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## Heavy traffic analysis of open processing networks with complete resource pooling: asymptotic optimality of discrete review policies (2005)

Venue: | ANN. APPL. PROBAB |

Citations: | 25 - 0 self |

### Citations

2191 | Data Networks - Bertsekas, Gallager - 1987 |

926 |
Stochastic Process
- Ross
- 1996
(Show Context)
Citation Context ...k ≤ t).36 B. ATA AND S. KUMAR It follows that P(˜σ (1) k ( ≤ t) ≤ P sup ∣ i=0,1,...,⌊λkt+εt⌋ Uk(i) − i ∣ λk ) εt ≥ . 2λk It is straightforward to conclude by Markov’s inequality (cf. page 39 of Ross =-=[43]-=-) that P(˜σ (1) k ≤ t) ≤ E(sup i=0,1,...,⌊λkt+εt⌋ |Uk(i) − i/λk|) 2+2ε1 (εt/(2λk)) 2+2ε1 We then use Doob’s inequality (cf. page 15 of [12]) to get the following: P(˜σ (1) k 2 + 2ε1 ≤ t) ≤ 1 + 2ε1 E|U... |

921 | Martingale limit theory and its application - Hall, Heyde - 1980 |

538 |
Tsitsiklis. Introduction to Linear Optimization. Athena Scientific
- Bertsimas, N
- 1997
(Show Context)
Citation Context ... is degeneracy in the solution of the static planning problem, then the set of basic variables as defined above is not the same as the “basic” solution as understood in linear programming theory; see =-=[3]-=-. We partition x∗ as x ∗ [ x∗ ] (12) = B , where x∗ B is the b-dimensional vector of nominal basic activity levels and x∗ N = 0. It will be also convenient for our later purposes to partition the inpu... |

463 |
Brownian Motion and Stochastic Flow Systems
- Harrison
- 1985
(Show Context)
Citation Context ... y1 ˜ X ∗ ) W )(t) > x , where the third step follows from Fatou’s lemma. Moreover, ˜P(ϕ( ˜ X ∗ W )(t) ≥ x) = 2N ( −xy1 h1σ √ t where N(·) is the standard normal cumulative distribution function; see =-=[13]-=-. □ Acknowledgments. We are grateful to J. M. Harrison for sharing his ideas with us generously throughout the project and to R. J. Williams for a number of discussions on her related work. We also th... |

216 |
Convergence of Probability Measures, 2nd ed
- Billingsley
- 1999
(Show Context)
Citation Context ...ts on (0, ∞). The identically zero function in D k will be denoted by 0. For ω ∈ D k and T ≥ 0, we let ‖ω‖T = sup t∈[0,T] |ω(t)|. Consider D k to be endowed with the usual Skorohod (J1) topology (see =-=[4, 11]-=-). Let M k denote the Borel σ-algebra on D k associated with this topology. This is the same σ-algebra generated by the coordinate maps, that is, M k = σ{ω(s):0 ≤ s < ∞}. Each continuous-time (stochas... |

163 | MaxWeight Scheduling in a Generalized Switch: State Space Collapse and Equivalent Workload Minimization in Heavy Traffic
- Stolyar
- 2004
(Show Context)
Citation Context ...tablish convergence to the desired limiting diffusion. The class of network control problems whose equivalent workload formulation is one dimensional has received considerable research attention, see =-=[1, 16, 19, 36, 44]-=-. Two recent and major contributions to this area are the papers by Stolyar [44] and Mandelbaum and Stolyar [36]. These papers consider parallel server systems under the complete resource pooling assu... |

140 |
Brownian models of queueing networks with heterogeneous customer populations
- Harrison
- 1985
(Show Context)
Citation Context ...communication networks and call-centers (see, e.g., [2, 9, 25, 40, 54]). One approach to designing control policies for such networks is the heavy traffic approximation approach pioneered by Harrison =-=[14, 17]-=-. This approach can be summarized by the following procedure; see [7, 15, 51]. (a) Formulate a stochastic network model and a notion of heavy traffic. (b) Formulate an approximating Brownian control p... |

