Decay rate for a PH/M/2 queue with shortest queue discipline

Abstract

In this paper, we consider a PH/M/2 queue in which each server has its own queue and arriving customers join the shortest queue. For this model, it has been conjectured that the decay rate of the tail probabilities for the shortest queue length in the steady state is equal to the square of the decay rate for the queue length in the corresponding PH/M/2 model with a single queue. We prove this fact in the sense that the tail probabilities are asymptotically geometric when the difference of the queue sizes and the arrival phase are fixed. Our proof is based on the matrix analytic approach pioneered by Neuts and recent results on the decay rates.Keywords Shortest queue discipline · Decay rate · Stationary distribution · Phase type arrival · Two parallel queues · Exponential server · Matrix analytic approach · Markov additive process · Quasi-birth and death process AMS subject classifications: 60K25 . 60K20 . 60F10 . 90B22

Light tail asymptotics in multidimensional reflecting processes for queueing networks

“…Even a finite background space is useful. There are some studied in this direction for d = 2 (see, e.g., Fujimoto et al 1998;Katou et al 2008;Sakuma et al 2006) for the so-called generalized Jackson network, which replaces the Poisson arrivals by the renewal arrivals and allows service times to be generally distributed. No satisfactory answer has been obtained, but Ozawa (2011) very recently solved this problem in a certain way using the framework presented in Sect.…”

“…Thus, the tail asymptotics have been studied. It has a long history from starting from Kingman (1961), but satisfactory answers are only available for two parallel queues (e.g., see Foley and McDonald 2001;Kurkova and Suhov 2003;Li et al 2007;Sakuma et al 2006 andTakahashi et al 2001). The large deviations principle is derived for the stationary distributions of joint queue lengths under very general assumption for general d ≥ 2 in Puhalskii and Vladimirov (2007).…”

Geometric decay in level-expanding QBD models

“…According to the standard result on the decay rate of the QBD process with infinitely many background states (e.g., see Sakuma et al 2006) under the conditions in (1) and (2) We now consider the decay rate in the direction of the original level. We convert the levelexpanding QBD process into a GI/G/1 type queue with infinitely many background states.…”

“…The redefined process (X t ,Ỹ t ) is again QBD type, if a −1 = c − 1 = c 1 = 0, for which the decay rate result for a QBD process (e.g., see Proposition 3.1 of Sakuma et al (2006)) can be applied.…”

“…We prove the main result on the decay rate by reformulating the level expanding QBD as a discrete time GI/GI/1 type queue with infinitely many background states and jumps of size at most two. Then, we apply the condition obtained in Miyazawa and Zhao (2004) or Sakuma et al (2006), and in Li et al (2007) to this GI/GI/1 type queue for the positive recurrent case, and for the case not necessarily positive recurrent, respectively. To successfully execute this idea, we show that the decay rate is not affected by perturbations in a finite number of transition probabilities for the original model.…”