A special discrete cyclic-waiting queuing system

Within the light of technological applications those type of queuing systems play a significant role, where the service time of entering entities cannot take up any value, it can only be multiples of a certain cycle–time. As an example to this mechanism landing of aeroplanes and optical buffers of internet networks can be mentioned. In this case the service of an entering customer can be started immediately, or in case of a busy server or waiting customers it joins a queue, so that it keeps moving along a closed path which can be completed within T units of time. Applications in digital technology induce the investigation of discrete systems. We give the mathematical description of systems serving two types of customers, where inter–arrival and service times are geometrically distributed. A Markov–chain is defined, generating functions of transition probabilites are calculated, as well as condition of ergodicity is established and equilibrium distribution is given. At last, the mean value of queue–length is given as a function of the input parameters.