## The Asymptotic Behavior of a Network Multiplexor with Multiple Time Scale and Subexponential Arrivals (1996)

Venue: | in Stochastic Networks: Stability and Rare Events |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Jelenkovic96theasymptotic,

author = {Predrag R. Jelenkovic and Aurel A. Lazar},

title = {The Asymptotic Behavior of a Network Multiplexor with Multiple Time Scale and Subexponential Arrivals},

booktitle = {in Stochastic Networks: Stability and Rare Events},

year = {1996},

pages = {215--235},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

Real-time traffic processes, such as video, exhibit multiple time scale characteristics, as well as subexponential first and second order statistics. We present recent results on evaluating the asymptotic behavior of a network multiplexer that is loaded with such processes.

### Citations

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Citation Context ...sult was shown using a random walk technique. Some of the rst applications of long-tailed distributions in queueing theory were made by Cohen [13], and Borovkov[6] for functions of regular variations =-=[26, 5]-=-. Recent results on long-tailed and subexponential asymptotics of a GI=GI=1 are given in [1, 38]. (Also, in [1] further motivation is given for the application of long-tailed distributions to communic... |

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Citation Context ...ation Qt+1 =(Qt + At ; Ct) + (1.1) completely de nes the queue length process fQt�t 0g. Queues of this type represent a natural model for ATM multiplexers. According to the classical result of Loynes =-=[31]-=-, if EAt < ECt (and fAt�Ct�t 0g are stationary and ergodic) fQtg couples with the unique stationary solution fQs tg of the recursion (1.1) for any initial condition Q0� in particular P[Qt x] ! P[Q s 0... |

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Citation Context ...h (EB) approximation (sometimes it is also called the dominant rootapproximation). Following this result admission control policies based on the concept of e ective bandwidth have been developed� see =-=[7, 16,18,17,27]-=-. However, as discussed in [9], the EB approximation may often be very inaccurate. This is usually the case when many sources (N) are multiplexed� under this assumption it was shown in [9] that e ; N ... |

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Citation Context ...l assumptions of the Gartner-Ellis (Cramer) type, one can show that lim x!1 ;logP[Q>x] = � (1.2) x for some positive constant , called the asymptotic decay rate (or the equivalent bandwidth constant) =-=[7, 17]-=-. Also, in some cases, like nite Markov arrival and service processes, the following stronger result holds: P[Q >x] e ; x as x !1, where and are positive constants� is the same as in (1.2). For simple... |

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Citation Context ... rst proved in [32]� in [36] the same result was shown using a random walk technique. Some of the rst applications of long-tailed distributions in queueing theory were made by Cohen [13], and Borovkov=-=[6]-=- for functions of regular variations [26, 5]. Recent results on long-tailed and subexponential asymptotics of a GI=GI=1 are given in [1, 38]. (Also, in [1] further motivation is given for the applicat... |

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Citation Context ...will attempt to answer some questions on the queue length asymptotics in the presence of multiple time scale arrivals and Gartner-Ellis (Cramer) assumptions. Using the Theory of Large Deviations (see =-=[37]-=-), under general assumptions of the Gartner-Ellis (Cramer) type, one can show that lim x!1 ;logP[Q>x] = � (1.2) x for some positive constant , called the asymptotic decay rate (or the equivalent bandw... |

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Citation Context ...� under this assumption it was shown in [9] that e ; N forsomeconstant . A more formal analysis of the multiplexing of a large number of sources and an improvement of the EB approximation is given in =-=[14, 15]-=-. Complementing the work done in [9], in [19] wehave shown that EB approximation may bevery inaccurate in the presence of multiple time scale1. The Asymptotic Behavior of a Network Multiplexer with M... |

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Citation Context ...l assumptions of the Gartner-Ellis (Cramer) type, one can show that lim x!1 ;logP[Q>x] = � (1.2) x for some positive constant , called the asymptotic decay rate (or the equivalent bandwidth constant) =-=[7, 17]-=-. Also, in some cases, like nite Markov arrival and service processes, the following stronger result holds: P[Q >x] e ; x as x !1, where and are positive constants� is the same as in (1.2). For simple... |

