Trade-off between inventory and repair capacity in spare part networks

The availability of repairable technical systems depends on the availability of (repairable) spare parts, to be influenced by (1) inventory levels and (2) repair capacity. In this paper, we present a procedure for simultaneous optimisation of these two factors. Our method is based on a modification of the well-known VARI-METRIC procedure for determining near-optimal spare part inventory levels and results for multi-class, multi-server queuing systems representing repair shops. The modification is required to avoid non-convexity problems in the optimisation procedure. To include part-time and overtime working, we allow for a non-integer repair capacity. To this end, we develop a simple approximation for queuing systems with a non-integer number of servers. Our computational experiments show that the near-optimal utilisation rate of the repair servers is usually high (0.80–0.98) and depends mainly on the relative price of the servers compared with inventory items. Further, the size of the repair shop (the minimal number of servers required for a stable system) plays its part. We also show that our optimisation procedure is robust for the choice of the step size for the server capacity.     

On Markovian Multi-Class,Multi-Server Queueing

Multi-class multi-server queueing problems are a generalisation of the well-known M/M/k queue to arrival processes with clients of N types that require exponentially distributed service with different average service times. In this paper, we give a procedure to construct exact solutions of the stationary state equations using the special structure of these equations. Essential in this procedure is the reduction of a part of the problem to a backward second order difference equation with constant coefficients. It follows that the exact solution can be found by eigenmode decomposition. In general eigenmodes do not have a simple product structure as one might expect intuitively. Further, using the exact solution, all kinds of interesting performance measures can be computed and compared with heuristic approximations (insofar available in the literature). We provide some new approximations based on special multiplicative eigenmodes, including the dominant mode in the heavy traffic limit. We illustrate our methods with numerical results. It turns out that our approximation method is better for higher moments than some other approximations known in the literature. Moreover, we demonstrate that our theory is useful to applications where correlation between items plays a role, such as spare parts management.