arXiv:math/0611526v1 [math.PR] 17 Nov 2006
A note on insensitivity in stochastic networks
Maxwell Institute for Mathematical Sciences, Heriot-Watt University
We give a simple and direct treatment of insensitivity in stochastic networks which
is quite general and which provides probabilistic insight into the phenomenon. In the
case of multi-class networks, the results generalise those of Bonald and Prouti` ere (2002,
It is well-known that many stochastic networks—notably queueing and loss networks—
have stationary distributions of their level of occupancy which depend on certain input
distributions only through the means of the latter. This phenomenon of insensitivity has
been studied by various authors over an extended period of time, in varying degrees of
generality and abstraction, and using a variety of techniques.
In the present paper we revisit this topic to develop an insight of Pechinkin (1983, 1987)
to give a very simple and direct treatment of insensitivity. In particular the approach
avoids those based on brute-force calculations, the consideration of phase-type distribu-
tions (Schassberger, 1978, Whittle, 1985, Bonald and Prouti` ere, 2002, 2003), or the use of
quite complex machinery for handling generalised semi-Markov processes (Burman, 1981,
Schassberger, 1986)—although such processes are implicit in the current approach. It fur-
ther avoids assumptions about, for example, continuity of distributions, necessary for some
of the above approaches, and also explicitly identifies the entire stationary distributions
of the networks concerned, showing that, where insensitivity obtains, these stationary dis-
tributions have a particularly simple and natural form. Pechinkin used his insight, which
involves what is in effect a coupling argument together with induction, to give probabilistic
proofs of the insensitivity of a number of single-class loss systems with state-dependent ar-
rival rates—results originally proved analytically by Sevastyanov (1957). He also indicated
the wider applicability of the approach in the single-class case. In the present paper we
give a substantial reformulation of the underlying idea, under more general conditions and
showing that its most natural expression is in terms of balance equations. This consider-
ably simplifies its application to single-class systems—notably the quite complex coupling
constructions are no longer needed. It further makes possible the extension of the idea to
the multi-class networks considered in Section 3. The main aim is to provide probabilistic
insight, notably for multi-class networks. Indeed it is shown that insensitivity is simply a
byproduct, under appropriate conditions, of probabilistic independence.
We study networks in which individuals arrive at various classes at rates which may
depend on the state of the entire system, bringing workloads which are independent and
identically distributed within classes and which have finite means. Within each class
0American Mathematical Society 1991 subject classifications. Primary 60K20
Key words and phrases. insensitivity, stochastic network, partial balance
workloads are reduced at rates which may again be state-dependent (when the rate is
constant workloads may be identified with lifetimes in classes), and on completion of its
workload an individual moves to a different class or leaves the system, with probabilities
which may yet again be state-dependent.
In order to obtain insensitivity we typically require that an individual joining a class is
immediately served, i.e. has its workload reduced, at a rate which is the same as that
of an individual immediately prior to leaving the class (where in each of these cases the
number of individuals in each class of the system is the same)—more generally that the
service discipline should define a network which is symmetric in the sense of Kelly (1979).
The most common example is that of processor-sharing networks, but other possibilities
are well-known, for example, “last-in-first-out preemptive resume” networks. We shall
concentrate on a very broad class of processor-sharing networks, introduced by Bonald
and Prouti` ere (2002) and including, for example, traditional loss networks and processor-
sharing Whittle and Jackson networks, as special cases). We shall also indicate the simple
modifications required to deal with other possibilities.
For the above class of processor-sharing networks, Bonald and Prouti` ere used phase-type
arguments to show that, under conditions which correspond to the satisfaction of the
appropriate partial balance equations, the stationary distribution of the number of indi-
viduals in each class is insensitive to the workload distributions, subject to the means of
the latter being fixed and to the distributions themselves being drawn from the broad class
of Cox-type distributions (dense in the class of all distributions on R+). In the present pa-
per we formally consider all workload distributions on R+with finite means, and identify
also the stationary residual workload distributions. However, as stated above our main
aim is to give a direct and probabilistically natural treatment. It turns out (and is in
many cases well-known) that, when the appropriate partial balance equations are satis-
fied for such a network, then the stationary distribution of the entire system, including
the specification of residual workloads, is such that departures from each class are exactly
balanced by arrivals to that class—in a sense again to be made precise below. Indeed, for
single-class systems, this is the essence of Pechinkin’s insight. What is of interest is that
same idea extends to establish insensitivity for the very much more general networks con-
sidered here, and indeed appears also to establish insensitivity in more abstract settings
such as that considered by Whittle (1985), though we do not formally consider this more
abstract environment here.
