minimise the average delay experienced by a customer. The obvious way to do this is to route the customer to a queue with the least number of waiting customers. But, since there are a large number of queues, polling all of them and finding the ones with the least number of customers is expensive. So a simple alternative is to pick a queue at random and route the incoming customer to that queue, which is studied in [@graham-00]. It turns out that, if we pick two queues at random and route the customer to the least loaded queue between the two (with ties broken uniformly at random), the delay decreases dramatically. This algorithm demonstrates the power of two choices, and the evolution of the state of each queue under this algorithm can be described using the mean-field model which has been used to analyse the delay performance [@mitzhenmaker-00]. For related problems on load balancing in networks, see Mukhopadhyay et al. [@mukhopadhyay-etal-16] who study heterogenous servers, Aghajani et al. [@aghajani-etal-17; @aghajani-ramanan-19] who study non-Markovian queues, etc., and the references therein. Note that one important difference with our setting is that the state space of a queue is countably infinite in this 