A Queueing System with Two Parallel Lines, Cost-Conscious Customers, and Jockeying

We consider a two-server queueing system with the special feature of cost-conscious and jockeying customers. It is assumed that a new arriving customer joins one of the two queues only if the associated costs for him do not exceed a given upper threshold c. These costs are given by a linear combination of the current queue length at the chosen server and a service fee. Thus, if N i are the current queue lengths, β i > 0 are the service fees and α i > 0 are certain prespecified weights for the two servers (i ∈ {1, 2}), the customer enters the system if and only if min [α1 N 1 + β1, α2 N 2 + β2] ≤ c, and in this case he goes to server 1 or 2 depending on which of the two terms is smaller; if equality occurs, he selects each server with probability 1/2. A customer waiting in line jockeys if, due to a departure from the other line during his waiting time, his residual costs become larger than the costs he will have after switching to the other server. In this model, new customers do not necessarily join the shortest queue (if they enter the system at all) and customers may jockey to a longer queue (if such a move reduces residual costs). Under the assumption that the arrival times form a Poisson process and the service times are exponential we derive the stationary distribution of the two-dimensional Markov process of queue lengths.