Analysis of the shortest relay queue policy in a cooperative random access network with collisions

Abstract

The scope of this work is twofold: On the one hand, strongly motivated by emerging engineering issues in multiple access communication systems, we investigate the performance of a slotted-time relay-assisted cooperative random access wireless network with collisions and with join the shortest queue relay-routing protocol. For this model, we investigate the stability condition, and apply different methods to derive the joint equilibrium distribution of the queue lengths. On the other hand, using the cooperative communication system as a vehicle for illustration, we investigate and compare three different approaches for this type of multi-dimensional stochastic processes, namely the compensation approach, the power series algorithm (PSA), and the probability generating function (PGF) approach. We present an extensive numerical comparison of the compensation approach and PSA, and discuss which method performs better in terms of accuracy and computation time. We also provide details on how to compute the PGF in terms of a solution of a Riemann-Hilbert boundary value problem.Cooperative communication is a new communication paradigm in which different terminals (i.e., nodes, devices) in a wireless network share their antennas and resources for distributed transmission and processing. Recent studies have shown that cooperative communications yield significant performance improvements for 5G networks, which need massive uncoordinated access, low latency, energy efficiency and ultra-reliability [45].The unprecedented growth of wireless networking, and the ever growing demand for higher data rates and capacity over the last decades, have already pushed the limits of current cellular systems [57]. By exploiting the spatial diversity inherent to wireless channels, which is an important tool to overcome the effects of fading (decrease in signal power due to path loss), shadowing and attenuation (decrease in signal strength), relay-based cooperative communications have been proposed as the appropriate solution to achieve the requirements of future needs; see e.g., [31,37]. * m.mayank@tue.nl † idimit@math.upatras.gr ‡ s.kapodistria@tue.nl arXiv:1810.00754v1 [math.PR] 1 Oct 2018 ContributionApplication oriented contribution. In this work, we consider a slotted-time relay-assisted cooperative communication system with a JSQ routing protocol at the relays and collisions. Such models have not been extensively analysed and very little is known regarding their delay performance. In particular, we provide insights on the characterisation of the delay and the performance of the system at hand. This is a particularly difficult task even in small scale systems, due to the strong interdependence/interaction between the queues. Furthermore, we demonstrate how to obtain the equilibrium distribution of the joint queue lengths and present an extensive numerical comparison of the techniques implemented, discussing which performs better in terms of accuracy and time complexity (computational time).

Analysis of the Symmetric Join the Shortest Orbit Queue

“…To conclude, the series in (25) is the unique up to a multiplicative constant, solution of the balance equations ( 1)-( 6). The following Theorem states the main result of this section:…”

“…The next task is to show that formal solution in (25) converges absolutely. To show that, we need some preliminary results.…”

“…In the following, we focus on the asymptotic behaviour of δ i /γ i and γ i+1 /δ i . This result is important to investigate the convergence of the series in (25). Lemma 7 a) Let γ i fixed and δ i the root in (14)…”

“…In this section we provide a numerical example using (25). In particular, in Table 1 we list the probabilities q 0,n := q 0,n (0) + q 0,n (1), n = 0, 1, 2, 3 that are computed with a precision of 10 −10 .…”

“…The queueing system in [2] is described by a multilayer random walk in the quarter plane. For other related important works, see also [1,9,26,25]. Alternatively, numerical/approximation methods were applied.…”