128 | State space collapse with application to heavy traffic limits for multiclass queueing networks - Bramson - 1998 |

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 ...ution as a control policy for the original processing network. Finally, there are very few proofs of asymptotic optimality of the interpreted policy even when such an interpretation has been advanced =-=[1, 16, 27, 28, 30, 35, 38, 42]-=-. In this paper we will carry out all four steps (a)–(d) for a large class of open network models. The crucial assumption for our analysis is what Harrison and Lopez [19] called complete resource pool... |

104 |
Stochastic-process limits: an introduction to stochastic-process limits and their application to queues
- Whitt
- 2002
(Show Context)
Citation Context ...measures induced by the W i on (Dk, Mk ) converge weakly to the probability measure induced on (Dk, Mk ) by W as i → ∞. For more on tightness and weak convergence of processes taking values in Dk see =-=[4, 11, 50]-=-. 3. Scheduling controls and network dynamics. We specify a scheduling policy or control by an n-dimensional continuous stochastic process T = {T(t),t ≥ 0}, where Tj(t) can be interpreted as the amoun... |

92 | Scheduling flexible servers with convex delay costs: Heavy-traffic optimality of the generalized cµ-rule
- Mandelbaum, Stolyar
- 2004
(Show Context)
Citation Context ...tablish convergence to the desired limiting diffusion. The class of network control problems whose equivalent workload formulation is one dimensional has received considerable research attention, see =-=[1, 16, 19, 36, 44]-=-. Two recent and major contributions to this area are the papers by Stolyar [44] and Mandelbaum and Stolyar [36]. These papers consider parallel server systems under the complete resource pooling assu... |

90 | Scheduling networks of queues: Heavy traffic analysis of a simple open network
- Harrison, Wein
- 1989
(Show Context)
Citation Context ...cy proposed in (c). In particular, determine whether it is asymptotically optimal in the heavy traffic limit. Even though the heavy traffic approach has proved successful in many particular examples, =-=[10, 21, 22, 29, 30, 31, 32, 37, 38, 46, 47, 48, 49]-=-, the complete procedure outlined above has not been resolved in general. The steps (a) and (b) have been resolved quite generally in the literature [7, 14, 17, 18, 20, 24, 32]. Steps (c) and (d) pres... |

86 | Brownian models of open processing networks: canonical representation of workload
- Harrison
- 2000
(Show Context)
Citation Context ...communication networks and call-centers (see, e.g., [2, 9, 25, 40, 54]). One approach to designing control policies for such networks is the heavy traffic approximation approach pioneered by Harrison =-=[14, 17]-=-. This approach can be summarized by the following procedure; see [7, 15, 51]. (a) Formulate a stochastic network model and a notion of heavy traffic. (b) Formulate an approximating Brownian control p... |

83 |
Linear Algebra and its Applications, 3rd ed
- Strang
- 1988
(Show Context)
Citation Context ...which contradicts (84). □ Proof of Lemma 2. We consider solving the following equation: [ ][ ] [ ] H 0 x e1 (87) = , B −e ρ 0 which has a unique solution by Lemma 5. By Cramér’s rule (see page 233 of =-=[45]-=-), we conclude We can write (87) as (88) (89) ρ = det[ H e1 B 0 ] H 0 det[ B −e ] Hx = e1, Bx = eρ. Premultiplying (88) by y ′ gives y ′ Hx = y1, and also by (89), (16) we have that y ′ Hx = π ′ Bx = ... |

68 |
Dynamic control of Brownian networks: state space collapse and equivalent workload formulations
- Harrison, Mieghem
- 1997
(Show Context)
Citation Context ...10, 21, 22, 29, 30, 31, 32, 37, 38, 46, 47, 48, 49], the complete procedure outlined above has not been resolved in general. The steps (a) and (b) have been resolved quite generally in the literature =-=[7, 14, 17, 18, 20, 24, 32]-=-. Steps (c) and (d) present several difficulties. First, the approximating Brownian control problem is not always analytically tractable. Second, even when the Brownian control problem is tractable, i... |

