UDC 004.75
Network Сalculus and its application to calculate the end-to-end delay
bounds in VPN
A.V. Roslyakov, A.A. Lysikov, V.D. Vitevskiy
Povolzhskiy State University of Telecommunications and Informatics,
Leo Tolstoy str. 23, Samara 443010, Russia
Abstract. Virtual private networks (VPNs) are one of the most popular services in the
Next Generation Networks. At the planning stage of VPN topologies, the NGN operator
must determine the optimal routes to ensure the maximum volume of realized services
and efficient allocation of network resources over the VPN. The theory of Network
Calculus (NC) allows you to get the boundary values of the characteristics of VPN. NC
theory is proposed for analyzing the performance characteristics of a VPN in
Recommendation ITU-T Y.1315. However, the recommendation provides only a basic
approach without specific methods for analyzing QoS in a VPN. The article proposes an
approach for computing end-to-end delay bounds in virtual private networks. The analysis
method uses the theory of deterministic network calculation and takes into account the
influence of cross-traffic of other networks.
Keywords: virtual private network, network calculus, end-to-end delay, cross-traffic.
1. Introduction
Virtual private networks (VPNs) [1] are one of the most popular services in Next
Generation Networks (NGN) [2]. When implementing VPN services to a telecom
operator, it is necessary to comply with the QoS characteristics prescribed in the Service
Level Agreement (SLA). Routes of different VPNs between client connection points often
overlap, so when implementing multiple VPNs, the problem arises of optimal allocation
of network resources. At the planning stage of VPN topologies, the NGN operator must
determine the optimal routes to ensure the maximum volume of services realized and
efficient allocation of network resources over the VPN. To analyze the characteristics of
QoS in such networks, the models based on the queuing theory, which was originally
developed for telephone networks with circuit switching. Queuing theory allows to
determine average values of QoS characteristics based on assumptions about specific
distributions of application flows and service disciplines, but in practice in multiservice
packet networks these distributions are often unknown due to the complex nature of traffic
(often self-similar) and complex disciplines of traffic servicing at nodes (the presence of
various types scheduler, shaper, etc.). Due to the complexity of the research object, it is
obvious that the models that allow to determine the exact and average values of the QoS
characteristics are rejected in favor of models operating with their boundary values. This
approach is used in the recommendations of international telecommunication
standardization organizations such as ITU-T, IETF, 3GPP, etc. for example,
Recommendation ITU-T G.1010 [3] specifies that the delay in two-way voice
communication should normally not exceed 150 ms, the maximum delay should be less
than 400 ms, and the loss should not exceed 1%. A promising theory that allows obtaining
boundary values of the characteristics of network models is the Network Calculus (NC).
NC theory is proposed for analyzing the performance characteristics of a VPN in
Recommendation ITU-T Y.1315 [4]. However, the recommendation provides only a basic
approach without specific methods for analyzing QoS in a VPN. Therefore, the
development of models and methods for assessing the boundary characteristics of QoS in
a VPN is an urgent task.
2. VPN model
Each fixed VPN route can be represented as a sequential network of N nodes
connecting two endpoints of the VPN (Fig. 1). Each node of the VPN route can serve
cross-flows of other networks that will occupy a certain node performance resource. At
the planning stage of VPN topology, it is necessary to evaluate the mutual influence of
different networks on the quality of services.
Planned VPN
Cross-flow 1
1
Cross-flow 2
2
….
....
Cross-flow 3
d1
N
Cross-flow N
dN
d2
Dend-to-end
Figure. 1 VPN model
Existing models and methods for evaluating QoS characteristics in VPN do not allow
defining end-to-end boundary estimates of the delay and do not take into account the
effect of cross-flows on them. Therefore, the work used the mathematical apparatus of the
theory of Network Calculus (NC), operating with boundary estimates of the characteristics
of networks [5].
3. Basic fundamentals of NC
The theory of Network Calculus (NC) is used in the study of networks providing
deterministic service guarantees for incoming packet flows [6-8]. NC can be used to
investigate systems that receive unknown traffic, but have deterministic limits on the
maximum number of packets received, and the service of incoming packets in the system
is also carried out with deterministic upper bounds of performance. Such systems include
local computer networks, specialized on-board networks (for example, aircraft), systems
on a chip, etc. Deterministic NC models, in contrast to stochastic models, are simpler and
more understandable for perception.
