Asymptotic properties of self-similar traffic models based on discrete-time and continuous-time martingales are considered. We discovered that their performance indicators are asymptotically equal at λ → ∞ to indicators for model based on Brownian motion

Pilipenko, Andriy Y.

Trach, Bohdan V.

Uryvskyi, Leonid O.

Asymptotic properties of self-similar traffic models based on discrete-time and continuous-time martingales are considered. We discovered that their performance indicators are asymptotically equal at to indicators for model based on Brownian motion

Pilipenko, Andriy

Uryvskyi, Leonid

Trach, Bogdan

Наукова періодика України

ASYMPTOTIC PROPERTIES OF SELF-SIMILAR TRAFFIC MODELS BASED ON DISCRETE-TIME AND CONTINUOUS-TIME MARTINGALES

Electronic Archive of Kyiv Polytechnic Institute

19  ISSN 2219-9454, Telecommunication Sciences, 2013, Volume 4, Number 2 © 2013, National Technical University of Ukraine “Kyiv Polytechnic Institute”  UDC 621.39  ASYMPTOTIC PROPERTIES OF SELF-SIMILAR TRAFFIC MODELS BASED ON DISCRETE-TIME AND CONTINUOUS-TIME MARTINGALES  Andriy Y. Pilipenko, Leonid O. Uryvskyi, Bohdan V. Trach  NTUU “KPI”, Institute of telecommunication system, Kyiv, Ukraine  Asymptotic properties of self-similar traffic models based on discrete-time and continuous-time martingales are considered. We discovered that their performance indicators are asymptotically equal at λ → ∞  to indicators for model based on Brown-ian motion.  Introduction Massive implementation of computer networks and increases in their productivity turned attention of scien-tists studying network traffic to the properties of the Internet traffic. In 1993, Willinger et al. [1] discovered that computer network traffic exhibits statistical self-similarity. This phenomenon can be described by three features — non-integer fractal dimension (hence, the object is fractal), scale invariance — statistical parame-ters are independent of the level of flow aggregation and time scale, and long-range dependence. Autocorre-lation function of process with long-range dependence decays slower than exponential function. This property is not present in telephone network traffic — it possess-es only scale invariance. Therefore, new approaches to network parameter estimation should be developed. The numerical parameter of self-similarity is the Hurst exponent [2], which lies between 0 and 1. When the Hurst exponent is between 0 and 0.5, the stochastic process is antipersistent. In this case, stochastic process exhibits no trends. When the Hurst exponent is greater than 0.5, it exhibits persistence, which means that long trends are formed in the process. It was estimated that the Hurst exponent for network traffic approximately equals 0.8. It is possible to accurately describe traffic structure using this parameter. Numerous stochastic processes are utilized for re-search and development of network traffic models. Martingales are one of the examples of such process. Martingale is a discrete or continuous process which satisfies the following condition  ( ) ( ){ } ( )|M X t X τ ,τ < s = X s , s t   ∀ ≤ .  One can derive the following property from this equation: the expectation of process is constant throughout the realization of the process. Wide range of stochastic processes exhibits this property; for example, Brownian motion is a martingale. Martingales are commonly used for analysis of time series.  Problem statement  The purpose of the research is the study of asymptot-ic properties of discrete and continuous models of net-work traffic with discrete and continuous time. The ob-ject of the research is asymptotic properties of network traffic models. The subject of the study is application of martingales for network traffic modeling.  Research of asymptotic properties  To describe our approach, let’s at first consider the М/М/1/m system with arrival intensity  and processing intensity	.  