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...

Existing approaches to reducing the low-frequency noise exposure of dwellings are not always sufficient. This study investigated the significance of dwelling layout design for low-frequency noise control. The sound distribution in six typical Chinese dwelling layouts was analysed using in-situ measurements under steady-state noise of various low frequencies. The results showed that among two-bedroom dwelling layouts, the overall average noise reduction varied considerably (6 dB). The noise reduction for room levels (number of rooms sound crosses) 1–2 and 2–3 varies by 5 and 3 dB, respectively, and the noise reduction at door openings varies by 5 dB. A model to approximate the low-frequency noise reduction of a layout was developed using the polyline distance from the noise source and the number of walls the polyline has to cross, which were clearly shown to influence low-frequency noise reduction and seem to be the strongest investigated factors.

No abstract