Open Queueing Networks with Blocking – a Personal Log

The analysis of queueing networks with blocking reached its peak in the 1980s and early 1990s. Several elegant decomposition-based approximation techniques were developed, and in this paper, the author gives a personal account of some of these techniques and also some of the players involved. The lessons learnt from analyzing such queueing networks have been useful in developing decomposition-based approximation algorithms for novel queueing networks which have been motivated by new networking technologies. 1. A not so simple open queueing network with blocking I am very pleased to participate in the festivities for the 60 birthday of my good friend, colleague, and almost compatriot, Dr. Günter Haring. XRONIA POLLA, KAI NA TA EKATOSTHSEIS! I first met Günter back in the summer of 1986, the year of the Chernobyl nuclear disaster, during a conference in Vienna. I remember spending a pleasant evening with Günter and my colleague Billy Stewart at a heuringen! I also remembered how astonished I was when I discovered that he spoke fluent Greek! At that time, there was still a lot of interest in the analysis of queueing networks, and also in the more specialized area of queueing networks with blocking. My first encounter with queueing networks with blocking was in 1972, when I started my PhD studies in Trinity College Dublin under the supervision of Professor F. G. Foster. Somewhere between Trinity College and my local pub, O’Neil’s The Crowing Cock, I got the idea to model a set of IBM 2314 disks as an open queueing network with blocking. This is a queueing network of finite capacity nodes, where a customer upon completion of its service at a node gets blocked (i.e., stuck) if its destination node is full. The customer remains blocked until a space becomes available at the destination node. During the time the customer is blocked, the server that had just finished serving this customer is also blocked and it cannot serve any other customers. (This type of blocking should not be confused with the notion of a customer getting blocked in teletraffic, which means that the customer gets lost if it arrives at a node at a time when the node is full.) 1 Department of Computer Science, North Carolina State University, Raleigh, NC 27695, USA The simplest open queueing network with blocking is the one shown in figure 1. The first node has an infinite capacity and the second node has a finite capacity of size m. Service at each node is provided by a single exponential server. External arrivals join the first node in a Poisson fashion at the rate λ. Let μ1, and μ2 be the service rate at the first and second server respectively. Customers in each node are served in a FIFO manner. We assume blocking-after-service. That is, the first server becomes blocked when a customer completes its service at a time the second node is full. The server remains blocked, and it cannot serve any other customer in its queue until a departure occurs from node 2.