The waiting requests for processing get into a queue with the buffer capacity of  requests and get lost at the buffer overflow. Let Χλ,µ(t) be a number of requests in a system at time	; t be a number of requests that arrived be-fore	; t be a number of requests that could be pro-cessed up to time  if there are no idle time. It is well-known [3] that t and t are independent Pois-son processes with intensities  and	, respectively.  If there were neither losses during period	[0, ], nor idle times, then  Χλ,µt = Χλ,µ0 + t − t         (1)  A number of requests in the system cannot be less than 0 physically: Χλ,µt ≥ 	0. During the period, when  Χλ,µ(0)+Nλ(t)-Nμ(t) <0 the system is non-busy. Calcu-lated below of the threshold “0” requests should be con-sidered as virtually processed, i.e. actually not served. Therefore, in general case we have the following representation: Χλ,µt = Χλ,µ0+ t − −t + 	λ,µt− 	λ,µt,   where 	t is the number of lost requests (above the threshold ), 	t is the number of virtually processed requests (below the threshold 0).  TELECOMMUNICATION SCIENCES  VOLUME 4  NUMBER 2 JULY ─ DECEMBER 2013  20Observe that processes Χ, 	, 	 can be considered as a solution of two-sided Skorokhod’s problem. Let us recall the corresponding definition. Definition [4]. Let  be a function, 0 ∈ [0,]. We say that functions , 	, 	 satisfy two-sided Skorokhod’s reflection problem with reflections at 0 and m if 1)  = t + 	 − 	,  ≥ 0; 2) 	0 = 	 = 0, 		 and 	 are non-decreasing; 3) 		 and 	 may increase only when  equals 0 or m, respectively, i.e.  1		()+  1		() = 0,	 ≥ 0, 4)  ∈ 0,,  ≥ 0. It is known [5] that there is a unique solution of the Skorokhod’s problem (functions , 	,	 are un-known) for a given function . This solution in some sense depends continuously on . In our model:  = Χλ,µ0 + t − t,  = Χλ,µ Let us assume that , , are large enough in such a way that   = 	 + √ + (√),  = 	√ + (√), 	Χλ,μ0 = √ + (√),	 → +∞  (2) By invariance principle [6] a process 	√ con-verges weakly to Brownian motion () as  → +∞. So Χλ,μ = Χλ,μ0 +  + √ − −  + √ +  + +ελ,μ + 	λ,μ − 	λ,μ,  (3) where  is a Brownian motion independent on , ελ,µ	√→ 0 as  → ∞. It follows from (3) that a process Υλ,µ = Χλ,µ	√  converges as  → ∞ to a solution of the following Skorokhod’s problem on [0,]: Υ = x + √2w − a + L − Lβ, where w is a Brownian motion.  Remark. A process  −  has the same distri-bution as √2w. Thus,  Χ, ≈ √Y; ,t ≈ √Lβ (4) Note that it is very easy to find a rejection probabil-ity for Χλ,µ [3]: π =(/)1 + /+. . . +(/) =/(1 − /)1 − (/)  It follows from (2) that π ∼aeβ1 − eβ1√ =aeβ − 11√ ,  → ∞ (5) If we consider more complicate model than М/М/1/m, then it could be very difficult to find the ex-plicit formula for rejection probability. However, the invariance principle holds true under very insignificant assumptions. For example, there is no need to assume the exponential distribution between arrivals (or pro-cessing) as it is for a Poisson process. It is sufficient to suppose that these times are i.i.d. with finite expecta-tion. We may also assume that each arrival time a group of requests arrive (the second moment of a group should be finite). The collective processing is permitted too. In any case a limit will be of the form Υ = x + w − a + L − Lβ (6) Remark. For the М/М/1/m model a constant  equals √2, but for a general model, (say for a group arrivals) a constant  can be arbitrary. It is much easier to investigate a simple continuous model (6), then a very general discrete model. The re-jection probability for a discrete model equals lim→∞λ,µ	. This approximately equals lim→∞β	√. So, let’s calculate this limit for a general model (6). It can be proved [7] that lim→∞()= lim→∞()=(), where  , ∈ [0,) is a stationary density for a process Υ. It can be found from the Fokker-Planck-Kolmogorov equation [8] for Υ: π′′  + π′  = 0. So, π  = ,  ∈ [0,];   (7) and lim→∞β	== .     (8) This completely agrees with (5), (4), when  = √2. It can be seen that that rejection probabilities for ex-treme case of discrete Markov process (5) and continu-A. PYLYPENKO, L. URYVSKYI, B. TRACH: ASYMPTOTIC PROPERTIES OF SELF-SIMILAR TRAFFIC MODELS   21 ous Markov process (7) are the same. Similar results can be obtained for other characteristics of the process-es.  