Queueing with Negative Network Effects

The usual negative effect of queue size on arrival to an M/M/c service facility is augmented, in this paper, by the negative network effect due to waiting with others, which increases as the exposure to others increases. The reward and cost structure acquires a new meaning, as the cost of exposure to later arrivals needs to be considered, on top of those already in the queue on arrival. Threshold joining strategies inducing M/M/c/n-type queues are studied in this context. The distributions of exposure size and exposure time of a customer joining the queue at a given position, are analytically derived. Equilibria under no reneging for a class of utility functions that grant a reward for completing service and charge a penalty that increases with the exposure to others in the queue in terms of their number or the exposure time, are identified as threshold strategies. If the penalty function is concave (such as an exponential model for the chance of infection during a pandemic), customers should not renege from the queue, even if allowed. The equilibrium threshold is contrasted with the socially-optimal threshold set by a regulator as the queue capacity which maximizes the social welfare in an M/M/c facility without reneging. A comparison between these two thresholds determines the toll imposed on society if customers are allowed into the facility based on their individual utility, with respect to regulated thresholds. We further introduce a novel threshold concept, the safe threshold, a queue size at which a customer joining the M/M/c facility and staying until completing service, has positive expected utility regardless of what other customers do