{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T21:40:04Z","timestamp":1750282804855,"version":"3.41.0"},"reference-count":12,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[1986,1,2]],"date-time":"1986-01-02T00:00:00Z","timestamp":505008000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["J. ACM"],"published-print":{"date-parts":[[1986,1,2]]},"abstract":"<jats:p>An explicit steady-state solution is given for any queuing loop made up of two general servers, whose distribution functions have rational Laplace transforms. The solution is in matrix geometric form over a vector space that is itself a direct or Kronecker product of the internal state spaces of the two servers. The algebraic properties of relevant entities in this space are given in an appendix. The closed-form solution yields simple recursive relations that in turn lead to an efficient algorithm for calculating various performance measures such as queue length and throughput. A computational-complexity analysis shows that the algorithm requires at least an order of magnitude less computational effort than any previously reported algorithm.<\/jats:p>","DOI":"10.1145\/4904.5391","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T11:25:57Z","timestamp":1027769157000},"page":"207-223","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":14,"title":["A matrix-algebraic solution to two\n            <i>\n              K\n              <sub>m<\/sub>\n            <\/i>\n            servers in a loop"],"prefix":"10.1145","volume":"33","author":[{"given":"Appie","family":"van de Liefvoort","sequence":"first","affiliation":[{"name":"Univ. of Kansas, Lawrence"}]},{"given":"Lester","family":"Lipsky","sequence":"additional","affiliation":[{"name":"Univ. of Nebraska, Lincoln"}]}],"member":"320","published-online":{"date-parts":[[1986,1,2]]},"reference":[{"key":"e_1_2_1_1_2","volume-title":"Introduction to Matrix Analysis","author":"BELLMAN R.","year":"1970","unstructured":"BELLMAN , R. Introduction to Matrix Analysis . 2 nd ed. McGraw-Hill , New York , 1970 . BELLMAN, R. Introduction to Matrix Analysis. 2nd ed. McGraw-Hill, New York, 1970.","edition":"2"},{"key":"e_1_2_1_2_2","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1287\/opre.30.3.490","article-title":"type loops with extension to 34\/(7\/1 and G 1\/M\/l queues","volume":"30","author":"CARROLL J.L.","year":"1982","unstructured":"CARROLL , J.L. , VAN DE LIEFVOORT , A. , AND LIPSKY , L. Solutions of M\/G\/1\/\/N type loops with extension to 34\/(7\/1 and G 1\/M\/l queues . Oper. Res. 30 ( 1982 ), 490 - 514 . CARROLL, J.L., VAN DE LIEFVOORT, A., AND LIPSKY, L. Solutions of M\/G\/1\/\/N type loops with extension to 34\/(7\/1 and G 1\/M\/l queues. Oper. Res. 30 (1982), 490-514.","journal-title":"Oper. Res."},{"key":"e_1_2_1_3_2","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1017\/S0305004100030437","article-title":"The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables","volume":"51","year":"1955","unstructured":"Cox, D.R . The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables . Proc. Camb. Phil. Soc. 51 ( 1955 ), 433 - 441 . Cox, D.R. The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables. Proc. Camb. Phil. Soc. 51 (1955), 433-441.","journal-title":"Proc. Camb. Phil. Soc."},{"key":"e_1_2_1_4_2","volume-title":"Management Sciences Research Project, Univ. of California","author":"EVANS R.V.","year":"1963","unstructured":"EVANS , R.V. The structure of some two-dimensional queueing systems. Res. Rep. 85 , Management Sciences Research Project, Univ. of California , Los Angeles , Calif ., 1963 . EVANS, R.V. The structure of some two-dimensional queueing systems. Res. Rep. 85, Management Sciences Research Project, Univ. of California, Los Angeles, Calif., 1963."},{"key":"e_1_2_1_5_2","volume-title":"Ellis Horwood","author":"GRAHAM A.","year":"1981","unstructured":"GRAHAM , A. Kronecker Products and Matrix Calculus . Ellis Horwood , Chicester, England , 1981 . GRAHAM, A. Kronecker Products and Matrix Calculus. Ellis Horwood, Chicester, England, 1981."