{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:14:06Z","timestamp":1760242446011,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,7,10]],"date-time":"2017-07-10T00:00:00Z","timestamp":1499644800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Future Internet"],"abstract":"<jats:p>Abstract: Queueing networks are used to model the performance of the Internet, of manufacturing and job-shop systems, supply chains, and other networked systems in transportation or emergency management. Composed of service stations where customers receive service, and then move to another service station till they leave the network, queueing networks are based on probabilistic assumptions concerning service times and customer movement that represent the variability of system workloads. Subject to restrictive assumptions regarding external arrivals, Markovian movement of customers, and service time distributions, such networks can be solved efficiently with \u201cproduct form solutions\u201d that reduce the need for software simulators requiring lengthy computations. G-networks generalise these models to include the effect of \u201csignals\u201d that re-route customer traffic, or negative customers that reject service requests, and also have a convenient product form solution. This paper extends G-networks by including a new type of signal, that we call an \u201cAdder\u201d, which probabilistically changes the queue length at the service center that it visits, acting as a load regulator. We show that this generalisation of G-networks has a product form solution.<\/jats:p>","DOI":"10.3390\/fi9030034","type":"journal-article","created":{"date-parts":[[2017,7,10]],"date-time":"2017-07-10T11:05:31Z","timestamp":1499684731000},"page":"34","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["G-Networks with Adders"],"prefix":"10.3390","volume":"9","author":[{"given":"Jean-Michel","family":"Fourneau","sequence":"first","affiliation":[{"name":"Laboratoire DAVID, Universit\u00e9 de Versailles-Saint-Quentin, 78000 Versailles, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Erol","family":"Gelenbe","sequence":"additional","affiliation":[{"name":"Intelligent Systems & Networks Group, Imperial College, London SW7 2AZ, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,7,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1145\/362342.362345","article-title":"Computational algorithms for closed queueing networks with exponential servers","volume":"16","author":"Buzen","year":"1973","journal-title":"Commun. ACM"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/0166-5316(85)90012-4","article-title":"The Tree MVA Algorithm","volume":"5","author":"Tucci","year":"1985","journal-title":"Perform. Eval."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1093\/comjnl\/bxq092","article-title":"A framework for energy-aware routing in packet networks","volume":"54","author":"Gelenbe","year":"2011","journal-title":"Comput. J."},{"key":"ref_4","unstructured":"(2017, July 07). IBM Research Performance Modelling and Analysis. Available online: researcher.watson.ibm.com\/researcher\/view_group.php?id=150."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1109\/36.823897","article-title":"Area based results for mine detection","volume":"38","author":"Gelenbe","year":"2000","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1093\/comjnl\/bxp101","article-title":"Levenberg-Marquardt training algorithms for Random Neural Networks","volume":"54","author":"Basterrech","year":"2011","journal-title":"Comput. J."},{"key":"ref_7","unstructured":"\u00c7ag\u0303layan, M.U. (2017). G-Networks and their applications to Machine Learning, Energy Packet Networks and Routing: Introduction to the Special issue. Probab. Eng. Inf. Sci., 1\u201315."},{"key":"ref_8","first-page":"63","article-title":"A queue with server of walking type (autonomous service)","volume":"16","author":"Gelenbe","year":"1980","journal-title":"Annales de l\u2019I.H.P. Probabilit\u00e9s et Statistiques,"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1287\/trsc.16.2.207","article-title":"Queueing models of classification and connection delay in railyards","volume":"16","author":"Turnquist","year":"1982","journal-title":"Transp. