PIK JOURNAL ISSUE 1 / 2009 1
Stochastic Packet Loss Model to Evaluate QoE
Impairments
Oliver Hohlfeld
Abstract—The estimation of quality for real-time services
over telecommunication networks requires realistic models for
impairments and failures during transmission. We focus on the
classical Gilbert-Elliott model whose second-order statistics is
derived over arbitrary time-scales. The model is used to fit packet
loss processes of backbone and DVB-H traffic traces. The results
show that simple Markov models are appropriate to capture the
observed loss pattern and to discuss how such models can be used
to examine the quality degradations caused by packet losses.
I. INTRODUCTIONW ITH provisioning of broadband access for massmarket—even in wireless and mobile networks—
multimedia content, especially real-time streaming of high-
quality audio and video, is extensively viewed and exchanged
over the Internet. Quality of Experience (QoE) aspects, de-
scribing the service quality perceived by the user, is a vital
factor in ensuring customer satisfaction in today’s networks.
Creating frameworks for accessing quality degradations in
streamed video currently is investigated as a complex multi-
layered research topic, involving network traffic load, codec
functions and measures of user perception of video quality.
However, the transfer of real-time data for multimedia ser-
vices over the Internet and channels in heterogeneous packet
networks is subject to errors of various types, which will affect
the QoS and QoE. On wireless and mobile links temporary
and long lasting reductions in the available capacity frequently
occur and even in fixed and wired network sectors packets may
be dropped at routers and switches in phases of overload. Lost
information will affect the perceived quality by impairing the
multimedia content. The QoE degradation not only depends
on the amount of lost packets, but also on the semantic of the
lost information at the application layer.
The impact of packet loss on the user’s perception of real-
time services can be investigated starting from measurement
traces of traffic or generated by finite-state stochastic models,
which have been adapted to the characteristics observed in
the measurement and thus produce statistically similar traces.
Using model based generators for loss processes has several
advantages:
• the amount of necessary storage capacity is reduced
significantly from several gigabyte to a set of model
parameters,
• stochastic models usually include a set of parameters with
a clear interpretation, which can be adapted to meet the
demands of a considered scenario in which the model
is used (e.g. a certain packet loss rate) and makes them
more flexible than a measurement trace,
• the length of the generated sequence is independent of
the measurement trace used for training,
• stochastic models produce random but statistically con-
sistent sequences.
Both, using real data loss traces—e.g. captured in backbone
links—and model generated loss traces have their benefits. The
main disadvantage of using model generated loss traces is that
statistical properties may not fit to those of a measured trace,
as they are likely to be biased by model limitations.
Measurement traces show characteristics on multiple time-
scales. Thus, we derive the second-order statistics of finite
Markov models, to be used as a parameter estimation tech-
nique. This is used to adapt the model to the second-order
statistics of the amount of packet losses observed in a given
traffic trace on multiple time-scales by moment matching.
The present paper gives a brief introduction on finite
Markov models and discusses how these models can be used
in the study of QoE impacts on video streams. The aim is to
provide a generator for packet loss pattern to be used in the
estimation of the degradation in the Quality of Experience for
Internet services.
II. STOCHASTIC PACKET LOSS MODELS
Finite-state Markov chains are widely used to characterise
error processes in telecommunication systems and for per-
formance evaluation of coding or other measures for error
resilience [8], [11]. In this paper, we will use finite-state
models to describe the packet loss process found in backbone
measurements (wired channels) and DVB-H traces (wireless
channels). A discrete Markov chain with a set of M states
S = {S1, S2, · · · , SM} characterises the course of the process
with regard to the current state, which may change over time
at predefined events, e.g. packet arrivals, based on transition
probabilities. Each state is associated with different error or
packet loss behaviour. Let qt denote the current state at event
time t, t ∈ N0. Then the probabilities aij to change from state
qt−1 = i to qt = j are give in the transition matrix A with
coefficients
aij = P (qt = Sj |qt−1 = Si), 1 ≤ i, j ≤M,
where
aij ≥ 0;
M∑
j=1
aij = 1.
We restrict our considerations to irreducible and aperiodic
Markov chains, where each state can be reached from each
other with positive probability after a number of transitions
and steady state probabilities pik exist for finding the process
to sojourn a state in a long term perspective. The steady
state probabilities are invariant with regard to a transition

PIK JOURNAL ISSUE 1 / 2009 2
with matrix A and thus can be computed from the system
of linear equations: pik =
∑M
j=1 pijajk, k = 1, · · · ,M and∑M
j=1 pij = 1. Finally, we define error or packet loss rates in
each state E = (e1, ..., eM ); 0 ≤ ej ≤ 1 for j ∈ [1,M ]
and the output of the process O(t) as a binary sequence
O(t) ∈ 0, 1 indicating an error or loss at an event with
O(t) = 1, whereas O(t) = 0 stands for error free events,
respectively. Thus we have P (O(t) = 1|qt = Sj) def= ej .
