Exponential Tail Bounds on Queues

A popular approach to computing performance measures of queueing systems (such as delay and queue length) is studying the system in an asymptotic regime. However, these results are only valid in the limit and often provide bounds for the pre-limit systems that are not optimized and, hence, give loose bounds for the tail probabilities. In this paper, we provide optimized bounds for the tail probabilities of the scaled total queue length in a load-balancing system under Join the Shortest Queue (JSQ). Our bounds characterize the rate of convergence of the tail probabilities to the corresponding heavy traffic values. For the tail probability of the JSQ system, our bounds yield a multiplicative error that arises from three factors: pre-limit tail, pre-exponent error, and State-Space Collapse (SSC). As an immediate corollary of our main theorem, we provide a bound to the tail probabilities of a single-server queue. In this case, the multiplicative error only consists of pre-limit tail and pre-exponent error, since there is no state-space collapse.