## Loss networks and Markov random fields (1999)

Venue: | Journal of Applied Probability |

Citations: | 13 - 3 self |

### BibTeX

@ARTICLE{Zachary99lossnetworks,

author = {Stan Zachary and Ilze Ziedins},

title = {Loss networks and Markov random fields},

journal = {Journal of Applied Probability},

year = {1999},

volume = {36},

pages = {403--414}

}

### OpenURL

### Abstract

This paper examines the connection between loss networks without controls and Markov random field theory. The approach taken yields insight into the structure and computation of network equilibrium distributions, and into the nature of spatial dependence in networks. In addition, it provides further insight into some commonly used approximations, enables the development of more refined approximations, and permits the derivation of some asymptotically exact results. 1

### Citations

273 |
Multiservice Loss Models for Broadband Telecommunication Networks”, ISBN 3-540-19918-8
- Ross
- 1995
(Show Context)
Citation Context ... more refined approximations. Finally, for some highly symmetric networks this approach yields asymptotically exact results. For excellent introductions to loss networks see Kelly [12], and also Ross =-=[14]-=-. A general loss network without controls can be described as follows. Denote by J the finite collection of resources in the network and let C = {Cj, j ∈ J}where Cj is the capacity of resource j. Let ... |

169 |
Blocking probabilities in large circuit-switched networks
- Kelly
- 1986
(Show Context)
Citation Context ...d and Erlang fixed point approximations coincide 2sin the case where the capacity requirements Ajr can only take the values 0 or 1. Here the fixed point equations do have a unique solution, see Kelly =-=[11]-=-, and also Ross [14]. The use of the EFPA has been justified by considering two limiting regimes. Kelly [11] (see also Hunt and Kelly [8]) shows that the EFPA is asymptotically exact in the Kelly limi... |

129 |
Loss networks
- Kelly
- 1991
(Show Context)
Citation Context ...s the development of more refined approximations. Finally, for some highly symmetric networks this approach yields asymptotically exact results. For excellent introductions to loss networks see Kelly =-=[12]-=-, and also Ross [14]. A general loss network without controls can be described as follows. Denote by J the finite collection of resources in the network and let C = {Cj, j ∈ J}where Cj is the capacity... |

115 |
Computational algorithms for the closed queueing networks with exponential servers
- Buzen
- 1973
(Show Context)
Citation Context ...see also Zachary [19]) give an exact recurrence based on consideration of the reduced state space which records only the total occupancy of each resource. Refinements of Buzen’s convolution algorithm =-=[4]-=- for closed queueing networks have also been applied to loss networks. Of particular interest are Choudhury et al. [5], who invert the generating function of the partition function, and Bean and Stewa... |

106 |
Stochastic Models of Computer Communication Systems
- Kelly
- 1985
(Show Context)
Citation Context ...d. For the general technique here (in the context of general Markov random fields on trees) see, for example, Zachary [17], and for an application to a simple linear cellular radio network, see Kelly =-=[10]-=-. We now show that this idea may be extended to a considerably more general class of network topologies by considering an appropriate neighbour relation between (possibly overlapping) groups of call t... |

78 |
Blocking when service is required from several facilities simultaneously
- Whitt
- 1985
(Show Context)
Citation Context ...limiting regime in which the network topology, defined by J, R and the matrix (Ajr), is held fixed, while both the arrival rates νr, r ∈ R, and the capacities Cj, j ∈ J, increase in proportion. Whitt =-=[16]-=-, and Ziedins and Kelly [20] consider what has come to be known as the diverse routing limit; here the numbers of resources and call types increase while the capacity of each resource and the total tr... |

43 |
Reduced Load Approximations for Multirate Loss Networks
- Chung, Ross
- 1993
(Show Context)
Citation Context ...ources block “as if” independently of each other. We are thus led to a set of fixed point equations in the probabilities Ljr, for which the existence—but not always the uniqueness, see Chung and Ross =-=[6]-=-—of a solution is guaranteed. In the same spirit, the probability Br that a call of type r is blocked is then taken to be given by 1 − Br = � (1 − Ljr). (2) j∈J The reduced load approximation is of co... |

37 |
Insensitivity of blocking probabilities in a circuit switching network
- Burman, Lehoczky, et al.
- 1984
(Show Context)
Citation Context ...of each type r ∈ R arrive as a Poisson process with rate νr and have identically distributed holding (or service) times, which we assume, without loss of generality, to have mean 1 (see Burman et al. =-=[3]-=-). Each call of type r requires (integer) capacity Ajr from resource j ∈ J for the duration of its holding time. If one or more of these resources does not have sufficient free capacity to carry the c... |

36 |
Congestion probabilities in a circuit- switched integrated services network
- Dziong, Roberts
- 1987
(Show Context)
Citation Context ...ing the normalising constant. However, a number of relatively fast and efficient methods have been suggested which do permit exact calculations to be made in certain circumstances. Dziong and Roberts =-=[7]-=- (see also Zachary [19]) give an exact recurrence based on consideration of the reduced state space which records only the total occupancy of each resource. Refinements of Buzen’s convolution algorith... |

