## Heavy Traffic Limits for Some Queueing Networks (2001)

Venue: | Annals of Applied Probability |

Citations: | 18 - 1 self |

### BibTeX

@ARTICLE{Bramson01heavytraffic,

author = {Maury Bramson and J. G. Dai},

title = {Heavy Traffic Limits for Some Queueing Networks},

journal = {Annals of Applied Probability},

year = {2001},

volume = {11},

pages = {49--88}

}

### OpenURL

### Abstract

Using a slight modification of the framework in Bramson [7] and Williams [52], we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single station systems operating under first-in first-out (FIFO), generalized head-of-the-line proportional processor sharing (GHLPPS) and static buffer priority (SBP) service disciplines. The next two families are re-entrant lines operating under first-buffer-first-serve (FBFS) and last-buffer-first-serve (LBFS) service disciplines; the last family consists of certain 2-station, 5-class networks operating under an SBP service discipline. Some of these heavy traffic limits have appeared earlier in the literature; our new proofs demonstrate the significant simplifications that can be achieved in the present setting.

### Citations

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Citation Context ...ffusion approximation. 1 Introduction Queueing networks have been extensively used to model computer systems, telecommunications networks, and manufacturing systems (see, e.g., Bertsekas and Gallager =-=[2]-=-, and Yao [54]). Classical queueing network theory imposes restrictive assumptions on the distributions of the interarrival and service times, and on the service disciplines employed in a queueing net... |

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Citation Context ...network theory imposes restrictive assumptions on the distributions of the interarrival and service times, and on the service disciplines employed in a queueing network (Jackson [35], Baskett et. al. =-=[1]-=- and Kelly [37]). These restrictions exclude the use of such theory for many practical systems. Brownian model approximations have been employed as an alternative tool for more general queueing networ... |

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Citation Context ...). Classical queueing network theory imposes restrictive assumptions on the distributions of the interarrival and service times, and on the service disciplines employed in a queueing network (Jackson =-=[35]-=-, Baskett et. al. [1] and Kelly [37]). These restrictions exclude the use of such theory for many practical systems. Brownian model approximations have been employed as an alternative tool for more ge... |

101 | Distributed scheduling based on due dates and buffer priorities
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Citation Context ... network counterparts determine the evolution of the corresponding queueing network uniquely. This is, for example, the case for the fluid model that corresponds to the well known Lu-Kumar network in =-=[40]-=-. (Dai and Weiss [22, Section 5] presented a divergent fluid solution with Z(0) = 0; another solution is given by Z(·) ≡ 0.) The FIFO fluid model equations consist of (4.16)-(4.21), together with for ... |

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State space collapse with application to heavy traffic limits for multiclass queueing networks, Queueing Syst
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Citation Context ...tems Engineering and School of Mathematics Georgia Institute of Technology, Atlanta, GA 30332-0205 dai@isye.gatech.edu August 30, 1999 Abstract Using a slight modification of the framework in Bramson =-=[7]-=- and Williams [52], we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single station systems operating under first-in first-out (FIFO... |

83 |
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Citation Context ...rovkov [3, 4] and Iglehart [32]. Heavy traffic limits, in the form of functional central limit theorems, were first studied by Iglehart and Whitt [33, 34]; a survey can be found in Whitt [49]. Reiman =-=[43]-=- proved a heavy traffic limit theorem for single-class networks; his proof was simplified by Johnson [36] by studying the corresponding fluid models. (Readers are referred to Chen and Mandelbaum [8] f... |

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Citation Context ...hich dates back to Kingman [38, 39], Prohorov [42], Borovkov [3, 4] and Iglehart [32]. Heavy traffic limits, in the form of functional central limit theorems, were first studied by Iglehart and Whitt =-=[33, 34]-=-; a survey can be found in Whitt [49]. Reiman [43] proved a heavy traffic limit theorem for single-class networks; his proof was simplified by Johnson [36] by studying the corresponding fluid models. ... |

