Document Zbl 1319.60178

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Methods Oper. Res.</a> <a href="/?q=in%3A345910" title="Articles in this Issue">81, No. 3, 299-315 (2015)</a>. </div> <div class="abstract">Summary: Consider the single-server queue in which customers are rejected if their total sojourn time would exceed a certain level $K$. A basic performance measure of this system is the probability $P_K$ that a customer gets rejected in steady state. This paper presents asymptotic expansions for $P_K$ as $K\to \infty$. If the service time $B$ is light-tailed and the inter-arrival times are exponential, it is shown that the loss probability has an exponential tail. The proof of this result heavily relies on results on the two-sided exit problem for Lévy processes with no positive jumps. For heavy-tailed (subexponential) service times and generally distributed inter-arrival times, the loss probability is shown to be asymptotically equivalent to the trivial lower bound $P(B>K)$.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A60K25" title="MSC2020">60K25</a> </td> <td class="space"> Queueing theory (aspects of probability theory) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60G51" title="MSC2020">60G51</a> </td> <td class="space"> Processes with independent increments; Lévy processes </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68M20" title="MSC2020">68M20</a> </td> <td class="space"> Performance evaluation, queueing, and scheduling in the context of computer systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90B22" title="MSC2020">90B22</a> </td> <td class="space"> Queues and service in operations research </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Asingle-server+queue">single-server queue</a>; <a href="/?q=ut%3Acomplete+rejection">complete rejection</a>; <a href="/?q=ut%3Aloss+probability">loss probability</a>; <a href="/?q=ut%3AL%C3%A9vy+processes">Lévy processes</a>; <a href="/?q=ut%3Atwo-sided+exit+problem">two-sided exit problem</a>; <a href="/?q=ut%3Aasymptotic+expansions">asymptotic expansions</a></div> <div class="software"> <h3>Software:</h3><a href="/software/5198">MCQueue</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1319.60178" data-ciurl="/ci/06457215" data-biburl="/bibtex/06457215.bib" data-amsurl="/amsrefs/06457215.bib" data-xmlurl="/xml/06457215.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06457215.pdf" title="Zbl 1319.60178 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s00186-015-0497-x" aria-label="DOI for “Loss rates in the single-server queue with complete rejection”" title="10.1007/s00186-015-0497-x">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Asmussen, S; Perry, D, Rejection rules in the $M/G/1$ queue, Queueing Syst, 19, 105-130, (1995) · <a href="/0835.60080" class="nowrap">Zbl 0835.60080</a> · <a href="https://doi.org/10.1007/BF01148942" class="nowrap">doi:10.1007/BF01148942</a></td> </tr><tr> <td>[2]</td> <td class="space">Asmussen, S; Sigman, K, Monotone stochastic recursions and their duals, Probab Eng Inf Sci, 10, 1-20, (1996) · <a href="/1095.60519" class="nowrap">Zbl 1095.60519</a> · <a href="https://doi.org/10.1017/S0269964800004137" class="nowrap">doi:10.1017/S0269964800004137</a></td> </tr><tr> <td>[3]</td> <td class="space">Asmussen, S; Klüppelberg, C, Stationary $M/G/1$ excursions in the presence of heavy tails, J Appl Probab, 33, 208-212, (1997) · <a href="/0876.60080" class="nowrap">Zbl 0876.60080</a> · <a href="https://doi.org/10.2307/3215187" class="nowrap">doi:10.2307/3215187</a></td> </tr><tr> <td>[4]</td> <td class="space">Asmussen, S, Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distributions and first passage times, Ann Appl Probab, 