31 |
Non-negative Matrices and Markov
- Seneta
- 1980
(Show Context)
Citation Context ...tive for m, n such that m+n ≤ C and are zero otherwise. It follows that the matrix Q 2 has entries which are all strictly positive. The result is now immediate from, for example, Theorem 1.2 of Seneta=-=[15]-=-. It follows that the blocking probabilities exhibit similar exponential convergence. In particular, as l, i and l − i tend to infinity, the single- and two-resource call blocking probabilities Bi and... |

24 | An algorithm to compute blocking probabilities in multi-rate multi-class multi-resource loss models
- Choudhury, Leung, et al.
- 1995
(Show Context)
Citation Context ...he total occupancy of each resource. Refinements of Buzen’s convolution algorithm [4] for closed queueing networks have also been applied to loss networks. Of particular interest are Choudhury et al. =-=[5]-=-, who invert the generating function of the partition function, and Bean and Stewart [1], who apply refined dimension reduction techniques to Buzen’s algorithm, and thus obtain considerable efficiency... |

24 |
Asymptotically optimal loss network control
- Hunt, Laws
- 1993
(Show Context)
Citation Context ...ally exact. (Further, in this case the EFPA may continue to perform well even when controls, such as trunk reservation or alternative routing are added to the network; see, for example, Hunt and Laws =-=[9]-=-, and MacPhee and Ziedins [13].) All these results extend easily to the more refined reduced load approximation. However, for networks with small capacities and/or highly linear topologies (see below)... |

19 |
Countable state space Markov random fields and Markov chains on trees
- Zachary
- 1983
(Show Context)
Citation Context ...d, an exact analysis of the stationary distribution π is relatively straightforward. For the general technique here (in the context of general Markov random fields on trees) see, for example, Zachary =-=[17]-=-, and for an application to a simple linear cellular radio network, see Kelly [10]. We now show that this idea may be extended to a considerably more general class of network topologies by considering... |

16 |
Limit theorems for loss networks with diverse routing
- Ziedins, Kelly
- 1989
(Show Context)
Citation Context ...e capacity. The approximation (3) reduces the number of variables in the fixed point equations referred to above. However, these may again have multiple solutions, see, for example, Ziedins and Kelly =-=[20]-=-. Blocking probabilities are calculated using (2) as before. In general the EFPA is not exact for a single-resource network. The reduced load and Erlang fixed point approximations coincide 2sin the ca... |

14 | On blocking in loss networks
- Zachary
- 1991
(Show Context)
Citation Context ...nstant. However, a number of relatively fast and efficient methods have been suggested which do permit exact calculations to be made in certain circumstances. Dziong and Roberts [7] (see also Zachary =-=[19]-=-) give an exact recurrence based on consideration of the reduced state space which records only the total occupancy of each resource. Refinements of Buzen’s convolution algorithm [4] for closed queuei... |

12 |
On critically loaded loss networks
- Hunt, Kelly
- 1989
(Show Context)
Citation Context ...he fixed point equations do have a unique solution, see Kelly [11], and also Ross [14]. The use of the EFPA has been justified by considering two limiting regimes. Kelly [11] (see also Hunt and Kelly =-=[8]-=-) shows that the EFPA is asymptotically exact in the Kelly limiting regime in which the network topology, defined by J, R and the matrix (Ajr), is held fixed, while both the arrival rates νr, r ∈ R, a... |

9 |
attractive and repulsive Markov specifications on trees and on the one-dimensional lattice. Stochastic Process
- Zachary, Bounded
- 1985
(Show Context)
Citation Context ... B (1) =1− B (2) =1− � C−1 n=0 λ(n)λ(n +1)νn 2 /n! � C n=0 λ(n)2 ν n 2 /n! (13) �C−1 n=0 λ(n +1)2νn 2 /n! �C n=0 λ(n)2ν n 2 /n! . (14) Some of the expressions found here have been seen before. Zachary=-=[18]-=- obtains the result (6) for the Markov random field which models the above network in the case ν1 = 0—corresponding to the presence of two-resource calls only. This latter model is also considered by ... |

8 | Improved fixed point methods for loss networks with linear structure
- Bebbington, Pollett, et al.
- 1997
(Show Context)
Citation Context ...a tree, the Markov random field approach once more permits an exact analysis. The expressions obtained in Section 5 for the blocking probabilities are suggested as approximations by Bebbington et al. =-=[2]-=-. The analyses we give also illustrate the exponential decay of spatial correlation in such networks. Although all the examples that we consider here have Ajr ∈{0,1}, there is no difficulty in impleme... |

5 |
Admission controls for loss networks with diverse routing
- MacPhee, Ziedins
- 1996
(Show Context)
Citation Context ... case the EFPA may continue to perform well even when controls, such as trunk reservation or alternative routing are added to the network; see, for example, Hunt and Laws [9], and MacPhee and Ziedins =-=[13]-=-.) All these results extend easily to the more refined reduced load approximation. However, for networks with small capacities and/or highly linear topologies (see below), neither of the above approxi... |

3 | A new perspective on the normalisation of invariant measures for loss networks and other product form systems
- Bean, Stewart
- 1998
(Show Context)
Citation Context ...r closed queueing networks have also been applied to loss networks. Of particular interest are Choudhury et al. [5], who invert the generating function of the partition function, and Bean and Stewart =-=[1]-=-, who apply refined dimension reduction techniques to Buzen’s algorithm, and thus obtain considerable efficiency gains. Due to the difficulties of calculating the normalising constant, various approxi... |