Analyticity of single-server queues in light traffic

Recently, several methods have been proposed to approximate performance measures of queueing systems based on their light traffic derivatives, e.g., the MacLaurin expansion, the Padé approximation, and interpolation with heavy traffic limits. The key condition required in all these approximations is that the performance measures be analytic when the arrival rates equal to zero. In this paper, the authors study the GI/G/1 queue. They show that if the c.d.f. of the interarrival time can be expressed as a MacLaurin series over [0,•), then the mean steady-state system time of a job is indeed analytic when the arrival rate to the queue equals to zero. This condition is satisfied by phase-type distributions but not c.d.f.’s without support [0,•), such as uniform and shifted exponential distributions. In fact, the authors show through two examples that the analyticity does not hold for most commonly used distribution functions which do not satisfy this condition.