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We consider a load balancing system comprised 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 receiveservice after some (random) amount of time. In this setting, we characterizethe centered and appropriately scaled steady state queue length distribution(hereafter referred to as limiting distribution), in the limit as theabandonment rate goes to zero at the same time as the load either converges toone or is larger than one.  Depending on the arrival, service, and abandonment rates, we observe threedifferent regimes of operation that yield three different limitingdistributions. The first regime is when the system is underloaded and its loadconverges relatively slowly to one. In this case, abandonments do not affectthe limiting distribution, and we obtain the same exponential distribution asin the system without abandonments. When the load converges to one faster, wehave the second regime, where abandonments become significant. Here, the systemundergoes a phase transition, and the limiting distribution is a truncatedGaussian. Further, the third regime is when the system is heavily overloaded,and so the queue lengths are very large. In this case, we show the limitingdistribution converges to a normal distribution.  To establish our results, we first prove a weaker form of State SpaceCollapse by providing a uniform bound on the second moment of the (unscaled)perpendicular component of the queue lengths, which shows that the systembehaves like a single server queue. We then use exponential Lyapunov functionsto characterize the limiting distribution of the steady state queue lengthvector.

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