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## TUIe technische universiteit eindhoven (1567)

### Citations

897 |
Applied Probability and Queues
- ASMUSSEN
- 2003
(Show Context)
Citation Context ...s). For the MIG/I queue, this result is well-known; see for example Takacs [21], Cohen [10] and Hooghiemstra [14J. For a rigorous proof of (33) in case of a general release rate, we refer to Asmussen =-=[1]-=-, Chapter XIII, Example 5.1. 10 Writing 1-PK = JID(wK,r(')+8 ~ K), conditioning on 8, applying (33), and deconditioning on 8 then results in = prO x = 1 - JID(wx,r(.) + 8 ~ x) JID(Wr(.) + 8> x) - JPl(... |

427 | Modelling Extremal Events - Embrechts, Klppelberg, et al. - 1997 |

231 |
The Single Server Queue
- Cohen
- 1982
(Show Context)
Citation Context ...in steady state. It is readily seen that (1) 1 with W K being the steady-state waiting time, and S a generic service time. Thus, information about PK can be derived from the distribution of WK. Cohen =-=[10]-=-, Chapter III.6, analyzed the distribution of W K in the case that both the interarrival times and service times have a rational Laplace transform. For the M/G/1 queue with p < 1, the distribution of ... |

94 |
Combinatorial Methods in the Theory of Stochastic Processes
- Takács
- 1967
(Show Context)
Citation Context ...ibution of W K can be written in an elegant form, i.e., in terms of the steady-state waiting-time distribution of the M /G/1 queue with infinite buffer size. This result is already known since Takacs =-=[21]-=-. Using this result, Zwart [22] showed that PK can be identified with Takacs' expression [21] for the tail distribution of the cycle-maximum in the M/G/1 queue, i.e., it is shown in [22] that PK = 1P'... |

56 |
Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities
- ASMUSSEN
- 1998
(Show Context)
Citation Context ...r the distribution of Cmax in general, we expect that this representation may be useful to obtain asymptotics and/or bounds. Asymptotic results in the light-tailed case are hardly known; see Asmussen =-=[6, 7]-=-. 5 Conclusion We have considered several queueing models which operate under the partial-rejection mechanism. For these models, we have shown that the loss probability of a customer can be identified... |

55 |
Extreme values in the GI/G/1 queue.
- Iglehart, L
- 1972
(Show Context)
Citation Context ...nder their assumptions) PK rv W(Cmax > K), where f(x) rv g(x) denotes f(x)lg(x) -+ 1 as x -+ 00. Asymptotics for the latter quantity are due to Iglehart [15J: Under certain regularity conditions (see =-=[15]-=-), it holds that (31 ) for certain positive constants D and 'Y. Using Theorem 2.1, the proof of the main result of [18] is now trivial: Just combine Theorem 2.1 with (31) to (re-)obtain (32) For more ... |

54 |
Applied Probability and Queues, Second Edition
- Asmussen
- 2003
(Show Context)
Citation Context ...lity theory for stochastic recursions, which has been developed by Asmussen & Sigman [5], and dates back to Lindley [16], Loynes [17J, and Siegmund [20]. For a recent textbook treatment, see Asmussen =-=[8]-=-. This type of duality, also known as Siegmund duality, relates the stationary distribution of a given model to the first passage time of another model, called the dual model. Thus, Siegmund duality p... |

30 |
On Regenerative Processes in Queueing Theory;
- Cohen
- 1976
(Show Context)
Citation Context ... - JPl(Wr(·) > x) ( max> ) - JID(WK,r(.) = 0) JPl(Wr(.) ~ x) This is an extension of the classical formula for the distribution of Cmax in the M/G/1 queue, which is due to Takacs [21] (see also Cohen =-=[9]-=-, and Asmussen & Perry [3] for alternative proofs). His result can be easily recovered from Corollary 4.3, since, for the M/G/1 queue, we have r(x) == f(x) == 1. This yields the well-known formula JID... |

30 |
The stationary distribution and first exit probabilities of a storage process with general release rule,
- Harrison, Resnick
- 1976
(Show Context)
Citation Context ...ing that state zero can be reached in a finite amount of time. This ensures that CIDax is well-defined. Note that 0(·) is strictly increasing and we can thus unambiguously speak of 0-1(t). Similar to =-=[13]-=- and [19], we define q{u, t) := 0-1(0(u) - t). (21) Then q(u, t) is the workload level at time t if we start from level u at time 0 and no arrivals have taken place in between. Denote the workload pro... |