74 | Personal Communication
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Citation Context ...Bramson [7] and Williams [52]. Presumably, this framework can be employed for further heavy traffic limit theorems. It has been reported to us that, in a contemporaneous independent work, Chen and Ye =-=[9]-=- have shown a heavy traffic limit theorem for LBFS re-entrant lines by using a related framework from Chen and Zhang [13]. The paper is organized as follows. Multiclass networks are introduced in Sect... |

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Citation Context ...and School of Mathematics Georgia Institute of Technology, Atlanta, GA 30332-0205 dai@isye.gatech.edu August 30, 1999 Abstract Using a slight modification of the framework in Bramson [7] and Williams =-=[52]-=-, we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single station systems operating under first-in first-out (FIFO), generalized hea... |

67 |
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Citation Context ...rownian motion minus the drift is always a martingale with respect to its own filtration. If the reflection matrix R satisfies an appropriate spectral radius condition, such as in Harrison and Reiman =-=[29]-=-, the strong solution always exists and is unique. 3.6 Heavy traffic limit theorems We state here the heavy traffic limit theorems 3.1-3.6, which are the main results of the paper. For these results, ... |

51 | Existence and Uniqueness of Semimartingale Reflecting Brownian Motion in Convex Polyhedrons. Theory of Probability and its Applications
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Citation Context ... jth column of R. The parameters θ, Γ and R are called the drift vector, covariance matrix and reflection matrix of the SRBM, respectively. Results of Reiman and Williams [46] and Taylor and Williams =-=[47]-=- show that a necessary and sufficient condition for the existence and uniqueness (in distribution) of the SRBM associated with (RJ +, θ, Γ, R, ν), for each initial distribution ν on (RJ +, B), is that... |

46 |
Brownian models of multiclass queueing networks: current status and open problems. Queueing Systems 13
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Citation Context ...s exclude the use of such theory for many practical systems. Brownian model approximations have been employed as an alternative tool for more general queueing networks (see, e.g., Harrison and Nguyen =-=[27]-=-). They share two distinctive features: (a) the analysis of a Brownian model is mathematically more tractable than that of the corresponding queueing network, since a complicated Markov chain is repla... |

44 |
A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Related Fields 77
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Citation Context ... + : xj = 0}, is given by the jth column of R. The parameters θ, Γ and R are called the drift vector, covariance matrix and reflection matrix of the SRBM, respectively. Results of Reiman and Williams =-=[46]-=- and Taylor and Williams [47] show that a necessary and sufficient condition for the existence and uniqueness (in distribution) of the SRBM associated with (RJ +, θ, Γ, R, ν), for each initial distrib... |

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Citation Context ... multi-dimensional semimartingale reflecting Brownian (SRBM). Many quantities for the SRBM, including the stationary distribution, can be computed either exactly or numerically (Harrison and Williams =-=[30]-=-, Dai and Harrison [16]). Ideally, these should provide 1 Research supported in part by NSF grant DMS-9971248 2 Research supported in part by NSF grants DMI-9457336 and DMI-9813345, and by MaPhySto—Ce... |

36 |
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Citation Context ...gth limit will be strongly influenced by the service discipline for the networks in the sequence. The study of heavy traffic limits of queueing systems has a long history, which dates back to Kingman =-=[38, 39]-=-, Prohorov [42], Borovkov [3, 4] and Iglehart [32]. Heavy traffic limits, in the form of functional central limit theorems, were first studied by Iglehart and Whitt [33, 34]; a survey can be found in ... |

35 |
Networks of queues with customers of different types
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Citation Context ... imposes restrictive assumptions on the distributions of the interarrival and service times, and on the service disciplines employed in a queueing network (Jackson [35], Baskett et. al. [1] and Kelly =-=[37]-=-). These restrictions exclude the use of such theory for many practical systems. Brownian model approximations have been employed as an alternative tool for more general queueing networks (see, e.g., ... |