},{"key":"e_1_2_1_6_2","unstructured":"LlpSKY L. FANG X. AND RAMASWAMI V. A unique minimal representation of Coxian service centers. (Unpublished manuscript).  LlpSKY L. FANG X. AND RAMASWAMI V. A unique minimal representation of Coxian service centers. (Unpublished manuscript)."},{"key":"e_1_2_1_7_2","volume-title":"Matrix-Geometric Solutions in Stochastic Models","author":"NEUTS M.F.","year":"1981","unstructured":"NEUTS , M.F. Matrix-Geometric Solutions in Stochastic Models . John Hopkins University Press , Baltimore, Md ., 1981 . NEUTS, M.F. Matrix-Geometric Solutions in Stochastic Models. John Hopkins University Press, Baltimore, Md., 1981."},{"key":"e_1_2_1_8_2","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1287\/opre.30.3.480","article-title":"Explicit steady-state solutions to some elementary queueing models","volume":"30","author":"NE","year":"1982","unstructured":"NE urs, M.F . Explicit steady-state solutions to some elementary queueing models . Oper. Res. 30 ( 1982 ), 480 - 489 . NEurs, M.F. Explicit steady-state solutions to some elementary queueing models. Oper. Res. 30 (1982), 480-489.","journal-title":"Oper. Res."},{"key":"e_1_2_1_9_2","volume-title":"Methods of Matrix Algebra","author":"PEASE M.C.","year":"1965","unstructured":"PEASE , M.C. Methods of Matrix Algebra . Academic Press , Orlando, Fla ., 1965 . PEASE, M.C. Methods of Matrix Algebra. Academic Press, Orlando, Fla., 1965."},{"key":"e_1_2_1_10_2","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1017\/S0305004100028620","article-title":"On the distribution ofqueueing times","volume":"49","author":"SMITH W.L","year":"1953","unstructured":"SMITH , W.L . On the distribution ofqueueing times . Proc. Camb. Phil. Soc. 49 ( 1953 ), 449 - 461 . SMITH, W.L. On the distribution ofqueueing times. Proc. Camb. Phil. Soc. 49 (1953), 449-461.","journal-title":"Proc. Camb. Phil. Soc."},{"key":"e_1_2_1_11_2","first-page":"275","volume-title":"Eds. Birkhiiuser","author":"STEWART W.J.","year":"1982","unstructured":"STEWART , W.J. On computing the stationary probability vector of a network of two Coxian servers. In Applied Probability--Computer Science, The Interface, vol I, R.L. Disney and T.J. Ott , Eds. Birkhiiuser , Boston , 1982 , pp. 275 - 296 . STEWART, W.J. On computing the stationary probability vector of a network of two Coxian servers. In Applied Probability--Computer Science, The Interface, vol I, R.L. Disney and T.J. Ott, Eds. Birkhiiuser, Boston, 1982, pp. 275-296."},{"key":"e_1_2_1_13_2","first-page":"141","volume-title":"AFIPS Spring Joint Computer Conference Proceedings. AFIPS Press, Reston, Va.","author":"WALLACE V.L.","year":"1966","unstructured":"WALLACE , V.L. , AND ROSENBERG , R.S. Markovian models and numerical analysis of computer system behavior . In AFIPS Spring Joint Computer Conference Proceedings. AFIPS Press, Reston, Va. , 1966 , pp. 141 - 148 . WALLACE, V.L., AND ROSENBERG, R.S. Markovian models and numerical analysis of computer system behavior. In AFIPS Spring Joint Computer Conference Proceedings. AFIPS Press, Reston, Va., 1966, pp. 141-148."}],"container-title":["Journal of the ACM"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/4904.5391","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/4904.5391","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T21:15:16Z","timestamp":1750281316000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/4904.5391"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,1,2]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1986,1,2]]}},"alternative-id":["10.1145\/4904.5391"],"URL":"https:\/\/doi.org\/10.1145\/4904.5391","relation":{},"ISSN":["0004-5411","1557-735X"],"issn-type":[{"type":"print","value":"0004-5411"},{"type":"electronic","value":"1557-735X"}],"subject":[],"published":{"date-parts":[[1986,1,2]]},"assertion":[{"value":"1986-01-02","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}