Head of Laboratory – Dr.Sc.(Math.) Valerii Polesskii

Tel: (095) 299-50-02; E-mail: poles@iitp.ru

The leading researchers of the laboratory include:

It is expected that in the third generation of CDMA mobile communication networks, which will be capable to transmit not only voice but also data and file messages, the traffic will be long-range dependent. For such traffic, it is important to find the following measures: the probability of heavy-traffic period of given length, the outage probability, the level crossing probability, and the average length of heavy traffic interval. These measures can be expressed via the multi-dimensional probability distribution of traffic.

B. S. Tsybakov suggested a method for finding the multi-dimensional probability distribution of a model of long-range dependence call traffic. The method allows to calculate the measures mentioned above. He showed that the level crossing probability depends on the average call length only. Further, this probability for traffic with dependent samples is lower than for traffic with independent samples. Also, it is shown that there is a linear dependence between the average heavy traffic interval and the average call length. These and other related results were put down in a paper, which will appear in the international journal "Mobile Networks and Applications" (Baltzer Science Publishers in cooperation with ACM) in 2001.

Fractal model of the network traffic is today one of the most appropriate model for mathematical analysis of the network performance. From practical point of view it is important to find characteristics of this model and to make queueing system analysis for it. As one approach, N. B. Likhanov proposed sufficiently general model of network traffic and computed asymptotic packet loss probability for the network node operated under proposed model.

N. D. Vvedenskaya continued to work on large queueing systems with dynamic routing that are investigated asymptotically as number of servers in increasing tending to infinity. The so called fast Jackson-type networks were investigated. In these models the nodes (of Jackson network) contain a large number of servers. The task selects a server with the shortest queue out of few servers pre-selected randomly (the pre-selected servers may be from different nods). The limit network is of interest – it’s performance is presented by a dynamic system. Both the exponential and non-exponential service time distribution is considered. The later brings more difficulties that the exponential one. Some modification of multiple access stack algorithm was considered in case where the packets are of two priority types. It is shown that the throughput of the algorithm can be higher if packets of first priority type do not complete with the packets of second priority type. N. D. Vvedenskaya attended the conference "Mathematical Problems of Statistical Physics" in Armenia (August 26-September 2, 2000).

One of the principal trends in the network reliability theory is search of effectively computable bounds for the network reliability characteristics. V. P. Polessky continued his investigations on constructing these bounds. The connectedness probability of any random graph from the random graph class generated by 2-connected multigraphs with given number of edges and given values of the first and second components of their acyclic spectrums was evaluated. This effectively computable bound can be used in order to estimate the network reliability, if the network structure is a 2-connected graph with a small number of edges.

D. L. Belotserkovsky investigated diverse aspects of open theoretic problems of graph theory motivated by the following problem: how many the number of edges does a graph which has not a cycle of length at most a positive integer k contain? He proved the theorem asserting that graphs with vertex number unequal 11, 50, 30 having a maximum number of edges over all graphs which have not a cycle of length at most 6 contain a cycle of length 7. Also other proved assertions concern the properties of extremal graphs with a girth of an arbitrary length. Based on the theorem the paper "On same open problems of extremal graphs of high girth" was submitted to "Graph theory". Meanwhile, researches related to finding the smallest number of edges in a 2-connected graph with the restriction on the graph diameter have been continuing. The main results are presented in paper "The smallest number of edges in a 2-connected graph with specified diameter" which was submitted to "Discrete Mathematics".

I. P. Levshin considered the problem underwater communication and demonstrated that it includes a wide spectrums of the scientific fields like to the hydrodynamic, h 24tion for optimal estimation of ocean media.

I. A. Orlov analyzed computer programme languages and architectural special features of the 0.S.Windows 95. It was shown that for development of the simulation model of underwater sound channel information transmission on computers, supplied 0.S.Windows 95 and similar, Fortran Power Station ver. 4.0 and Visual C++ ver. 6.0 are the most proper programme languages.