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## Two Workload Properties for Brownian Networks (2003)

Citations: | 21 - 3 self |

### Citations

2191 | Data Networks
- Bertsekas, Gallager
- 1987
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Citation Context ...Graduate School of Business, Stanford University. 1s1 Introduction Open stochastic processing networks are used as models for complex manufacturing, telecommunications and computer systems (see e.g., =-=[5, 40]-=-). Some of these networks allow for flexible scheduling through dynamic (that is, state-dependent) alternate routing and sequencing, hereafter collectively referred to as dynamic scheduling. Usually, ... |

538 |
Tsitsiklis. Introduction to Linear Optimization. Athena Scientific
- Bertsimas, N
- 1997
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Citation Context ..., the standard form of (LP ) has a constraint matrix with I + K linearly independent rows and I + K − (B + 1) is the degree of degeneracy of the basic solution (ρ ∗ ,x ∗ ) of the linear program (cf., =-=[6]-=-, page 59). The quantity in the right side of (1) is then the degree of degeneracy plus one. Examples. A simple way to satisfy the hypothesis of Corollary 6.2 is to require that the matrix A have at m... |

163 | MaxWeight Scheduling in a Generalized Switch: State Space Collapse and Equivalent Workload Minimization in Heavy Traffic
- Stolyar
- 2004
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Citation Context ...ces may be possible. This feature may result in a reduction in dimension considerably beyond that which is achievable in Brownian networks associated with open multiclass queueing networks, see e.g., =-=[15, 20, 39, 2, 3, 37, 33, 1]-=-. It is a natural problem to determine conditions under which the dimension reduction is significant and to obtain other interesting properties of workload processes. This paper is a contribution in t... |

139 |
Brownian Models of Queueing Networks with Heterogeneous Customer Populations
- Harrison
- 1988
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Citation Context ...ly and Laws [23] have argued that in this regime “important features of good control policies are displayed in sharpest relief”. In a series of successively more general papers, J. M. Harrison et al. =-=[13, 14, 21, 16]-=- have proposed and developed various aspects of a scheme for using Brownian networks (formal heavy traffic diffusion approximations) and associated control problems, called Brownian control problems (... |

128 |
State space collapse with application to heavy traffic limits for multiclass queueing networks
- Bramson
- 1998
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Citation Context ...unterpart for controlled networks of the state-space collapse that has played a fundamental role in analyzing the performance of open multiclass queueing networks with fixed scheduling policies (cf., =-=[7, 38]-=-). Furthermore, in networks with alternate routing capabilities, a buffer may be served by more than one server and pooling of server resources may be possible. This feature may result in a reduction ... |

118 | 2001. Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: Asymptotic optimality of a threshold policy
- Bell, Williams
(Show Context)
Citation Context ...ces may be possible. This feature may result in a reduction in dimension considerably beyond that which is achievable in Brownian networks associated with open multiclass queueing networks, see e.g., =-=[15, 20, 39, 2, 3, 37, 33, 1]-=-. It is a natural problem to determine conditions under which the dimension reduction is significant and to obtain other interesting properties of workload processes. This paper is a contribution in t... |

109 |
Diffusion Approximations for Open Multi Class Queueing Networks: Sufficient Conditions Involving State Space Collapse
- Williams
- 1998
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Citation Context ...unterpart for controlled networks of the state-space collapse that has played a fundamental role in analyzing the performance of open multiclass queueing networks with fixed scheduling policies (cf., =-=[7, 38]-=-). Furthermore, in networks with alternate routing capabilities, a buffer may be served by more than one server and pooling of server resources may be possible. This feature may result in a reduction ... |

92 | Scheduling flexible servers with convex delay costs: Heavy-traffic optimality of the generalized cµ-rule
- Mandelbaum, Stolyar
- 2004
(Show Context)
Citation Context ...ces may be possible. This feature may result in a reduction in dimension considerably beyond that which is achievable in Brownian networks associated with open multiclass queueing networks, see e.g., =-=[15, 20, 39, 2, 3, 37, 33, 1]-=-. It is a natural problem to determine conditions under which the dimension reduction is significant and to obtain other interesting properties of workload processes. This paper is a contribution in t... |

89 | Scheduling Networks of Queues: Heavy Traffic Analysis of a TwoStation Network With Controllable Inputs
- M
- 1990
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Citation Context ...n to some essential aspects of parts (I)–(III) of the scheme for approximating controlled stochastic processing networks. Although much of this material could be gleaned by careful study of the works =-=[13, 16, 21, 20, 22]-=-, the presentation given here is intended to provide a helpful introduction to those works. As a guide to the reader, we now elaborate on the content of Sections 2–7. In Section 2, we introduce the ne... |

86 | Brownian models of open processing networks: canonical representation of workload
- Harrison
- 2000
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Citation Context ...we derive a formula for the dimension of such processes. For a given Brownian network, this dimension is unique. However, there are infinitely many possible choices for the workload process. Harrison =-=[16]-=- has proposed a “canonical” choice, which reduces the possibilities to a finite number. Our second result provides sufficient conditions for this canonical choice to be valid and for it to yield a non... |

