Veszteséges rendszerek
Takács György
6. Előadás
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Forgalmi alapfogalmak - 1
Forgalom intenzitása
(Traffic intensity, traffic per time unit)
Definition: The instantaneous traffic intensity in a pool of
resources is the number of busy resources at a given instant
of time.
The pool of resources may be a group of servers, e.g. trunk lines, registers,
buffers.
T
1
Y (T)  ·  n t  dt.
T 0
n(t) = a t időpillanatban foglalt eszközök száma
Egység: erlang (E) – Erlang dán matematikusról elnevezve
Nincs dimenziója
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Forgalmi alapfogalmak - 2
Felajánlott forgalom
(Offered traffic)
Definition: The traffic that would be carried by an infinitely large
pool of resources
A  .s
  hívás intenzitás, egységnyi idő alatt
A  .sfelajánlott hívások/igények száma
s = átlagos kiszolgálási idő
Esetleg több csatornát lefoglaló N forgalom típus esetében:
N
A   i si di
i 0
si = az i-dik forgalomtípus átlagos kiszolgálási ideje
di = az i-dik forgalomtípus hívásai/igényei által
lefoglalt csatornák száma
Nem mérhető !
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Forgalmi alapfogalmak - 3
Lebonyolított forgalom
(Traffic carried)
Definition: The traffic served by a pool of resources.
Gyakorlatban az átlagos forgalom intenzitás
Forgalom mennyisége
(Traffic volume)
Definition:Traffic volume is equivalent to the sum of the holding
times in the given time interval.
Egység: erlangóra (Eh)
Elveszett/visszautasított forgalom
(Lost/rejected traffic)
Definition: The difference between offered traffic and carried
traffic
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Forgalmi alapfogalmak - 4
Kihasználtság
(Utilisation)
.s


 a feladatok (job) beérkezési intenzitása
 az adatátviteli sebesség (pl. egységnyi adatmennyiség/sec)
s a feladat adatmennyisége (egység pl. bit, byte, packet, frame)
0   1
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Forgalmi alapfogalmak - 5
Forgalmas óra
(Busy hour)
Többféle meghatározás lehetséges
Time consistent busy hour, (TCBH): those 60 minutes (determined
with an accuracy of 15 minutes) which during a long period on the
average has the highest traffic.
It may happen that the traffic during the busiest hour is larger than the
time consistent busy hour, but on the average over several days, the
TCBH traffic will be the largest.
We also distinguish between busy hour for the total telecommunication
system, an exchange, and for a single group of servers, e.g. a trunk
group. In practice, for measurements of traffic, dimensioning, and
other aspects it is an advantage to have a predetermined well–
defined busy hour.
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Summary
Incoming demands (intensity, holding time)
Simplified scheme:
human factors,
queue management,
etc.
are missing.
no free resource
service principle:
redirection
limited delay
delay
no waiting
place
no overflow
overflow
loss
free
waiting place
loss
waiting
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Erlang’s model –1.

Structure: n identical channels (servers, trunks, slots) –
homogeneous group

Strategy:




full accessibility, one demand – one channel
if all channels are busy the demand is lost without any
after effect (lost calls cleared)
Erlang’s loss model – Lost Calls Cleared (LCC model)
Traffic:



exp. holding time distribution. μ intensity (1/μ mean
value, „holding time”)
arrival rate:  intensity (Poisson process)
pure birth and death process Pure Chance Traffic
type One  PCT-1
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Erlang’s model –2.

Offered traffic:

offered traffic = carried traffic, if n∞
that is:
mean arrival rate x mean
holding time

Considered cases:



(n = ∞  Poisson distribution)
n < ∞  truncated Poisson distribution
Performance measures



