A service system where junior servers approach a senior server on behalf of customers

Abstract

Many service systems in various type of industry operate under a policy according to which customers are not allowed to directly approach the senior server of the system. If the senior server's intervention in the service is required, the junior server who handles the request approaches the senior server on behalf of the customer. Thus, the service is divided into two phases, where the first phase (e.g., obtaining information or performing an initial service) is conducted by one of the junior servers, after which the junior server approaches the senior server and they together complete the service. However, the senior server may be occupied by another junior server, in which case the junior server joins a queue of such servers who are waiting for the senior server to become available. That is, the servers of the first phase (junior servers) act as customers in the second phase. We formulate and analyze this stochastic queueing system as a two-dimensional quasi-birth-and-death (QBD) process and derive its steady-state probabilities using matrix geometric methods. By means of economic analysis, we provide a scheme for deriving an optimal work division policy, which dictates how much work should be assigned to each phase in order to minimize the system's overall cost or maximize the system's utility. The economic analysis is carried out with an eye to three practical examples: (i) a bank office model; (ii) an app development model; and (iii) a system subject to the constraint that a certain amount of work is assigned to the senior server. Sensitivity analyses are conducted for all three models.