Annals of Operations Research 93 (2000) 15–40 15 AMVA-based solution procedures for open queueing networks with population constraints Ronald Buitenhek a,* , Geert-Jan van Houtum b and Henk Zijm b a Faculty of Mechanical Engineering, University of Twente, Enschede, The Netherlands b Faculty of Technology Management, Eindhoven University of Technology, Eindhoven, The Netherlands We propose a new method for the performance evaluation of Open Queueing Networks with a Population Constraint (represented by a set of tokens). The method is based on the application of Approximate Mean Value Analysis (AMVA) algorithms. We present procedures for single class networks and for multiple class networks, subject to either a common constraint (shared tokens) or to class-based constraints (dedicated tokens). In fact, the new method is a unified framework into which all procedures for the different types of networks fit. We show how the new method relates to well-known methods and present some numerical results to indicate its accuracy. 1. Introduction We consider the performance evaluation of Open Queueing Networks with a Population Constraint (OQNs-PC). In a single job class OQN-PC the total number of jobs in service or waiting for service at the set of service stations is never larger than the maximum number N . The set of service stations constitute the inner system. Jobs that arrive when the inner system is full are forced to wait outside in an external queue. Once a job is inside the inner system, it follows its Markovian routing until it has completed its final operation. Upon the departure of a job, the job in the first position of the external queue, if any, is allowed to enter the inner system. Due to the presence of the external queue, the OQN-PC fundamentally differs from a loss system, in which arriving jobs that find the network full are lost. The population constraint of the inner system can be represented by a set of tokens. To be allowed in the inner system a job must first get a token. The job keeps the token during its stay in the inner system and releases it upon departure. An obvious generalization of the single class OQN-PC is the multiclass OQN- PC. For the multiclass OQNs-PC, we distinguish three cases. Either each job class has a dedicated population constraint represented by a set of class-specific tokens, or the classes have a common population constraint represented by a set of tokens that * Corresponding author. Current address: KPN Research, P.O. Box 421, 2260 AK Leidschendam, The Netherlands. E-mail: r.buitenhek@research.kpn.com.  J.C. Baltzer AG, Science Publishers