Asymptotically Optimal Solution for TSF-Constrained Staffing Problem

Finding the right balance between the customer service and personnel expenditure has been an important task both in practice and academia and in academia this problem was usually approached using a total cost problem. However, it is difficult to directly apply this formulation in practice and hence recent works in queueing systems have investigated this problem with staffing minimization with constraints on quality-of-service constraints. Our work contributes on this stream of works by studying TSF measure, which shows the percentage of customers who waited beyond the prespecified threshold. TSF measures the proportion of customers whose waiting time exceed a certain threshold. It is one the most prevalent kind of quality of service measure, along with ASA, which measures the average of waiting time. TSF measure is intuitive to understand and hence widely been used but has not received sufficient attention in academia. Part of the reason comes from the unnatural property of TSF measure. Once a customer waits more than the threshold, the incentive to serve that customer starkly diminishes since decreasing the eventual waiting time of him does not enhance the performance of the system. Hence a policy that is far from FIFO turns out to be more efficient. Especially we are interested in context where service level differentiation among various classes of customers. Hence, we work on V-model where multiple classes of customer and homogeneous pool of servers exist. Our task is to come up with the minimum possible number of servers while maintaining TSF measure on the pre-defined level. The decision variables are two-kind: number of servers and prioritization policy. The prioritization policy defines which class of customer to server first when a server becomes available. The nature of TSF measure, which tends to give lower priority to the customers who had already waited beyond the threshold, makes the problem more difficult to solve and hence it was usually treated in literature more added conditions or constraints added. We devise an optimal solution of TSF constrained problem without adding more structures. Since the problem is difficult to solve in exact analysis, we apply the heavy traffic technique to solve the problem. First, we show that our suggested staffing level is the lower bound for all the feasible solutions. Second, it is proved that the proposed prioritization policy combined with our staffing level actually is feasible and hence asymptotically optimal.