Faro, Portugal - January 22, 1999
Routing Algorithms for Robustness
Arianna Bertollini, Ugo Mocci, Caterina Scoglio Fondazione Ugo Bordoni, Roma, Italy
e-mail: mocci@fub.it, caterina@fub.it
Abstract
In this paper different routing policies are studied and evaluated from the point of view of the network robustness, i.e. the capability of the network to well react to overloads and uncertain variations of the offered traffic. A class of simple dynamic overflow routing schemes among which Dynamic Adaptive Routing (DAR) and Optimal overflow Load Sharing (OLS) has been chosen for the analysis. The efficiency of such routing schemes has been evaluated both in nominal traffic conditions and in a set of significant overload conditions referring to the stability of the performance with respect to the Erlang Bound. Our simulative results show that efficiency is just lightly reduced in many overload conditions by using DAR and OLS, confirming the robustness of these routing schemes provided that trunk reservation is applied.
1. Introduction
Traffic routing has been subject of study for a long time and many techniques have been proposed and implemented. The basic objective of any routing technique is mostly to find the “optimal” mapping of connection demands into routes through the network, i.e. to achieve the efficient use of resources, meeting at the same time the Quality of Service constraints.
In relation to this objective a problem can arise from the heavy uncertainties of the future scenarios about the possibility to monitor the current traffic level and to have a full continuous knowledge of the resources available in the network. To cope with this problem we could modify the main objectives of the network design and management moving from “optimality” to “robustness”, i.e. to achieve good performances in a wide range of situations. In this view, the demand routing can play a new role, as one of the ways to improve robustness in uncertain environment.
The main goal of the paper is to evaluate the robustness of different routing techniques. In section 2
Work carried out in the framework of the agreement between Telecom Italia and Fondazione Ugo Bordoni
we select some routing techniques, which seem more promising for their behaviour. In section 3 the performances achieved with each routing algorithm are evaluated and compared with reference to a case study. Some general conclusions are reported at the end.
2. Routing techniques and robustness
The performance evaluation of a routing scheme can be made from different points of view: the fairness of QoS distribution, the resource usage, the behaviour in overloads.
Hereinafter the attention is focused on robustness. As robustness is a rather wide concept a quantitative analysis can be carried out only after a precise definition. To this aim we define robustness of a routing scheme as the capability to maintain the total carried traffic above a given threshold with respect to the maximum in a given set of significant “overload” conditions. In a context where connection requests arrive according to a Poisson process and generate offered traffic with size O, the maximal carried traffic is bounded by the Erlang Bound carried traffic TE(O) [GK]. The relevant Erlang Bound blocking probability represents the lower-bound on the connection blocking probability of any dynamic routing scheme under any stationary pattern of the Poissonian offered traffic with size O.
The definition of a set of significant overload conditions is a prerequisite for the robustness evaluation;
it is very complex because there exist infinite offered traffic matrices which can represent a traffic variation with respect to the nominal traffic Oref considered in the dimensioning. For analysis purposes we focused on the following four classes of overloads:
– Relation overload RO(x, r): traffic offered to the traffic relation r is incremented by x ;
– Node overload NO(x, n): traffic offered to each relation from/to the node n is incremented by x;
– General overload GO(x): traffic offered to each relation is incremented by x;
– Balanced overload BO ( x , r): traffic offered to the relation r is incremented by x and the traffic offered to the remaining relations are fairly decreased to maintain constant the total offered traffic.
The above definitions are applied to the sample network of figure 1. In each overload class, it is possible to order the different overload conditions according to the increasing offered traffic of the whole network or of a subset of network elements.
Finally we can give the following definition of robustness :
The robustness I(RS) of the routing scheme RS, defined with respect to a given overload class and the discrete set of increasing offered traffic size O ∈ [Oref, ..., ON] is the maximum traffic overload Omax (Oref ≤ Omax ≤ ON) normalised with respect to Oref, such that the corresponding carried traffic T(Omax) is not less than a fixed fraction f(0 ≤ f ≤ 1) of the Erlang Bound traffic. More formally:
I(RS) = [(Omax – Oref) / Oref] × 100
where:
Omax : ∀Ο , Oref ≤ O ≤ Omax T(O) ≥ fTE
Before using this definition for a quantitative analysis, we first analyse the robustness of different routing schemes from a qualitative view.