67 |
Heavy traffic resource pooling in parallel server systems
- Harrison, Lopez
- 1999
(Show Context)
Citation Context ...ed [1, 16, 27, 28, 30, 35, 38, 42]. In this paper we will carry out all four steps (a)–(d) for a large class of open network models. The crucial assumption for our analysis is what Harrison and Lopez =-=[19]-=- called complete resource pooling (CRP), extended to a more general network setting similar to that considered by Bramson and Williams [8]. Roughly speaking, the CRP assumption requires enough overlap... |

62 | Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling
- Kelly, Laws
- 1993
(Show Context)
Citation Context ...10, 21, 22, 29, 30, 31, 32, 37, 38, 46, 47, 48, 49], the complete procedure outlined above has not been resolved in general. The steps (a) and (b) have been resolved quite generally in the literature =-=[7, 14, 17, 18, 20, 24, 32]-=-. Steps (c) and (d) present several difficulties. First, the approximating Brownian control problem is not always analytically tractable. Second, even when the Brownian control problem is tractable, i... |

55 | Sequencing and routing in multiclass queueing networks. part i: Feedback regulation
- Meyn
- 2001
(Show Context)
Citation Context ... for the case where the system is near heavy traffic as well. This point will be elaborated on later. All discrete review policies described to date in the literature of heavy traffic network control =-=[15, 33, 34, 39]-=- require the system manager to solve a new linear programming problem at each review point. However, by fully exploiting the special structure of CRP networks, we arrive at a far simpler type of polic... |

54 | Heavy traffic analysis of a system with parallel servers: Asymptotic optimality of discrete-review policies - Harrison - 1998 |

51 | Dynamic scheduling with convex delay costs: The generalized c
- Mieghem
- 1995
(Show Context)
Citation Context ...cy proposed in (c). In particular, determine whether it is asymptotically optimal in the heavy traffic limit. Even though the heavy traffic approach has proved successful in many particular examples, =-=[10, 21, 22, 29, 30, 31, 32, 37, 38, 46, 47, 48, 49]-=-, the complete procedure outlined above has not been resolved in general. The steps (a) and (b) have been resolved quite generally in the literature [7, 14, 17, 18, 20, 24, 32]. Steps (c) and (d) pres... |

50 |
Discrete-review policies for scheduling stochastic networks: trajectory tracking and fluid-scale asymptotic optimality
- Maglaras
- 2000
(Show Context)
Citation Context ...l at zero while keeping all servers busy, is not easy to interpret in the original stochastic network. We provide an interpretation based on the discrete review approach of Harrison [15] and Maglaras =-=[33, 34]-=-. Our interpretation provides a policy that reviews the contents of the buffers at discrete points in time, computes a processing plan based on the observed contents (interpreting zeros in the Brownia... |

45 |
The BIGSTEP approach to flow management in stochastic processing networks, Stochastic Networks: Theory and Applications
- Harrison
- 1996
(Show Context)
Citation Context ...ne approach to designing control policies for such networks is the heavy traffic approximation approach pioneered by Harrison [14, 17]. This approach can be summarized by the following procedure; see =-=[7, 15, 51]-=-. (a) Formulate a stochastic network model and a notion of heavy traffic. (b) Formulate an approximating Brownian control problem for the network control problem, and reduce the dimension of this prob... |

42 |
Resource pooling in queueing networks with dynamic routing
- Laws
- 1992
(Show Context)
Citation Context ...cy proposed in (c). In particular, determine whether it is asymptotically optimal in the heavy traffic limit. Even though the heavy traffic approach has proved successful in many particular examples, =-=[10, 21, 22, 29, 30, 31, 32, 37, 38, 46, 47, 48, 49]-=-, the complete procedure outlined above has not been resolved in general. The steps (a) and (b) have been resolved quite generally in the literature [7, 14, 17, 18, 20, 24, 32]. Steps (c) and (d) pres... |

42 |
An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Systems Theory Appl
- Williams
- 1998
(Show Context)
Citation Context ...tructure of this policy, and fully exploiting the CRP assumption, we are able to prove asymptotic optimality in a conceptually simple fashion. Our proof uses the roadmap given by Bramson and Williams =-=[5, 52, 53]-=-, but it does not invoke fluid limits. Rather, the proof establishes state space collapse (i.e., all buffer levels but one are zero in the diffusion limit) directly, and then uses the continuity of th... |

