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## Conditional Dynamics of Non-Markovian, Infinite-Server Queues (2005)

Citations: | 2 - 2 self |

### Citations

5092 | Convergence of Probability Measures - Billingsley - 1968 |

1151 |
Nonlinear Programming
- Bertsekas
- 1995
(Show Context)
Citation Context ...de variety of assumptions (controlled service rate, controlled arrival rate, etc..). One can find classical examples of the use of exponential service times in Bertsekas [6], Bertsekas and Tsitsiklis =-=[7]-=-. Other authors use the exponential service times to derive structural results (monotonicity of the policy with the queue length, convexity of the value function, etc..) for various queueing problems ... |

465 | Stochastic modeling and the theory of queues - Wolff - 1989 |

463 |
Brownian motion and stochastic flow systems
- Harrison
- 1985
(Show Context)
Citation Context ... be found. Fleischer and Sethuraman develops approximately optimal control of queues using optimization technique in [16]. Optimal control of the Brownian motion has its theory covered by Harrison in =-=[19]-=-, [21] and by Harrison and Van Mieghem in [22]. The current status of the field is surveyed by Harrison and Nguyen in [18]. It is applied successfully in the case of tandem queues (most of the time, 2... |

165 | Constrained Markov decision processes
- Altman
- 1999
(Show Context)
Citation Context ...that we are trying to solve a dynamic programming problem with constraints. Several papers deal with this field; an extensive summary and presentation of Lagrange-based methods can be found in Altman =-=[3]-=-; sample-path constrained dynamic programming is considered by Ross in [39] and [40]. Finally, Duffield and Whitt in [12] study a problem fairly similar problem to ours (control and recovery from rare... |

90 | Scheduling networks of queues: Heavy traffic analysis of a simple open network - Harrison, Wein - 1989 |

86 | Brownian models of open processing networks: canonical representation of workload
- Harrison
- 2000
(Show Context)
Citation Context ...und. Fleischer and Sethuraman develops approximately optimal control of queues using optimization technique in [16]. Optimal control of the Brownian motion has its theory covered by Harrison in [19], =-=[21]-=- and by Harrison and Van Mieghem in [22]. The current status of the field is surveyed by Harrison and Nguyen in [18]. It is applied successfully in the case of tandem queues (most of the time, 2 queue... |

68 |
Dynamic control of Brownian networks: state space collapse and equivalent workload formulations
- Harrison, Mieghem
- 1997
(Show Context)
Citation Context ...pproximately optimal control of queues using optimization technique in [16]. Optimal control of the Brownian motion has its theory covered by Harrison in [19], [21] and by Harrison and Van Mieghem in =-=[22]-=-. The current status of the field is surveyed by Harrison and Nguyen in [18]. It is applied successfully in the case of tandem queues (most of the time, 2 queues) by Wein ([36], [44]), Veatch and Wein... |

67 | Server staffing to meet timevarying demand - Jennings, Mandelbaum, et al. - 1996 |

66 |
Networks of infinite-server queues with nonstationary Poisson input. Queueing Systems 13
- Massey, Whitt
- 1993
(Show Context)
Citation Context ... transient dynamic results of infinite-server queueing systems; Eick, Massey and Whitt wrote a seminal paper [13] about the Mt/G/oo queue. Other examples are given by Nelson and Taafe [32], or Massey =-=[30]-=-. However, most of these paper consider that the system starts in an empty state. If it is not the case, they usually assume we know what the remaining time distribution for customers in system is. Tw... |

60 |
Brownian models of multiclass queueing networks: current status and open problems. Queueing Systems Theory Appl
- Harrison, Nguyen
- 1993
(Show Context)
Citation Context ... Optimal control of the Brownian motion has its theory covered by Harrison in [19], [21] and by Harrison and Van Mieghem in [22]. The current status of the field is surveyed by Harrison and Nguyen in =-=[18]-=-. It is applied successfully in the case of tandem queues (most of the time, 2 queues) by Wein ([36], [44]), Veatch and Wein ([44]) and Reiman and Wein [36]; other control families, such as trajectory... |

