Exact tail asymptotics: revisit of a retrial queue with two input streams and two orbits

Abstract

We revisit a single-server retrial queue with two independent Poisson streams (corresponding to two types of customers) and two orbits. The size of each orbit is infinite. The exponential server (with a rate independent of the type of customers) can hold at most one customer at a time and there is no waiting room. Upon arrival, if a type i customer (i = 1, 2) finds a busy server, it will join the type i orbit. After an exponential time with a constant (retrial) rate µ i , an type i customer attempts to get service. This model has been recently studied by Avrachenkov, Nain and Yechiali [3] by solving a Riemann-Hilbert boundary value problem. One may notice that, this model is not a random walk in the quarter plane. Instead, it can be viewed as a random walk in the quarter plane modulated by a two-state Markov chain, or a two-dimensional quasi-birth-and-death (QBD) process. The special structure of this chain allows us to deal with the fundamental form corresponding to one state of the chain at a time, and therefore it can be studied through a boundary value problem. Inspired by this fact, in this paper, we focus on the tail asymptotic behaviour of the stationary joint probability distribution of the two orbits with either an idle or busy server by using the kernel method, a different one that does not require a full determination of the unknown generating function. To take advantage of existing literature results on the kernel method, we identify a censored random walk, which is an usual walk in the quarter plane. This technique can also be used for other random walks modulated by a finite-state Markov chain with a similar structure property.Keywords: Retrial queue · Random walks in the quarter plane · Random walks in the quarter plane modulated by a finite-state Markov chain · Censored Markov 1 exponential rate for the customers of type i. Such a queueing system could serve as a model for two competing job streams in a carrier sensing multiple access system, and it has an application in a local area computer network (LAN) as explained in [3].Retrial queueing systems have been attracting researchers' attention for many years (e.g., [1,2,5,25] and references therein). Much of the previous work lays the emphasis on performance measures, such as the mean size of the orbit, the average number of the customers in the system, the average waiting time among others. We also notice that stationary tail asymptotic analysis has recently become one of the central research topics for retrial queues due to not only its own importance, but also its applications in approximation and performance bounds. For example, in [24], Shang, Liu and Li proved that the stationary queue length of the M/G/1 retrial queue has a subexponential tail if the queue length of the corresponding M/G/1 queue has a tail of the same type. Kim, Kim and Kim extended the study on the M/G/1 retrial queue in [11] by Kim, Kim and Ko to a M AP/G/1 retrial queue, and obtained tail asymptotics for the queue size distribution in [12]. By adopting matrix-analytic th...

A Two-Class Retrial System With Coupled Orbit Queues

“…The dominant singularity of π (1) 1 (x) is either a branch point of the kernel equation k (1) (x, y) = 0 or a pole of the function π (1) 1 (x) [42,43]. Clearly, k (1) (x, y) = −αK(x, y), and thus, the branch points of k (1) (x, y) = 0 coincides with the branch points of K(x, y) = 0, thoroughly investigated in Section 3.2 (see also [52] pp. [10][11].…”

“…Following the discussion in [52], p. 17, it can be seen that the Tauberian-like theorem can be applied for the marginal probability π (1) i :…”

“…As indicated above, our model it is seen as a RWQP modulated by a finite two-state Markov process. In the following, using the special structure of our model we will convert it, using the approach in [52], to a usual RWQP [30] and investigate its stability condition.…”

“…For more properties on the zero pairs of the kernel the interested reader is referred to [7] (Proposition 3, p. 14), and [52] (Lemmas 4.1, 4.2, pp. 10-11).…”

“…In [42,43], the kernel method was applied to study the exact tail asymptotic properties for random walks in the quarter plane. Song, Liu, and Zhao [52] performed asymptotic analysis for the model in [7]. In this section, we fill the gap and provide asymptotic results for a retrial model with two input streams and coupled orbit queues, based on the random walk approach presented in Section 3.1.…”