About Analysis and Modelling of the Open Message Switching System

The modern queueing theory is one of the powerful tools for a quantitative and qualitative analysis of communication systems, computer networks, transportation systems, and many other technical systems. The paper is designated to the analysis of queueing systems, arising in the networks theory and communications theory (called open queueing network). The authors of this research in the sphere of queueing theory present the theorem about the law of the iterated logarithm (LIL) for the queue length of a customers in open queueing network and its application to the mathematical model of the open message switching system. Keywords—models of information systems, open message switching system, open queueing network, queue length of a customers, heavy traffic, a law of the iterated logarithm. I. PROBLEM FORMULATION At first the authors continues the research in queueing theory about LIL in queueing systems and presents theorem for the queue length of a customers in open queueing network. We note that papers on the queue length in heavy traffic for a multiphase queue and open queueing network are sparse. Kogan et al [6], [7] investigated tandem queues with blocking in heavy traffic. In Minkevičius [8], functional limit theorems for the queue length of customers in multiphase queues in various conditions of heavy traffic are proved. [3] surveyed some results on the queue length in multiphase queues. [9] presented laws of the iterated logarithm for a total queue length of customers and a queue length of customers in multiphase queues. In [10], functional limit theorems on complex transient phenomena for a queue length in multiphase queues are proved. Knessl et al [4], [5] presented diffusion approximations for a two-node tandem queue with general renewal process input and independent exponential service times at each node. Note that the research of the LIL in more general systems than the queueing system GI/G/1 or multiphase queueing systems and open queueing network has just started (see [1]). In [8], [9], the LIL is proved in heavy traffic for the queue length of customers, waiting time of a customer, a virtual waiting time of a customer in a multiphase queueing system. In [8], [9], [12], also gives the proof of the theorem on the S. Minkevičius is with the Department of System Analysis, Institute of Mathematics and Informatics, Akademijos 4, 08663 Vilnius, Lithuania and Vilnius Gediminas Technical University, Sauletekio 11, 10223 Vilnius, Lithuania e-mail: ([email protected]). Genadijus Kulvietis is with Department of Information Technology, Vilnius Gediminas Technical University, Sauletekio 11, 10223 Vilnius, Lithuania email: ([email protected]). Manuscript received September 20, 2010. LIL under the conditions of heavy traffic for a virtual waiting time of a customer in the open Jackson network. In this paper, we investigated an open queueing network model in heavy traffic. The LIL for the queue length of customers in an open queueing network has been presented. The main tool for the analysis of these queueing systems in heavy traffic is a functional LIL for renewal process (the proof can be found in [13] and [2]). The service discipline is ”first come, first served” (FCFS). We consider open queueing networks with the FCFS service discipline at each station and general distributions of interarrival and service times. The queueing network we studied has k single server stations, each of which has an associated infinite capacity waiting room. Every station has an arrival stream from outside the network, and the arrival streams are assumed to be mutually independent renewal processes. Customers are served in the order of arrival and after service they are randomly routed to either another station in the network, or out of the network entirely. Service times and routing decisions form mutually independent sequences of independent identically distributed random variables. The basic components of the queueing network are arrival processes, service processes, and routing processes. In particular, there are mutually independent sequences of independent identically distributed random variables { z (j) n , n ≥ 1 } , { S (j) n , n ≥ 1 } and { Φ n , n ≥ 1 } for j = 1, 2, . . . , k; defined on the probability space. Random variables z n and S (j) n are strictly positive, and Φ n have support in {0, 1, 2, . . . , k}. We define μj = ( M [ S (j) n ])−1 > 0, σj = D ( S (j) n ) > 0 and λj = ( M [ z (j) n ])−1 > 0, aj = D ( z (j) n ) > 0, j = 1, 2, ..., k; with all of these terms assumed finite. Denote pij = P ( Φ n = j ) > 0, j = 1, 2, . . . , k. In the context of the queueing network, the random variables z n function as interarrival times (from outside the network) at the station j, while S n is the nth service time at the station j, and Φ (j) n is a routing indicator for the nth customer served at the station j. If Φ n = j (which occurs with probability pij), then the nth customer served at the station i is routed to the station j. When Φ n = 0, the associated customer leaves the network. The matrix P is called a routing matrix. First let us define Qj(t) as the queue length of customers at the jth station of the queueing network in time t; σ̂ j =