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## Asymptotic expansions for the conditional sojourn time distribution

Venue: | in the M/M/1-PS queue, Queueing Syst |

Citations: | 3 - 3 self |

### Citations

260 |
Asymptotic Approximations of Integrals
- Wong
- 1989
(Show Context)
Citation Context ...Note that s− < s0 < ∞ whenever 1 < t/x < ∞, and s0 → ∞ as t/x ↓ 1, while s0 → s− as t/x → ∞. If we shift Br in (3.2) to Br ′ , on which ℜ(s) = s0, and use the standard Laplace method (see, e.g., Wong =-=[21]-=-) we get p(t|x) ∼ (1 − ρ)e−ρx [1 − ρr2 (s0; ρ)] exϕ(s0) √ 2π [1 − ρr(s0; ρ)] 2 √ xϕ ′′ . (3.4) (s0) But from (3.1) and (2.2) we get √ t − x r(s0; ρ) = , (3.5) ρt and xϕ(s0) = (x − t)(1 + ρ) + 2 √ t(t ... |

88 | Time-shared systems: A theoretical treatment
- Kleinrock
- 1967
(Show Context)
Citation Context ...r FIFO is that customers requiring only short amounts of service get through the system more rapidly than with other service disciplines. The PS discipline was apparently introduced by Kleinrock [1], =-=[2]-=-, and has been the subject of much further investigation over the past forty years. In [3], Coffman, Muntz, and Trotter derived an expression for the Laplace transform of the distribution of the waiti... |

66 |
Waiting time distributions for processor sharing systems
- Coman, Muntz, et al.
- 1970
(Show Context)
Citation Context ...re rapidly than with other service disciplines. The PS discipline was apparently introduced by Kleinrock [1], [2], and has been the subject of much further investigation over the past forty years. In =-=[3]-=-, Coffman, Muntz, and Trotter derived an expression for the Laplace transform of the distribution of the waiting time in the M/M/1-PS model. We shall denote the waiting time (in the steady state) by W... |

56 | Stochastic service systems - Riordan - 1962 |

54 | Sojourn time asymptotics in the M/G/1 processor sharing queue.” Queueing Systems 35
- Zwart, Boxma
- 2000
(Show Context)
Citation Context ...vestigations into tail behaviors of PS models that are “heavy tailed”, in that the service time distribution has an algebraic tail. The M/G/1 model with this assumption is analyzed by Zwart and Boxma =-=[19]-=-, where it is shown that the sojourn time has a similar algebraic tail. A thorough recent survey of sojourn time asymptotics in PS models can be found in [20], where both heavy and light tailed distri... |

24 |
Response-time distribution for a processor-sharing system
- Morrison
- 1985
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Citation Context ...large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison =-=[4]-=- and Flatto [8]. ∗ Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan (M/C 249), Chicago, IL 60607-7045, USA. Email: qzhen2@uic.edu. † Dep... |

21 |
Sojourn time asymptotics in processor-sharing queues.” Queueing Systems 53
- Borst, Núñez-Queija, et al.
- 2006
(Show Context)
Citation Context ...assumption is analyzed by Zwart and Boxma [19], where it is shown that the sojourn time has a similar algebraic tail. A thorough recent survey of sojourn time asymptotics in PS models can be found in =-=[20]-=-, where both heavy and light tailed distributions are discussed, as well as various approaches to obtaining the asymptotics. The remainder of the paper is organized as follows. In section 2 we summari... |

18 |
The sojourn time distribution in the M/G/1 queue with processor sharing
- Ott
- 1984
(Show Context)
Citation Context ...results all of the expansion in [4] can be recovered. We mention some other work on PS queues. The G/M/1-PS model was studied by Ramaswami [13] and the M/G/1-PS model by Yashkov [14], [15] and by Ott =-=[16]-=-. For the latter there is a complicated expression for the Laplace transform of the conditional sojourn time distribution. the special case of M/D/1-PS this expression simplifies considerably, and the... |

18 |
Large deviations of sojourn times in processor sharing queues.” Queueing Systems 52
- Mandjes, Zwart
- 2006
(Show Context)
Citation Context ...o (1.4) with (1.3) the interesting question arises as to what are the variety of possible tail behaviors for the general M/G/1-PS model. For the G/G/1-PS queue the tail was shown by Mandjes and Zwart =-=[18]-=- to be roughly exponential, in that log{Pr[V > t]} ∼ −A0t as t → ∞. This assumes that the arrival and service distributions have expoFor 3nentially small tails, so that their moment generating functi... |

17 |
The waiting time distribution for the random order service M/M/1 queue.” Annals of Applied Probability 7
- Flatto
- 1997
(Show Context)
Citation Context ...or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison [4] and Flatto =-=[8]-=-. ∗ Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan (M/C 249), Chicago, IL 60607-7045, USA. Email: qzhen2@uic.edu. † Department of Math... |

