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  <doc>
    <id>1319</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2011-06-22</completedDate>
    <publishedDate>2011-06-22</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">On sojourn times for an infinite-server system in random environment and its application to processor sharing systems</title>
    <abstract language="eng">We deal with an infinite-server system where the&#13;
service speed is governed by a stationary and ergodic&#13;
process with countably many states. Applying a random&#13;
time transformation such that the service speed&#13;
becomes one, the sojourn time of a class of virtual&#13;
requests with given required service time is equal&#13;
in distribution to an additive functional defined&#13;
via a stationary version of the time-changed process.&#13;
Thus bounds for the expectation of functions of additive&#13;
functionals yield bounds for the expectation&#13;
of functions of virtual sojourn times, in particular&#13;
bounds for fractional moments and the distribution&#13;
function. Interpreting the $GI(n)/GI(n)/\infty$ system or&#13;
equivalently the $GI(n)/GI$ system under state-dependent&#13;
processor sharing as an infinite-server system with&#13;
random states given by the number $n$ of requests&#13;
in the system provides results for sojourn times&#13;
of virtual requests. In case of $M(n)/GI(n)/\infty$,&#13;
the sojourn times of arriving and added requests are&#13;
equal in distribution to sojourn times of virtual&#13;
requests in modified systems, which yields many results&#13;
for the sojourn times of arriving and added requests.&#13;
In case of integer moments, the bounds generalize&#13;
earlier results for $M/GI(n)/\infty$. In particular,&#13;
the mean sojourn times of arriving and added requests&#13;
in $M(n)/GI(n)/\infty$ are proportional to the required&#13;
service time, generalizing Cohen's famous result&#13;
for $M/GI(n)/\infty$.</abstract>
    <identifier type="serial">11-28</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-13190</identifier>
    <enrichment key="SourceTitle">Appeared under the title "Additive functionals with application to sojourn times in infinite-server and processor sharing systems" in: Queuing Systems 70 (2012) 369-409</enrichment>
    <author>Brandt Manfred</author>
    <submitter>Brandt Manfred</submitter>
    <author>Brandt Andreas</author>
    <series>
      <title>ZIB-Report</title>
      <number>11-28</number>
    </series>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>infinite-server</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>random environment</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>time transformation</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>sojourn times</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>state-dependent processor sharing</value>
    </subject>
    <collection role="msc" number="60-XX">PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX)</collection>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="institutes" number="aopt">Applied Optimization</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/1319/ZR-11-28.pdf</file>
  </doc>
</export-example>