We note that usually the simplest routing methods are defined choosing fixed paths among the shortest or the “optimal” with respect to a given global objective function [Gir]. The drawback of this class of methods is the inability to adapt the routing paths to the state of the network. For this purpose, dynamic routing schemes were introduced with the aim to adjust the routing paths in accordance with variable and uncertain offered traffic, providing, in principle, extra flexibility and robustness in case of overloads and failures. However, even dynamic routing schemes show some drawbacks since they can allocate a quantity of resources higher than the minimum that implies, in some traffic conditions, a global performance reduction. The above considerations show that there is not a simple answer to the question
“which routing is better for robustness?”. Nevertheless in this view, it is possible to recognise what are some “good” features for a robust routing scheme. We can summarise:
– the fresh traffic should be protected from the overflow traffic by providing trunk reservation;
– in case of traffic variations at constant total traffic, the available path set should be suitably adapted in order to route calls through the underloaded part of the network;
– in case of light local overload the available path set should be enlarged compatibly with QoS and alternative paths freely used;
– in case of general overload, an extensive use of alternate paths leads to a waste of resources due to the use of longer paths, therefore it should be avoided.
– in case of general overload, restrictive admission policy should be also adopted;
Several routing schemes with a reduced complexity offer at least some of the features previously described. In order to maintain the simplicity of the implementation, calls are usually offered to the first choice route, which is the shortest path according to the number of hops. In this way a wide number of calls are carried using the minimum quantity of resources. Calls blocked on the first choice route can overflow on a unique but variable second-choice path, characterised by a number of hops close to the minimum; the overflow possibility is limited to one short path to keep low the level of complexity and to avoid the waste of resources related to longer paths. If a call is rejected by both the first choice and the alternative path, it is lost from the system. Sometime in order to distribute QoS and to avoid the overflow in case of general overload, trunk reservation is introduced as a simple and effective method to improve resource usage in such conditions [Son]. It has been proved [GK] that the determination of trunk reservation parameters is not a critical factor: the performance achieved keeping constant and equal to few bandwidth units the parameters are only lightly worse than performance achievable adopting optimal values of trunk reservation.
the following four types of routing:
– Dynamic Alternative Routing (DAR), where the overflow path is kept fixed until a call is blocked and lost; when this event occurs a new overflow path is randomly chosen.
– Least Loaded Routing (LLR), in which the overflow path with the maximum available capacity is selected call-by-call;
– Equiprobable overflow Load Sharing (ELS), where the overflow traffic is equally shared among the overflow paths;
– Optimal overflow Load Sharing (OLS), where overflow traffic is shared among the overflow paths according to the optimal load distribution determined solving a multicommodity flow problem by the flow deviation method [FGK].
Note that in case of complete symmetric network ELS and OLS coincide. All this routing differ just for the determination of the second choice route.
3. A case study: traffic scenarios, evaluation & comparison of the routing schemes
For the comparison, we consider the four node full-connected network shown in figure 1, assuming all link capacities equal to 100 capacity units.
In this network the first choice path is the direct path and the overflow paths are two-link long. The network carries one class of service and each call requires one capacity unit (c.u.). The nominal traffic is equal to 90 Erlang for each relation (540 Erlang for the total traffic in the network).
To perform the numerical evaluations we resorted to simulation and results are reported in three subsections with the following headers: trunk reservation and traffic overload, routing schemes and GoS in overload condition, routing efficiency and robustness.
Trunk reservation and traffic overload
In order to use trunk reservation it is necessary firstly to fix a proper trunk reservation threshold (tr).
In figure 2 call blocking probability is reported for different values of tr and offered traffic equal to 150 Erlang/relation (66% general overload). The network performance improves increasing tr, but the rate of improvement decreases rapidly. Almost the same performances are obtained for all the selected routing schemes when tr is greater than 2.
