IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 3, MAY 2004 929
Correspondences______________________________________________________________________
Packet Loss Probability for Bursty Wireless Real-Time
Traffic Through Delay Model
Kelvin K. Lee and Samuel T. Chanson
Abstract—In this work, two Markov chains modeling packet delay have
been developed to obtain closed-form solutions for packet loss probabili-
ties and packet delay distributions in a real-time wireless communication
environment. Packets with delay constraints are queued at the source and
are transmitted one by one with possible retransmissions to correct errors.
The first model models packet delay and correlated packet arrivals using a
one-dimensionalMarkov chain. Usually, at least two dimensions are needed
in queuing models. The second model is a two-dimensional Markov chain
modeling packet delay, correlated channel errors, and correlated arrivals.
Both correlated errors and arrivals are known to have significant impact
on packet loss. Possible applications of the results are also discussed.
Index Terms—Automatic repeat request (ARQ), correlated arrivals, cor-
related errors, delay model, packet loss probability.
I. INTRODUCTION
Wireless communication is becoming more and more popular nowa-
days. To provide quality services to their clients, service providers need
to know the behaviors and performance of their systems in a continu-
ously changing environment with variable traffic load. It is important
to be able to react swiftly to varying conditions so that the quality of
service (QoS) can be maintained. This can be achieved if a formula re-
lating system performance to different parameters is available.
Wireless traffic may be classified by their delay and loss character-
istics. Two common types of traffic are data traffic, which cannot tol-
erate any loss but has no delay constraint, and real-time traffic, which
can tolerate some small amount of loss but has time constraints. It is
challenging to handle the second type since both the delay constraint
and the loss requirements need to be satisfied.
Since the wireless environment is harsh, errors in packet transmis-
sion cannot be avoided and bursty errors are common. Two common
techniques to correct the errors are forward error correction (FEC) and
automatic repeat request (ARQ). FEC is often used in either real-time
communications or in environments in which feedback channels are not
available. A suitable interleaving method, such as block interleaving,
which inevitably introduces some delay, is needed to reduce error cor-
relation in FEC.
Traditionally, ARQ has been used to deliver packets in applications
that cannot tolerate any error but there is no delay constraint. As the
bandwidth of the wireless channel increases, the number of slots avail-
able for possible retransmissions within a fixed time period increases.
This has created room for ARQ to be used even in delay-constrainted
packet transmission, even for applications with delay constraints. Zorzi
[1] showed that ARQ can perform better than FEC in terms of packet
loss probability when the tolerable delay is large, while FEC’s per-
formance is better when the delay limit is very small. The break-even
Manuscript received March 31, 2002; revised July 24, 2003 and December
15, 2003.
K. K. Lee is with TheOpenUniversity of HongKong, Homantin, HongKong,
China (e-mail: kwlee@ouhk.edu.hk).
S. T. Chanson is with The Hong Kong University of Science and Technology,
Clear Water Bay, Hong Kong, China (e-mail: chanson@cs.ust.hk).
Digital Object Identifier 10.1109/TVT.2004.825769
point, where the performance of the two methods is roughly the same,
is around 10–25 ms for a 200-kb/s channel. The exact point depends
on the error-correction method used and the packet size. Other papers
using ARQ as the error correction method include [2] and [3].
There are few papers that use ARQ to study the loss probability of
delay-constrained wireless traffic at the packet level. Zorzi and Rao
[4] calculated the packet-lateness probability (probability that a packet
is received later than a time limit) as an approximation to packet loss
probability using two-state correlated error models with independent
Bernoulli arrivals. In their Markov models, packets were not discarded
at the sender; instead, they were discarded at the receiver (we shall refer
to it as “late discard”). This consumes some bandwidth and may delay
other packets. The accuracy of lateness probability, which is an approx-
imation to the packet loss probability, decreases as the load increases.
Moreover, the arrival assumption is not realistic since expired packets
would be discarded at the source in real systems.
The random arrival problem was tackled by Krunz and Kim in [5] by
using fluid analysis. Apart from modeling correlated errors, the paper
included an ON–OFF fluid process (which was continuous) to model dis-
crete arrivals approximately. This is an enhancement over the afore-
mentioned model, since traffic sources can be correlated. However, the
delay was estimated by the number of retransmissions, which did not
include queuing delay. Similar to [4], expired packets were discarded
only at the receiver, which introduced inaccuracy. Turin and Zorzi [6]
generalized the arrival process in [4] to an n-state Markov model, but
still did not address the “late discard” issue. Thus, the “late discard”
issue remains unresolved until a delay model is introduced in [7]. Since
this new model was the first attempt to find a solution for the “late dis-
card” issue, only correlated errors and geometric interarrival timeswere
considered.
In this paper, we use the approach developed in [7] to address the
“late discard” issue by modeling the packet-discard mechanism at
the sender instead of at the receiver. We also replace the independent
Bernoulli arrivals in [7] by bursty arrivals without increasing the
number of dimensions of the Markov chain. This is achieved by
exploiting the capability of delay models in tracking simple arrivals by
using their interarrival time distribution. Furthermore, exact solutions
are obtained by including a discrete arrival process instead of using
fluid approximation. We present the solution of our first Markov
model with bursty arrivals in Section II. Formulas for packet loss
probability and delay distribution, as well as some numerical results,
are included. The second model with more general correlated errors is
described in Section III. Closed-form solutions for delay distribution
and packet loss probability have been derived. Section IV discusses
the properties of the formula with respect to packet loss probability in
the second model. Numerical results and applications are presented in
Section V. Finally, Section VI concludes the paper.
II. DELAY MODEL WITH BURSTY ARRIVALS ONLY
A. Packet-Transmission Process
Awireless communication channel delivering packets from a sender
to a receiver is considered. This channel is one of the many channels
using the same frequency band supported by a wireless system. Time is
divided into fixed-size slots and a packet can be transmitted in each slot.
Time-divisionmultiplexing is used and the channel under consideration
is regularly allocated a time slot after a fixed (and nonzero) period of
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