Join-the-Shortest Queue with Abandonment: Critically Loaded and Heavily Overloaded Regimes

We consider a load-balancing system composed of a fixed number of singleserver queues operating under the well-known join-the-shortest queue policy and where jobs/customers are impatient and abandon if they do not receive service after some (random) amount of time. In this setting, we characterize the centered and appropriately scaled steady-state queue-length distribution (hereafter referred to as limiting distribution) in the limit as the abandonment rate goes to zero at the same time as the load either converges to one or is larger than one. Depending on the arrival, service, and abandonment rates, we observe three different regimes of operation that yield three different limiting distributions. The first regime is when the system is underloaded, and its load converges relatively slowly to one. In this case, abandonments do not affect the limiting distribution, and we obtain the same exponential distribution as in the system without abandonments. When the load converges to one faster, we have the second regime, where abandonments become significant. Here, the system undergoes a phase transition, and the limiting distribution is a truncated Gaussian. Further, the third regime is when the system is heavily overloaded, and so, the queue lengths are very large. In this case, we show that the limiting distribution converges to a normal distribution. To establish our results, we first prove a weaker form of state space collapse by providing a uniform bound on the second moment of the (unscaled) perpendicular component of the queue lengths, which shows that the system behaves like a single-server queue. We then use exponential Lyapunov functions to characterize the limiting distribution of the steady-state queue-length vector.