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Citation Context ...i def i t = Yt (B i t)� where fY i (1)�::: �Yi (Ki)g�Ki 1 are stationary ergodic processes that are stochastically ordered such thatY i (j) st Y i (Ki)� 1 j < Ki (for stochastic ordering see [34], or =-=[4]-=-, chapter 4). Further, the processes Bi = �t 0g are stationary, ergodic, discrete time process with a nite state fBi t space Si = f1�:::�Kig. All processes Y i (j), and Bi are assumed independent of e... |

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Citation Context ...e [36]). Let K be the d.f. of At. Theorem 3.5 (i) F1 2S() K1 2S and limx!1 (ii) If K� K1 2S,then P[Qt >x] 1 ECt ; EAt Z 1 x ^F (x) ^K(x) =1. P[At >u]du� as x !1: (1.11) This theorem was rst proved in =-=[32]-=-� in [36] the same result was shown using a random walk technique. Some of the rst applications of long-tailed distributions in queueing theory were made by Cohen [13], and Borovkov[6] for functions o... |

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Citation Context ...istribution (IV) for 0 < <1. is the standard normal distribution. F (x) =1; e ;x ;x(log x);a F (x) =e � for a>0. This class was proven to be subexponential in [33]. (V) Benktander Type I distribution =-=[28]-=- F (x) =1; cx ;a;1 x ;b log x (a +2b log x)� a>0�b>0� and c appropriately chosen. (V) Benktander Type II distribution [28] F (x) =1; cax ;(1;b) expf;(a=b)x b g� a>0� 0 <b<1, and c appropriately chosen... |

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Citation Context ...butions in queueing theory were made by Cohen [13], and Borovkov[6] for functions of regular variations [26, 5]. Recent results on long-tailed and subexponential asymptotics of a GI=GI=1 are given in =-=[1, 38]-=-. (Also, in [1] further motivation is given for the application of long-tailed distributions to communication networks.)1. The Asymptotic Behavior of a Network Multiplexer with Multiple Time Scale an... |

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Citation Context ...) This theorem was rst proved in [32]� in [36] the same result was shown using a random walk technique. Some of the rst applications of long-tailed distributions in queueing theory were made by Cohen =-=[13]-=-, and Borovkov[6] for functions of regular variations [26, 5]. Recent results on long-tailed and subexponential asymptotics of a GI=GI=1 are given in [1, 38]. (Also, in [1] further motivation is given... |

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Citation Context ...onential asymptotics of a Markov-modulated M/G/1 queue. However the constant of proportionality was left in a complex form. Full extension of Theorem 3.5 to Markov-modulated G/G/1 queues was given in =-=[23]-=- (preliminary results were reported in [22]), where it was proved that the queue length asymptotics are invariant under Markov modulation. A precise statement of this result follows. Let fJtg be a sta... |

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Citation Context ... processes (sources) that arise in ATM networks (like voice and video) have avery complex statistical structure� an especially troublesome characteristic is the high statistical dependency (e.g., see =-=[25, 30]-=-). Modeling of this high dependency usually leads to analytically very complex statistical characteristics, typically making the associated evaluation of the queue length distribution intractable. How... |

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Citation Context ...mptotic Behavior of a Network Multiplexer with Multiple Time Scale and Subexponential Arrivals 13 An extensive treatment of subexponential distributions (and further references) can be found in Cline =-=[11, 12]-=-. Before we proceed any further, let us try to understand some of the basic properties of the sequence fXn�n 1g of subexponentially distributed i.i.d. random variables. One of the main sample path cha... |

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Citation Context ...ls. Similar observations of inaccuracy of the EB approximation in the presence of multiple time scales (in the context of nearly decomposable Markov-modulated arrivals) were independently obtained in =-=[35]-=-. From a mathematical point of view, the inaccuracy of the EB approximation is due to fact that two processes that are \close" in the distribution sense may be far apart in the cumulant sense. Recall ... |

