condition. The result is extended by Gamarnik =-=[13]. Des-=-pite the progress, no explicit or algorithmic characterization is known for checking stability of a given policy in a given network. That is no algorithm is available which on an input "queueing ... |

43 | First come, first served can be unstable
- Seidman
- 1994
(Show Context)
Citation Context ...by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simple priority policy can lead to instability in some queueing networks even if the load condition is met. Bramson [5] and Seidman =-=[25]-=- showed that even FIFO policy can be unstable in stochastic networks. Instability of FIFO was later demonstrated in an adversarial queueing setting by Andrews et al. [1]. Dai [6] and Stolyar [27] esta... |

40 | Stability of Networks and Protocols in the Adversarial Queueing Model for Packet Routing
- Goel
(Show Context)
Citation Context ...he results were obtained primarily in the context of stochastic networks ([5],[6],[10],[20],[24]), deterministic fluid networks ([2],[6],[7],[8],[9]), deterministic adversarial networks ([1],[4],[12],=-=[14]-=-). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simple priority policy can lead to instability in some queueing networks eve... |

36 | Piecewise linear test functions for stability and instability of queueing networks
- Down, Meyn
- 1997
(Show Context)
Citation Context ...n model) is not bigger than one. Many results have demonstrated that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks ([5],[6],=-=[10]-=-,[20],[24]), deterministic fluid networks ([2],[6],[7],[8],[9]), deterministic adversarial networks ([1],[4],[12],[14]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] ... |

32 |
Stability and instability of fluid models for certain re-entrant lines
- Weiss
- 1996
(Show Context)
Citation Context ... that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks ([5],[6],[10],[20],[24]), deterministic fluid networks ([2],[6],[7],[8],=-=[9]-=-), deterministic adversarial networks ([1],[4],[12],[14]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simple priority poli... |

30 | Stability of Adversarial Queues via Fluid Models
- Gamarnik
- 1998
(Show Context)
Citation Context ...ty. The results were obtained primarily in the context of stochastic networks ([5],[6],[10],[20],[24]), deterministic fluid networks ([2],[6],[7],[8],[9]), deterministic adversarial networks ([1],[4],=-=[12]-=-,[14]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simple priority policy can lead to instability in some queueing network... |

27 |
The Undecidability of the Turing Machine Immortality Problem
- Hooper
- 1966
(Show Context)
Citation Context ...r [26]. Recently Blondel, et al. [3] demonstrated that global stability of a piece-wise affine dynamical system is undecidable. This was established via reduction from so called counter machines (see =-=[16]-=-,[17]), which is a variation of a Turing machine. We will also use counter machines as a reduction tool. We will show how a counter machine can be embedded into a single station queueing system, and h... |

22 | Deciding stability and mortality of piecewise affine systems
- Blondel, Bournez, et al.
- 2001
(Show Context)
Citation Context ...ine+input word" will tell whether the Turing machine halts on the input word. For a definition of a Turing machine and discussion of a Turing halting problem see Sipser [26]. Recently Blondel, et=-= al. [3]-=- demonstrated that global stability of a piece-wise affine dynamical system is undecidable. This was established via reduction from so called counter machines (see [16],[17]), which is a variation of ... |

18 | Stability conditions for multiclass fluid queueing networks
- Bertismas, Gamarnik, et al.
- 1996
(Show Context)
Citation Context ...ave demonstrated that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks ([5],[6],[10],[20],[24]), deterministic fluid networks (=-=[2]-=-,[6],[7],[8],[9]), deterministic adversarial networks ([1],[4],[12],[14]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simp... |

16 |
On random walks arising in queueing systems: Ergodicity and transience via quadratic forms as Lyapunov functions—Part II. Queueing Systems 5
- Fayolle
- 1989
(Show Context)
Citation Context ...ing network model stability of homogeneous random walks in a nonnegative orthant Z d + (Z+ is the set of nonnegative integers) was considered in several papers: Malyshev [21], Menshikov [23], Fayolle =-=[11]-=-, Ignatyuk and Malyshev [18], Malyshev [22]. The transition vectors \Delta have deterministically bounded length in max norm and the transition probability p(; \Delta) along the direction \Delta depen... |

16 |
Computability properties of low-dimensional dynamical systems
- Koiran, Cosnard, et al.
- 1994
(Show Context)
Citation Context ...esults in the area of control theory. It is known that certain dynamical systems like hybrid systems or piecewise affine systems can simulate a Turing machine, see Henzinger et al.[15], Koiran et al. =-=[19]. The-=- stability of these systems is then reduced to the halting problem of the Turing machine which is a 2 classical example of an undecidable problem: no algorithm exists which given "Turing machine+... |

11 |
Classification of two-dimensional positive random walks and almost linear semimartingales
- Malyshev
- 1972
(Show Context)
Citation Context ...policies. Motivated by a queueing network model stability of homogeneous random walks in a nonnegative orthant Z d + (Z+ is the set of nonnegative integers) was considered in several papers: Malyshev =-=[21]-=-, Menshikov [23], Fayolle [11], Ignatyuk and Malyshev [18], Malyshev [22]. The transition vectors \Delta have deterministically bounded length in max norm and the transition probability p(; \Delta) al... |

11 |
Networks and dynamical systems
- Malyshev
- 1993
(Show Context)
Citation Context ...random walks in a nonnegative orthant Z d + (Z+ is the set of nonnegative integers) was considered in several papers: Malyshev [21], Menshikov [23], Fayolle [11], Ignatyuk and Malyshev [18], Malyshev =-=[22]-=-. The transition vectors \Delta have deterministically bounded length in max norm and the transition probability p(; \Delta) along the direction \Delta depends only on the facesthat the random walk is... |

6 |
Classification of random walks in �d
- Ignatyuk, Malyshev
- 1993
(Show Context)
Citation Context ...of homogeneous random walks in a nonnegative orthant Z d + (Z+ is the set of nonnegative integers) was considered in several papers: Malyshev [21], Menshikov [23], Fayolle [11], Ignatyuk and Malyshev =-=[18]-=-, Malyshev [22]. The transition vectors \Delta have deterministically bounded length in max norm and the transition probability p(; \Delta) along the direction \Delta depends only on the facesthat the... |

4 |
The stability of two-station fluid networks
- Dai, VandeVate
- 1997
(Show Context)
Citation Context ...nstrated that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks ([5],[6],[10],[20],[24]), deterministic fluid networks ([2],[6],=-=[7]-=-,[8],[9]), deterministic adversarial networks ([1],[4],[12],[14]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simple prior... |

4 |
Ergodicity and transience conditions for random walks in the positive octant of space
- Menshikov
- 1979
(Show Context)
Citation Context ...ted by a queueing network model stability of homogeneous random walks in a nonnegative orthant Z d + (Z+ is the set of nonnegative integers) was considered in several papers: Malyshev [21], Menshikov =-=[23]-=-, Fayolle [11], Ignatyuk and Malyshev [18], Malyshev [22]. The transition vectors \Delta have deterministically bounded length in max norm and the transition probability p(; \Delta) along the directio... |

1 | Stability of a three-station fluid network
- Dai, Vate, et al.
- 1999
(Show Context)
Citation Context ...ated that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks ([5],[6],[10],[20],[24]), deterministic fluid networks ([2],[6],[7],=-=[8]-=-,[9]), deterministic adversarial networks ([1],[4],[12],[14]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simple priority ... |