2
Traffic Issues and Solutions
short circuits, short packets
This chapter is the executive summary for the book: it provides a quick
way to find a range of analytical solutions for a variety of design and
performance issues relating to IP and ATM traffic problems. If you are
already familiar with performance evaluation and want a quick overview
of what the book has to offer, then read on. Otherwise, you’ll probably
find that it’s best to skip this chapter, and come back to it after you have
read the rest of the book – you’ll then be able to use this chapter as a
ready reference.
DELAY AND LOSS PERFORMANCE
In cell- or packet-based networks, the fundamental behaviour affecting
performance is the queueing experienced by cells/packets traversing the
buffers within those switches or routers on the path(s) from source to
destination through the network. This queueing behaviour means that
cells/packets experience variations in the delay through a buffer and
also, if that delay becomes too large, loss.
At its simplest, a buffer has a fixed service rate, a finite capacity
for the temporary storage of cells or packets awaiting service, and
a first-in–first-out (FIFO) service discipline. Even in this simple case,
the queueing behaviour depends on the type and mix of traffic being
multiplexed through the buffer. So let’s first look at the range of source
models covered in the book, and then we’ll summarize the queueing
analysis results.
Introduction to IP and ATM Design Performance: With Applications Analysis Software,
Second Edition. J M Pitts, J A Schormans
Copyright © 2000 John Wiley & Sons Ltd
ISBNs: 0-471-49187-X (Hardback); 0-470-84166-4 (Electronic)
16
TRAFFIC ISSUES AND SOLUTIONS

Location: Chapter 6, page 86
Model: binomial distribution
Use: number of arrivals (in time, or from a number of inputs) or amount of
work, for octets, cells, packets, bursts, flows, calls
Formula: Prfk arrivals in N time slotsgD
N!
N  k! Ð k!
Ð 1  p
Nk
Ð p
k
Parameters: k – number of arrivals, or amount of work
p – probability of an arrival, in a time slot or from an input
N –numberoftimeslots,ornumberofinputs
Location: Chapter 6, page 86
DELAY AND LOSS PERFORMANCE
17
Model: Batch distribution
Use: number of arrivals, or amount of work, for octets, cells, packets,
bursts, flows, calls
Formulas: a0 D 1  p
a1 D p Ð b1
a2 D p Ð b2
.
.
.
ak D p Ð bk
.
.
.

υ
x

˛
Fx D 1 

υ
x

˛
f x D
˛
υ
Ð

υ
x

˛C1
E[x] D υ Ð
˛
˛  1
(continued overleaf )
18
TRAFFIC ISSUES AND SOLUTIONS
Model: Pareto distribution
Parameters: υ – minimum amount of work
x –numberofarrivals,oramountofwork
˛ – power law decay
Location: Chapter 17, page 289

served)
t
q
– mean time a customer spends in the system
Location: Chapter 4, page 61
Model: M/M/1
Use: classical continuous-time queueing model; NB: assumes variable-
size customers, so more appropriate for IP, but has been used for
ATM
Formulas: q D

1  
(continued)
DELAY AND LOSS PERFORMANCE
19
Model: M/M/1
t
w
D
 Ð s
1  
Prfsystem size D xgD1  
x
Prfsystem size > xgD
xC1
Parameters:  – utilization; load (as fraction of service rate) offered to system
q – mean number in the system (waiting or being served)
t
w
– mean time spent waiting for service

iD0
ai
E[a]
T
d
k D
k

jD1
U
d
j Ð B
d
k  j
T
d,n
k D
k

jD1
T
d,n1
j Ð T
d,1
k  j
for M/D/1: t
w
D
 Ð s
2 Ð 1  

u0 D 1
uk D
uk  1  ak  1 
k1

iD1
ui Ð ak  i
a0
uX D AX
s0 D
1
X

iD0
ui
sk D s0 Ð uk
CLP D
E[a]  1  s0
E[a]
Parameters: ak – probability there are k arrivals in a time slot
Ak – probability there are at least k arrivals in a time slot
E[a] – mean number of arrivals per time slot
sk – probability there are k cells in the system at the end of any slot
 – utilization; load (as fraction of service rate) offered to system
CLP – probability of loss (whether cells or packets)
Location: Chapter 7, page 105
DELAY AND LOSS PERFORMANCE
21
Model: N·D/D/1
Use: multiple constant-bit-rate (CBR) sources into deterministic server –