50 |
Discrete-review policies for scheduling stochastic networks: trajectory tracking and fluid-scale asymptotic optimality
- Maglaras
- 2000
(Show Context)
Citation Context ...se of tandem queues (most of the time, 2 queues) by Wein ([36], [44]), Veatch and Wein ([44]) and Reiman and Wein [36]; other control families, such as trajectory tracking, are developed by Maglaras (=-=[31]-=-). For applications of these techniques, see Plambeck ([33] and [34]), Carr and Duenyas [11], or Kushner [28]. In a sense, there are two cases that seem tractable: the low volumes systems, "atomic 14s... |

40 | Mt/G/∞ queues with sinusoidal arrival rates - Eick, Massey, et al. - 1993 |

40 | Structural results for the control of queueing systems using event-based dynamic programming
- Koole
- 1998
(Show Context)
Citation Context ...notonicity of the policy with the queue length, convexity of the value function, etc..) for various queueing problems (controlled service rate, controlled arrival rate, etc..): see for instance Koole =-=[27]-=-. Stidham and Weber [43] also present general results of monotonicity of the optimal policy for average-cost exponential queues. They also approach general-service times in specific frame13sworks and ... |

36 | Dynamic staffing in a telephone call center aiming to immediately answer all calls
- Whitt
- 1999
(Show Context)
Citation Context ...From a practical point of view, infinite server queues model well systems with large number of servers, as argued in [12]; obviously, they model even better systems like the one described by Whitt in =-=[49]-=- or the one we study in this paper, as the controller's objective is precisely to be able to immediately serve any arriving customer; making the queue equivalent to an infinite-server one. We call thi... |

34 |
Optimal operating policies for M/G/1 queueing systems
- Heyman
- 1968
(Show Context)
Citation Context ...ted from a computational point of view. By considering specific queues (one server) and restricting the control to certain families, one can find closed form solution for the optimal control. Heynman =-=[23]-=-, and later Bell ([4],[5]) look at this type of problems, when the arrivals are Poisson and the policy is restricted to be stopping or re-starting the server at customers arrival or departure. In [42]... |

26 |
Optimal admission control and sequencing in a maketo-stock/make-to-order production system
- Carr, Duenyas
(Show Context)
Citation Context ...) and Reiman and Wein [36]; other control families, such as trajectory tracking, are developed by Maglaras ([31]). For applications of these techniques, see Plambeck ([33] and [34]), Carr and Duenyas =-=[11]-=-, or Kushner [28]. In a sense, there are two cases that seem tractable: the low volumes systems, "atomic 14slevel" situations that can be solved using classical dynamic programming, and the high-volum... |

25 |
Superposition of point processes
- Cinlar
- 1972
(Show Context)
Citation Context ..., in the case of medium to high volumes, the depleting binomial process will be the superposition of several point processes; the theory of superposition of point processes, as explained by Cinlar in =-=[9]-=-, is approximately Poisson. We can therefore make the approximation that the output process will be Poisson, and therefore the theory approximately applies to networks of Mt/G/oo. 55sA second extensio... |

24 |
Optimal leadtime differentiation via diffusion approximations
- Plambeck
(Show Context)
Citation Context ...), Veatch and Wein ([44]) and Reiman and Wein [36]; other control families, such as trajectory tracking, are developed by Maglaras ([31]). For applications of these techniques, see Plambeck ([33] and =-=[34]-=-), Carr and Duenyas [11], or Kushner [28]. In a sense, there are two cases that seem tractable: the low volumes systems, "atomic 14slevel" situations that can be solved using classical dynamic program... |

