ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 3, pp. 289–303.
Pleiades Publishing, Inc., 2011.
Original Russian Text
N.D. Vvedenskaya, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 3, pp. 80–95.
COMMUNICATION NETWORK THEORY
Conﬁguration of Overloaded Servers
with Dynamic Routing
N. D. Vvedenskaya
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow
Received May 7, 2009; in ﬁnal form, June 10, 2011
Abstract—We consider overload of servers in a network with dynamic routing of messages.
The system consists of k servers and independent Poisson input ﬂows. Messages from each
ﬂow are directed to m servers, and each message is directed to a server that is the least loaded
at the moment of its arrival. In such a system, conﬁguration of overloaded servers depends on
the intensity of input ﬂows. A similar eﬀect was considered in  for a system with another
We consider networks with dynamic routing of messages. In such a system a message is directed
so that it is served by the least loaded server. One of the problems related to such systems is the
probability of large message delay.
In many papers devoted to problems of large delay ﬂuctuations in networks with dynamic rout-
ing, it was proposed that distributions of message length have “light tails.” For example, [2–8]
considered systems with two servers and three independent Poisson ﬂows where only one ﬂow is
distributed between two servers, depending on workloads of servers or on their queue lengths. In 
a system with a group of servers and several nonconstant Poisson ﬂows was considered. Each ﬂow is
served by some subgroup of servers among which an arriving message chooses one with the shortest
queue, and the serving time is distributed exponentially.
In  a cyclic network is considered with k servers and k Poisson ﬂows. Each ﬂow is served by
two servers, and each arriving message is directed to a server where its delay will be the least. It is
shown that, under the condition that a pair of servers is overloaded and under light intensity of
input ﬂows, other servers are most probably not overloaded. But if the intensity of input ﬂows is
close to critical, it is most probable that all servers are overloaded.
Here we consider a symmetric system consisting of k similar servers at which Poisson ﬂows
, k ≥ 3. Messages of each ﬂow can be served by m servers that are ﬁxed for
this ﬂow, 2 ≤ m<k. Each arriving message is directed to a server where its preservice waiting
time is minimal. It turns out that the eﬀect noticed in  also takes place for this system: under
the condition that m servers are overloaded and the intensity of input ﬂows is light, other servers
are most probably not overloaded; but if the intensity of input ﬂows is close to critical, it is most
probable that all k servers are overloaded. We compute the asymptotics of the probability that
servers are heavily overloaded and show that this probability exponentially decreases as the load
Supported in part by the Russian Foundation for Basic Research, project no. 2011-01-00485a.