A Fluid Model for an Overloaded Bipartite Queueing System with Heterogeneous Matching Utility

We consider a bipartite queueing system (BQS) with multiple types of servers and customers, where different customer-server combinations may generate different utilities. Whenever a server is available, it serves the customer with the highest index, which is the sum of a customer's waiting index and the matching index. We call this an {\em M W} index. We assume that the waiting index is an increasing function of a customer's waiting time and the matching index depends on both the customer's and the server's types. We develop a fluid model to approximate the behavior of such a BQS system, and show that the fluid limit process can be computed over any finite horizon. We develop an efficient algorithm to check whether a steady state of the fluid process exists or not. When a steady state exists, the algorithm also computes one efficiently. We prove that there can be at most one steady state, and that the fluid limit process converges to the steady state under mild conditions. These results enable a policy designer to predict the behavior of a BQS when using an M W index, and to choose an indexing formula that optimizes a given set of performance metrics. We derive a closed-form M W index that optimizes the steady-state performance according to some well-known efficiency and fairness metrics