Some first passage time problems for the shortest queue model

We consider the symmetric shortest queue (SQ) problem. Here we have a Poisson arrival stream of rate λ feeding two parallel queues, each having an exponential server that works at rate μ. An arrival joins the shorter of the two queues; if both are of equal length the arrival joins either with probability 1/2. We consider the first passage time until one of the queues reaches the value m ₀, and also the time until both reach this level. We give explicit expressions for the first two first passage moments, conditioned on the initial queue lengths, and also the full first passage distribution. We also give some asymptotic results for m ₀→∞ and various values of ρ=λ/μ. H. Yao work was partially supported by PSC-CUNY Research Award 68751-0037. C. Knessl work was supported in part by NSF grants DMS 02-02815 and DMS 05-03745.