15 |
Mathematical problems in the theory of processor-sharing queueing systems
- Yashkov
- 1992
(Show Context)
Citation Context ...ane. From these results all of the expansion in [4] can be recovered. We mention some other work on PS queues. The G/M/1-PS model was studied by Ramaswami [13] and the M/G/1-PS model by Yashkov [14], =-=[15]-=- and by Ott [16]. For the latter there is a complicated expression for the Laplace transform of the conditional sojourn time distribution. the special case of M/D/1-PS this expression simplifies consi... |

14 |
On queues in which customers are served in random order
- Kingman
- 1962
(Show Context)
Citation Context ...ne seemingly unrelated to PS is random order service (ROS), where customers are chosen for service at random. The M/M/1ROS model has been studied by many authors, see Vaulot [5], Riordan [6], Kingman =-=[7]-=- and Flatto [8]. In [8] an explicit integral representation is derived for the waiting time distribution, from which the following tail behavior is computed: Pr [WROS > t] ∼ e −At e −Bt1/3 Ct −5/6 , t... |

12 |
The sojourn time in the GI/M/1 queue with processor sharing
- Ramaswami
- 1984
(Show Context)
Citation Context ...tions for several ranges of the space-time plane. From these results all of the expansion in [4] can be recovered. We mention some other work on PS queues. The G/M/1-PS model was studied by Ramaswami =-=[13]-=- and the M/G/1-PS model by Yashkov [14], [15] and by Ott [16]. For the latter there is a complicated expression for the Laplace transform of the conditional sojourn time distribution. the special case... |

10 | Délais d’attente des appels téléphoniques traités au hasard.” Comptes Rendus de l’Acadmie des Sciences Paris 222 - Vaulot - 1946 |

10 |
La loi d’attente des appels téléphoniques.” Comptes Rendus de l’Acadmie des Sciences Paris 222
- Pollaczek
- 1946
(Show Context)
Citation Context ...3) Here A = (1 − √ ρ) 2 if we scale time to make the service rate µ = 1, and ρ < 1. This formula appeared previously in the book of Riordan [9] (pg.105) and was apparently first obtained by Pollaczek =-=[10]-=-. Cohen [11] established a relationship between the sojourn time in the PS model and the waiting time in the ROS model, Pr [V > t] = 1 ρ Pr[WROS > t], (1.4) which extends also to the more general G/M/... |

9 |
Analysis of a time-shared processor.” Naval Research Logistics Quarterly 11
- Kleinrock
- 1964
(Show Context)
Citation Context ...e over FIFO is that customers requiring only short amounts of service get through the system more rapidly than with other service disciplines. The PS discipline was apparently introduced by Kleinrock =-=[1]-=-, [2], and has been the subject of much further investigation over the past forty years. In [3], Coffman, Muntz, and Trotter derived an expression for the Laplace transform of the distribution of the ... |

9 |
On processor sharing and random service” Letter to the editor
- Cohen
- 1984
(Show Context)
Citation Context ...(1 − √ ρ) 2 if we scale time to make the service rate µ = 1, and ρ < 1. This formula appeared previously in the book of Riordan [9] (pg.105) and was apparently first obtained by Pollaczek [10]. Cohen =-=[11]-=- established a relationship between the sojourn time in the PS model and the waiting time in the ROS model, Pr [V > t] = 1 ρ Pr[WROS > t], (1.4) which extends also to the more general G/M/1 case. In [... |

8 |
Queija. “The equivalence between processor sharing and service
- Borst, Boxma, et al.
(Show Context)
Citation Context ...] established a relationship between the sojourn time in the PS model and the waiting time in the ROS model, Pr [V > t] = 1 ρ Pr[WROS > t], (1.4) which extends also to the more general G/M/1 case. In =-=[12]-=- relations of the form (1.4) are explored for other models, such as finite capacity queues, repairman problems and networks. Later in this paper we discuss the relationship of the tail formula in (1.3... |

8 | Sojourn Time Tails in the M/D/1 Processor Sharing Queue
- Egorova, Zwart, et al.
(Show Context)
Citation Context ...r the Laplace transform of the conditional sojourn time distribution. the special case of M/D/1-PS this expression simplifies considerably, and then the tail behavior was obtained by Egorova, et. al. =-=[17]-=- in the form Pr[V > t] ∼ B ′ e −A′ t . Comparing this to (1.4) with (1.3) the interesting question arises as to what are the variety of possible tail behaviors for the general M/G/1-PS model. For the ... |

3 |
Delay curves for calls served at random
- Riordan
- 1953
(Show Context)
Citation Context ...vice discipline seemingly unrelated to PS is random order service (ROS), where customers are chosen for service at random. The M/M/1ROS model has been studied by many authors, see Vaulot [5], Riordan =-=[6]-=-, Kingman [7] and Flatto [8]. In [8] an explicit integral representation is derived for the waiting time distribution, from which the following tail behavior is computed: Pr [WROS > t] ∼ e −At e −Bt1/... |