A Stability Analysis of Randomized Scheduling Algorithm for Input-Queued Switches

The delay and throughput characteristics of a packet switch depend mainly on the queueing scheme and the scheduling algorithm deployed at the switch. Early research on scheduling algorithms has mainly focused on maximum weight matching scheduling algorithms. It is known that maximum weight matching algorithms guarantee the stability of input-queued switches, but are impractical due to their high computational complexity. Later research showed that the less complex maximal matching algorithms can stabilize input-queued switches when they are deployed with a speed-up of two. For practical purposes, neither a high computational complexity nor a speed-up of two is desirable.In this paper, we investigate the application of matching algorithms that approximate maximum weight matching algorithms to scheduling problems. We show that while having a low computational complexity, they guarantee the stability of input queued switches when they are deployed with a moderate speed-up.In particular, we show that the improve_matching algorithm stabilizes input-queued switches when it is deployed with a speed-up of 3<over>2+ε.In a second step, we further improve on these results by proposing a class of maximal weight matching algorithms that stabilize an input-queued switch without any speed-up.Whereas initial research has only focused on scheduling algorithms that guarantee the stability of a single switch, recent work has shown how scheduling algorithms for single switches can be modified in order to design distributed scheduling algorithms that stabilize networks of input-queued switches. Using those results, we show that the switching algorithms proposed in this paper do not only stabilize a single switch, but also networks of input-queued switches.

Scheduling algorithms for input-queued packet switches have been widely researched. It has been shown that various classes of scheduling algorithms provide guarantees on stability and on average delay for single switches. However, recent research has demonstrated that most of these scheduling algorithms do not guarantee stability for networks of switches. Most of the research that treats networks of switches proposes switching policies that require coordination among the single switches. The problem to find <i>distributed</i> scheduling policies that guarantee the stability of a network of switches has so far only been investigated for a policy based on a computationally very complex maximum weight matching algorithm. In this paper, we investigate <i>distributed</i> scheduling algorithms of <i>low complexity</i> that belong to the classes of maximal weight matching algorithms, <i>p</i>-maximal weight matching algorithms for switch architectures based on a space-division multiplexing extension, and MNCM algorithms. For these scheduling algorithms, we prove the stability of networks of input-queued switches where all switches deploy the same scheduling policy. We also show that networks of input-queued switches in which specific classes of different switching policies are deployed simultaneously at different switches are stable.

Random unitary beamforming is one of the schemes that can reduce the amount of feedback information in multiuser diversity techniques with multiple-antenna downlink transmission. In Multiple-Input Multiple-Output (MIMO) systems, throughput performance is greatly improved using AMC (Adaptive Modulation and Coding). Throughput performance is also improved by allocating power among streams appropriately. In random unitary beamforming, the transmitter has only partial channel state information (CSI) of each receiver. Thus, it is difficult for random unitary beamforming to use conventional power allocation methods that assumes that all receivers has full CSI. In this paper, we propose a new scheduling algorithm with power allocation for downlink random unitary beamforming that improves throughput performance without full CSI. We provide numerical results of the proposed scheduling algorithm and compare them to those of the conventional random unitary beamforming scheduling algorithm. We show that random unitary beamforming achieves the best system throughput performance with two transmit antennas. We also show that the proposed algorithm attains higher throughput with the small increase of feedback than the random unitary beamforming scheduling algorithm.

Currently the research on input-queued ATM switches is one of the most active research fields. Many achievements have been made in the research on scheduling algorithms for unicast input-queued ATM switches and also applied in commerce. But the goal of the research on scheduling algorithms for multicast input-queued ATM switches only focuses on providing high throughput and inadvertently ignoring its undesired effects on QoS of the multicast traffic. In this paper we present a design scheme of input-queued ATM switches supporting multicast and corresponding scheduling algorithm, referred to as multicast longest normalized queue first (MLNQF). The MLNQF algorithm has the characteristics of improving throughput, satisfying QoS requirements and providing service fairly.

In input-queued scheduling, iSLIP algorithm is a milestone for its high throughput and low implementation complexity. However under bursty arrivals, its average input latency increases dramatically and quite a few cells will be dropped as the input buffer overflows. Motivated by cutting down the cell loss rate without increasing average input latency, we present a new scheduling algorithm called iTFF (iterative time to leave and full or not first), which has better performance than iSLIP under bursty traffic conditions. Simulation results indicate that iTFF restrains the cell loss rate significantly and yet maintains low average input latency. Moreover, iTFF can be implemented in hardware to achieve high speed.