23 |
Characterization and Computation of Optimal Policies for Operating an M/G/1 Queuing System with Removable
- Bell
- 1971
(Show Context)
Citation Context ...oint of view. By considering specific queues (one server) and restricting the control to certain families, one can find closed form solution for the optimal control. Heynman [23], and later Bell ([4],=-=[5]-=-) look at this type of problems, when the arrivals are Poisson and the policy is restricted to be stopping or re-starting the server at customers arrival or departure. In [42], Sobel looks at a very s... |

22 |
Optimal Control of a Two-Station Tandem Production/Inventory System
- Veatch, Wein
- 1994
(Show Context)
Citation Context ...nd Van Mieghem in [22]. The current status of the field is surveyed by Harrison and Nguyen in [18]. It is applied successfully in the case of tandem queues (most of the time, 2 queues) by Wein ([36], =-=[44]-=-), Veatch and Wein ([44]) and Reiman and Wein [36]; other control families, such as trajectory tracking, are developed by Maglaras ([31]). For applications of these techniques, see Plambeck ([33] and ... |

18 |
Multichain Markov decision processes with a sample-path constraint: A decomposition approach
- Ross, Varadarajan
- 1991
(Show Context)
Citation Context ... papers deal with this field; an extensive summary and presentation of Lagrange-based methods can be found in Altman [3]; sample-path constrained dynamic programming is considered by Ross in [39] and =-=[40]-=-. Finally, Duffield and Whitt in [12] study a problem fairly similar problem to ours (control and recovery from rare congestion events in a large multi-server systems). While the objective (recovering... |

17 | Optimal Control of a Two-Station Brownian Network - Wein - 1988 |

14 |
Markov Decision Processes with a Sample Path Constraints: The Communicating Case
- Ross, Varadarajan
- 1989
(Show Context)
Citation Context .... Several papers deal with this field; an extensive summary and presentation of Lagrange-based methods can be found in Altman [3]; sample-path constrained dynamic programming is considered by Ross in =-=[39]-=- and [40]. Finally, Duffield and Whitt in [12] study a problem fairly similar problem to ours (control and recovery from rare congestion events in a large multi-server systems). While the objective (r... |

14 |
Optimal Average-Cost Policy for a Queue with Start-Up and Shut-Down Costs," Opns
- Sobel
- 1969
(Show Context)
Citation Context ...[23], and later Bell ([4],[5]) look at this type of problems, when the arrivals are Poisson and the policy is restricted to be stopping or re-starting the server at customers arrival or departure. In =-=[42]-=-, Sobel looks at a very similar problem, except that inter-arrival times are now drawn from a general distribution; the resulting problem being a controlled random walk. All of these papers find solut... |

13 | Decomposition Approximation for Time-Dependent Markovian Queueing Networks - Whitt |

12 | Sample path methods in the control of queues
- Liu, Nain, et al.
- 1995
(Show Context)
Citation Context ...hey also approach general-service times in specific frame13sworks and show insensitivity of the optimal policy to the service-time distribution for these special cases. Finally, Liu, Nain and Towsley =-=[29]-=- also present general properties of optimal policies using a different method, the sample path proofs, but once again, they require specific assumptions concerning the problem. The control of a queue ... |

11 | On the convergence of multiclass queueing networks in heavy traffic - Nguyen - 1994 |

10 | Control and recovery from rare congestion events in a large multi-server system
- Dueld, Whitt
- 1997
(Show Context)
Citation Context ...an empty state. If it is not the case, they usually assume we know what the remaining time distribution for customers in system is. Two natural approaches are commonly used (see for instance Duffield =-=[12]-=-) One assumes we know the age ri of the customer i in the system, in which case we can write the conditional residual time distribution using Bayes' rule. This distribution depends on zi, which for op... |

9 |
The Bilateral Laguerre Transform
- Keilson, Nunn, et al.
- 1981
(Show Context)
Citation Context ...composition that allows to have a finite state space for the dynamic programming problem, and try to limit the growth of the state-space as the order of the approximation increases. Keilson and Nunn (=-=[25]-=-, [26]) present the mathematical tools for the Erlang basis decomposition, while Schassberger presents the needed approximation proof [41].Whitt and 15sAbate ([1] and [2]) give practical methods to de... |