74 |
Activity Analysis of Production and Allocation
- Koopmans
- 1951
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Citation Context ...dels for describing such processing networks have a long history in economics and operations research. For example, the book “Activity Analysis of Production and Allocation”, edited by T. C. Koopmans =-=[25]-=-, provides an excellent summary of the early stages of development of such models. A prominent role is played there by the notion of a processing activity, which consumes certain kinds of material, pr... |

71 | 2005. Dynamic scheduling of a parallel server system in heavy traffic with complete resource pooling: Asymptotic optimality of a threshold policy
- Bell, Williams
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68 |
Mieghem. Dynamic control of brownian networks: State space collapse and equivalent workload formulations
- Harrison, Van
- 1996
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Citation Context ... Definition Harrison [16] has described a procedure for formally approximating an open stochastic processing network, whose first-order data (R, A, α) satisfy Assumption HT, by a Brownian network. In =-=[21]-=-, Harrison and Van Mieghem showed that a Brownian network can be reduced to an equivalent model of typically lower dimension by replacing the “queue length” process by a “workload” process. In essence... |

67 |
Heavy traffic resource pooling in parallel server systems
- Harrison, Lopez
- 1999
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62 | Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling
- Kelly, Laws
- 1993
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Citation Context ...els are relevant in the heavy traffic (or critically loaded) regime where the load placed on the system is approximately balanced by the system’s ability to process that load. In fact, Kelly and Laws =-=[23]-=- have argued that in this regime “important features of good control policies are displayed in sharpest relief”. In a series of successively more general papers, J. M. Harrison et al. [13, 14, 21, 16]... |

55 | Sequencing and routing in multiclass queueing networks. part i: Feedback regulation
- Meyn
- 2001
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Citation Context ...er (or law of large number) approximations to the original open stochastic processing networks. Asymptotic regimes in which these models are relevant include those of large initial queue lengths (see =-=[31, 35]-=- and references therein) and large numbers of buffers or servers (see [30] and references therein). Frequently, these models are used for analyzing “subcritical” systems where the load placed on the s... |

54 |
Heavy traffic analysis of a system with parallel servers: Asymptotic optimality of discrete-review policies
- Harrison
- 1998
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50 |
Discrete-review policies for scheduling stochastic networks: trajectory tracking and fluid-scale asymptotic optimality
- Maglaras
- 2000
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Citation Context ...er (or law of large number) approximations to the original open stochastic processing networks. Asymptotic regimes in which these models are relevant include those of large initial queue lengths (see =-=[31, 35]-=- and references therein) and large numbers of buffers or servers (see [30] and references therein). Frequently, these models are used for analyzing “subcritical” systems where the load placed on the s... |

45 | The BIGSTEP approach to flow management in stochastic processing networks, Stochastic Networks: Theory and Applications - Harrison - 1996 |

38 |
A broader view of brownian networks
- Harrison
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Citation Context ...rly general modeling framework for heavily loaded open stochastic processing networks that is used in Harrison’s canonical workload paper [16], although with some modifications of notation. (Harrison =-=[18, 19]-=- has recently proposed an even more general model framework with a broader notion of heavy traffic than that in [16]. Since relevant supporting equivalent workload theory has not yet been developed fo... |

25 | Heavy traffic analysis of open processing networks with complete resource pooling: Asymptotic optimality of discrete review policies
- Ata, Kumar
- 2005
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20 |
Heavy traffic convergence of a controlled, multi-class queuing system
- Martins, Shreve, et al.
- 1996
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Citation Context ...routed next to buffer i. A specific example of an open multiclass queueing network is depicted in Figure 1. This is the socalled criss-cross network. It has been studied by various authors, see e.g., =-=[22, 10, 12, 34, 29, 32]-=-. This network is arguably the first example where the Brownian network methodology was successfully applied to obtain a “good” control policy, cf. [22]. For this example, α =(α1,α2,0), ⎡ β1 ⎢ R = ⎢ 0... |

13 | Optimal control of assignment of jobs to processors under heavy traffic
- Kushner, Chen
- 1998
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Citation Context ... routing control, is exercised before incoming material enters a buffer. Kelly and Laws [23] have summarized the literature on such systems up through 1993. For more recent work, see Kushner and Chen =-=[28]-=- and the references therein.) A primary feature of the parallel server model of Harrison and López [20] is that material is processed only once before it exits the system. In this model, each activity... |

12 | Heavy traffic analysis of controlled multiplexing systems, Queueing systems
- KUSHNER
- 1998
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Citation Context ...routed next to buffer i. A specific example of an open multiclass queueing network is depicted in Figure 1. This is the socalled criss-cross network. It has been studied by various authors, see e.g., =-=[22, 10, 12, 34, 29, 32]-=-. This network is arguably the first example where the Brownian network methodology was successfully applied to obtain a “good” control policy, cf. [22]. For this example, α =(α1,α2,0), ⎡ β1 ⎢ R = ⎢ 0... |