ABSTRACTFor negatively associated (NA) random variables, we obtain two general strong laws of large numbers (SLLN) in which the coefficient of sum and the weight are both general functions. As corollaries, we obtain Marcinkiewicz-type SLLN, the logarithmic SLLN and Marcinkiewicz SLLN for NA random variables.

Strong laws of large numbers for t-norm-based addition of fuzzy set-valued random variables

A strong law of large numbers for sub-linear expectation under a general moment condition

In this paper, we shall prove the strong law of large numbers (SLLN) for set-valued random variables in the sense of dH, and the basic space being Rademacher type p(1lesples2) Banach space. This kind of SLLN is the extension of classical SLLN's for Xi-valued random variables and it also implies previous SLLN's results for set-valued random variables

Due to its simplicity and scalability, the differentiated services (DiffServ) model is expected to be widely deployed across the Internet. For each DiffServ compliant router, the scheduling algorithm is critical in implementing per hop behaviors (PHBs), according to which packets are forwarded. We propose a hierarchical DiffServ scheduling (HDS) algorithm to support DiffServ classes on input-queued switches. The proposed HDS algorithm features in a hierarchical scheduling scheme that consists of two levels of schedulers. One level is the central scheduler which is designed to maximize the switch throughput by computing a maximal size matching between input ports and output ports. The other level is formed by input port schedulers which provide differentiated services by serving cells belonging to different classes dynamically. Using such a hierarchical scheme, the implementation complexity and the amount of information needed to be transmitted between input ports and the central scheduler are dramatically reduced compared with existing maximal weight matching based DiffServ scheduling algorithms. The tradeoff of its slightly worse delay performance is acceptable.

The employment of "strong laws of large numbers" is instrumental to the analysis of system estimation and identification strategies. However, the vast bulk of such laws, as presented in the wider literature, assume independence or at least uncorrelatedness of random components, and these assumptions are quite restrictive from an engineering point of view. By way of contrast, the paper shows how to establish strong laws for possibly nonstationary random processes with very general dependence structure. A brief example is provided that illustrates the utility of the strong law of large numbers presented.

High-speed routers and high-performance data centers share a common system-level architecture in which multiple processing nodes are connected by an interconnection network for high-speed communications. Load balancing is an important technique for maximizing throughput and minimizing delay of the interconnection network. In this thesis, efficient load balancing schemes are designed and analyzed for next-generation routers and data centers.
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\nIn high-speed router design, two preferred switch architectures are input-queued switch and load-balanced switch. In an input-queued switch, time-domain load balancing can be carried out by an iterative algorithm that schedules packets for sending in different time slots. The complexity of an iterative algorithm increases rapidly with the number of scheduling iterations. To address this problem, a single-iteration scheduling algorithm called D-LQF is designed, in which exhaustive service policy is adopted for reusing the matched input-output pairs in the previous time slots to grow the match size. 
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\nUnlike an input-queued switch, a load-balanced switch consists of two stages of crossbar switch fabrics, where load balancing is carried out in both time and space domains. Among various load-balanced switches, the feedback-based switch gives the best delay-throughput performance. In this thesis, the feedback-based switch is enhanced in three aspects. Firstly, we focus on reducing its switch fabric complexity. Instead of using crossbars, a dual-banyan network is proposed. The complexity of dual-banyan can be further reduced by merging the two banyans to form a Clos network, resulting in a Clos-banyan network. Secondly, we target at improving the delay performance of the feedback-based switch. A Clos-feedback switch architecture is devised where each switch module in the Clos network is a small feedback-based switch. With application-flow based load balancing, packet order is ensured and the average packet delay is reduced from O(N) to O(n), where N and n are the switch and switch module sizes, respectively. Thirdly, we extend the feedback-based switch to support multicast traffic. Based on the notion of pointer-based multicast VOQ, an efficient multicast scheduling algorithm with packet replication at the middle-stage ports only is proposed. In order to provide close-to-100% throughput for any admissible multicast traffic patterns, a three-stage implementation of feedback-based switch is also designed.
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\nIn designing load balancing schemes for data centers, we focus on the most popular fat-tree based data centers. Notably, packet-based load balancing is widely considered infeasible for data centers. This is because the associated packet out-of-order problem will cause unnecessary TCP fast retransmits, and as a result, severely undermine TCP performance. In this thesis, we show that if packet-based load balancing is performed properly, the packet out-of-order problem can be easily addressed by slightly increasing the number of duplicate ACKs required for triggering fast retransmit. Admittedly, in case of a real packet loss, the loss recovery time will be increased. But our simulation results show that such an increase is far less than the reduction in the network queueing delay (due to a better load-balanced network). As compared to a flow-based load balancing scheme, our packet-based scheme consistently provides significantly higher goodput and noticeably smaller delay.

