Service capacity competition with peak arrivals and delay sensitive customers

We study capacity competition in a service environment where the arrival rates are highly seasonal (e.g., lunch time rushes) and customers are time sensitive, so may depart without receiving service if the waiting time is too long. As a stepping stone for the competitive model, we begin by studying a monopolist's capacity decision, where the key trade-off is between the cost of extra capacity for low demand periods and the loss of revenue for high demand periods; we provide an attractive rule of thumb for capacity decisions in this setting. We then study a duopoly model, where lost demand for one firm may become increased demand for the competitor. In both models we use a fluid model for the analysis, which allows us both to provide explicit insights into the trade-offs when setting capacity and to solve for the Nash equilibrium (when it exists) in the duopoly. The canonical environment we have in mind for our modeling is a food court, but any service environment where the peak arrival rate will likely exceed available capacity is similarly appropriate.