 
Summary: Queueing Systems, 5 (1989) 7798 77
THRESHOLD PHENOMENA IN THE TRANSIENT BEHAVIOUR
OF MARKOVIAN MODELS OF COMMUNICATION NETWORKS
AND DATABASES
V. ANANTHARAM *
School of Electrical Engineering. Cornell University, Ithaca, NY 14853, U.S.A.
Received 4 August 1988; revised 3 March 1989
Abstract
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 aN
proportional to N such that dU(taN) ~1for 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 aN 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