33 |
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Citation Context ...els. (Readers are referred to Chen and Mandelbaum [8] for a survey on single-class networks.) There have been a number of heavy traffic limits for multiclass queueing networks in Whitt [48], Peterson =-=[41]-=-, Reiman [45], Dai and Kurtz [17], Chen and Zhang [11, 12, 10], Bramson [7] and Williams [52]; Williams [51] provides a survey. Examples of non-existence of heavy traffic limits were given in Whitt [5... |

31 | A fluid-limit model criterion for instability of multiclass queueing networks
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Citation Context ...m convergence on compact intervals. One can show that each limit point ¯ X, of { ¯ X r (·, ω)}, is a fluid model solution to (4.16)-(4.21). (The reasoning is now quite standard, see, for example, Dai =-=[15]-=- for an analogous argument.) We claim that the queue length | ¯ Z(t)| grows linearly in t. For this, we observe that m2( ¯ Z1(t) + ¯ Z2(t)) + m5( ¯ Z1(t) + · · · + ¯ Z5(t)) = m2(t − ¯ D2(t)) + m5(t − ... |

31 | Reflected Brownian motion in an orthant: numerical methods for steady-state analysis
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Citation Context ...martingale reflecting Brownian (SRBM). Many quantities for the SRBM, including the stationary distribution, can be computed either exactly or numerically (Harrison and Williams [30], Dai and Harrison =-=[16]-=-). Ideally, these should provide 1 Research supported in part by NSF grant DMS-9971248 2 Research supported in part by NSF grants DMI-9457336 and DMI-9813345, and by MaPhySto—Centre for Mathematical P... |

31 |
Some diffusion approximations with state space collapse
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Citation Context ...with dimension equal to the number of classes in each network, will be a constant multiple of the workload limit. This last property is an example of state space collapse, a term first used in Reiman =-=[44]-=-, although such phenomena were observed earlier in Whitt [48] and Foschini and Salz [26]. The nature of the queue length limit will be strongly influenced by the service discipline for the networks in... |

28 |
An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Systems 30
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Citation Context ... R is not of Harrison and Reiman type, and the i.i.d. assumption is needed for the proof in this case. (For the last observation, see the appendix of Dai, Yeh and Zhou [23].) Bramson [6] and Williams =-=[53]-=- considered more general initial data than that assumed in Section 3.3. They allowed the service times and routing vectors for the initial jobs to have distributions 12sthat are different from those f... |

24 |
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(Show Context)
Citation Context ... are referred to Chen and Mandelbaum [8] for a survey on single-class networks.) There have been a number of heavy traffic limits for multiclass queueing networks in Whitt [48], Peterson [41], Reiman =-=[45]-=-, Dai and Kurtz [17], Chen and Zhang [11, 12, 10], Bramson [7] and Williams [52]; Williams [51] provides a survey. Examples of non-existence of heavy traffic limits were given in Whitt [50], and Dai a... |

21 |
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Citation Context ... re-entrant line, R is not of Harrison and Reiman type, and the i.i.d. assumption is needed for the proof in this case. (For the last observation, see the appendix of Dai, Yeh and Zhou [23].) Bramson =-=[6]-=- and Williams [53] considered more general initial data than that assumed in Section 3.3. They allowed the service times and routing vectors for the initial jobs to have distributions 12sthat are diff... |

21 | Nonexistence of Brownian models of certain multiclass queueing networks. Queueing Systems: Theory and Applications
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Citation Context ...r Mathematical Physics and Stochastics, funded by The Danish National Research Foundation 1sestimates for the corresponding queueing networks. Unfortunately, this is not always the case (Dai and Wang =-=[21]-=-); it is thus essential to determine when a Brownian model can be used for the analysis of a queueing network. This task is often carried out by establishing a heavy traffic limit for a sequence of re... |

