A stochastic analysis of goods allocation by queuing and the prevention of violence

When demand for a good exceeds its supply, queuing mechanisms are commonly used to allocate the good in question to citizens. However, very long queues result in excessive wait times and this can lead to violence. As such, the purpose of this paper is to analyze two stochastic models of goods allocation with queuing and the possibility of violence. In the first model, there is no capacity constraint. Using this model, we compute the long run delay per citizen in allocating the pertinent good. Next, we discuss the computation of the equilibrium probabilities for our discrete-time Markov chain theoretic model. In the second model, we capture the violence aspect of the underlying story explicitly with a capacity constraint. Then, we compute the long run fraction of citizens who are not provided the relevant good and the long run fraction of time the good allocating public official is busy.