Problems of Information Transmission, Vol. 38, No. 2, 2002, pp. 136–153. Translated from Problemy Peredachi Informatsii, No. 2, 2002, pp. 44–63.
Original Russian Text Copyright
2002 by Suhov, Vvedenskaya.
COMMUNICATION NETWORK THEORY
Fast Jackson Networks with Dynamic Routing
Yu. M. Suhov and N. D. Vvedenskaya
Received December 11, 2001
Abstract—A new class of models of queueing networks with load-balanced dynamic routing
is considered. We propose a suﬃcient condition for positive recurrence of the arising Markov
process and a limiting mean-ﬁeld approximation where the process becomes deterministic and
is described by a system of nonlinear ordinary diﬀerential equations.
This paper proposes a class of queueing network models with dynamic routing based on the
principle of balanced load. The idea of dynamic routing is to select a path across a network in
such a way that it minimizes (1) the delivery time (or the end-to-end delay) of a given task and
(2) the occupancy of buﬀers in the network. These goals do not always agree; moreover, a decision
is taken on the basis of a limited amount of available information. To our knowledge, until recently,
there was no mathematical model of a queueing network with dynamical routing proposed in the
literature that could be studied rigorously.
The ﬁrst example of such a model was a queueing
system introduced in  (and independently but somewhat later and without rigorous proofs,
in ; see also [5–8]). The original model was then modiﬁed in  to include a class of Jackson-type
networks; however, the dynamic routing principle was still reduced in  to the choice between
servers from a given station. We refer the reader to the introductory section of  for a discussion
of the approach adopted in the above-mentioned papers. A review of available literature can also
be found in .
The principle of dynamic routing proposed in the above papers is to select a server with the
shortest queue among a sample collection of servers chosen at random. Thus, in the model con-
sideredin,thereareN identical exponential servers, each with an inﬁnite buﬀer and service
rate one. The exogenous ﬂow is Poisson with rate Nλ. Service times and arrival times are all
independent. Upon arrival, each task chooses m servers at random (m>1) and joins the one
whose queue is the shortest. If λ<1, the system is described by a positive recurrent Markov pro-
cess, whose state is represented by a tail histogram, or distribution function, identifying, for each
n =0, 1, 2,..., a proportion r(n) of servers with at least n tasks in the queue. This Markov process
has a unique invariant distribution π
. The main result of  is that the expected value E
of proportion r(n) converges as N →∞to λ
, which gives a superexponential decay
as n →∞. This result contrasts with a model where each task selects a server independently and
completely at random (which corresponds to the previous scheme with m =1):here,E
(a geometric, or exponential, decay).
Supported in part by the Institut des Hautes
Etudes Scientiﬁques (Bures-sur-Yvette), Dublin Institute
for Advanced Studies, INTAS, Grant no. 93-820, EC Grant (contract no. ERBMRXT-CT 960075A), and
Russian Foundation for Basic Research, project no. 02-01-00068.
See  for references to works on two-server systems. However, these do not include networks with
Loss networks with dynamic routing have been discussed in . However, loss networks do not pose such
challenging problems as existence of a stationary distribution.
2002 MAIK “Nauka/Interperiodica”