E (time congestion)
B (call congestion)
C (traffic congestion)
The model is insensitive to the
holding time distribution
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Erlang's model –3.
The mathematical model is insensitive to the
holding time distribution
Insensitivity:
A system is insensitive to the
holding time distribution
if the state probabilities of the system
only depend on the
mean value of the holding time.
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Erlang's distribution -1.
Traffic: PCT-1
Erlang’s distribution
(truncated Poisson)
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állapottér
állapotvalószínűség
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Erlang's distribution -2.
Time congestion
All n channels are occupied in a random point of time
Erlang B formula
Call congestion
Rejection of a random demand
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Erlang's distribution - 3.
Carried traffic
Mean value or
expectation
Lost traffic
Traffic congestion
E=B=C
since the intensity of demands is state independent
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Erlang's distribution - 4.
For large state spaces
numerical difficulties may
occur in calculating state
probabilities.
Easily applicable methods
and recursion form are
available.
Tabular calculation aid:
GG Honlap, Gyakorlatok
Erlang B táblázat
A (traffic),
from any
N (number of channels two the
Erlang B (congestion)
third
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Erlang's distribution - 5.
For large state spaces
numerical difficulties may
occur in calculating state
probabilities.
Easily applicable methods
and recursion form are
available.
Tabular calculation aid:
GG Honlap, Gyakorlatok
Erlang B táblázat
A (traffic),
from any
N (number of channels two the
Erlang B (congestion)
third
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Erlang's distribution - 6.
For large state spaces
numerical difficulties may
occur in calculating state
probabilities.
Easily applicable methods
and recursion form are
available.
Tabular calculation aid:
GG Honlap, Gyakorlatok
Erlang B táblázat
A (traffic),
from any
N (number of channels two the
Erlang B (congestion)
third
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Erlang's distribution - 7.
http://www.erlang.com/calculator/erlb/
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Erlang B formula is robust
Generalisation of Erlang B
– It is valid for any holding time distribution
(formulas depend only on the average holding time
which is included in A, the offered traffic).
– The deduction assumed a Poisson arrival process.
According to Palm’s theorem this is fulfilled, if the
traffic is offered by many indpendent sources.
– Mathematical generalization is possible for
fractional number of channels.
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Engset’s model -1.
•
Structure: n identical channels (servers, trunks, slots) –
•
Strategy:
homogeneous group


•
full accessibility, one demand – one channel
if all channels are busy the demand is lost without any after
effect – LCC (lost calls cleared) model
Traffic:
exp. holding time distribution. μ intensity (1/μ mean value,
„holding time”)

offered traffic, A = carried traffic, if the number of channels is
not limited (independent of the number of channels)

pure birth and death process
Pure Chance Traffic type Two  PCT-2
Results are independent from the holding time distribution they
depend on its’ average value.


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Engset’s model -2.
S traffic sources offer demands to n fully available channels.
The arrival intensity of new demands is: (S-i)
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Engset distribution - 1
Normalization:
offered traffic
of free traffic
source
Distribution:
Engset, 1918 !!
(truncated
binomial)
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Engset distribution - 2
Time congestion
Call congestion
After some
transformations:
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Engset distribution - 3
Interpretation:
As if the remaining S-1 traffic sources had occupied
all channels.
When S increases E is increasing too, therefore:
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Engset distribution - 4
Carried
traffic:
transformation with
cut equations
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Engset distribution - 5
Traffic congestion:
designation:
Relationship
applied:
A  Sa
Number of calls
(traffic demands)
per time unit
(S – Y) the number
of free traffic sources
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Engset distribution - 6
Lost traffic:
Duration of state [i]
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Engset distribution - 7
Relations between E, B and C
Designation:
Already
derived
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Evaluation of Engset’s formula - 1
There are numerical problems for large values
of S and n. Various numerically stable recursive
formulae have been elaborated.
Tabular calculation aid:
GG Honlap, Gyakorlatok
Engset táblázat
S (number of sources),
n (number of channels
γ (call intesity)
μ (release intensíty)
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Engset E, B, C
A
A-Y
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Evaluation of Engset’s formula - 2
http://www.erlang.com/calculator/engset/
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Summary
Incoming demands (intensity, holding time)
Simplified scheme:
human factors,
queue management,
etc.
are missing.
no free resource
service principle:
redirection
limited delay
delay
no waiting
place
no overflow
overflow
loss
free
waiting place
loss
waiting
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Overflow traffic - model
Basic problem: traffic from node A to nodes B or C are directed on
different „first choice” routes and if these are fully occupied the
overflow traffic might use the „overflow” route
Nowadays these type of arrangements are used only in networks.
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Overflow traffic – Example 1a.
16
10 erl
PCT-I
……
8
8
1. 10 erl, 16 channels, E16=2,23%,
lost traffic 0,223 erl.
Could this be calculated in two steps ??
PCT-I
If yes,
how ?
8
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Overflow traffic – Example 1b.
PCT-I
??
8
8
2. 10 erl, 8 channels, E8 =33,832%, Alost = 3,3832 erl
A’ =3,3832 erl, 8 channels, E8’=0,1457
A’lost= 3,3832 x 0,1457 = 0,0483 erl.
0,223 erl = 0,0483 erl
What is the reason ???
Overflow traffic does not have
PCT-I/PCT-II character
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Overflow traffic – Z peakedness – 2
Peakedness (Z)
The peakedness has dimension: [number of channels]
„Number representation”
Index of Dispersion for Counts – IDC
= peakedness
 2 


