ISSN 0032-9460, Problems of Information Transmission, 2008, Vol. 44, No. 4, pp. 370–384.
Pleiades Publishing, Inc., 2008.
Original Russian Text
N.D. Vvedenskaya, E.A. Pechersky, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 4, pp. 92–108.
COMMUNICATION NETWORK THEORY
Circle of Interacting Servers: Spontaneous Collective
Behavior in the Case of Large Fluctuations
N. D. Vvedenskaya and E. A. Pechersky
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Received July 3, 2007; in ﬁnal form, September 10, 2008
Abstract—We consider large ﬂuctuations and overload of servers in a network with dynamic
routing of messages. The servers form a circle. The number of input ﬂows is equal to the
number of servers; the messages of a ﬂow are distributed between two neighboring servers;
upon its arrival, a message is directed to the least loaded of these servers. Under the condition
that at least two servers are overloaded, the number of overloaded servers in such a network
depends on the rate of input ﬂows. In particular, there exists a critical level of the input rate
above which all servers are most probably overloaded.
This work describes an eﬀect in a network with interacting servers that can be called spontaneous
collective behavior in the case of large ﬂuctuations.
We consider networks with dynamic routing of messages. In such networks, the server to which
a message is directed depends on the network state at the message arrival moment. One of the
problems is to analyze the probability of large ﬂuctuations, for example, the probability of large
There are many works investigating large ﬂuctuations in networks with dynamic routing.
In [1–7], networks with two servers and three independent input ﬂows were considered where only
one ﬂow is divided between two servers, depending on either the workload of servers or the queue
lengths. In , a network with a group of servers and several ﬂows was considered where each ﬂow
is assigned to some subgroup of servers; upon its arrival, a message selects a server with the shortest
queue (i.e., a queue with the least number of messages). In the same work, the large deviation
principle for ﬂows to servers is proved.
Here we consider circular networks formed by k servers and k identical independent input Poisson
ﬂows. Messages of a ﬂow are assigned to two neighboring servers (see Fig. 1). A message direction
depends on workloads of these two servers; namely, upon its arrival, a message is directed to the
server with the smallest workload. Each server operates with a constant rate equal to 1, and the
discipline is ﬁrst-in-ﬁrst-out (FIFO). If a message ﬁnds the server busy, it is put into an inﬁnite
buﬀer to wait for service. We consider networks that work stationary. This means that, with
probability 1, the queues do not increase inﬁnitely. However, there may appear large ﬂuctuations;
for example, one of the ﬂows may bring a very large amount of work during a short period. We say
that during this period the ﬂow is overheated. Assume, for example, that the ﬂow f
Then the buﬀers of the servers s
, to which this ﬂow is assigned, contain a large amount of
work. What is the behavior of other ﬂows under such condition?
We shown that in the case where a message of the ﬂow f
waits for service long there are at least
two scenarios of the network performance. Which one is more probable and is realized depends