A Wireless Sensor Network (WSN) is a network system consisting of wireless sensor nodes that collect and transmit data. One crucial factor that affects WSN performance is latency, which is the time required to send data to the gateway. High latency can cause data delivery delays, resulting in data becoming irrelevant. At the same time, receiving real-time data quickly is the primary monitoring method. In addition, high latency can cause greater energy consumption because the sensor node device must be active longer. One way to overcome latency is to implement traffic management techniques such as scheduling data delivery or using priority delivery methods, which can reduce congestion and latency. This paper discusses the effect of latency on WSN using scheduling algorithms, namely Round-Robin and Random scheduling. The results were then taken 30 times to obtain a statistical comparison. The results obtained in the experiment received a Round-Robin scheduling algorithm that was superior to the random scheduling algorithm with a mean and standard deviation value of 12.68 and 2.33, respectively.

The goal of this paper has two-folds. First, we establish skeleton and spine decompositions for superprocesses whose underlying processes are general symmetric Hunt processes. Second, we use these decompositions to obtain weak and strong law of large numbers for supercritical superprocesses where the spatial motion is a symmetric Hunt process on a locally compact metric space $E$ and the branching mechanism takes the form $$\psi_{\beta}(x,\lambda)=-\beta(x)\lambda+\alpha(x)\lambda^{2}+\int_{(0,{\infty})}(e^{-\lambda y}-1+\lambda y)\pi(x,dy)$$ with $\beta\in\mathcal{B}_{b}(E)$, $\alpha\in \mathcal{B}^{+}_{b}(E)$ and $\pi$ being a kernel from $E$ to $(0,{\infty})$ satisfying $\sup_{x\in E}\int_{(0,{\infty})} (y\wedge y^{2}) \pi(x,dy)<{\infty}$. The limit theorems are established under the assumption that an associated Schr\"{o}dinger operator has a spectral gap. Our results cover many interesting examples of superprocesses, including super Ornstein-Uhlenbeck process and super stable-like process. The strong law of large numbers for supercritical superprocesses are obtained under the assumption that the strong law of large numbers for an associated supercritical branching Markov process holds along a discrete sequence of times, extending an earlier result of Eckhoff, Kyprianou and Winkel [Ann. Probab.,43(5), 2594--2659, 2015] for superdiffusions to a large class of superprocesses. The key for such a result is due to the skeleton decomposition of superprocess, which represents a superprocess as an immigration process along a supercritical branching Markov process.

In this note, we revisit the fundamental question of the strong law of large numbers and central limit theorem for processes in continuous time with conditional stationary and independent increments. For convenience, we refer to them as Markov additive processes, or MAPs for short. Historically used in the setting of queuing theory, MAPs have often been written about when the underlying modulating process is an ergodic Markov chain on a finite state space, cf. (Asmussen in Applied probability and queues, Springer-Verlag, New York, 2003; Asmussen and Albrecher in Ruin probabilities, Hackensack, 2010), not to mention the classical contributions of Pacheco and Prabhu (Markov additive processes of arrivals, CRC, Boca Raton, 1995), Prabhu (Stochastic storage processes, Springer-Verlag, New York, 1998). Recent works have addressed the strong law of large numbers when the underlying modulating process is a general Markov processes; cf. as reported (Kyprianou et al. Entrance laws at the origin of self-similar Markov processes in high dimensions, 2019; Yaran and Çağlar in ALEA Lat Am J Probab Math Stat 22:991–1010, 2025) . We add to the latter with a different approach based on an ergodic theorem for additive functionals and on the semimartingale structure of the additive part. This approach also allows us to deal with the setting that the modulator of the MAP is either positive or null-recurrent. The methodology additionally inspires a CLT-type result.