In order to fix ideas, it is convenient to consider first, in Section 2, single-class networks.
Here the extension of previous ideas is not too difficult. Nevertheless it is desirable to give
a careful treatment of this case, avoiding notational complexity while preserving rigour, so
as both to establish the underlying principle and also to set the scene for the multi-class
networks which we consider in Section 3.
Consider an open system with a single class of individual (customer, call, or job). Indi-
viduals arrive as a Poisson process with state-dependent rate α(n), where n is the number
of individuals currently in the system. Arriving individuals have workloads which are
independent of each other and of the arrivals process with a common distribution µ on
R+which we assume to have a finite mean m(µ). While there are n individuals in the
where κi= νi/σifor each i, and where a is naturally chosen to be a normalising constant.
As was originally shown by Burman et al (1984), we therefore again have insensitivity of
the occupancy distribution π of the network. The stationary distribution of the residual
call durations is as identified by Theorem 2.
Other examples of processor-sharing networks with no internal transitions are given by
those used to model connections in communications networks with simultaneous resource
requirements and variable bandwidth requirements—see, for example, Bonald and Mas-
souli´ e (2001) and de Veciana et al (2001). Here it is far from automatic that the detailed
balance equations (17) are satisfied.
The author is most grateful to Serguei Foss, Takis Konstantopoulos and Ilze Ziedins for
some very helpful discussions, and to the referee for a careful reading of the manuscript
and some most helpful suggestions.
 Bonald, T. and Massouli´ e, L. (2001). Impact of fairness on Internet performance. In
Proceedings of ACM SIGMETRICS 2001. Cambridge, Massachusetts.
 Bonald, T. and Prouti` ere, A. (2002). Insensitivity in processor-sharing networks. Per-
formance Evaluation, 49, 193–209.
 Bonald, T. and Prouti` ere, A. (2003). Insensitive bandwidth sharing in data networks.
Queueing Systems, 44, 69–100.
 Burman, D. Y. (1981). Insensitivity in queueing systems. Adv. Appl. Prob., 13, 846–
 Burman, D. Y., Lehoczky, J. P. and Lim, Y. (1984). Insensitivity of blocking proba-
bilities in a circuit switching network. J. Appl. Prob. 21, 850–859.
 Davis, M. H. A. (1984). Piecewise deterministic Markov processes: a general class of
non-diffusion stochastic models. J Royal Stat Soc, Ser B 46, 353–388.
 Davis, M. H. A. (1993). Markov Models and Optimization. Chapman and Hall, Lon-
 de Veciana, G., Lee, T.-J. and Konstantopoulos, T. (2001). Stability and performance
analysis of networks supporting elastic services. IEEE/ACM Trans. on Networking,
 Kelly, F. P. (1979). Reversibility and Stochastic Networks. Wiley, Chichester.
 Kelly, F. P. (1986). Blocking probabilities in large circuit-switched networks. Adv.
Appl. Prob. 18, 473–505.
 Pechinkin, A. V. (1983). An invariant queueing system. Math. Operationsforsch. und
Statist., ser. optimization, 14, No. 3, 433–444.
 Pechinkin, A. V. (1987). A new proof of Erlang’s formula for a lossy multichannel
queueing system. Soviet J. Comput. System Sci. 25, 165–168. Translated from: Izv.
Akad. Nauk. SSSR. Techn. Kibernet, (1986), No. 6, 172–175 (in Russian).
 Schassberger, R. (1978). Insensitivity of steady-state distributions of generalized semi-
Markov processes with speeds. Adv. Appl. Prob. 10, 836–851.
 Schassberger, R. (1986). Two remarks on insensitive stochastic models. Adv. Appl.
Prob. 18, 791–814.
 Serfozo, R. F. (1999). Introduction to Stochastic Networks. Springer, Berlin.
 Sevastyanov, B. A. (1957). An ergodic theory for Markov processes and its application
to a telephone system with refusals. Theory Probability. Appl., 2, 104–112.
 Whittle, P. (1985). Partial balance and insensitivity. J. Appl. Prob., 22, 168–176.