8, 354-374, (1998) · <a href="/0942.60034" class="nowrap">Zbl 0942.60034</a> · <a href="https://doi.org/10.1214/aoap/1028903531" class="nowrap">doi:10.1214/aoap/1028903531</a></td> </tr><tr> <td>[5]</td> <td class="space">Asmussen S (2003) Appl Probab Queues, 2nd edn. Springer, New York · <a href="/1029.60001" class="nowrap">Zbl 1029.60001</a></td> </tr><tr> <td>[6]</td> <td class="space">Asmussen, S; Pihlsgaard, M, Loss rates for Lévy processes with two reflecting barriers, Math Oper Res, 32, 308-321, (2007) · <a href="/1341.60036" class="nowrap">Zbl 1341.60036</a> · <a href="https://doi.org/10.1287/moor.1060.0226" class="nowrap">doi:10.1287/moor.1060.0226</a></td> </tr><tr> <td>[7]</td> <td class="space">Bekker, R; Zwart, AP, On an equivalence between loss rates and cycle maxima in queues and dams, Probab Eng Inf Sci, 19, 241-255, (2005) · <a href="/1097.90017" class="nowrap">Zbl 1097.90017</a> · <a href="https://doi.org/10.1017/S0269964805050138" class="nowrap">doi:10.1017/S0269964805050138</a></td> </tr><tr> <td>[8]</td> <td class="space">Bertoin, J; Doney, R, Cramér’s estimate for Lévy processes, Stat Probab Lett, 21, 363-365, (1994) · <a href="/0809.60085" class="nowrap">Zbl 0809.60085</a> · <a href="https://doi.org/10.1016/0167-7152(94)00032-8" class="nowrap">doi:10.1016/0167-7152(94)00032-8</a></td> </tr><tr> <td>[9]</td> <td class="space">Bertoin, J, Exponential decay and ergodicity of completely asymmetric Lévy processes on a finite interval, Ann Appl Probab, 7, 156-169, (1997) · <a href="/0880.60077" class="nowrap">Zbl 0880.60077</a> · <a href="https://doi.org/10.1214/aoap/1034625257" class="nowrap">doi:10.1214/aoap/1034625257</a></td> </tr><tr> <td>[10]</td> <td class="space">Boots, NK; Tijms, HC, A multiserver queueing system with impatient customers, Manag Sci, 45, 444-448, (1999) · <a href="/1231.90134" class="nowrap">Zbl 1231.90134</a> · <a href="https://doi.org/10.1287/mnsc.45.3.444" class="nowrap">doi:10.1287/mnsc.45.3.444</a></td> </tr><tr> <td>[11]</td> <td class="space">Cohen, JW, Single-server queues with restricted accessibility, J Eng Math, 3, 265-284, (1969) · <a href="/0186.24503" class="nowrap">Zbl 0186.24503</a> · <a href="https://doi.org/10.1007/BF01535299" class="nowrap">doi:10.1007/BF01535299</a></td> </tr><tr> <td>[12]</td> <td class="space">Cohen JW (1976) Regen processes queueing theory. Springer, Berlin · <a href="/0323.60083" class="nowrap">Zbl 0323.60083</a> · <a href="https://doi.org/10.1007/978-3-642-95281-4" class="nowrap">doi:10.1007/978-3-642-95281-4</a></td> </tr><tr> <td>[13]</td> <td class="space">Cohen JW (1982) The single server queue. North Holland, Amsterdam · <a href="/0481.60003" class="nowrap">Zbl 0481.60003</a></td> </tr><tr> <td>[14]</td> <td class="space">Debicki, K; Mandjes, M, Lévy-driven queues, Surv Oper Res Manag Sci, 17, 15-37, (2012)</td> </tr><tr> <td>[15]</td> <td class="space">Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling extremal events. Springer, Berlin · <a href="/0873.62116" class="nowrap">Zbl 0873.62116</a> · <a href="https://doi.org/10.1007/978-3-642-33483-2" class="nowrap">doi:10.1007/978-3-642-33483-2</a></td> </tr><tr> <td>[16]</td> <td class="space">Foss, S; Zachary, S, The maximum on a random time interval of a random walk with long-tailed increments and negative drift, Ann Appl Probab, 13, 37-53, (2003) · <a href="/1045.60039" class="nowrap">Zbl 1045.60039</a> · <a href="https://doi.org/10.1214/aoap/1042765662" class="nowrap">doi:10.1214/aoap/1042765662</a></td> </tr><tr> <td>[17]</td> <td class="space">Gavish, B; Schweitzer, P, The Markovian queue with bounded waiting time, Manag Sci, 23, 1349-1357, (1977) · <a href="/0372.60134" class="nowrap">Zbl 0372.60134</a> · <a