38 | A broader view of Brownian networks - Harrison - 2001 |

33 | Heavy traffic limits for some queueing networks
- Bramson, Dai
- 2001
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Citation Context ...h column and at least one nonzero entry in each row. For an example of a network that fits in our modeling framework; see [26]. 2.1. Stochastic primitives. Following the exposition of Bramson and Dai =-=[6]-=-, we associate with each buffer k = 1,...,m a sequence of independent and identically distributed strictly positive random variables ūk = {ūk(i),i ≥ 1} and a λk ≥ 0, where it is assumed that E(ūk(1)) ... |

23 |
Routing and singular control for queueing networks in heavy traffic
- Martins, Kushner
- 1990
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Citation Context |

21 | Two workload properties for Brownian networks. Queueing Syst
- Bramson, Williams
- 2003
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Citation Context ...ucial assumption for our analysis is what Harrison and Lopez [19] called complete resource pooling (CRP), extended to a more general network setting similar to that considered by Bramson and Williams =-=[8]-=-. Roughly speaking, the CRP assumption requires enough overlap in the processing capabilities of the various servers to ensure that their capacities are exchangeable or transferable in the heavy traff... |

21 |
Optimal and approximately optimal control policies for queues in heavy traffic
- Kushner, Ramachandran
- 1989
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Citation Context |

20 |
Dynamic scheduling in multiclass queueing networks: Stability under discrete-review policies. Queueing Systems Theory Appl
- Maglaras
- 1999
(Show Context)
Citation Context ...l at zero while keeping all servers busy, is not easy to interpret in the original stochastic network. We provide an interpretation based on the discrete review approach of Harrison [15] and Maglaras =-=[33, 34]-=-. Our interpretation provides a policy that reviews the contents of the buffers at discrete points in time, computes a processing plan based on the observed contents (interpreting zeros in the Brownia... |

20 |
Heavy traffic convergence of a controlled, multi-class queuing system
- Martins, Shreve, et al.
- 1996
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Citation Context |

17 |
The Equivalence of Functional Central Limit Theorems of Counting Processes and Associated Partial Sums
- Iglehart, Whitt
- 1969
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Citation Context ...ribution to a nondegenerate limit. This can be proved by using the continuous mapping theorem, random time change theorem (cf. [4]) and the fact that S r j (s) → µjs as r → ∞ for every s ∈ [0,T] (cf. =-=[23]-=- for a proof). On the other hand, since X r = 1 r ̂ Xr , we have that We also conclude by (74) that ‖X r ‖T ⇒ 0 as r → ∞. ‖X r W ‖T ⇒ 0 as r → ∞. Since the one-dimensional regulator map commutes with ... |

13 | Optimal control of assignment of jobs to processors under heavy traffic
- Kushner, Chen
- 1998
(Show Context)
Citation Context ...ution as a control policy for the original processing network. Finally, there are very few proofs of asymptotic optimality of the interpreted policy even when such an interpretation has been advanced =-=[1, 16, 27, 28, 30, 35, 38, 42]-=-. In this paper we will carry out all four steps (a)–(d) for a large class of open network models. The crucial assumption for our analysis is what Harrison and Lopez [19] called complete resource pool... |

12 | Heavy traffic analysis of controlled multiplexing systems, Queueing systems
- KUSHNER
- 1998
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Citation Context |

12 | A multiclass queue in heavy traffic with throughput time constraints: Asymptotically optimal dynamic controls. Queueing Systems
- Plambeck, Kumar, et al.
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Citation Context ...ution as a control policy for the original processing network. Finally, there are very few proofs of asymptotic optimality of the interpreted policy even when such an interpretation has been advanced =-=[1, 16, 27, 28, 30, 35, 38, 42]-=-. In this paper we will carry out all four steps (a)–(d) for a large class of open network models. The crucial assumption for our analysis is what Harrison and Lopez [19] called complete resource pool... |

