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On the Model Transform in Stochastic Network
Calculus
Kui Wu∗, Yuming Jiang†, and Jie Li‡
Abstract—Stochastic network calculus requires special care
in the search of proper stochastic traffic arrival models and
stochastic service models. Tradeoff must be considered between
the feasibility for the analysis of performance bounds, the use-
fulness of performance bounds, and the ease of their numerical
calculation. In theory, transform between different traffic arrival
models and transform between different service models are
possible. Nevertheless, the impact of the model transform on
performance bounds has not been thoroughly investigated. This
paper is to investigate the effect of the model transform and to
provide practical guidance in the model selection in stochastic
network calculus.
Index Terms—Stochastic Network Calculus, Model Transform,
Performance
I. INTRODUCTION
Performance has always been one of the major concerns
in networking systems. Mathematical models for quantitative
evaluation of network performance, however, have remained
as a slow-paced research area. A.K. Erlang published the first
paper on queuing theory in 1909 [1], and since then queuing
theory has been developed and applied in a wide variety
of applications. In particular, it has been the foundation in
performance modeling and evaluation of telecommunication
systems and has been applied broadly in the performance
analysis of computer networks. Nevertheless, with the advance
of the Internet technology, the assumptions behind the tractable
queuing models may not hold anymore. Despite the research
efforts in the last one hundred years, the tractable models with
traditional queueing theory consist of only a minority of practi-
cal network problems. The research community is in dire need
of new mathematical models for network-wide performance
evaluation where the Markovian property in traffic arrivals or
services may not hold.
Network calculus is one of such new analytical techniques.
The theory of network calculus was introduced in early
1990s [5] for network performance evaluation. Unlike the
traditional queueing theory which aims at obtaining exact
analytical results, network calculus focuses on the analysis
of performance bounds using the cumulative amount of traffic
arrivals or services. Since network calculus usually does not
assume particular distributions on traffic arrivals or service
times, it can obtain broadly applicable performance results.
Network calculus has evolved along two tracks– determinis-
tic [2], [16] and stochastic [10], [15], [17], [20]. The determin-
istic network calculus is to obtain the worst case performance
bounds, which may be too loose to be useful in practice. Due
to this reason, research on this direction gradually fades out.
To overcome the problem, stochastic network calculus was
developed. Nevertheless, due to some special difficulties [10],
[17], basic properties of stochastic network calculus have been
proved only in recent years [4], [10], [11]. Although the major
theoretical barriers have been cleared, it is still unclear whether
or not stochastic network calculus will be broadly accepted by
network practitioners.
Without the driving force from real applications, broad ac-
ceptance of stochastic network calculus as a valuable technique
for performance evaluation may not be optimistic. One of the
major practical challenges is the lack of effective algorithms
to calculate and compare the performance bounds. After all,
what really matter to network engineers are the meaningful
numerical results instead of the complex equations. Although
there are some efforts using Legendre transform [7], [9]
to simplify the calculation of major operations in network
calculus, the treatments are far from sufficient to tackle the
difficulties in the stochastic network calculus, where we are
often faced with multiple tradeoffs.
Specifically, three tradeoffs must be considered in stochastic
network calculus. First, there is a tradeoff between the sim-
plicity of deriving performance bounds and the difficulty in the
numerical calculation of the bounds. It is known that in order
to derive performance bounds easily, we need to put extra con-
straints on the traffic and the service models [11]. For instance,
we may need to put some constraints in the cumulative traffic
arrivals/services, e.g, we may change the calculation from the
form of Prob{ft > 0} to the form of Prob{supt ft > 0},
which is usually not equal to supt Prob{ft > 0}. Note
that sup is the supremum (i.e., least upper bound) operation.
Prob{supt ft > 0} thus represents an instantaneous property
and is generally hard to calculate. Second, there is a tradeoff
between the usefulness of the traffic (service) models and the
hardness of searching for these models in real applications.
This tradeoff is closely related to the first one. In general,
it is easy to obtain the traffic model (or service model)
directly from the distribution of packet inter-arrival times
(or the distribution of the service times). Introducing extra
constraints on the traffic arrival or service model, e.g., adding
the sup operation in the model [10], requires that we either
perform model transform [11] or search for the models using
queueing analysis methods [8], [13]. Third, we must consider
the tightness of performance bounds in a stochastic sense. In
stochastic network calculus, we need to weigh a performance
bound regarding its tightness and its bounding function, e.g.,
we need to avoid poor claims like “the probability that the
delay is larger than 30 seconds is less than 90%,” which is
not helpful in practice.
Handling the above problems has been a very tricky and
intimidating task. Without a clear guideline, it may not be easy
to use stochastic network calculus in real-world problems. We
are thus motivated in this paper to analyze the above tradeoffs