Only in simple cases, e.g. for a 2-state Markov process with
e1 = 0 and e2 = 1, the current state can be recovered from the
output. The term Hidden Markov Models expresses that O(t)
in general leaves uncertainty about the corresponding states
S(t) of the Markov chain. A Markov process is completely
defined by the transition matrix A, the state specific error
rates E and initial state S0 or, more generally an initial state
distribution pi0 = P (S0 = j). We continue with a brief
discussion of two state Markov models before the second-
order statistics of finite Markov models is derived in general.
A. Gilbert-Elliot: The Classical 2-State Markov Model for
Error Processes
In 1960, Gilbert [3] proposed a 2-state Markov chain to
characterise a burst-noise channel. The usual notation of the
Gilbert model distinguishes at first a good (G) and secondly
a bad (B) state with different loss rates eG = 1 − k < eB =
1−h. Gilbert [3] started with the special case of an error-free
good state (k = 1) and left the extension to include losses
generated in both states to Elliott [1]. Dwell times in the states
are geometrically distributed with mean 1/p for the good and
1/r for the bad state, where p and r are the probabilities to
change from the good state to the bad and vice versa. The
Gilbert-Elliott 2-state Markov approach as depicted in Figure 1
is widely used for describing error patterns in transmission
channels [12], [8] and for analysing the efficiency of coding
for error detection and correction. For applications in data loss
processes, we interpret an event as the arrival of a packet and
an error as a packet loss. Thus, the transition matrix and the
steady state probabilities are
A =
(
1− p p
r 1− r
)
; piG =
r
p+ r ; piB =
p
p+ r
with a total error rate e = piGeG + piBeB = r(1−k)+p(1−h)p+r .
G B
p
1−p
r
1−r1−h1−k
Fig. 1. The 2-state Markov model introduced by Gilbert and Elliott
III. VARIANCE OF THE ERROR PROCESS OVER MULTIPLE
TIME-SCALES
Second-order statistics in multiple time-scales are a stan-
dard approach to capture and to describe traffic variability
including long-range dependencies and self-similarity [10],
[7]. Following this trend, we next derive the second-order
statistics of the number of packet losses over a range of
relevant time frames. Based on our work in [4], [5], where
we discussed the derivation in-depth, we will briefly discuss
the concrete result for the Gilbert-Elliott model. We discussed
the general eigenvalue solution for N -state Markov models
in [6]. Although Markov models do not exhibit self-similar
properties, they have been successfully adapted to self-similar
traffic [2], [9] and are still popular since they often lead to
simple analytical treatment.
In order to capture a packet loss process generated by an
M -state Markov model, we can set up recursive equations for
the distribution function of losses in a considered sequence of
packets. Let GN (z) (BN (z)) denote the generating function
X(z) def= ∑
i
P{X = i}zi for the number of packet drops in
a sequence of N packet arrivals, leaving the Markov chain in
the last step at state G (B). Iterative relationships can be set
up to compute GN+1(z) from GN (z), taking into account the
state transitions and factors (k+(1−k)z) and (h+(1−h)z)
for possible drop of the (N+1)-th packet with state dependent
probabilities 1− k and 1− h, respectively:
GN+1(z) = (1−p)(k+(1−k)z)GN (z)+r(h+(1−h)z)BN (z)
BN+1(z) = p(k+(1−k)z)GN (z)+(1−r)(h+(1−h)z)BN (z)
Starting in steady state conditions, we initialise G0(z) =
r/(p+r) and B0(z) = p/(p+r). The corresponding distribu-
tions GN (z), BN (z) remain defective GN (1) = r/(p + r)
and BN (1) = p/(p + r) ∀N ∈ N. We finally evalu-
ate complete distributions given by GN (z) + BN (z), where
GN (1) +BN (1) = 1 independent of the final state.
The k-th moment can be derived from the generat-
ing function by considering the k-th derivative: E[Xk] =
∂
∂zkX(z)|z=1. The mean, given by µGN = G′N (1) and
µBN = B′N (1), and the second moment E(X2) are sufficient
to derive the second-order statistics involving the first and
second derivative of the generating functions. Due to the
symmetry of both states G and B, GN (z) can be obtained
from BN (z) by swapping the parameters p ↔ r and h ↔ k
and vice versa. Thus, GN (p, r, h, k, z) = BN (r, p, k, h, z) and
µGN (p, r, h, k) = µBN (r, p, k, h).
The second-order statistics—expressed by the coefficient of
variation cv = σ/µ—of the number of packet losses in a
sequence of length N generated by the Gilbert-Elliott model
is
cv(N) =
1√
N
√
hp+ kr
ω
+ α2pr(1− p− r)(h− k)
2
ω2(p+ r) (1)
with
α def=
(
1− 1− (1− p− r)
N
N(p+ r)
)
; ω def= (1−h)p+(1−k)r
The solution is comprehensible enough to interpret the in-
fluence of the model parameters and leads to a new param-
eter adaptation technique, where model parameters can be
obtained by numerically fitting (1) to the empirical cv-curve
of a particular trace. Note that the evaluation of the term