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"MacGregor Smith, J., and Tan, B. (2013). Queueing network models of material handling and transportation systems. Handbook of Stochastic Models and Analysis of Manufacturing System Operations, Springer.","DOI":"10.1007\/978-1-4614-6777-9_8"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Henry, K.D., Wood, N.J., and Frazier, T.G. (2016). Influence of road network and population demand assumptions in evacuation modelling for distant tsunamis. Nat. Hazards, 1\u201323.","DOI":"10.1007\/s11069-016-2655-8"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Bi, H., and Abdelrahman, O.H. (2016). Energy-aware navigation in large-scale evacuation using G-Networks. Probab. Eng. Inf. Sci.","DOI":"10.1017\/S0269964816000115"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1017\/S0269964800002539","article-title":"Stability of product form G-Networks","volume":"6","author":"Gelenbe","year":"1992","journal-title":"Probab. Eng. Inf. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/S0377-2217(97)00371-8","article-title":"G-networks with multiple classes of signals and positive customers","volume":"108","author":"Gelenbe","year":"1998","journal-title":"Eur. J. Oper. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/S0377-2217(99)00476-2","article-title":"G-networks: A versatile approach for work removal in queuing networks","volume":"126","author":"Artalejo","year":"2000","journal-title":"Eur. J. Oper. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1287\/mnsc.10.1.131","article-title":"Jobshop-like queueing systems","volume":"10","author":"Jackson","year":"1963","journal-title":"Manag. Sci."},{"key":"ref_17","first-page":"54","article-title":"G-network with the route change","volume":"Special Issue","author":"Manzo","year":"2008","journal-title":"Sist. I Sredstva Inform."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1017\/S0269964800002953","article-title":"G-Networks with signals and batch removal","volume":"7","author":"Gelenbe","year":"1993","journal-title":"Probab. Eng. Inf. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1023\/A:1010960804669","article-title":"On the existence of invariant measures for networks with string transitions","volume":"38","author":"Schassberger","year":"2001","journal-title":"Queueing Syst."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Medhi, J. (2003). Stochastic Processes in Queueing Theory, Elsevier Science Direct.","DOI":"10.1016\/B978-012487462-6\/50001-1"},{"key":"ref_21","unstructured":"Ross, S.M. (1996). Stochastic Processes, Wiley Series in Probability and Statistics, Wiley."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1007\/BF02033314","article-title":"G-networks: An unifying model for queuing networks and neural networks","volume":"48","author":"Gelenbe","year":"1994","journal-title":"Ann. Oper. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/0304-3975(95)00018-6","article-title":"G-networks with multiple classes of positive and negative customers","volume":"155","author":"Fourneau","year":"1996","journal-title":"Theor. Comput. Sci."},{"key":"ref_24","first-page":"28","article-title":"Modelling and analysis of gene regulatory networks based on the G-network","volume":"6","author":"Kim","year":"2014","journal-title":"J. Int. Adv. Intell. Paradig."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1162\/neco.1993.5.1.154","article-title":"Learning in the recurrent random network","volume":"5","author":"Gelenbe","year":"1993","journal-title":"Neural Comput."},{"key":"ref_26","unstructured":"Gelenbe, E., Gellman, M., Lent, R., Liu, P., and Su, P. (2004, January 17\u201318). Autonomous smart routing for network QoS. Proceedings of the 2004 International Conference on Autonomic Computing, New York, NY, USA."},{"key":"ref_27","unstructured":"Gelenbe, E., Lent, R., Montuori, A., and Xu, Z. (2002, January 11\u201316). Cognitive packet networks: QoS and performance. Proceedings of the 10th IEEE Computer Society: International Symposium on Modeling, Analysis, and Simulation of Computer and Telecommunications Systems (MASCOTS 2002), Fort Worth, TX, USA."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"155","DOI":"10.17512\/jamcm.2014.3.16","article-title":"Investigation of G-Networks with random delay of signals at non-stationary behaviour","volume":"13","author":"Matalytski","year":"2014","journal-title":"J. Appl. Math. Comput. Mech."}],"container-title":["Future Internet"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-5903\/9\/3\/34\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:42:07Z","timestamp":1760208127000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-5903\/9\/3\/34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,10]]},"references-count":28,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,9]]}},"alternative-id":["fi9030034"],"URL":"https:\/\/doi.org\/10.3390\/fi9030034","relation":{},"ISSN":["1999-5903"],"issn-type":[{"type":"electronic","value":"1999-5903"}],"subject":[],"published":{"date-parts":[[2017,7,10]]}}}