Location: Chapter 8, page 116
Model: M/D/1 heavy-traffic approximation
Use: cell-scale queueing in ATM, basic packet queueing in IP (with fixed
packet sizes); NB: below ³80% load, underestimates loss
Formulas: Qx D e
2ÐxÐ

1


x D
1
2
Ð lnQx Ð


1  

 D
2 Ð x
2 Ð x  lnQx
Parameters: x – buffer capacity (in cells or packets)
 – utilization; load (as fraction of service rate) offered to system
Qx – probability that queue exceeds x (estimate for loss probability)
Location: Chapter 8, page 117
Model: N·D/D/1 heavy-traffic approximation
Use: multiple constant-bit-rate (CBR) sources into deterministic server – this
can be applied to ATM, and to IP (with fixed packet sizes); NB: below
³80% load, underestimates performance
Formulas: Qx D e

1  q
Ð

1  q
1  p

k
Qk D
p
q
Ð

1  q
1  p

k
Qx D
p
q
Ð

1  q
1  p

x/q
Parameters: q – probability a packet completes service at the end of an octet slot
p – probability a packet arrives in an octet slot
sk – probability there are k octets in system
Qk – probability that queue exceeds k octets
Qx – probability that queue exceeds x packets



k
Qk D

 Ð e

 e

 
2
C  C e

  1 C e


kC1
Parameters:  – arrival rate of Poisson process
pk – probability an arriving excess-rate cell/packet finds k in the
system
(continued)
DELAY AND LOSS PERFORMANCE
23
Model: excess-rate, Geometrically Approximated Poisson Process (GAPP),
M/D/1
Qk – probability an arriving excess-rate cell/packet finds more than k
in the system
Location: Chapter 14, page 245
Model: excess-rate GAPP analysis for bi-modal service distributions
Use: accurate approximation to M/bi-modal/1 – suitable for IP, with


E[a] Ð 1  a1  1 C a1 C a0
2
a0 Ð E[a]  1 C a0

k
Qk D

E[a] Ð 1  a1  1 C a1 C a0
2
a0 Ð E[a]  1 C a0

kC1
Parameters: ak – probability there are k arrivals in a packet service time
E[a] – mean number of arrivals per packet service time
 – packet arrival rate of Poisson process (i.e. per time unit D short
packet)
p
s
– proportion of short packets
n – length of long packets (multiple of short packet)
pk – probability an arriving excess-rate packet finds k in the system
Qk – probability an arriving excess-rate packet finds more than k
in the system
Location: Chapter 14, page 249
Model: excess-rate GAPP analysis for M/G/1
Use: accurate approximation to M/G/1 – suitable for IP, with general
service time distribution to model variable-length packets
(continued overleaf )
24

a0 Ð E[a]  1 C a0

k
Qk D

E[a] Ð 1  a1  1 C a1 C a0
2
a0 Ð E[a]  1 C a0

kC1
Parameters: Ak – probability there are k arrivals in a packet service time
E[a] – mean number of arrivals per packet service time
 – packet arrival rate of Poisson process (i.e. per unit time)
gk – probability a packet requires k units of time to be served
pk – probability an arriving excess-rate packet finds k in the system
Qk – probability an arriving excess-rate packet finds more than k in
the system
Location: Chapter 14, page 249
Model: ON–OFF/D/1/K
Use: basic continuous-time queueing model for IP or ATM, suitable for
per-flow or per-VC scenarios
Formulas: ˛ D
T
on
T
on
C T
off
CLP
excess-rate

T
off
–meandurationinOFFstate
˛ – activity factor of source (probability of being ON)
CLP – loss probability
Location: Chapter 9, page 130
Model: ON–OFF/D/1/K
Use: basic discrete-time queueing model for IP or ATM, suitable for per-flow
or per-VC scenarios
Formulas: a D 1 
1
T
on
Ð R  C
s D 1 
1
T
off
Ð C
pX D
1
1 C


s
a

X
 1


Formulas: ˛ D
m
h
D
T
on
T
on
C T
off
N
0
D
C
h
(continued overleaf )