 m1 
Gives a characterization for the probability distribution of occupied
servers (lines, channels).
Poisson
distribution:
Erlang
distribution:
Binomial and Engset
distribution:
In the case of binomial and Engset distribution β (offered traffic of free traffic
sources), takes congestion already into account.
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Overflow traffic – Z peakedness – 2
1. Peakedness Z is a good indicator for the
relative loss probabilities of traffics with the
same average value (A).
2. For a given A traffic Z has a maximum as a
function of n, the number of channels.
3. For PCT-I Z = 1.
4. If Z < 1, the traffic is smooth.
5. If Z > 1, the traffic is bursty.
6. Congestion: smooth < PCT-I < bursty !!.
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Overflow traffic – Z peakedness – 3
Peakedness Z
of overflow
traffic as a
function of the
offered PCT-1
traffic (A) and
the number of
channels (n)
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Dimensioning methods of overflow systems
• ERT (Equivalent Random Theory)
– an equivalent random traffic is applied which is derived
from the average value (expected value) and the variance
of the overflow traffic
• Modified ERT
– calculation is based on a Z peakedness value which is
derived from the average value (expected value) and the
variance of the overflow traffic
• IPP (Interrupted Poisson Process)
– If the primary route is occupied, a random (Poisson) traffic
appears temporarily (in an interrupted way) on the
secondary route.
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Multi-dimensional loss systems – 1
Example: multi-dimensional Erlang-B loss formula
•
Structure: n uniform channels (trunks,
slots) – homogenous group
•
Strategy:


•
full accessibility
LCC - lost calls cleared
Input process:


•
two independent PCT-I traffic streams with 1 and 2
intensity
holding times: exp. distribution. μ1 and μ2 intensity
Offered traffic

