This paper presents an efficient way to calculate a good approximation to the blocking probability in a tree-shaped loss network consisting of exchanges and a server. The method applies to situations where the network supports a service generating traffic which has a non-stationary arrival process and creates a focused load to the server. Examples of such situations are mass calls, which appear in televoting and hot spots of certain application services in the internet. The situation appears also in a military command and control system if services are concentrated to service hubs. The traffic load is focused on the service hubs, strongly depends on the state of operation, and can create mass call type traffic.

This paper presents a blocking probability for a network where a loss system is fed by a number of exchanges or concentrators. We can treat exchanges and concentrators as loss systems where the service time is the call holding time for dimensioning purposes. An exchange in Public Switched Telephone Network (PSTN) is actually not a loss system but a combined loss and queueing system. Still, dimensioning can be made using the loss model. The same argument applies to Internet Protocol (IP) traffic. Even though an IP network is a queueing network, we can dimension services which require fixed capacity using a loss system network model.

If the loss system network is in a stationary state, blocking can be calculated by solving the state equations of the entire system. However, there are reasons to look for a simpler solution: if traffic volumes are strongly time-dependent, stationary state approaches are not valid; modeling the whole network is not precise since the network elements are in reality more complicated than simple loss or queueing systems; analytical methods are not always very precise because they use approximations; and typically there are not enough measurements to give input data to precise calculations. We present here a simple combinatorial method that does not make assumptions on the arrival process and does not require us to model the whole network.

The classical way to treat a flow through an exchange in Syski (1960) Chapter 10 is also a combinatorial solution but it assumes that the arrival process is Poissonian. We cannot assume to have a Poisson arrival process in a very dynamic
situation, such as televoting, hot spot, or a currently interesting case of a military network with centralized servers. At the beginning of a military operation, this type of a network is likely to generate mass call traffic. At the most, the arrival process could be a non-homogeneous Poisson process. Palm’s theorem (1987) and its extensions (Carrillo, 1991) give reasons why the arrival often should be non-homogeneous Poisson process, and some measurements of internet traffic indicate that dial-up traffic is almost Poissonian (Iversen et al., 2000), and IP-level traffic may also be seen this way (Karagiannis et al., 2004), instead of as self-similar traffic. Still, traffic need not be non-homogeneous Poissonian. We will use the assumption of non-homogeneous Poisson arrivals as little as possible in the combinatorial method that is derived.

Telecommunications Journal, Internet Protocol, Public Switched Telephone Network, Military Networks, Non Homogeneous Poisson Arrival Process, Discrete-time Analysis, Time-Dependent Poisson Process, Discrete Recursion Equations, Loss Systems, TCP/IP Technology, Non-stationary Poisson Process.