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The maximum steady-state waiting time in this queueing system scales around <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\frac{1}{\\gamma }\\log N$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mfrac>\n                      <mml:mn>1<\/mml:mn>\n                      <mml:mi>\u03b3<\/mml:mi>\n                    <\/mml:mfrac>\n                    <mml:mo>log<\/mml:mo>\n                    <mml:mi>N<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, where <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\gamma $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b3<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is determined by the cumulant generating function <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Lambda $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u039b<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> of the service times distribution and solves the Cram\u00e9r\u2013Lundberg equation with stochastic service times and deterministic interarrival times. This value <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\frac{1}{\\gamma }\\log N$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mfrac>\n                      <mml:mn>1<\/mml:mn>\n                      <mml:mi>\u03b3<\/mml:mi>\n                    <\/mml:mfrac>\n                    <mml:mo>log<\/mml:mo>\n                    <mml:mi>N<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is reached at a certain hitting time. The number of arrivals until that hitting time satisfies the central limit theorem, with standard deviation <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\frac{\\sigma _A}{\\sqrt{\\Lambda '(\\gamma )\\gamma }}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mfrac>\n                    <mml:msub>\n                      <mml:mi>\u03c3<\/mml:mi>\n                      <mml:mi>A<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:msqrt>\n                      <mml:mrow>\n                        <mml:msup>\n                          <mml:mi>\u039b<\/mml:mi>\n                          <mml:mo>\u2032<\/mml:mo>\n                        <\/mml:msup>\n                        <mml:mrow>\n                          <mml:mo>(<\/mml:mo>\n                          <mml:mi>\u03b3<\/mml:mi>\n                          <mml:mo>)<\/mml:mo>\n                        <\/mml:mrow>\n                        <mml:mi>\u03b3<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:msqrt>\n                  <\/mml:mfrac>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. By using the distributional form of Little\u2019s law, we can extend this result to the maximum queue length. Finally, we extend these results to a fork-join queue with different classes of servers.\n<\/jats:p>","DOI":"10.1007\/s11134-025-09937-2","type":"journal-article","created":{"date-parts":[[2025,2,22]],"date-time":"2025-02-22T20:30:58Z","timestamp":1740256258000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Extreme values for the waiting time in large fork-join queues"],"prefix":"10.1007","volume":"109","author":[{"given":"Dennis","family":"Schol","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0457-2925","authenticated-orcid":false,"given":"Maria","family":"Vlasiou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bert","family":"Zwart","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,2,22]]},"reference":[{"key":"9937_CR1","unstructured":"ASML Holding N.V. 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