Transient Analysis of a Single-Channel Service System Operation with Positive and Negative Messages

In the paper, a transient analysis of a single-channel service system with independent Poisson input flows of positive and negative messages is presented. Only positive messages are subject to the service and accumulation process in a finite capacity buffer. Each time a negative message appears, a positive message is removed from the queue waiting to be serviced. Assuming that the service time of a single positive message is exponentially distributed, a system of integral equations is derived for the transient conditional distribution of the number of positive messages present in the system, where the condition is defined by the number of positive messages present in the system at the time of its opening. Using linear algebra and matrix calculus, an explicit-form solution is obtained for the corresponding system of equations written for Laplace transforms. The representation for the stationary queue-size distribution is obtained via the Tauberian theorem. Some performance measures, namely the mean queue size and the mean sojourn time, are also found. The numerical analysis of the sensitivity of the transient conditional queue size distribution to changes in the input parameters describing the system is attached.