1.2. TRAFFIC ARRIVAL MODELS 3
It turns out that using the tools of probability and stochastic processes, queueing theory in
particular, gives us very a powerful framework for predicting network behavior and thus leads to
good rules for designing them. One of the key assumptions will be that of stationarity (in a statistical
sense). Although, stationarity is very special; it turns out that one can indeed identify long periods
when network performance is well predicted using stationary models. Thus, this will be an implicit
assumption in the sequel unless specifically mentioned to the contrary.
Network performance is usually defined in terms of statistical quantities called the QoS
parameters:
Call blocking probabilities in the context of flows and circuit-switched architectures–often
called the Grade of Service (GoS).
Packet or bit loss probabilities.
Moments and distributions of packet delay, denoted D, such as the mean delay E[D], the
variance var(D) which is related to the jitter, the tail distribution P(D t), etc.
The mean or average throughput (average number of packets or bits transmitted per sec),
usually in kbits/sec, Mbits/sec, etc.
The sources of randomness in the internet arise from the following:
Call, session, and packet arrivals are unpredictable (random).
Holding times, durations of calls, file sizes, etc are random.
Transmission facilities, switches, etc are shared.
Numbers of users are usually not known a priori, and they arrive and depart randomly.
Statistical variation, noise, and interference on wireless channels.
Queueing arises because instantaneous speeds exceed server or link capacities momentarily
because bit flows or packet flows depend on the devices or routers feeding a link. Queueing leads to
delays, and these delays depend not only on traffic characteristics but the way packets are processed.
The goal of performance analysis is to estimate the statistical effects of variations. In the following,
we begin by first presenting several useful models of traffic, in particular ways of describing packet
or bit arrivals.
We first begin by introducing models for traffic arrivals.
1.2 TRAFFIC ARRIVAL MODELS
It must be kept in mind that the primary purpose of modeling is to obtain insights and qualitative
information. There are some situations when the models match empirical measurements. In that
case, the analytical results also provide quantitative estimates of performance. To paraphrase the
4 INTRODUCTION TO TRAFFIC MODELS AND ANALYSIS
words of the famous statistician Box, models are always wrong but the insights they provide can be
useful. It is with this caveat that one must approach the issue of traffic engineering.
As far as studying the performance of networks, the key is to understand how traffic arrives
and how packets or bits are processed. Modeling these processes is the basic building blocks of
queueing theory. However, even in this case, there is no one way to do it. It depends on the effect
we are studying and the time scale of relevance.
Let us look at this issue a bit further. Packet arrivals can be as discrete events when packets
arrive on a link or router buffer, or in the case, when a source transmits at very high speed and viewed
at the time-scale of bits, one could think of arrivals as fluid with periods of activity punctuated with
periods of inactivity. The former leads to so-called point process models of arrivals while the latter
are called fluid inputs. In essence, we are just viewing arrivals at different time-scales. Both models
are relevant in modeling real systems, but we will restrict our selves to point process models for the
most part. The two arrival patterns are illustrated below.
T
1
T
3
T
6
T
5
T
4
T
7
T
2
t
0
Point process model
Fluid arrival model
0
t
Figure 1.1: Discrete and fluid traffic arrival models.
Definition 1.1
A stochastic process is called a simple point process if it is characterized as follows:
Let {T
n
}
n=−∞
be a real sequence of random variables (or points) in R with ···<T
1
<T
0
0 <T
1
<T
2
< ··· and lim
n→∞
T
n
=∞a.s..
Define N
t
as :
N
t
=
n
1I
(0<T
n
t)
(1.1)
Then N
t
is the counting process of {T
n
} and is commonly called a point process.
Remark 1.2
1. N
t
counts the number of points T
n
that occur in the interval (0,t]. The notation N(a,b] is
also used to indicate the number of points in (a, b]. Thus, N
t
= N(0,t].
2. lim
t→∞
N
t
=∞a.s. by definition.
3. The random variables S
n
= T
n
T
n1
are called the inter-arrival times.

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