J.A.C. Resing

Scaling low-latency optical packet switches to a thousand ports

Optical packet switches that scale to thousands of input/output ports might find their application in next-generation datacenters (DCs). They will allow interconnecting the servers of a DC in a flat topology, providing higher bandwidth and lower latency in comparison with currently applied electronic switches. Using a simple analytic model that allows computing end-to-end latency and throughput, we show that optical interconnects that employ a centralized (electronic) controller cannot scale to thousands of ports while providing end-to-end latencies below 1 μs and high throughput. We therefore investigate architectures with highly distributed control. We present astrictly non-blocking wavelength division multiplexing architecture with contention resolution based on wavelength conversion. We study the packet loss probability of such architecture for different implementations of the contention resolution functionality. Furthermore, we show that the proposed architecture, applied in a short link with flow control, provides submicrosecond end-to-end latencies and allows high load operation, while scaling over a thousand ports.

The M/G/1 processor sharing queue as the almost sure limit of feedback queues

In the paper a probabilistic coupling between the M/G/ 1 processor sharing queue and the M/M/ 1 feedback queue, with general feedback probabilities, is established. This coupling is then used to prove the almost sure convergence of sojourn times in the feedback model to sojourn times in the M/G/ 1 processor sharing queue. Using the theory of regenerative processes it follows that for stable queues the stationary distribution of the sojourn time in the feedback model converges in law to the corresponding distribution in the processor sharing model. The results do not depend on Poisson arrival times, but are also valid for general arrival processes.

A finite‐source queuing model for the IEEE 802.11 DCF

The most mature medium access control protocol for wireless local area networks is the IEEE 802.11 standard. The primary access mode of this protocol is based on the mechanism of carrier sense multiple access with binary exponential backoff. We develop a finite-source feedback queueing model for this access mode. In this model, we derive expressions for the throughput and the first moment of the delay and a set of equations to compute the Laplace Stieltjes Transform and higher moments of the delay. The accuracy of the model is verified by simulation.

Discrete event systems with stochastic processing times

Discrete-event dynamic systems (DEDS) are studied in which the underlying algebra is the max-algebra and the coefficients in the system, referring to processing times in practice, are stochastic. The authors note that until now the study of DEDS in the context of max-algebra has been purely deterministic; in the present work, the authors discuss certain stochastic extensions. The theory given deals with the analysis of stochastic DEDS in which the processing times and/or the transportation times within a network show stochastic fluctuations.<<ETX>>

Bounds for a discrete-time multi-server queue with an application to cable networks

In this paper we consider a discrete-time multi-server queue. We present simple, explicit, and sharp bounds on the mean queue content and mean delay. These bounds have two considerable advantages over the exact expressions. Firstly, they apply quite generally as they depend on the distribution of the number of arrivals in a time slot only through the first two moments. Secondly, they do not require the numerical procedures that must usually be employed to study this model. We give an application of our model to upstream data transport in cable networks. Cable networks are characterised by a request-grant procedure in which actual data transmission follows after a reservation procedure. Although the delay structure is rather complicated, bounds on the mean queue content and mean packet delay are obtained. The bounds can be used to investigate the impact of the reservation procedure on the mean delay.

STOCHASTIC DISCRETIZATION FOR THE LONG-RUN AVERAGE REWARD IN FLUID MODELS

Stochastic discretization is a technique of representing a continuous random variable as a random sum of i.i.d. exponential random variables. In this article, we apply this technique to study the limiting behavior of a stochastic fluid model. Specifically, we consider an infinite-capacity fluid buffer, where the net input of fluid is regulated by a finite-state irreducible continuous-time Markov chain. Most long-run performance characteristics for such a fluid system can be expressed as the long-run average reward for a suitably chosen reward structure. In this article, we use stochastic discretization of the fluid content process to efficiently determine the long-run average reward. This method transforms the continuous-state Markov process describing the fluid model into a discrete-state quasi-birth–death process. Hence, standard tools, such as the matrix-geometric approach, become available for the analysis of the fluid buffer. To demonstrate this approach, we analyze the output of a buffer processing fluid from K sources on a first-come first-served basis.

Queues and Risk Processes with Dependencies

We study the generalization of the G/G/1 queue obtained by relaxing the assumption of independence between inter-arrival times and service requirements. The analysis is carried out for the class of multivariate matrix exponential distributions introduced in Ref.[13]. In this setting, we obtain the steady-state waiting time distribution, and we show that the classical relation between the steady-state waiting time and workload distributions remains valid when the independence assumption is relaxed. We also prove duality results with the ruin functions in an ordinary and a delayed ruin process. These extend several known dualities between queueing and risk models in the independent case. Finally, we show that there exist stochastic order relations between the waiting times under various instances of correlation.

The single server queue with synchronized services

We consider a single server queueing system with generally distributed synchronized services. More specifically, customers arrive according to a Poisson process and there is a single server that provides service, if there is at least one customer present in the system. Upon the initialization of a service, all present customers start to receive service simultaneously. We consider the gated version of the model, that is, customers who arrive during a service time do not receive service immediately but wait for the beginning of the next service time. At service completion epochs, all served customers decide simultaneously and independently whether they will leave the system or stay for another service. The probability that a served customer gets another service is the same for all customers. We study the model and derive its main performance measures that include the equilibrium distribution of the number of customers at service completion epochs and in continuous time, the busy period and the sojourn time distributions. Moreover, we prove some limiting results regarding the behavior of the system in the extreme cases of the synchronization level. Several variants and extensions of the model are also discussed.

An M / G /1 Queueing Model with Gated Random Order of Service

We analyse an M/G/1 queueing model with gated random order of service. In this service discipline there are a waiting room, in which arriving customers are collected, and a service queue. Each time the service queue becomes empty, all customers in the waiting room are put instantaneously and in random order into the service queue. The service times of customers are generally distributed with finite mean. We derive various bivariate steady-state probabilities and the bivariate Laplace–Stieltjes transform (LST) of the joint distribution of the sojourn times in the waiting room and the service queue. The derivation follows the line of reasoning of Avi-Itzhak and Halfin [4]. As a by-product, we obtain the joint sojourn times LST for several other gated service disciplines.

Asymptotic behavior of random discrete event systems

Various aspects of the asymptotic behavior of discrete-events dynamic systems (DEDS) in which the activity times are random variables are discussed. The main result is that the central limit theorem holds for DEDS and consequently that the cycle time of the system is asymptotically normally distributed. Calculations of the expectation and variance of the cycle time are given. Reducible random DEDS are considered, and the behavior of random DEDS is compared with that of deterministic DEDS.<<ETX>>