## Subexponential Loss Rates in a GI/GI/1 Queue with Applications (1999)

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Citations: | 17 - 4 self |

### BibTeX

@MISC{Jelenkovic99subexponentialloss,

author = {Predrag R. Jelenkovic},

title = {Subexponential Loss Rates in a GI/GI/1 Queue with Applications},

year = {1999}

}

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### Abstract

Consider a single server queue with i.i.d. arrival and service processes, fA; A n ; n 1g and fC; C n ; n 1g, respectively, and finite buffer B. The queue content process fQ B n ; n 0g is recursively defined as Q B n+1 = min((Q B n + A n+1 \Gamma C n+1 ) + ; B), q + = max(0; q). When E(A \Gamma C) ! 0, and A has a subexponential distribution, we show that the stationary expected loss rate for this queue E(Q B n + A n+1 \Gamma C n+1 \Gamma B) + has the following explicit asymptotic characterization E(Q B n +A n+1 \Gamma C n+1 \Gamma B) + ¸ E(A \Gamma B) + as B !1; independently of the server process C n . For a fluid queue with capacity c, M/G/1 arrival process a t , characterized by intermediately regularly varying On periods ø on , that arrive with Poisson rate , the average loss rate B loss satisfies B loss ¸ E(ø on j \Gamma B) + as B !1; where j = r + ae \Gamma c, ae = Ea t ! c, and r; r c, is the rate at which the fluid is arriving during an On per...

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Citation Context ...ubexponential asymptotic behavior of the waiting time distribution in a GI/GI/1 queue [12,35,43] (these results were used in [2,21]). Asymptotic expansion refinements of these results can be found in =-=[1,44]-=-. Generalizations to queueing processes (random walks) with dependent increments were investigated in [4,5,24]. Queueing models with multiple long-tailed arrival streams are of particular interest for... |

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Citation Context ...xponential loss rates in a GI/GI/1 queue with applications Theorem 5. If Ae ∈S, EA <EC, then E ( W B n + An+1 − B ) + = E(A − B) + ( 1 + o(1) ) as B →∞. Remark. This theorem generalizes a result from =-=[45]-=- which is true for A being regularly varying, and also for the case of C being exponential and A subexponential. Proof. The proof of the lower bound is immediate: E ( W B n + An+1 − B ) + � E(An+1 − B... |

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Citation Context ...els with a single long-tailed arrival stream are the classical results on subexponential asymptotic behavior of the waiting time distribution in a GI/GI/1 queue [12,35,43] (these results were used in =-=[2,21]-=-). Asymptotic expansion refinements of these results can be found in [1,44]. Generalizations to queueing processes (random walks) with dependent increments were investigated in [4,5,24]. Queueing mode... |

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Citation Context ...32]. In [8] a limiting process obtained by multiplexing an infinite number of on–off sources with regularly varying on periods was analyzed. This limiting arrival process, the so-called M/G/∞ process =-=[37]-=-, appears to be quite promising for the analysis. In [25] an explicit asymptotic formula for the behavior of the infinite buffer queue length distribution with M/G/∞ arrivals was derived. In the same ... |

5 | Long-tailed loss rates in single server queue
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Citation Context ... + /EA, which in conjunction with (10) can be expressed in the following compact form p(B) ∼ P[Ae >B] as B →∞. (iii) This theorem is an improvement of a theorem from the original version of the paper =-=[23]-=- which was proved under the assumption of A being regularly varying P[A >x] = l(x)/x α with index α>2. □98 P.R. Jelenković / Subexponential loss rates in a GI/GI/1 queue with applications Proof. Assu... |

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