8 | Modeling servicetime distributions with non-exponential tails: beta mixtures of exponentials. Stochastic Models
- Abate, Whitt
- 1999
(Show Context)
Citation Context ...increases. Keilson and Nunn ([25], [26]) present the mathematical tools for the Erlang basis decomposition, while Schassberger presents the needed approximation proof [41].Whitt and 15sAbate ([1] and =-=[2]-=-) give practical methods to decompose the service time distributions on the Erlang basis. Another difficulty that arises in our case is that we are trying to solve a dynamic programming problem with c... |

7 |
Dynamic Programming and Optimal Control Vol 1
- Bertsekas
- 1995
(Show Context)
Citation Context ...n each queue The arrival times of the customers do not affect the future. Second, the state of such a system is discrete, and changes -discretely- at random times. Using uniformization (see Bertsekas =-=[6]-=- or Gallager [17]), which amounts to scaling time, one can write a discrete-time dynamic program that solves the continuous-time optimal control of the queue under a wide variety of assumptions (contr... |

6 |
Stationary poisson departure processes from non-stationary queues
- Foley
- 1986
(Show Context)
Citation Context ...liminaries: General Poisson splitting In this part, we present theorems that we will use in the proofs of all the other parts. These theorems are already known and can be found in Ross [38], or Foley =-=[15]-=- for the time-heterogeneous versions. Unless stated otherwise, we assume that the queue started at time 0. Theorem 1 (Classical Poisson Splitting) Let A(t) be a Poisson process with arrival rate A. Le... |

6 | Approximately optimal control of fluid networks
- Fleischer, Sethuraman
- 2003
(Show Context)
Citation Context ... Ricard and Bertsimas in [37]. Several papers in the domain by the same authors can be found. Fleischer and Sethuraman develops approximately optimal control of queues using optimization technique in =-=[16]-=-. Optimal control of the Brownian motion has its theory covered by Harrison in [19], [21] and by Harrison and Van Mieghem in [22]. The current status of the field is surveyed by Harrison and Nguyen in... |

4 |
Control and optimal control of assemble-to-order manufacturing systems under heavy traffic
- Kushner
- 1999
(Show Context)
Citation Context ...Wein [36]; other control families, such as trajectory tracking, are developed by Maglaras ([31]). For applications of these techniques, see Plambeck ([33] and [34]), Carr and Duenyas [11], or Kushner =-=[28]-=-. In a sense, there are two cases that seem tractable: the low volumes systems, "atomic 14slevel" situations that can be solved using classical dynamic programming, and the high-volume systems solved ... |

4 | Heavy traffic analysis of polling systems in tandem
- Reiman, Wein
- 1999
(Show Context)
Citation Context ...ison and Van Mieghem in [22]. The current status of the field is surveyed by Harrison and Nguyen in [18]. It is applied successfully in the case of tandem queues (most of the time, 2 queues) by Wein (=-=[36]-=-, [44]), Veatch and Wein ([44]) and Reiman and Wein [36]; other control families, such as trajectory tracking, are developed by Maglaras ([31]). For applications of these techniques, see Plambeck ([33... |

3 |
Optimization of queueing networks; an optimal control approach
- Ricard
- 1995
(Show Context)
Citation Context ...) are high. These approximations allow to apply the theory of optimal control much more easily. Optimal control of the fluid approximation of a queueing system is developed by Ricard and Bertsimas in =-=[37]-=-. Several papers in the domain by the same authors can be found. Fleischer and Sethuraman develops approximately optimal control of queues using optimization technique in [16]. Optimal control of the ... |

