Threshold Phenomena in the Transient Behaviour of Markovian Models of Communication Networks and Databases

This paper accompanies a talk given at the Workshop on Mathematical Methods in Queueing Networks held at the Mathematical Sciences Institute at Cornell University in August 1988. In earlier work we had exhibited a threshold phenomenon in the transient behaviour of a closed network of . / M / 1 nodes: When there are N customers circulating, and the initial state is x, let d~(t ) denote the total variation distance between the distribution at time t and the stationary distribution. Let dU(t )= maxxdff(t). We explicitly found a N proportional to N such that dU(taN) ---~1 for every t <1, and dr( taN) --~ 0 for every t >1. Thus it appears that the network has not yet converged to stationarity upto aN, but has converged to stationarity after aN, SO a N can be naturally interpreted as the settling time of the network. Here we briefly deal with some other similar models closed networks of . / M / m nodes, a well studied model for circuit switched networks, and a model of Mitra for studying concurrency control in databases. Similar threshold phenomena are established in the transient behaviour of these models.