21 |
Weak convergence theorems for priority queues: Preemptive-resume discipline
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Citation Context ..., will be a constant multiple of the workload limit. This last property is an example of state space collapse, a term first used in Reiman [44], although such phenomena were observed earlier in Whitt =-=[48]-=- and Foschini and Salz [26]. The nature of the queue length limit will be strongly influenced by the service discipline for the networks in the sequence. The study of heavy traffic limits of queueing ... |

19 |
Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks, Queueing Syst
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Citation Context ...ommon in telecommunication networks. When the weight vector β = (1, . . . , 1), the GHLPPS discipline becomes the head-of-the-line proportional processor sharing (HLPPS) service discipline in Bramson =-=[5, 7]-=- and Williams [52]. Under an SBP discipline, the classes at each station are assigned a fixed ranking. When the server switches from one job to another, the new job will be taken from the leading (or ... |

18 | A multiclass network with non-linear, non-convex, non-monotonic stability conditions. Queueing Systems: Theory and Applications
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Citation Context ...se of this behavior, classes 2 and 5 are said to form a virtual station; (3.21) is thus a virtual station condition. This type of behavior was first observed in Harrison and Nguyen [28], and in Dumas =-=[24]-=- for certain networks; it has been systematically employed in Dai and Vande Vate [19, 20]. 32sWe provide an abbreviated argument for Theorem 8.1. This is the only proof in the paper where we need to w... |

18 | Re Brownian motion on an orthant. The Annals of Probab - Harrison, Reiman - 1981 |

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Citation Context ...le of the workload limit. This last property is an example of state space collapse, a term first used in Reiman [44], although such phenomena were observed earlier in Whitt [48] and Foschini and Salz =-=[26]-=-. The nature of the queue length limit will be strongly influenced by the service discipline for the networks in the sequence. The study of heavy traffic limits of queueing systems has a long history,... |

17 |
The heavy traffic approximation in the theory of queues. Proc. Sympos. Congestion Theory (Chapel
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(Show Context)
Citation Context ...gth limit will be strongly influenced by the service discipline for the networks in the sequence. The study of heavy traffic limits of queueing systems has a long history, which dates back to Kingman =-=[38, 39]-=-, Prohorov [42], Borovkov [3, 4] and Iglehart [32]. Heavy traffic limits, in the form of functional central limit theorems, were first studied by Iglehart and Whitt [33, 34]; a survey can be found in ... |

17 | Distributed scheduling based on due dates and bu er priorities - Lu, Kumar - 1991 |

16 |
Some badly behaved closed queueing networks. Stochastic Networks. Editors
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Citation Context ...eive service. Because of this behavior, classes 2 and 5 are said to form a virtual station; (3.21) is thus a virtual station condition. This type of behavior was first observed in Harrison and Nguyen =-=[28]-=-, and in Dumas [24] for certain networks; it has been systematically employed in Dai and Vande Vate [19, 20]. 32sWe provide an abbreviated argument for Theorem 8.1. This is the only proof in the paper... |

16 |
Diffusion Approximations for Optimal Filtering of Jump Processes and for Queueing Networks
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Citation Context ...were first studied by Iglehart and Whitt [33, 34]; a survey can be found in Whitt [49]. Reiman [43] proved a heavy traffic limit theorem for single-class networks; his proof was simplified by Johnson =-=[36]-=- by studying the corresponding fluid models. (Readers are referred to Chen and Mandelbaum [8] for a survey on single-class networks.) There have been a number of heavy traffic limits for multiclass qu... |

14 | A Multiclass Station with Markovian Feedback in Heavy Traffic
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Citation Context ...n and Mandelbaum [8] for a survey on single-class networks.) There have been a number of heavy traffic limits for multiclass queueing networks in Whitt [48], Peterson [41], Reiman [45], Dai and Kurtz =-=[17]-=-, Chen and Zhang [11, 12, 10], Bramson [7] and Williams [52]; Williams [51] provides a survey. Examples of non-existence of heavy traffic limits were given in Whitt [50], and Dai and Nguyen [18]. Unco... |