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

10 |
Production and Operations Analysis," 3 rd ed
- Nahmias
- 1997
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Citation Context ...f of Proposition 2 1. Introduction. Stochastic processing networks have been extensively used to model manufacturing systems, computer systems, telecommunication networks and call-centers (see, e.g., =-=[2, 9, 25, 40, 54]-=-). One approach to designing control policies for such networks is the heavy traffic approximation approach pioneered by Harrison [14, 17]. This approach can be summarized by the following procedure; ... |

10 |
Stochastic Modeling and Analysis of Manufacturing Systems
- Yao, Ed
- 1994
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Citation Context ...f of Proposition 2 1. Introduction. Stochastic processing networks have been extensively used to model manufacturing systems, computer systems, telecommunication networks and call-centers (see, e.g., =-=[2, 9, 25, 40, 54]-=-). One approach to designing control policies for such networks is the heavy traffic approximation approach pioneered by Harrison [14, 17]. This approach can be summarized by the following procedure; ... |

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 ...ne approach to designing control policies for such networks is the heavy traffic approximation approach pioneered by Harrison [14, 17]. This approach can be summarized by the following procedure; see =-=[7, 15, 51]-=-. (a) Formulate a stochastic network model and a notion of heavy traffic. (b) Formulate an approximating Brownian control problem for the network control problem, and reduce the dimension of this prob... |

9 |
Brownian networks with discretionary routing
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7 |
Limit theorems for pathwise average cost per unit time problems for controlled queues in heavy traffic
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6 | Continuous-review tracking policies for dynamic control of stochastic networks. Queueing Systems
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5 |
Stochastic Analysis of Manufacturing Systems
- Buzacott, Shantikumar
- 1993
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Citation Context ...f of Proposition 2 1. Introduction. Stochastic processing networks have been extensively used to model manufacturing systems, computer systems, telecommunication networks and call-centers (see, e.g., =-=[2, 9, 25, 40, 54]-=-). One approach to designing control policies for such networks is the heavy traffic approximation approach pioneered by Harrison [14, 17]. This approach can be summarized by the following procedure; ... |

3 | Scheduling open queueing networks with sufficiently flexible resources
- Kumar
- 1999
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Citation Context ...ces of zeros and ones such that each contains exactly one nonzero entry in each column and at least one nonzero entry in each row. For an example of a network that fits in our modeling framework; see =-=[26]-=-. 2.1. Stochastic primitives. Following the exposition of Bramson and Dai [6], we associate with each buffer k = 1,...,m a sequence of independent and identically distributed strictly positive random ... |

1 |
Queueing Systems: Computer Applications II
- Kleinrock
- 1976
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Citation Context |

1 |
Reflecting Brownian motions and queueing networks. Unpublished manuscript
- Nguyen, Williams
- 1996
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Citation Context ... (iii)(c). Thus, τε = ∞ for each ε > 0, and, hence, y(t) ≤ ˜y(t) ∀t ∈ [0,T] as desired. We now prove (102). To this end, we first define ˆw(t) = x(t) + ˆy(t) for t ∈ [0,T]. It is well known that (cf. =-=[41]-=-) ˆw,x, ˆy jointly satisfy: (i)′ (ii)′ ˆw(t) = x(t) + ˆy(t) ∀t ∈ [0,T], ˆw(t) ≥ 0 ∀t ∈ [0,T], (iii) ′ (a) ˆy(0) = 0, (b) ˆy is nondecreasing, (c) ∫ [0,T] (0,∞)(w(t))dˆy(t) = 0.34 B. ATA AND S. KUMAR ... |

1 |
Diffusion approximations for open multiclass networks: Sufficient conditions for state space collapse. Queueing Systems Theory Appl
- Williams
- 1998
(Show Context)
Citation Context ...tructure of this policy, and fully exploiting the CRP assumption, we are able to prove asymptotic optimality in a conceptually simple fashion. Our proof uses the roadmap given by Bramson and Williams =-=[5, 52, 53]-=-, but it does not invoke fluid limits. Rather, the proof establishes state space collapse (i.e., all buffer levels but one are zero in the diffusion limit) directly, and then uses the continuity of th... |