The optimal value of tr depends on the type and size of network overload; however, the performance improvement achieved adapting tr is not balanced by the involved complexity. To confirm this choice, In figure 3, the blocking probability obtained with DAR is reported for different overload conditions assuming tr = 2 or adapting tr according to the link occupation. The overload condition reported in the
x-axis correspond respectively to: Oref, RO(33%; a), RO(33%; a, b), NO(33%; 1), RO(33%; a, b, c, d), RO(33%; a, b, c, d, e), GO(33%).
Routing schemes and GoS in overload conditions
In all the evaluations reported in the section, we fixed tr = 2. In figure 4 the network blocking probability is evaluated in nominal traffic condition and in the same overload conditions of figure 3 for all the four selected routing schemes. The relevant Erlang Bounds have been calculated by the analytic method reported in [GK]. For a given overload condition, the network blocking probability has its minimum for the Erlang bound and it remains almost constant for the examined routing schemes. This behaviour is due to the fact that such routing schemes have many similarities and in a full-connected network calls are always routed either on the direct path or can overflow on a two-link path. The overflow path is chosen differently in each routing scheme, but this difference has no significant impact on the network blocking probability. Lightly better performance is provided by OLS; however it requires
1 2
4 3
RO (x,a): relation overload
1 2
4 3
a b c
d e
f
NO (x,1): node overload
1 2
4 3
GO (x): general overload
1 2
4 3
BO (x,a): balanced overload
1 2
4 3 Traffic offered to relation a
increased by x
Traffic offered to relations a, b, c increased by x
Traffic offered to relation a increased by x Traffic offered to relations
(b, c, d, e, f) decreased to maintain the traffic constant Traffic offered to all relations
increased by x
Figure 1 - Sample network and overload conditions
an off-line computing of the optimal sharing of the overflow traffic. Lightly worse performances are achieved by LLR because more often it uses two-link alternative paths, reducing the acceptance of future calls on direct paths. Obviously, for a fixed routing policy the blocking probability increases considering the overload conditions ordered, from the nominal traffics (Oref) to the 5 relation overload (RO(33%; a, b, c, d, e)), since they dominate each other in the order of increasing total offered traffic.
In figure 5 DAR and OLS=ELS routing schemes are compared in two general overloads, GO(33%) and GO(66%). Even in these cases results confirm that network blocking probabilities are almost equal
0,30 0,32 0,34 0,36 0,38 0,40 0,42 0,44 0,46 0,48
0 1 2 26 100
Trunk Reservation Parameter
Overall Blocking Probability
DAR ELS LLR Erlang Bound
Figure 2 - Network blocking probability vs. constant Trunk Reservation parameters (66% general overload)
0 0,05 0,1 0,15 0,2 0,25
Nominal condition
1 relation overload
2 relation overload
3 relation overload
4 relation overload
5 relation overload
6 relation overload Overload condition
Overall Blocking Probability
Adaptive Trunk Res.
Constant Trunk Res.
Figure 3 - Blocking probability vs. overload conditions (routing scheme DAR, single relation overload = 33%)
for the examined routing schemes. In spite of this behaviour the routing schemes differ each other for the robustness, as it is shown in the next section.
Routing efficiency and robustness
Let us recall that the robustness I(RS) of the routing scheme RS, introduced in section 2, is represented by the maximum offered traffic overload Omax for which the carried traffic T(Omax) is not less than a fixed fraction f (0 ≤ f ≤ 1) of the corresponding Erlang Bound carried traffic (Erlang Traffic). In
0 0,05 0,1 0,15 0,2 0,25
DAR ELS LLR OLS Erlang Bound
Routing scheme
Overall Blocking Probability
Nominal condition 1 relation overload 2 relation overload 3 relation overload 4 relation overload 5 relation overload
Figure 4 - Blocking probability vs. routing schemes for different overload conditions (single relation overload = 33%)
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4
DAR OLS Erlang bound
Routing scheme
Overall Blocking Probability Nominal traffic
General overload 30%
General overload 60%
Figure 5 - Blocking probability vs. routing schemes for two general overload conditions
the following numerical examples it has been assumed f = 0.9. Since we have seen in the previous analysis that the performance of the considered routing scheme are quite similar, in the robustness analysis we consider uniquely the policy DAR evaluated for different trunk reservation thresholds and the routing policy Direct in which only direct paths are used (this policy is optimal in case of severe general overload).
