# Impact of Bimodal Traffic on Latency in   Optical Burst  Switching Networks

Authors: Yuhua Chen; Pramode K. Verma
Journal: Research Letters in Communications
 (2008)

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## Abstract

This paper analyzes the impact of bimodal traffic composition on latency in optical burst switching networks. In particular, it studies the performance degradation to short-length packets caused by longer packets, both of which are part of a heterogeneous traffic model. The paper defines a customer satisfaction index for each of the classes of traffic, and a composite satisfaction index. The impact of higher overall utilization of the network as well as that of the ratio of the traffic mix on each of the customer satisfaction indices is specifically addressed.


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## Body

## 1. Introduction

Optical transmission  technologies
have been widely adopted in the core telecommunication network today. The
adoption of optical transmission technology has been far more pervasive than
the switching technology. Optical burst switching [1, 2] effectively fills
the gap between coarse lambda switching and the still-distant optical packet
switching [3, 4], and is considered the most promising form of optical
switching technology [5].Optical burst switching relies on
relatively large bursts of data assembled at the ingress node to be transported
over an end-to-end optical path to the destination or egress node. Since it is
economically infeasible to form a burst in response to each customer packet,
the packets entering the ingress node will likely originate from different
sources and thus have varying statistical characteristics. In this paper, we
consider this traffic to have a bimodal distribution, each component of which
is negative exponentially distributed with a separate mean. This assumption is
justifiable in the instance when the ingress traffic is a mixture
of short packets and relatively long packets, for example, packets containing dense payloads that
allow sharing of resources of a grid network to a remote point
[6].It is generally known that
heterogeneous traffic leads to poor performance, specifically, as far as the
smaller-length traffic is concerned. However, it is imperative from a service
provider point of view to quantify and understand the impact of the heterogeneous
traffic on the network as a whole as well as on each of the constituent
customer groups. In this paper, we introduce the concept of a customer
satisfaction index for each of the two classes of traffic, and a composite
customer satisfaction index for the network as a whole. We then derive a closed
form mathematical solution for the individual and composite customer
satisfaction indices as a function of the ratio of the mean lengths and further
as a function of the overall traffic intensity.


## 2. Bimodal Traffic Model

In this paper, we are interested
in evaluating the impact of the bimodal traffic on the latency of the two
traffic types, as well as on the system latency as a whole. More specifically,
we want to understand how the mixed traffic affects the latency experienced by
individual traffic type. We define theindividual
customer satisfaction index as follows: η1=T1/Tc,
and η2=T2/Tc,
where T1 and T2 are the latency experienced by type 1 an type
2 traffic in systems carrying single type of traffic, respectively, and Tc is the latency experienced by the composite
traffic. We further define the composite
customer satisfaction index as η=δ⋅η1+(1−δ)⋅η2,
where  δ(0≤δ≤1) is the portion of traffic type 1 in the
composite traffic load ρc.In the rest of the section, we
derive a mathematical model for the individual and composite customer satisfaction
indices in terms of the composite load and the ratio of the mean message
lengths of the two traffic types.We assume that each component of
the bimodal traffic follows a Poisson arrival process. Since the message
lengths of the two streams are negative exponential distributed, we have the
distribution of the composite stream as(1)f(x)=λ1λcμ1e−μ1x+λ2λcμ2e−μ2x.Equation (1) follows from the
fact that the probability density function of the composite stream is the sum
of the density functions of the two streams weighted by the relative
probabilities of their occurrence. Note that the combined stream is not
negative exponentially distributed unless μ1=μ2.We derive the average latency of
the composite stream as follows. For evaluating the mean delay suffered by an
individual   message, we have [7] (2)T=1μC+ρ2+λ2σS22λ(1−ρ), where 1/μ is the mean message length, σS2 is the variance of the service time and C is the channel rate at which the messages are being emptied.  Equation
(2) holds for any message length distribution. Let σc2 be the variance of the message length
distribution.For the combined stream, the mean
length can be evaluated as(3)1μc=xc¯=∫0∞xf(x)dx=1λc⋅(λ1μ1+λ2μ2).Letg1(x)=∫0∞μ1e−μ1xdx    and g2(x)=∫0∞μ2e−μ2xdx.
The Laplace transform of g1(x) is G1(s)=μ1/(μ1+s).
The second moment of g1(x) can be calculated using (4)xg12¯=d2G1*(s)ds2|s=0=d2ds2(μ1μ1+s)|s=0=2μ12.We also havexg22¯=2/μ22.
The second moment of the combined stream can be calculated as (5)xc2¯=∫0∞x2f(x)dx=λ1λcxg12¯+λ2λcxg22¯=2λc(λ1μ12+λ2μ22).The variance of the combined
stream can be derived as(6)σc2=xc2¯−(xc¯)2=2λc(λ1μ12+λ2μ22)−1λc2⋅(λ1μ1+λ2μ2)2.
From (6)
σS2 can be calculated using σS2=σc2/C2.
We then define the ratio of type 2 message-length to type 1 message-length as γ=μ1/μ2. Let χ=δ+(1−δ)/γ.
Equation  (3) becomes (7)1μc=1μ1⋅1δ+(1−δ)/γ=1χμ1.We can rewrite the variance of
the service time of the combined stream as(8)σS2=1μ12C2(2(δ+γ(1−δ))χ−1χ2).Substituting (7) and
(8) in (2),
we have (9)T=1χμ1C(1+ρc(2χ(δ+γ(1−δ))−1)2χ2μ12C2(1−ρc)⋅(1+χ2μ12C2)).Using the mean service times of
the messages from the two streams, we have(10)T1=1μ1C+ρ1μ1C(1−ρ1).T2 can be similarly calculated. Thus, we have the
individual customer satisfaction index (11)η1=χ(1+ρc(2χ(δ+γ(1−δ))−1)2χ2μ12C2(1−ρc)⋅(1+χ2μ12C2))−1,η2=γ⋅χ(1+ρc(2χ(δ+γ(1−δ))−1)2χ2μ12C2(1−ρc)⋅(1+χ2μ12C2))−1. The composite
customer satisfaction index η can be calculated based on definition.