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Citation Context ... intherest of the paper for any d.f. G, we de ne its corresponding ^ G(x) and G1(x). Then the following result on the waiting time distribution asymptotics of the GI=GI=1 queue holds (see Veraverbeke =-=[36]-=-). Let K be the d.f. of At. Theorem 3.5 (i) F1 2S() K1 2S and limx!1 (ii) If K� K1 2S,then P[Qt >x] 1 ECt ; EAt Z 1 x ^F (x) ^K(x) =1. P[At >u]du� as x !1: (1.11) This theorem was rst proved in [32]� ... |

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Citation Context ...mptotic Behavior of a Network Multiplexer with Multiple Time Scale and Subexponential Arrivals 13 An extensive treatment of subexponential distributions (and further references) can be found in Cline =-=[11, 12]-=-. Before we proceed any further, let us try to understand some of the basic properties of the sequence fXn�n 1g of subexponentially distributed i.i.d. random variables. One of the main sample path cha... |

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Citation Context ...De nition 3.6 F 2S if Z x where mF = R 1 0 yF(dy). 0 F (x ; y) F(x) F(y)dy ! 2mF < 1� as x !1� This class has the property thatS S, and that F 2S ) F1 2S. Su cient conditions for F 2S can be found in =-=[29]-=-, where it was explicitly shown that lognormal, Pareto, and certain Weibull distributions are in S . An extension of Theorem 3.5 was investigated in [2]. In that paper the authors established the sube... |

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Citation Context .... Su cient conditions for F 2S can be found in [29], where it was explicitly shown that lognormal, Pareto, and certain Weibull distributions are in S . An extension of Theorem 3.5 was investigated in =-=[2]-=-. In that paper the authors established the subexponential asymptotics of a Markov-modulated M/G/1 queue. However the constant of proportionality was left in a complex form. Full extension of Theorem ... |

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E ective bandwidth of general Markovian tra c sources and admission control of high speed networks
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Citation Context ...h (EB) approximation (sometimes it is also called the dominant rootapproximation). Following this result admission control policies based on the concept of e ective bandwidth have been developed� see =-=[7, 16,18,17,27]-=-. However, as discussed in [9], the EB approximation may often be very inaccurate. This is usually the case when many sources (N) are multiplexed� under this assumption it was shown in [9] that e ; N ... |

21 |
Asymptotic expansion for waiting time probabilities in an M/G/1 queue with long-tailed service time, Queueing Systems 10
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Citation Context ...butions in queueing theory were made by Cohen [13], and Borovkov[6] for functions of regular variations [26, 5]. Recent results on long-tailed and subexponential asymptotics of a GI=GI=1 are given in =-=[1, 38]-=-. (Also, in [1] further motivation is given for the application of long-tailed distributions to communication networks.)1. The Asymptotic Behavior of a Network Multiplexer with Multiple Time Scale an... |

20 | Multiple time scales and subexponentiality in MPEG video streams
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Citation Context ...he large peaks tend to be isolated in time, as suggested by (1.10). This is unlike the case of the geometrically distributed process. (For the description of MPEG data and the de nition of scenes see =-=[24]-=-.) In terms of video tra c, subexponentiality can also manifest itself in the time-dependent (autocorrelation) structure. As shown in Figure 8, the autocorrelation function of MPEG video (17 streams m... |

19 |
Subexponential distribution functions
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Citation Context ... log x ; � 2 R� >0� where (III) Weibull distribution (IV) for 0 < <1. is the standard normal distribution. F (x) =1; e ;x ;x(log x);a F (x) =e � for a>0. This class was proven to be subexponential in =-=[33]-=-. (V) Benktander Type I distribution [28] F (x) =1; cx ;a;1 x ;b log x (a +2b log x)� a>0�b>0� and c appropriately chosen. (V) Benktander Type II distribution [28] F (x) =1; cax ;(1;b) expf;(a=b)x b g... |

13 | Algorithmic modeling of TES processes
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Citation Context ... processes (sources) that arise in ATM networks (like voice and video) have avery complex statistical structure� an especially troublesome characteristic is the high statistical dependency (e.g., see =-=[25, 30]-=-). Modeling of this high dependency usually leads to analytically very complex statistical characteristics, typically making the associated evaluation of the queue length distribution intractable. How... |