It can be inferred that in extreme case of discrete and continuous processes all output parameters of the model are described by the same equations. As a result, simpler continuous model can be used to calculate pa-rameters of network traffic in queuing systems with complex structure or traffic with priorities. Dependence between rejection probability and the main parameters has been investigated. The main pa-rameters are a  — coupling coefficient between λ  and µ , and b  — buffer scaling factor. As it is assumed that λ →∞ , changing λ  won’t have any noticeable effect on the process.   Fig. 1. Graphical dependence of rejection probability on the value of λ  for different values of a : 1a =  and 2a = , with parameters 1b = , 1m = .  The dependence of rejection probability on the arri-val intensity λ  for different values of coupling factor a  is shown on Figure 1. When the service probability in-creases, the rejection probability decreases. This behav-ior is consistent with physical sense of processes in the buffer and results of modeling for continuous and dis-creet Markov processes.   Conclusions  Our study of discrete and continuous Markov pro-cesses has shown that in extreme case of arrival intensi-ty λ →∞  and values of service intensity, which are of the same order of magnitude, the output parameters of the model (service, rejection probabilities, etc.) are de-scribed by the same equations.  The practical result of this work is the fact that a lot less complicated continuous queuing system model based on the Brownian motion can be used to calculate parameters of network traffic of complex queuing sys-tems. For example, in order to find state probabilities for queuing system featuring traffic with priorities using Poisson model, a large system of linear equations must be solved. Brownian model gives us ability to find these probabilities by solving one equation. We have conducted the research for rejection proba-bility. But other output parameters of performance and latency can be obtained in similar fashion. Also, more complex queuing systems can be considered (for exam-ple, with multiple servers). In addition to that, further research must be conducted with the use of fractional Brownian motion, which would give us ability to study properties of processes with long-range dependence. All of this will give us practical models of communication networks [9].  References  1. Will E. Leland, Murad S. Taqqu, Walter Willinger, Daniel V. Wilson: On the Self-Similar Nature of Ethernet Traffic. –  SIGCOMM, 1993. 2. O. Shelukhin, A. Tenyakshev, A. Osin Fractal process-es in telecommunications. – M.: Radio Engineering, 2003. – 480 p. 3. L. Klyaynrok. Queueing Theory. Translated from Eng-lish. – M.: “Engineering”, 1979 – 432 p. 4. A.V. Skorohod, Stochastic equations for diffusion pro-cesses in a bounded region 1, 2, Theor. Veroyatnost. i Prime-nen. 6 (1961), 264-274; 7 (1962), 3-23. 5. P. L. Lions, A. S. Sznitman. Stochastic differential equations with reflecting boundary conditions. Communica-tions on Pure and Applied Mathematics. Volume 37, Issue 4, (1984), 511–537.  6. Billingsley, P., Convergence of probability measures. –John Wiley & Sons, 1968. 7. Peter W. Glynn, Rob J. Wang, Central Limit Theorems and Large Deviations for Additive Functionals of Reflecting Diffusion Processes, arXiv:1307.1574 (2013). 8. V.I. Tikhonov, M.A. Mironov Markov processes. – M.: “Sov. Radio”, 1977. – 488 p. 9. L.A. Uryvskyy, B.V. Trach The queuing model in the queuing system using the method of penalties. “Info-communications – present and prospective”: Materials of the Second Intern. Science-pr. conf. of Young Scientists – Odes-sa ONAT, 2012, pp. 40-44.  Received in final form October 2, 2013  

Asymptotic properties of self-similar traffic models based on discrete-time and continuous-time martingales

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