11 | Two-server closed networks in heavy traffic: Diffusion limits and asymptotic optimality - Kumar |

10 |
Control and scheduling in a two-station queueing network: optimal policies, heuristics, and conjectures
- Chen, Yang, et al.
- 1991
(Show Context)
Citation Context ...routed next to buffer i. A specific example of an open multiclass queueing network is depicted in Figure 1. This is the socalled criss-cross network. It has been studied by various authors, see e.g., =-=[22, 10, 12, 34, 29, 32]-=-. This network is arguably the first example where the Brownian network methodology was successfully applied to obtain a “good” control policy, cf. [22]. For this example, α =(α1,α2,0), ⎡ β1 ⎢ R = ⎢ 0... |

9 | On dynamic scheduling of stochastic networks in heavy traffic and some new results for the workload process
- Bramson, Williams
- 2000
(Show Context)
Citation Context ...work (in the sense of Harrison [16]) whose firstorder data is unitary according to the following definition. (Those seeking a more detailed description of a unitary network may consult our prior work =-=[8]-=- which provides a detailed stochastic model for a class of unitary networks, where each activity consumes material from exactly one buffer, the processing of each activity is performed by exactly one ... |

8 | Data networks. Prentice-Hall, Englewood Cliffs - Bertsekas, Gallager - 1987 |

6 |
The Programing of Interdependent Activities: Mathematical Model. In: T.C. Coopmans, Activity Analys is of Production and Allocation
- Dantzig
- 1951
(Show Context)
Citation Context ...ved from 6shis earlier work [13] on sequencing problems for multiclass queueing networks) is a stochastic analogue of dynamic deterministic production models such as those first considered by Dantzig =-=[11]-=-. (See Harrison [19] for more on this connection.) An especially desirable feature of Harrison’s model is that it is broad enough to include several familiar categories of processing networks as speci... |

6 | Continuous-review tracking policies for dynamic control of stochastic networks. Queueing Systems
- MAGLARAS
- 2002
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Citation Context |

5 |
Stochastic Analysis of Manufacturing Systems
- Buzacott, Shantikumar
- 1993
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Citation Context ...etwork. Such networks arise in modeling systems where material may be split up, or combined with other material, at various stages of processing. Common applications occur in manufacturing (see e.g., =-=[9]-=-); the term disassembly-assembly is sometimes used instead of fork-join in this context. (This model does not require synchronization, i.e., a joining activity need not (re)combine units of material t... |

4 |
Threshold based priority policies for parallel-server systems with affinity scheduling
- Squillante, Xia, et al.
- 2001
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Citation Context ...r Systems The simplest unitary networks in which a buffer can be served by more than one server are the parallel server systems introduced in the heavy traffic context by Harrison and López [20]; see =-=[1, 2, 3, 33, 36, 37, 39]-=- for subsequent related work. (Other authors have studied similar systems where the model setup is slightly different; in those systems, the scheduling control, also called alternate routing control, ... |

3 |
Optimization over Leontief Substitution Systems. North Holland/American
- KOEHLER, WHINSTON, et al.
- 1975
(Show Context)
Citation Context .... For further details on the mathematical theory of Leontief matrices, we refer the interested reader to Chapter 9 of Berman and Plemmons [4], for square matrices, and to Koehler, Whinston and Wright =-=[24]-=-, more generally. In the following assumption, the positive part H + of the matrix H is defined by replacing all of the negative entries in H by zeros. Assumption DCH (Decomposition of H). The matrix ... |

3 | Scheduling open queueing networks with sufficiently flexible resources
- Kumar
- 1999
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Citation Context ...side it represents the routing of material. The details of the routing mechanism are not apparent from the figure; they are described more fully below.) This example is taken from a paper by S. Kumar =-=[27]-=- where motivation is provided by a job shop application. For this network, α =(α1,α2,0). It is assumed that material from buffer 2, after being processed at either server 1 or 2, is routed to buffer 3... |

1 |
Dynamic scheduling of a multiserver system with complete resource pooling: asymptotic optimality of a threshold policy in heavy traffic, preprint
- Bell, Williams
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1 |
Scheduling in a multi-class series of queues with deterministic service times, Queueing Systems
- Hariharan, Moustafa, et al.
- 1996
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1 |
Stochastic networks and activity analysis, to appear
- Harrison
- 2002
(Show Context)
Citation Context ...rly general modeling framework for heavily loaded open stochastic processing networks that is used in Harrison’s canonical workload paper [16], although with some modifications of notation. (Harrison =-=[18, 19]-=- has recently proposed an even more general model framework with a broader notion of heavy traffic than that in [16]. Since relevant supporting equivalent workload theory has not yet been developed fo... |

1 |
Alternative routeing in fully-connected queueing networks
- Laws, Teh
(Show Context)
Citation Context ...cessing networks. Asymptotic regimes in which these models are relevant include those of large initial queue lengths (see [31, 35] and references therein) and large numbers of buffers or servers (see =-=[30]-=- and references therein). Frequently, these models are used for analyzing “subcritical” systems where the load placed on the system is less than the capacity of the system to process that load. While ... |