A1= 1/μ1 and A2 = 2/μ2
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Multi-dimensional loss systems – 2
In state (i,j)
i channels are occupied by the first,
j channels are occupied by the second
traffic stream.
One demand occupies one channel.
Restrictions:
Statistical equilibrium,
(n+1)(n+2)/2 node equations.
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Multi-dimensional loss systems – 3
State space:
Number of states:
(n+1)(n+2)/2
Example of
node equation:
p(0,1)[1+2+μ2]=
p(0,0) 2 +
p(1,1) μ1 +
p(0,2)2μ2
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Multi-dimensional loss systems – 4
The state space diagram depicts a reversible Markov process
with local balance and with product form solution.
It can be shown that the solution is:
where: p(i) and p(j) are one dimensional,
truncated Poisson distributions and Q is the
normalisation constant
Time congestion
Call congestion
Traffic congestion
P(i+j=n)
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Multi-dimensional loss systems – 5
It can be shown, that:
This is a truncated Poisson distribution,
with offered traffic
i.e.
this is a
two dimensional
Erlang distribution
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Teletraffic Engineering (TTE) in networks – 1.
Traffic engineering functions
ITU-T Rec. E.360.1 (02/05) – Framework for QoS routing and related traffic engineering methods for IP ......
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TTE in networks – 2.
Input
instantaneous
hour-to-hour
day-to-day
week-to week
seasonal
load variations
predicted average demand
„noisy”
traffic load
unkown
forecast
error
Feedback
the time constants of the feedback controls are
matched to the load variations
regulates the service provided by the network
through capacity and routing adjustments.
ITU-T Rec. E.360.1 (02.05) – Framework for QoS routing and related traffic engineering methods for IP ......
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TTE in networks – 3.
Traffic engineering functions include:
traffic management,
capacity management, and
network planning.
Traffic management ensures that network performance is maximized
under all conditions, including load shifts and failures.
Capacity management ensures that the network
is designed and provisioned to meet performance objectives for network
demands at minimum cost.
Network planning ensures that node and transport capacity is planned
and deployed in advance of forecasted traffic growth. Figure 1 illustrates
traffic management, capacity management, and network planning as
three interacting feedback loops around the network.
ITU-T Rec. E.360.1 (02.05) – Framework for QoS routing and related traffic engineering methods for IP ......
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TTE in networks – 4.
3.35 traffic engineering: Encompasses traffic management,
capacity management, traffic measurement and modelling,
network modelling, and performance analysis.
3.36 traffic engineering methods: Network functions which
support traffic engineering and include call routing; connection
routing, QoS resource management, routing table management,
and capacity management.
3.37 traffic stream: A class of connection requests with the
same traffic characteristics
ITU-T Rec. E.360.1 (02.05) – Framework for QoS routing and related traffic engineering methods for IP ......
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TTE in networks – 5.
3.27 QoS (Quality of Service): A set of service requirements to be
met by the network while transporting a Connection or flow; the
collective effect of service performance which determine the Degree of
satisfaction of a user of the service.
3.28 QoS resource management: Network functions which include
class-of-service identification, routing table; derivation, connection
admission, bandwidth allocation, bandwidth protection, bandwidth
reservation, priority routing, and priority queuing.
3.29
QoS routing: See QoS Resource Management.
3.30 QoS variable: Any performance variable (such as congestion,
delay, etc.) which is perceivable by a user.
ITU-T Rec. E.360.1 (02.05) – Framework for QoS routing and related traffic engineering methods for IP ......
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TTE in IP networks - example a-1.
Rec. ITU-T Y.1543 (2007.11.)
Measurements in IP networks for inter-domain
performance assessment
The performance attributes that are used to characterize the
network performance (inter-domain QoS) of a path are:
• Mean one-way delay.
• One-way packet delay variation.
• Packet loss ratio.
• Path unavailability.
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TTE in IP networks - example a-2.
ITU-T Y.1543 (2007.11.) Measurements in IP networks for inter-domain performance assessment
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TTE in NGN networks - example b-1.
Recommendation ITU-T Y.2173 (2008.09.)
Management of performance measurement for NGN
Summary
This Recommendation specifies requirements, reference measurement
network model, high-level and functional architectures, and procedures
for performance measurement management. This Recommendation
together with [Recommendation ITU-T Y.1543] provides overall
consistency for performance measurement and management of NGN.
Scope
This document specifies the management aspects of performance measurement:
- Requirements for management of performance measurement....
- A reference measurement network model....
- A general and functional architecture for the management of performance
measurement....
- Management procedures covering various management scenarios....
- Application scenarios for management of performance measurement (MPM)
use cases.....
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TTE in NGN networks - example b-2.
ABG =
Access
Border Gateway
IBG =
Interconnection
Border Gateway
CPNE =
Customer
Premises
Network Edge
Recommendation ITU-T Y.2173 (2008.09.)
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Traffic routing (PSTN, ISDN) – 1.
• Traffic routing may be:
– fixed
(Fixed Routing –FR)
– time-dependent
(Time dependent Routing – TDR)
– state dependent
(State Dependent Routing – SDR)
– event dependent
(Event Dependent Routing – EDR)
ITU-T Rec. E.350 (2000.03.) – Dynamic Routing Interworking
(Framework for dynamic routing interworking in circuit-switched
PSTN, narrow-band ISDN, and broadband ISDN networks)
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Traffic routing (PSTN, ISDN) – 2
Event-dependent routing (EDR)
In event-dependent routing (EDR), the routing tables are
updated locally on the basis of whether calls succeed or fail
on a given route choice. In EDR, for example, a call is offered
first to a fixed, preplanned route often encompassing only a
direct route, if it exists. If no circuit is available on the
preplanned routes, the overflow traffic is offered to a
currently selected alternate route. If a call is blocked on the
current alternate route choice, another alternate route is
selected from a set of available alternate routes for the traffic
stream according to the given EDR routing table rules. For
example, the current alternate route choice can be updated
randomly, cyclically, or by some other means, and may be
maintained as long as a call is established successfully on the
route.
ITU-T Rec. E.350 (2000.03.) – Dynamic Routing Interworking
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Traffic routing - MPLS (IP, ….) - 1
• Multiprotocol Label Switching (MPLS) is a mechanism in highperformance telecommunications networks which directs and carries
data from one network node to the next. MPLS makes it easy to create
"virtual links" between distant nodes. It can encapsulate packets of
various network protocols.
• MPLS is a highly scalable, protocol agnostic, data-carrying mechanism.
In an MPLS network, data packets are assigned labels. Packetforwarding decisions are made solely on the contents of this label,
without the need to examine the packet itself. This allows one to
create end-to-end circuits across any type of transport medium, using
any protocol. The primary benefit is to eliminate dependence on a
particular Data Link Layer technology, such as ATM, frame relay, SONET
or Ethernet, and eliminate the need for multiple Layer 2 networks to
satisfy different types of traffic. MPLS belongs to the family of packetswitched networks.
http://en.wikipedia.org/wiki/MPLS - 2011.09.
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Traffic routing MPLS (IP, ….) - 2
• MPLS operates at an OSI Model layer that
is generally considered to lie between
traditional definitions of Layer 2 (Data Link
Layer) and Layer 3 (Network Layer), and
thus is often referred to as a "Layer 2.5"
protocol. It was designed to provide a
unified data-carrying service for both
circuit-based clients and packet-switching
clients which provide a datagram service
model. It can be used to carry many
different kinds of traffic, including IP
packets, as well as native ATM, SONET, and
Ethernet frames.
http://en.wikipedia.org/wiki/MPLS - 2011.09.
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