In figure 6 some general overloads (GO(11%), GO(33%), GO(66%)) are considered by increasing the offered traffic per relation from 90 to 150. In this range of overloads, DAR tr = 2 and Direct are always above the Erlang Traffic threshold; Direct approaches the maximal Erlang carried traffic at the highest overload. In such case I(DAR tr = 2) = UL (Upper Limit in the considered range of traffic). On the contrary, with DAR tr = 0, when the offered traffic increases beyond 540 Erl the carried traffic decreases and it becomes lower than the Erlang threshold above a total offered traffic equal to around 600 Erlang; in correspondence it results I(DAR tr = 0) = 10%. This negative behaviour is due to the inefficient use of resources when calls are accepted and routed on overflow paths. A similar behaviour is found also in case of node overloads NO(66%, 1) and NO(133%, 1) (figure 7), where we find I(DAR tr = 2) = I(Direct) = UP, I(DAR tr = 0) = 22%, and in case of relation overloads RO(200%, a) and RO(400%, a) (figure 8) where we find I(DAR tr = 2) = I(Direct)=UP and I(DAR tr = 0) = 18%.
To show some differences between the robustness of Direct and DAR tr = 2 we need to consider balanced overloads BO(200%, a) and BO(433%, a). Still the index I (DAR tr = 2) = UP, while Direct is no more optimal and its robustness index is around a half with respect to DAR tr = 2 in the considered load range (in this particular overload condition the use of the overflow path increases the carried traffic having a positive effect on performances).
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450 500 550 600 650 700 750 800 850 900
total offered traffic
total carried traffic
DAR tr=0 DAR tr=2 Direct Erlang Traffic 90% Erlang Traffic
Figure 6 - Robustness index evaluation for different routing policies (general overload)
5. Conclusions
Robustness has become one of the main objectives of the network design. Many routing schemes show a good robustness as they adopt the available routing paths according to the network state.
However, robustness depends on the distribution and the size of traffic offered to the network. For analysis and comparison purposes we have introduced a proper definition of robustness as a function of both the operating traffic conditions and the capability of the network to maintain “good” performances, i.e. within a given margin with respect to the “optimal” compatible with each traffic condition.
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450 500 550 600 650 700 750 800 850 900
total offered traffic
total carried traffic
DAR tr=0 DAR tr=2 Direct Erlang Traffic 90% Erlang Traffic
Figure 7 - Robustness index evaluation for different routing policies (node overload)
400 450 500 550 600
450 500 550 600 650 700 750 800 850 900
total offered traffic
total carried traffic
DAR tr=0 DAR tr=2 Direct Erlang Traffic 90% Erlang Traffic
Figure 8 - Robustness index evaluation for different routing policies (relation overload)
In this way we have given a metric to measure robustness in general and to discriminate different routing schemes from the robustness point of view.
Numerical results show that the selected routing schemes (DAR, LLR, ELS, OLS) perform equally well when Trunk Reservation is adopted, with a light preference for DAR and OLS. The efficiency in both cases is not reduced in overload cases, providing a high level of robustness.
References
[Gir] A. Girard “Routing and dimensioning in circuit-switched networks” Addison-Wesley (1990).
[GK] R. Gibbens, F. Kelly “Dynamic Routing in Fully Connected Networks” IMA Journal of Mathematical Control & Information (1990) vol. 7: 77-111.
[Son] D. Songhurst “Protection against traffic overload in hierarchical networks employing alternative routing” Network Planning Symposium (1987) Paris.
[FGK] L. Fratta, M. Gerla, L. Klainrock “The flow deviation method: an approach to store-and-forward communication network design” Networks (1973), 3: 97-133.
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100 200 300 400 500 600 700 800
relation traffic overload
total carried traffic
DAR tr=2 Direct Erlang Traffic 90% Erlang Traffic
Figure 9 - Robustness index evaluation for different routing policies (relation overload with constant total offered traffic)