21 |
Extreme value theory for queues via cycle maxima
- Asmussen
- 1998
(Show Context)
Citation Context ...parent proofs of existing results. 4.1 Exact expressions for PK In the literature, there are several studies devoted to the distribution of Cmax for a variety of queueing models. We refer to Asmussen =-=[7]-=- for a survey of these results. The M/G/1 case has already been treated in Zwart [22]. Here, we give an analogous result for the GI/M/1 queue. Corollary 4.1. Consider the finite GI/M/l dam with p < 1 ... |

19 |
The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes
- SIEGMUND
- 1976
(Show Context)
Citation Context ...oth proof techniques strongly rely on a powerful duality theory for stochastic recursions, which has been developed by Asmussen & Sigman [5], and dates back to Lindley [16], Loynes [17J, and Siegmund =-=[20]-=-. For a recent textbook treatment, see Asmussen [8]. This type of duality, also known as Siegmund duality, relates the stationary distribution of a given model to the first passage time of another mod... |

18 |
Monotone stochastic recursions and their duals.
- Asmussen, Sigman
- 1996
(Show Context)
Citation Context ...ill also give another proof based on a regenerative argument. Both proof techniques strongly rely on a powerful duality theory for stochastic recursions, which has been developed by Asmussen & Sigman =-=[5]-=-, and dates back to Lindley [16], Loynes [17J, and Siegmund [20]. For a recent textbook treatment, see Asmussen [8]. This type of duality, also known as Siegmund duality, relates the stationary distri... |

18 |
A fluid queue with a finite buffer and subexponential input.
- Zwart
- 2000
(Show Context)
Citation Context ...n an elegant form, i.e., in terms of the steady-state waiting-time distribution of the M /G/1 queue with infinite buffer size. This result is already known since Takacs [21]. Using this result, Zwart =-=[22]-=- showed that PK can be identified with Takacs' expression [21] for the tail distribution of the cycle-maximum in the M/G/1 queue, i.e., it is shown in [22] that PK = 1P'(Cmax > K). (2) For the GI/G/1 ... |

13 |
On cycle maxima, first passage problems and extreme value theory for queues. Stochastic Models 8
- Asmussen, Perry
- 1992
(Show Context)
Citation Context ... - JID(WK,r(.) = 0) JPl(Wr(.) ~ x) This is an extension of the classical formula for the distribution of Cmax in the M/G/1 queue, which is due to Takacs [21] (see also Cohen [9], and Asmussen & Perry =-=[3]-=- for alternative proofs). His result can be easily recovered from Corollary 4.3, since, for the M/G/1 queue, we have r(x) == f(x) == 1. This yields the well-known formula JID(C <) = JPl(W +8 ~ x) max ... |

11 | Ruin probabilities expressed in terms of storage processes - Asmussen, Petersen, et al. - 1988 |

11 | Rate modulation in dams and ruin problems - Asmussen, Kella - 1996 |

11 | and Siegmund duality relations for birth and death chains with reflecting - Dette, Fill, et al. - 1997 |

4 | Duality of dams via mountain processes.
- Perry, Stadje
- 2003
(Show Context)
Citation Context ...cle maximum of a regeneration cycle, or, more formally, Cmax := sup{D(t),O:::; t:::; TO}. (6) From the workload process in the finite GIIG/I dam we construct a dual risk process {R(t), t E lR}, as in =-=[19]-=-, by defining R(t) := K - D(t). (7) The risk process is also regenerative and regeneration points in the risk process correspond to downward jump epochs from level K. Hence, TO can be alternatively de... |

2 |
Asymptotic results for buffer systems under heavy loads
- Ommeren, Kok
- 1987
(Show Context)
Citation Context ... for the tail distribution of the cycle-maximum in the M/G/1 queue, i.e., it is shown in [22] that PK = 1P'(Cmax > K). (2) For the GI/G/1 queue with light-tailed service times, Van Ommeren and De Kok =-=[18]-=- derived exact asymptotics for PK as K -+ 00. From their main result, it immediately follows that PK rv 1P'(Cmax > K), (3) as K -+ 00. This naturally leads to the conjecture that (2) can be extended t... |

1 | A path construction for the virtual waiting time of an MIG/l queue - Hooghiemstra - 1987 |

1 |
Discussion of a paper by C.B. Winsten
- Lindley
- 1959
(Show Context)
Citation Context ...ed on a regenerative argument. Both proof techniques strongly rely on a powerful duality theory for stochastic recursions, which has been developed by Asmussen & Sigman [5], and dates back to Lindley =-=[16]-=-, Loynes [17J, and Siegmund [20]. For a recent textbook treatment, see Asmussen [8]. This type of duality, also known as Siegmund duality, relates the stationary distribution of a given model to the f... |

1 | On a property of the random walks describing simple queues and dams - Loynes - 1965 |