Testing the Null Hypothesis of Stationarity of Internet Traffic

The statistical analyses indicate that the measured traffic traces from the packet networks often contain non-stationary effects. In these cases several popular tests for long-range dependence and/or stationarity can result in wrong conclusions and unreliable estimate of the Hurst parameter. In this paper non-stationarities are modeled as the trends and/or level shifts in Internet traffic data. MMPP-Based Hierarchical Model simulation data are used for stationarity tests. Application of testing results are integrated into network resource allocation function as a Partially Observable Markov Decision process. Ill. 4, bibl. 7, tabl. 1 (in English; abstracts in English and Lithuanian). http://dx.doi.org/10.5755/j01.eee.112.6.439


Introduction
Network management techniques have long been of interest to the networking research community.Networks can be viewed as a distributed system in which coordinated and informed decision making is crucial for optimal resource allocation.In this paper we study the problem of finding an optimal policy for Network resource allocation as a Partially Observable Markov Decision Process (POMDP).Testing the stationarity of Network traffic is one of the keystone problem.Partially Observable Markov Decision Process (POMDP) is a basic framework for Multi-Agent planning, when the traffic model is not perfectly known and may change over time and is a wellstudied framework for sequential decision-making in partially observable domains.This paper restricts itself to two network management techniques: admission control and the partitioning of transmission and buffer resources among two or more classes of traffic using a common transmission path.The Decision Policy Agent (DPA) model and Network model are presented in fig. 1.For Network resource allocation, we are interested in the performance of a queue which represents the bottleneck of a network.In this paper we will use a simple hierarchical MMPP traffic model from [1] and queue model MMPP/GI/1/m.The aim of this paper is to estimate the various stationarity testing procedures for intergation into Network resource allocation agents.

Problem statement
The problem of the testing of the stationarity hypothesis for real traffic measurements is caused by the fact the mean of traffic with LRD does not exist.But, statistical analyses of measured traffic traces often contain non-stationary effects like level shifts or polynomial trends.The testing of the stationarity hypothesis is particularly difficult in the presence of LRD, where many classical statistical approaches cease to hold [2].In these cases several popular tests for long-range dependence can result in wrong conclusions and unreliable estimate of the Hurst parameter.On the other hand for decision making in CP MDP it is significant to detect the level shifts and/or polynomial trends with reasonable computational complexity.On longer time scales we can observe also a regular character of the traffic due to daily or weekly variations.Three types of trend models are used in our experiments, e.g.linear trend, parabolic trend, and level shift model.Level shift model can be observed when during our traffic measurements suddenly a new source starts to generate the traffic to the network nodes and the linear and parabolic trends, which can be observed in daily traffic variations.For example, when people start to work in their office between 8 and 10 am a monotonic increase of the total load of the aggregated traffic can be observed.These traffic trends should be identified by Decision Policy Agent.

MMPP traffic model
In this paper for traffic generation we will use a MMPP traffic model proposed by [1].This is hierarchical MMPP traffic model, capable of generating traffic that accurately emulates the aggregate Network traffic measured at an edge router.The model is based on a layered architecture of sessions, that generate flows, that finally generate packets.MMPP model is completely described by these five parameters:  Then: For network planning and dimensioning, we are typically interested in the performance of a queue which represents the bottleneck of a network.Besides the advantages of being simple to implement and efficient, a synthetic Markovian source as the one we propose has the additional advantage of allowing a Markovian model of a queue.In general, the buffer model can be described by a MMPP/GI/1/m queue, where the service time represents the transmission time of a packet, and can be easily derived from the capacity of the link and the distribution of the packet length.The general service time distribution can be approximated by a phase type distribution.By adopting an exponential service time distribution, we obtain an MMPP/M/1/m queuing system.[1] The infinitesimal generator (IG) of such a (Continuos-Time Markov Chain) CTMC is matrix A where and  is the rate matrix , and I n is the identity matrix.More detailed description of this IG can find in [1].

Decision Policy Agent model
Decision policy agent (DPA) and Network model are presented in Figure 1.  , . We use a following definition: policies that do not dependent on stages are called stationary policies.We are study over the infinite horizon policies and agent`s goal is to find a policy π by exacuting at each step (state) the actions that would maximize value function (cumulative reward over the horizon).Finding the optimal policies traditionally needs significant computational resources and is limited in time.Our idea is to reduce these resources by computing a new decision policy only when traffic stationarity has changed.For this we needed proportionate stationarity testing procedure of incoming traffic.Fully observable part of the system is process , where O(t) is number of packets in queue.In this paper we are testing several known stationarity test and to look for an appropriate policy π are not resesrch object of this paper.-common action space as the cross product between A and space of observation mappings O N , f

Used data set
General network simul Parameters we MMPP traffic λ s = 1.42, the f arrival rate pe session N f = 1 10.Generated and is depicted

Conclusions
The tests used in our experiments does not enable to decide between non-stationarities and LRD.We have shown that the presence of different non-stationarities such as level shifts, linear and polynomial trends (parabolic in such case) in the observations can deceive classical LRD methods.These simulation results confirm that short-range dependent (SRD) process with non-stationarities can produce the same variance-time plot as LRD processes.In the case of LRD processes trends can significantly destroy the accuracy of the estimation of the H parameter.These results show that granular computing methods could be more acceptable for selection of conditional observation strategy.

s
-the arrival rate of new sessions,  f  -the flow arrival rate per active session,  p  -packet arrival rate per active flow,  f N -average numbers of flow per session,  p N -average numbers of packets per flow.

Fig. 1 .
Fig. 1.Decision policy agent (DPA) and Network model DPA model consists of three objects: Partially Observable Markov Decision Process (POMDP), Finite state Controller (FSC) and Cross Product Markov Decision

FSC:
Finite state controller model as a deterministic policy graph π is a triple   , , N, where:  N is set of controller nodes n , can define the following FSC controller nodes:  n1 -state with stationary observations in POMDP,  n2 -state of testing of stationarity in POMDP,  n3 -state of finding (learning) new decision policy in POMDP.Cross Product MDP: POMDP with Z as the Cartesian product of external system states and internal memory nodes, which consists of pairs S

Table 1 .
Statistical analysis was performed using the R statistics software with tseries package used for stationarity tests.The test results are listed in Table1.Test of stationarity using three different kinds of tests by three different traffic models (Accepted = stationary) Level of significance in all cases are chosen 1% (α=0.01)