## 3. Numerical Results

Figures1 through 3 present plots that capture
the variations of the three different customer satisfaction indices as a
function of the ratio of the two mean message lengths at various composite
loads. The composite traffic, in each case, consists of 50% of the bits from
the shorter and 50% bits from the longer messages.Figure 1
Customer satisfaction index at low load.As can be seen from Figure1,
with a composite load of 0.4, the satisfaction index of the customer group
generating the shorter messages rapidly declines as the ratio of the mean
message lengths increases. This decline can be observed to be true for all
levels of the composite load. The latter observation follows from Figures 2 and
3 which address higher levels of load. This is consistent with the fact that a
preponderance of longer messages increasingly affects the satisfaction index of
the shorter messages.Figure 2
Customer satisfaction index at medium load.Figure 3
Customer satisfaction index at heavy load.It can be observed from all the
three figures taken together that all the customer satisfaction indices decline
as a function of the composite load. The intersection of the three indices when
the two  customer groups generate statistically identical messages verifies
the fact that satisfaction indices cannot differ when the
messages from the two groups are statistically identical.Simultaneously, it can be
observed that  when the ratio of the mean lengths increases, the composite
satisfaction index holds largely constant while the satisfaction index of the
users of longer messages shows an increase. This is attributed to the fact that as the ratio of
the message lengths increases, the mean latency in systems carrying only longer
messages also increases, while longer messages in composite systems benefit
from the existence of shorter messages, resulting in an increase in
satisfaction indices.


## 4. Conclusion

This paper has defined parameters
that reflect the satisfaction of customer groups generating bimodal traffic that
constitutes the payload of the OBS burst. It has derived analytical results
relating the customer satisfaction indices to variations in the ratio of the
average message lengths of the two groups and to the increase in the overall
utilization factor. The results clearly indicate the reducing satisfaction of
all the indices as the load is increased, and especially that of the customer
group generating shorter messages, as the ratio of the mean lengths of the
larger to smaller messages is increased. However, surprisingly, the network as
a whole, as indicated by the composite satisfaction index, is largely not
affected by the ratio of the mean lengths due to the interplay between the
longer and shorter messages in the composite system.



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*Source: 175359-2009-01-04.xml*