2 |
Monotonic and insensitive policies for control of queues with undiscounted costs
- Weber
- 1989
(Show Context)
Citation Context ... with the queue length, convexity of the value function, etc..) for various queueing problems (controlled service rate, controlled arrival rate, etc..): see for instance Koole [27]. Stidham and Weber =-=[43]-=- also present general results of monotonicity of the optimal policy for average-cost exponential queues. They also approach general-service times in specific frame13sworks and show insensitivity of th... |

1 |
Infinite series representation of Laplace transforms of probability density functions for numerical inversion
- Abate, Whitt
- 1999
(Show Context)
Citation Context ...imation increases. Keilson and Nunn ([25], [26]) present the mathematical tools for the Erlang basis decomposition, while Schassberger presents the needed approximation proof [41].Whitt and 15sAbate (=-=[1]-=- and [2]) give practical methods to decompose the service time distributions on the Erlang basis. Another difficulty that arises in our case is that we are trying to solve a dynamic programming proble... |

1 |
Optimal Operation of an M / G / 1 Priority Queue with Removable Server
- Bell
- 1973
(Show Context)
Citation Context ...al point of view. By considering specific queues (one server) and restricting the control to certain families, one can find closed form solution for the optimal control. Heynman [23], and later Bell (=-=[4]-=-,[5]) look at this type of problems, when the arrivals are Poisson and the policy is restricted to be stopping or re-starting the server at customers arrival or departure. In [42], Sobel looks at a ve... |

1 |
The Physics of the Mt/G/oo queue
- Eick, Massey, et al.
- 1993
(Show Context)
Citation Context ...hod and approach is. Chapter 4 deals with non-Markovian queues with infinite servers: in 4.1, we present general theorems that will be used in the rest of the paper. In 4.2 we recall the results from =-=[13]-=- concerning the analysis of the unobserved Mt/G/oo queue (and provide a different proof). In 4.3, we present our first results about the dynamics of the observed Mt/G/oo; we define the state and the c... |

1 |
The Pht/Pht/oo Queueing system: Part I - The single node(2004
- Nelson, Taafe
(Show Context)
Citation Context ...al papers derive transient dynamic results of infinite-server queueing systems; Eick, Massey and Whitt wrote a seminal paper [13] about the Mt/G/oo queue. Other examples are given by Nelson and Taafe =-=[32]-=-, or Massey [30]. However, most of these paper consider that the system starts in an empty state. If it is not the case, they usually assume we know what the remaining time distribution for customers ... |

1 |
Optimal control of high-volume assemble to order systems
- Plambeck, Ward
- 2003
(Show Context)
Citation Context ...36], [44]), Veatch and Wein ([44]) and Reiman and Wein [36]; other control families, such as trajectory tracking, are developed by Maglaras ([31]). For applications of these techniques, see Plambeck (=-=[33]-=- and [34]), Carr and Duenyas [11], or Kushner [28]. In a sense, there are two cases that seem tractable: the low volumes systems, "atomic 14slevel" situations that can be solved using classical dynami... |

1 |
The linear-programming approach to dynamic programming
- Farias, D, et al.
(Show Context)
Citation Context ...s systems. We believe that our results could be used in a medium to high volume framework, using neuro-dynamic programming techniques (see Bertsekas and Tsitsklis [7], and Pucci de Farias and Van Roy =-=[35]-=-) if necessary to further address the large-scale of the system. Several papers derive transient dynamic results of infinite-server queueing systems; Eick, Massey and Whitt wrote a seminal paper [13] ... |

1 |
Laplace Transform of probability density functions with series representations
- Whitt
- 1998
(Show Context)
Citation Context ... which order is limited at n. 47sIf coefficients are allowed to be negative (generalized distributions), the sets are respectively denoted hE's, hE Another method (and several others) is presented in =-=[47]-=-, [1] and [2]; the methods to truncate the infinite sum are also discussed. Proposition 9 (Abate, Whitt) Let g(t) be an analytic density on [0, +oo). We denote h(t) = t.et.g(t) which is also analytic ... |