12 | On the dependence of the queue tail distribution on multiple time scales of ATM multiplexers
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Citation Context ...e ; N forsomeconstant . A more formal analysis of the multiplexing of a large number of sources and an improvement of the EB approximation is given in [14, 15]. Complementing the work done in [9], in =-=[19]-=- wehave shown that EB approximation may bevery inaccurate in the presence of multiple time scale1. The Asymptotic Behavior of a Network Multiplexer with Multiple Time Scale and Subexponential Arrival... |

8 |
A theorem on sums of independent positive random variables and its application to branching random processes,” Theor
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Citation Context ...d Subexponential Arrivals 12 where F 2 denotes the 2-nd convolution of F with itself, i.e., F 2 (x) = R [0�1) F (x ; y)F (dy). The class of subexponential distributions was rst introduced by Chistakov=-=[8]-=-. The de nition is motivated by the simpli cation of the asymptotic analysis of the convolution tails. Some examples of distribution functions in S are: (I) the Pareto family x> >0� >0. F (x) =1; (x ;... |

6 |
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Citation Context ...O Markov sources) it turns out that the constant is of the order one. This led many authors to believe that the simple approximation P[Q> x] e ; x holds� this approximation is commonly referred to as =-=[9]-=- the e ective bandwidth (EB) approximation (sometimes it is also called the dominant rootapproximation). Following this result admission control policies based on the concept of e ective bandwidth hav... |

6 |
Bu er over ow asymptotics for a switch handling many tra c sources
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Citation Context ...� under this assumption it was shown in [9] that e ; N forsomeconstant . A more formal analysis of the multiplexing of a large number of sources and an improvement of the EB approximation is given in =-=[14, 15]-=-. Complementing the work done in [9], in [19] wehave shown that EB approximation may bevery inaccurate in the presence of multiple time scale1. The Asymptotic Behavior of a Network Multiplexer with M... |

5 |
Evaluating the queue length distribution of an atm multiplexer with multiple time scale arrivals
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(Show Context)
Citation Context ...h three exponentials). For that reason we have investigated a perturbation theory based approach for approximating all queue probabilities in the presence of multiple time scale (nearly decomposable) =-=[20]-=- arrival processes (a comprehensive treatment of a discrete time queue with multiple time scale arrivals can be found in [21]). In that work we developed a recursive asymptotic expansion method for ap... |

5 | Subexponential asymptotics of a network multiplexer
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- 1995
(Show Context)
Citation Context ...M/G/1 queue. However the constant of proportionality was left in a complex form. Full extension of Theorem 3.5 to Markov-modulated G/G/1 queues was given in [23] (preliminary results were reported in =-=[22]-=-), where it was proved that the queue length asymptotics are invariant under Markov modulation. A precise statement of this result follows. Let fJtg be a stationary irreducible aperiodic Markov chain ... |

5 |
Sur un mode de croissance réguilière des fonctions,” Mathematica
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Citation Context ...sult was shown using a random walk technique. Some of the rst applications of long-tailed distributions in queueing theory were made by Cohen [13], and Borovkov[6] for functions of regular variations =-=[26, 5]-=-. Recent results on long-tailed and subexponential asymptotics of a GI=GI=1 are given in [1, 38]. (Also, in [1] further motivation is given for the application of long-tailed distributions to communic... |

2 |
Asymptotic properties of a discrete time queue with multiple time scale arrivals. Submited to Queueing Systems
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(Show Context)
Citation Context ...probabilities in the presence of multiple time scale (nearly decomposable) [20] arrival processes (a comprehensive treatment of a discrete time queue with multiple time scale arrivals can be found in =-=[21]-=-). In that work we developed a recursive asymptotic expansion method for approximating the queue length distribution and investigated the radius of convergence of the queue asymptotic expansion series... |

2 |
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Citation Context ...) Benktander Type II distribution [28] F (x) =1; cax ;(1;b) expf;(a=b)x b g� a>0� 0 <b<1, and c appropriately chosen. The general relation between S and L is the following. Lemma 3.3 (Athrey and Ney, =-=[3]-=-) S L. The following lemma [8] clearly shows that for long-tailed distributions Cramer type conditions are not satis ed. Lemma 3.4 If F 2Lthen (1 ; F (x))e x !1as x !1, for all >0. �1. The Asymptotic... |