A constrained access repairman problem

A variant of the M/M/2/3/3 queueing system is considered in which one of the repairmen can limit the access of the other to a machine in need of repair. A practical application is to a system of three machines arranged in a row and serviced by two cranes which ride a shared track. If while the left crane is servicing the middle machine, the leftmost machine becomes ready for service, the right crane, even though it may be idle, will not be able to move into position to supply service. The system thus constrained is shown to have a total of 13 unique states comprising various combinations of crane assignments and machine states. The state probabilities are shown to satisfy a system of linear equations whose solution depends only on the ratio u of the mean service time and the mean time until a serviced machine next requires service. The queueing performance characteristics are computed for various values of u. They are shown to be intermediate to those of 1) the system in which a repairman is dedicated exclusively to one machine while the second shares the servicing of the other two, and 2) the three machine/two repairmen system with unrestricted repairman access.