Some analytical results for congestion in subscriber line modules

In modern telephone exchanges, subscriber lines are usually connected to the so-called subscriber line modules. These modules serve both incoming and outgoing traffic. An important difference between these two types of calls lies in the fact that in the case of blocking due to all channels busy in the module, outgoing calls can be queued whereas incoming calls get busy signal and must be re-initiated in order to establish the required connection. The corresponding queueing model was discussed recently by Lederman, but only the model with losses has been studied analytically. In the present contribution, we study the model which takes into account subscriber retrials and investigate some of its properties such as existence of stationary regime, derive explicit formulas for the system characteristics, limit theorems for systems under high repetition intensity of blocked calls and limit theorems for systems under heavy traffic.