Contents Telektronikk - Telenor

86
load control mechanisms. It is usually
more difficult to obtain non-stationary
solutions, but several powerful methods
can be used.
Discrete event simulation is a standard
technique for studying the steady-state
behaviour of queuing models, but it can
also be used to study transients. Suppose
that A(t) is the value of some quantity at
time t (e.g. the length of some queue or
the load of some server) and that we
want to study A(t) when t ∈ P where P is
a finite set of time points. The simulation
program is run several times with different
random number generator seeds.
Each time the program is run we sample
A(t) at the points in P. To get a high
accuracy we need many samples of A(t)
at each point in P, which means that we
have to run the program many times.
When we use simulation to obtain
steady-state results we often have to
make one long simulation run, when the
transient behaviour is studied we instead
make many short runs of the simulation
program.
In some cases it is possible to obtain
explicit analytical expressions for A(t). A
well-known example is P k (t) = probability
of k customers at time t for an
M/M/1 queuing system. In this case
P k (t) can be expressed using infinite
sums of Bessel functions. However,
explicit analytical solutions showing the
transient behaviour of queuing models
are very difficult to obtain and thus
almost never used in practice.
The derivation of P k (t) for the M/M/1
queuing system takes its starting point in
the well-known Chapman-Kolmogorov
differential-difference equations. It is
possible to use e.g. Runge-Kuttas method
to get numerical values of P k (t) without
explicitly solving the Chapman-Kolmogorov
equations. In many cases it is
possible to write down differential-difference
equations similar to those used for
describing M/M/1 and then solve these
equations numerically. We have named
such methods Markov approximation
methods. We often have to simplify the
state space considerably to be able to
handle the equations numerically.
If we only keep track of the mean values
of all stochastic quantities we get a very
simple state space. Often simple difference
equations that approximately describe
the time-dependent behaviour of
the system can be obtained. We call such
methods flow approximations. Such
methods are rather crude but they can
nevertheless give important insights into
the transient behaviour of queuing models.
Simulation has many advantages. It is in
principle easy to write and to modify
simulation programs. We can use any
statistical distributions, but execution
times can get long if we want accurate
results. The results are numerical, i.e. it is
hard to see the impact of different parameters
on system performance. Explicit
analytical solutions showing the transient
behaviour of queuing models are very
difficult to obtain and are thus never used
in practice. The numerical methods described
above can give somewhat shorter
execution times than simulation programs
and in many cases standard
numerical methods can be used. However,
the state space usually has to be
much simplified and it is often difficult
to modify the model to test a new overload
control mechanism. Fluid flow
methods can only be used to study transients
but has proven very useful not at
least in modelling overload control
schemes.
5 Conclusions
Though we foresee SPC-systems later
during this decade with very powerful
control systems, the need for efficient
overload control mechanisms will increase.
In this paper, we have briefly described
the environment in which these
switches will work. Though system complexity
will increase in general, it is
important to keep the overload control
mechanisms predictable, straightforward
and robust. We have stressed that the
design of overload control mechanisms
must be done in conjunction with the
design of the control system.
We would like to point out the importance
of studying time dependent
behaviour of modern SPC-systems in
general and especially their overload
control mechanisms. This, indeed, seems
to be a very urgent task. We have so far
only reached a point of basic understanding
of these mechanisms. It is also
important to have the performance,
derived from various analytic and simulation
based models, supported by measurements
from systems in use. Traditional
stochastic models do suffer from
limitations. We need to develop efficient
numerical solution methods for many of
today’s models as well as to develop new
models based on perhaps non-traditional
approaches.
Acknowledgement
Many thanks to Dr. Christian Nyberg for
valuable discussions and suggestions that
have led to many improvements throughout
this paper. The two of us have worked
together for a number of years on the
topic. Our research work is supported by
Ericsson Telecom, Ellemtel Telecommunication
Systems Laboratories and The
National Board for Industrial and Technical
Development (grant no. 9302820).
References
1 Nyberg, C, Wallström, B, Körner, U.
Dynamical effects in control systems.
I: GLOBECOM ’92, Orlando, 1992.
2 Løvnes, K et al. IN control of a B-
ISDN trial network. Telektronikk,
88(2), 52–66, 1992.
3 Erramilli, A, Forys, L J. Traffic synchronization
effect in teletraffic systems.
I: The 13th international teletraffic
congress, ITC 13, Copenhagen,
1991.
4 Körner, U. Overload control of SPC
systems. I: The 13th international
teletraffic congress, ITC 13, Copenhagen,
1991.
5 Kihl, M, Nyberg, C, Körner, U.
Methods for protecting an SCP in
Intelligent Networks from overload.
To be published.
6 Borcheing, J W et al. Coping with
overload. Bell Laboratories Record,
July/August 1981.
7 Skoog, R A. Study of clustered arrival
processes and signaling link
delays. I: The 13th international teletraffic
congress, ITC 13, Copenhagen,
1991.
8 Erramilli, A, Forys, L J. Oscillations
and chaos in a flow model of a
switching system. IEEE journal on
selected areas in communication, 9,
171–178, 1991.