Performance Analysis for a Two-stage Queuing System with Online Service

We consider a two-stage queueing system where the first stage represents an online channel (official website or apps) and the second stage represents a face-to-face service. There are two streams of customers, one is passionate to experience two stages of service and the other is only willing to go through the second stage. This kind of customer behavioral preference may be due to different personality or availability of the first stage service (for example, elder people may not be familiar with smartphone and thus cannot access the first stage service). We characterize the joining equilibrium of customers with different preference and investigate the impact of the online channel on system performance. We consider two measures to evaluate system performance: throughput and social welfare. We show that the impact of online channel on system performance is mixed: it may improve throughput and social welfare; it may also impair throughput and social welfare. In addition, we provide sufficient conditions under which service provider is better off or worse off by operating the online channel.