Let the arrival packet flow entering the node be characterized by a cumulative function
(curve) A(t). The departure packet flow at the node output is characterized by a
cumulative function (curve) D(t).
To calculate the boundary values of QoS characteristics, it is necessary to specify the
distribution laws describing the arrival packet flow and the discipline of its servicing in
the node. To do this, the NC uses an arrival curve (t) describing the traffic flow arrival
the node, and a service curve (t) describing the service discipline in the node.
The function will be the arrival curve for the traffic flow at the node's input, provided
that for some observable period of time [0,t] the following inequality holds:
A(t )  A( s )   (t  s), 0  s  t .
Therefore, the actual number of arrival packets in the stream will always be less than
or equal to the number of packets determined based on the arrival curve, i.e. it determines
the smallest upper bound of the number of packets in the arrival packet flow.
A function (t) is a service curve for a network node, provided that for some
observable period of time [0, t]:
D(t )  inf  A(t  s)   ( s)   ( A   )(t ), 0  s  t ,
 – (min,+)-convolution.
The basic idea of the theory of NC is that the boundary values of QoS characteristics
can be obtained by determining the maximum horizontal and vertical distances between
the curves (t) and (t).
The most important feature of NC is the extension of the concept of the service curve
from a single node to an arbitrary number of nodes in a tandem [9]. This allows obtaining
end-to-end characteristics of the whole network using the technique of estimating the
boundary characteristics for a single node. So the flow at the output of two tandem
network nodes with the service curves 1(t) and 2(t), respectively, can be determined by
recursion
D(t) ≥ (A ⊗ 1) ⊗ 2(t).
Since (min,+)-convolution is an associative operation, we can write
D(t) ≥ A ⊗ (1 ⊗ 2) (t).
Hence, in the general case, a network consisting of N tandem nodes with service
curves i, i = 1, ..., N is equivalent to one node with a network service curve
net (t) = 1 ⊗ 2 ⊗ ··· ⊗ N (t).
4. Application of NC for calculate the end-to-end delay bounds in VPN
A tandem of VPN nodes can be converted into one equivalent node with an equivalent
service curve using (min, +) convolution of the maintenance curves of nodes:
VPN (t )
1 (t )
2 (t )
...
N (t ) .
Let's introduce the notion of the residual service curve of the VPN stream, which
allows you to take into account the node resources consumed by the cross-flow of another
network. If the lossless node serves in the FIFO order the VPN flow with the arrival curve
and the cross-flow from the arrival curve, the residual VPN service flow curve is defined
as:
1 (t ,
)
(t )
2 (t
) 1{t } ,
where 1{t } is the indicator function, [ x] denotes max 0,x .
The method for estimating the upper delays for a planned VPN, taking into account
cross flows, is based on the LUDB methodology [10]. The LUDB methodology allows
you to determine the lowest upper delay limit for individual traffic flows passing through
a tandem of VPN nodes.
The concepts of nested and non-nested cross-flows are introduced. Nested we will call
cross-flows that do not intersect or nest each other; otherwise cross-flows will be called
non-nested. An algorithm is developed for determining the border of the delay of VPN
traffic transmission from end to end, taking into account nested and non-nested crossflows. The essence of the algorithm consists in iterative removal of cross-flows with
determination of residual service curves for the flow of the planned VPN.
The software package VPN Designer NC [11] was developed for carrying out
experiments. The purpose of the study was to assess the effect of cross-flow and node
characteristics on boundary of the end to end delay in planned VPN. It is proved that the
burst of cross-flows has the greatest effect on the delay of the planned VPN.
5. Conclusions
It was revealed that existing models and methods of QoS performance assessment in
VPN do not allow to determine their boundary values of QoS characteristics and do not
take into account the influence of cross-traffic on end-to-end delay bounds of the planned
VPN. The developed VPN model allows to take into account the performance of the
nodes used by the cross-traffic of other networks. A method is proposed for estimating the
end-to-end delays in the transmission of traffic of a planned VPN in tandem of nodes with
nested and non-nested cross-flows. Traffic burstiness and cross-traffic routes of other
networks have the greatest impact on the end-to-end delay bounds in planned VPN. In
particular, an increase in the burst of cross-flows can lead to an increase in the delay
boundaries by several times. The intensity of cross-flows effects on the end-to-end delay
bounds to a lesser extent. The difference in the boundary values of delays for different
cross-traffic routes can be more than 50%.
Acknowledgments
The work is partially supported by RFBR grant No 16–37–00363 mol_а.
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