Analytical modelling of systems with a ticket queue

A queuing system of MAP/GPH/N/K type as a model of a ticket queue is herein considered. It is assumed that arriving users, after receiving a service ticket (place in the queue), can leave the system with a probability based on the number of users in front of them if they find the queue too long. In addition, users may leave the system during waiting due to impatience. The system does not know about the presence (absence) of the called users for service and spends some time servicing them, even if the corresponding user has already left the system. The stationary distribution of the system under consideration is calculated. Formulas for finding the main characteristics of the system performance are given. The presented numerical experiment shows the possibility of using the results for optimisation purposes.