href="https://doi.org/10.1287/mnsc.23.12.1349" class="nowrap">doi:10.1287/mnsc.23.12.1349</a></td> </tr><tr> <td>[18]</td> <td class="space">Iglehart, DG, Extreme values in the $GI/G/1$ queue, Ann Math Stat, 43, 627-635, (1972) · <a href="/0238.60072" class="nowrap">Zbl 0238.60072</a> · <a href="https://doi.org/10.1214/aoms/1177692642" class="nowrap">doi:10.1214/aoms/1177692642</a></td> </tr><tr> <td>[19]</td> <td class="space">Kok, AG; Tijms, HC, A two-moment approximation for a buffer design problem requiring a small rejection probability, Perform Eval, 5, 77-84, (1985) · <a href="https://doi.org/10.1016/0166-5316(85)90040-9" class="nowrap">doi:10.1016/0166-5316(85)90040-9</a></td> </tr><tr> <td>[20]</td> <td class="space">Kyprianou, AE; Palmowski, Z, A martingale review of some fluctuation theory for spectrally negative Lévy processes, Sémin Probab, XXXVIII, 16-29, (2004) · <a href="/1063.60071" class="nowrap">Zbl 1063.60071</a></td> </tr><tr> <td>[21]</td> <td class="space">Ommeren, JCW, Exponential bounds for excess probabilities in systems with a finite capacity, Stoch Process Appl, 24, 143-149, (1987) · <a href="/0614.60091" class="nowrap">Zbl 0614.60091</a> · <a href="https://doi.org/10.1016/0304-4149(87)90034-2" class="nowrap">doi:10.1016/0304-4149(87)90034-2</a></td> </tr><tr> <td>[22]</td> <td class="space">Norvang Andersen, L, Subexponential loss rate asymptotics for Lévy processes, Math Methods Oper Res, 73, 91-108, (2011) · <a href="/1209.93139" class="nowrap">Zbl 1209.93139</a> · <a href="https://doi.org/10.1007/s00186-010-0335-0" class="nowrap">doi:10.1007/s00186-010-0335-0</a></td> </tr><tr> <td>[23]</td> <td class="space">Pakes, AG, On the tails of waiting-time distributions, J Appl Probab, 12, 555-564, (1975) · <a href="/0314.60072" class="nowrap">Zbl 0314.60072</a> · <a href="https://doi.org/10.2307/3212870" class="nowrap">doi:10.2307/3212870</a></td> </tr><tr> <td>[24]</td> <td class="space">Schmidli, H, On the distribution of the surplus prior and at ruin, ASTIN Bull, 29, 227-244, (1999) · <a href="/1129.62425" class="nowrap">Zbl 1129.62425</a> · <a href="https://doi.org/10.2143/AST.29.2.504613" class="nowrap">doi:10.2143/AST.29.2.504613</a></td> </tr><tr> <td>[25]</td> <td class="space">Suprun, VN, Problem of destruction and resolvent of terminating processes with independent increments, Ukr Math J, 28, 39-45, (1976) · <a href="/0349.60075" class="nowrap">Zbl 0349.60075</a> · <a href="https://doi.org/10.1007/BF01559226" class="nowrap">doi:10.1007/BF01559226</a></td> </tr><tr> <td>[26]</td> <td class="space">Takács L (1967) Combinatorial methods in the theory of stochastic processes. Wiley, New York · <a href="/0162.21303" class="nowrap">Zbl 0162.21303</a></td> </tr><tr> <td>[27]</td> <td class="space">Tijms HC (2003) A first course in stochastic models. Wiley, New York · <a href="/1088.60002" class="nowrap">Zbl 1088.60002</a> · <a href="https://doi.org/10.1002/047001363X" class="nowrap">doi:10.1002/047001363X</a></td> </tr><tr> <td>[28]</td> <td class="space">Veraverbeke, N, Asymptotic behaviour of Wiener-Hopf factors of a random walk, Stoch Process Appl, 5, 27-37, (1977) · <a href="/0353.60073" class="nowrap">Zbl 0353.60073</a> · <a href="https://doi.org/10.1016/0304-4149(77)90047-3" class="nowrap">doi:10.1016/0304-4149(77)90047-3</a></td> </tr><tr> <td>[29]</td> <td class="space">Zwart, AP, A fluid queue with a finite buffer and subexponential input, Adv Appl Probab, 32, 221-243, (2000) · <a href="/0964.60081" class="nowrap">Zbl 0964.60081</a> · <a href="https://doi.org/10.1239/aap/1013540031" class="nowrap">doi:10.1239/aap/1013540031</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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