Quasi-stationary analysis for queues with temporary overload

Motivated by the high variation in transmission rates for document transfer in the Internet and file down loads from web servers, we study the buffer content in a queue with a fluctuating service rate. The fluctuations are assumed to be driven by an independent stochastic process. We allow the queue to be overloaded in some of the server states. In all but a few special cases, either exact analysis is not tractable, or the dependence of system performance in terms of input parameters (such as the traffic load) is hidden in complex or implicit characterizations. Various asymptotic regimes have been considered to develop insightful approximations. In particular, the so-called quasistationary approximation has proven extremely useful under the assumption of uniform stability. We refine the quasi-stationary analysis to allow for temporary instability, by studying the “effective system load” which captures the effect of accumulated work during periods in which the queue is unstable

Analysis of a M/M/c queue with single and multiple synchronous working vacations

We consider a M/M/c queuing system with synchronous working vacation and two different policies of working vacation i.e. a multiple working vacation policy and a single working policy. During a working vacation the server does not completely halts the service rather than it will render service at a lower rate. In synchronous vacation policy all the servers leave for a vacation simultaneously, when the server finds the system empty after finishing serving a customer. In multiple working vacation (MWV) policy the servers continue to take vacation till they find the system nonempty at a vacation completion instant. Single working vacation (SWV) policy is different from the multiple working vacation policy in a way that, when the working vacation ends and servers find the system empty, they remains idle until the first arrival occurs rather than taking another vacation. We have derived explicit expressions for some performance measures in terms of two indexes by using PGF method. We derived some results regarding the limiting behavior of some performance measures based on these two indexes. A comparison between the models is carried out and numerical results are provided to illustrate the effects of various parameters on system performance measures

Performance Modelling and Optimisation of Multi-hop Networks

A major challenge in the design of large-scale networks is to predict and optimise the total time and energy consumption required to deliver a packet from a source node to a destination node. Examples of such complex networks include wireless ad hoc and sensor networks which need to deal with the effects of node mobility, routing inaccuracies, higher packet loss rates, limited or time-varying effective bandwidth, energy constraints, and the computational limitations of the nodes. They also include more reliable communication environments, such as wired networks, that are susceptible to random failures, security threats and malicious behaviours which compromise their quality of service (QoS) guarantees. In such networks, packets traverse a number of hops that cannot be determined in advance and encounter non-homogeneous network conditions that have been largely ignored in the literature. This thesis examines analytical properties of packet travel in large networks and investigates the implications of some packet coding techniques on both QoS and resource utilisation. Specifically, we use a mixed jump and diffusion model to represent packet traversal through large networks. The model accounts for network non-homogeneity regarding routing and the loss rate that a packet experiences as it passes successive segments of a source to destination route. A mixed analytical-numerical method is developed to compute the average packet travel time and the energy it consumes. The model is able to capture the effects of increased loss rate in areas remote from the source and destination, variable rate of advancement towards destination over the route, as well as of defending against malicious packets within a certain distance from the destination. We then consider sending multiple coded packets that follow independent paths to the destination node so as to mitigate the effects of losses and routing inaccuracies. We study a homogeneous medium and obtain the time-dependent properties of the packet’s travel process, allowing us to compare the merits and limitations of coding, both in terms of delivery times and energy efficiency. Finally, we propose models that can assist in the analysis and optimisation of the performance of inter-flow network coding (NC). We analyse two queueing models for a router that carries out NC, in addition to its standard packet routing function. The approach is extended to the study of multiple hops, which leads to an optimisation problem that characterises the optimal time that packets should be held back in a router, waiting for coding opportunities to arise, so that the total packet end-to-end delay is minimised

Fluid Queue Driven by an M

This paper deals with the stationary analysis of a fluid queue driven by an M/M/1 queueing model subject to Bernoulli-Schedule-Controlled Vacation and Vacation Interruption. The model under consideration can be viewed as a quasi-birth and death process. The governing system of differential difference equations is solved using matrix-geometric method in the Laplacian domain. The resulting solutions are then inverted to obtain an explicit expression for the joint steady state probabilities of the content of the buffer and the state of the background queueing model. Numerical illustrations are added to depict the convergence of the stationary buffer content distribution to one subject to suitable stability conditions

Heavy-traffic limits for Polling Models with Exhaustive Service and non-FCFS Service Order Policies

We study cyclic polling models with exhaustive service at each queue under a variety of non-FCFS local service orders, namely Last-Come-First-Served (LCFS) with and without preemption, Random-Order-of-Service (ROS), Processor Sharing (PS), the multi-class priority scheduling with and without preemption, Shortest-Job-First (SJF) and the Shortest Remaining Processing Time (SRPT) policy. For each of these policies, we rst express the waiting-time distributions in terms of intervisit-time distributions. Next, we use these expressions to derive the asymptotic waiting-time distributions under heavy-trac assumptions, i.e., when the system tends to saturate. The results show that in all cases the asymptotic wait

Reflections on queue modelling from the last 50 years

Queueing theory continues to be one of the most researched areas of operational research, and has generated numerous review papers over the years. The phrase 'queue modelling' is used in the title to indicate a more practical emphasis. This paper uses work taken predominantly from the last 50 years of pages of the Operational Research Quarterly and the Journal of the Operational Research Society to offer a commentary on attempts of operational researchers to tackle real queueing problems, and on research foci past and future. A new discipline of 'queue modelling' is proposed, drawing upon the combined strengths of analytic and simulation approaches with the responsibility to derive meaningful insights for managers