Queueing with Negative Network Effects

Problem definition: In a Markovian queueing system with strategic customers, a reward is gained from completing service, and a loss is incurred while waiting to be served. The common assumption in the queueing literature is that such loss is a function of the customer's waiting time. This paper takes a different and novel approach in that it models the customer's loss incurred because of negative network effects while waiting with others, which increases as the exposure to others increases. Methodology: Waiting time is complemented by two innovative measures that capture negative effects on a tagged customer joining an M/M/c queue: the total number of customers the tagged customer meets and person-time exposure to these customers while waiting to be served. Threshold joining strategies inducing M/M/c/n-type queues are studied in this context. Results: The distributions of exposure size and exposure time of a customer joining the queue at a given position are analytically derived. Equilibria under conditions of no reneging are identified as threshold strategies. If the customer's loss function is concave (such as an exponential model for the chance of infection during a pandemic), there is an equilibrium threshold strategy under which customers do not renege from the queue, even if reneging is allowed. The price of anarchy caused by lack of coordination among the individuals acting is identified. Unlike the equilibrium threshold built under the restrictive assumption that all potential customers have the same utility function, a novel safe threshold concept is introduced, a queue size at which a customer who joins the facility and stays until completing service has positive expected utility regardless of the actions of the other customers. Managerial implications: The implications of negative network effects caused by congestion in a queueing system are of interest to queue managers and, in particular, affect the optimal size of the waiting area. Safe and equilibrium thresholds are contrasted with the socially optimal threshold set by a regulator, and the safe threshold is suggested as a managerial tool to design the waiting room size.