13 |
Large fluctuations in a deterministic multiclass network of queues
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Citation Context ...1], Reiman [45], Dai and Kurtz [17], Chen and Zhang [11, 12, 10], Bramson [7] and Williams [52]; Williams [51] provides a survey. Examples of non-existence of heavy traffic limits were given in Whitt =-=[50]-=-, and Dai and Nguyen [18]. Unconventional heavy traffic limits were obtained in Harrison and Williams [31], and Coffman et. al. [14]. In this paper, we establish heavy traffic limits for six families ... |

13 |
On the approximation of queueing networks in heavy traffic. Pages 35–56 of
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Citation Context ...n a number of heavy traffic limits for multiclass queueing networks in Whitt [48], Peterson [41], Reiman [45], Dai and Kurtz [17], Chen and Zhang [11, 12, 10], Bramson [7] and Williams [52]; Williams =-=[51]-=- provides a survey. Examples of non-existence of heavy traffic limits were given in Whitt [50], and Dai and Nguyen [18]. Unconventional heavy traffic limits were obtained in Harrison and Williams [31]... |

12 | The stability of two-station multitype fluid networks
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Citation Context ...us a virtual station condition. This type of behavior was first observed in Harrison and Nguyen [28], and in Dumas [24] for certain networks; it has been systematically employed in Dai and Vande Vate =-=[19, 20]-=-. 32sWe provide an abbreviated argument for Theorem 8.1. This is the only proof in the paper where we need to work with random quantities; on account of Theorems 5.1-5.3, it sufficed to work with flui... |

12 | Open queueing networks in heavy trac - Reiman - 1984 |

10 | On the convergence of multiclass queueing networks in heavy traffic
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Citation Context ...Kurtz [17], Chen and Zhang [11, 12, 10], Bramson [7] and Williams [52]; Williams [51] provides a survey. Examples of non-existence of heavy traffic limits were given in Whitt [50], and Dai and Nguyen =-=[18]-=-. Unconventional heavy traffic limits were obtained in Harrison and Williams [31], and Coffman et. al. [14]. In this paper, we establish heavy traffic limits for six families of multiclass queueing ne... |

10 | The qnet method for re-entrant queueing networks with priority disciplines
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(Show Context)
Citation Context ...ype. For a LBFS re-entrant line, R is not of Harrison and Reiman type, and the i.i.d. assumption is needed for the proof in this case. (For the last observation, see the appendix of Dai, Yeh and Zhou =-=[23]-=-.) Bramson [6] and Williams [53] considered more general initial data than that assumed in Section 3.3. They allowed the service times and routing vectors for the initial jobs to have distributions 12... |

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9 | Global Stability of Two-Station Queueing Networks
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Citation Context ...guments used here also give a better idea of the type of reasoning that is often needed to verify the conditions of Theorem 5.3. (The networks in Theorem 3.6 were first examined in Dai and Vande Vate =-=[19]-=-.) Throughout the section, we assume that α1 = 1, except when specified otherwise. Also, as in the previous section, we set dk(t) = ˙ Dk(t) for the departure rate from a class k. Proof of Theorem 3.6.... |

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Citation Context ...for a survey on single-class networks.) There have been a number of heavy traffic limits for multiclass queueing networks in Whitt [48], Peterson [41], Reiman [45], Dai and Kurtz [17], Chen and Zhang =-=[11, 12, 10]-=-, Bramson [7] and Williams [52]; Williams [51] provides a survey. Examples of non-existence of heavy traffic limits were given in Whitt [50], and Dai and Nguyen [18]. Unconventional heavy traffic limi... |

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Citation Context ...[51] provides a survey. Examples of non-existence of heavy traffic limits were given in Whitt [50], and Dai and Nguyen [18]. Unconventional heavy traffic limits were obtained in Harrison and Williams =-=[31]-=-, and Coffman et. al. [14]. In this paper, we establish heavy traffic limits for six families of multiclass queueing networks. The first three families are single station systems operating under FIFO,... |

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