The first performance models of telecommunication systems were based
on the assumption that arrival processes follow Poisson processes. It became,
however, quickly apparent that this characterization ignores the significant
correlation present in network traffic. For this purpose, Markov modulated
processes have been introduced for representing a variety of traffic models
(see [Sch96], Chapter 3, for a thorough
overview of these processes). In such processes, transitions between states
are governed by an underlying continuous-time Markov chain while marginal
distributions of arrival processes are state-dependent. For example, a
N
states Markov modulated Poisson process is a process behaving as a Poisson
process, while entering and staying in some state
i, 
, with state-dependent rate 
. The Markov modulated fluid process [AMS82]
is another well-known Markov modulated process. It generates data at a
constant (fluid) rate
, when the system is in state i. This model is widely used (see
[GAN91] and references therein), due to
its simplicity. An additional important special case of a Markov modulated
process is the exponentially distributed ``On/Off'' process which has been
used to model the behavior of a voice source. A voice source can be characterized
as an alternating sequence of active and inactive (silence) intervals.
When the voice source is in its active state, it behaves as a periodic
source transmitting packets at a fixed rate. In the inactive state, no
packets are transmitted.
The characterization of the traffic arising from the multiplexing of a large number of voice sources has brought much interest in the teletraffic literature [HeL86, SrW86, Bai91]. It turns out that the behavior of such kind of traffic depends on the time-scale under consideration. At a fast time scale, commonly termed ``cell'' time scale, each individual voice source behaves as a locally periodic source. In this regime, the fluctuations in the aggregate traffic rate emerge from the independence of the ``phases'' of the sources. At a slow time scale, termed ``burst'' time scale, fluctuations arise as a consequence of the transitions in the traffic rates of individual sources (inactive sources becoming active and vice versa).
The complex statistical behavior of multiplexed voice traffic has a
profound impact on queueing performance (see [Sch96],
Section 3.8 and references therein). Indeed, the buffer occupancy probability
distribution of a buffer fed by such kind of traffic is characterized,
on a logarithmic scale, by two distinct linear regions. These two regions
are referred to as cell and burst regions. The cell region corresponds
to small value of buffer size where the buffer occupancy distribution is
found to drop rapidly. Queueing in the cell region is due to possible excess
of the input traffic rate over the output rate at the cell time scale.
Such an event may happen when several sources transmit cells more or less
simultaneously. The burst region begins from some point, depending on the
utilization and the traffic characteristics, where the buffer occupancy
distribution decreases more slowly as the buffer size increases. In this
region, large queues are due to possible excess of the input traffic rate
over the output rate at the burst time scale. This phenomenon occurs when
there is a large number of active sources. In the sequel, we will show
that the proper characterization of multiplexed voice traffic has important
implications for the performance of communication networks.