A Mathematical Modelling Approach for Systems Where the Servers Are Almost Always Busy

The design and implementation of new configurations of mental health services to meet local needs is a challenging problem. In the UK, services for common mental health disorders such as anxiety and depression are an example of a system running near or at capacity, in that it is extremely rare for the queue size for any given mode of treatment to fall to zero. In this paper we describe a mathematical model that can be applied in such circumstances. The model provides a simple way of estimating the mean and variance of the number of patients that would be treated within a given period of time given a particular configuration of services as defined by the number of appointments allocated to different modes of treatment and the referral patterns to and between different modes of treatment. The model has been used by service planners to explore the impact of different options on throughput, clinical outcomes, queue sizes, and waiting times. We also discuss the potential for using the model in conjunction with optimisation techniques to inform service design and its applicability to other contexts

Exact Solutions for M/M/c/Setup Queues

Recently multiserver queues with setup times have been extensively studied because they have applications in power-saving data centers. The most challenging model is the M/M/$c$/Setup queue where a server is turned off when it is idle and is turned on if there are some waiting jobs. Recently, Gandhi et al.~(SIGMETRICS 2013, QUESTA 2014) present the recursive renewal reward approach as a new mathematical tool to analyze the model. In this paper, we derive exact solutions for the same model using two alternative methodologies: generating function approach and matrix analytic method. The former yields several theoretical insights into the systems while the latter provides an exact recursive algorithm to calculate the joint stationary distribution and then some performance measures so as to give new application insights.Comment: Submitted for revie

Analysis of a class of distributed queues with application

Recently we have developed a class of media access control algorithms for different types of Local Area Networks. A common feature of these LAN algorithms is that they represent various strategies by which the processors in the LAN can simulate the availability of a centralized packet transport facility, but whose service incorporates a particular type of change over time known as 'moving sever' overhead. First we describe the operation of moving server systems in general, for both First-Come - First-Served and Head-of-the-Line orders of service, together with an approach for their delay analysis in which we transform the moving server queueing system into a conventional queueing system having proportional waiting times. Then we describe how the various LAN algorithms may be obtained from the ideal moving server system, and how a significant component of their performance characteristics is determined by the performance characteristics of that ideal system. Finally, we evaluate the compatibility of such LAN algorithms with separable queueing network models of distributed systems by computing the interdeparture time distribution for M/M/1 in the presence of moving server overhead. Although it is not exponential, except in the limits of low server utilization or low overhead, the interdeparture time distribution is a weighted sum of exponential terms with a coefficient of variation not much smaller than unity. Thus, we conjecture that a service centre with moving server overhead could be used to represent one of these LAN algorithms in a product form queueing network model of a distributed system without introducing significant approximation errors

User behaviour modelling for resource management in a hybrid UMTS/DVB-T network

Third generation mobile networks such as UMTS are designed to enhance the deployment of multimedia services providing high data rates and new flexible communication capabilities. However, these systems are interference limited and as such their performance is lowered in the case of a large number of users generating heavy traffic. A solution to this problem is to interconnect the UMTS network to a Digital Video Broadcasting-Terrestrial (DVB-T) network, so that the lack of capacity of UMTS during busy periods can be offset by the high bit rate available on the broadcast network. In order to justify this choice a prediction of the number of subscribers requesting the new multimedia applications designed for this scenario is needed. This paper focuses on the user behaviour modelling for multimedia services in a hybrid UMTS/DVB-T platform. The aim of the paper is to provide operators with a forecast of the demand for new multimedia services showing how they can be subject to a very high number of subscriptions, which UMTS would hardly be able to handle

A self-adapting latency/power tradeoff model for replicated search engines

For many search settings, distributed/replicated search engines deploy a large number of machines to ensure efficient retrieval. This paper investigates how the power consumption of a replicated search engine can be automatically reduced when the system has low contention, without compromising its efficiency. We propose a novel self-adapting model to analyse the trade-off between latency and power consumption for distributed search engines. When query volumes are high and there is contention for the resources, the model automatically increases the necessary number of active machines in the system to maintain acceptable query response times. On the other hand, when the load of the system is low and the queries can be served easily, the model is able to reduce the number of active machines, leading to power savings. The model bases its decisions on examining the current and historical query loads of the search engine. Our proposal is formulated as a general dynamic decision problem, which can be quickly solved by dynamic programming in response to changing query loads. Thorough experiments are conducted to validate the usefulness of the proposed adaptive model using historical Web search traffic submitted to a commercial search engine. Our results show that our proposed self-adapting model can achieve an energy saving of 33% while only degrading mean query completion time by 10 ms compared to a baseline that provisions replicas based on a previous day's traffic

A survey of the machine interference problem

This paper surveys the research published on the machine interference problem since the 1985 review by Stecke & Aronson. After introducing the basic model, we discuss the literature along several dimensions. We then note how research has evolved since the 1985 review, including a trend towards the modelling of stochastic (rather than deterministic) systems and the corresponding use of more advanced queuing methods for analysis. We conclude with some suggestions for areas holding particular promise for future studies.Natural Sciences and Engineering Research Council (NSERC) Discovery Grant 238294-200

Analysis of a batch-service queue with variable service capacity, correlated customer types and generally distributed class-dependent service times

Queueing models with batch service have been studied frequently, for instance in the domain of telecommunications or manufacturing. Although the batch server's capacity may be variable in practice, only a few authors have included variable capacity in their models. We analyse a batch server with multiple customer classes and a variable service capacity that depends on both the number of waiting customers and their classes. The service times are generally distributed and class-dependent. These features complicate the analysis in a non-trivial way. We tackle it by examining the system state at embedded points, and studying the resulting Markov Chain. We first establish the joint probability generating function (pgf) of the service capacity and the number of customers left behind in the queue immediately after service initiation epochs. From this joint pgf, we extract the pgf for the number of customers in the queue and in the system respectively at service initiation epochs and departure epochs, and the pgf of the actual server capacity. Combined with additional techniques, we also obtain the pgf of the queue and system content at customer arrival epochs and random slot boundaries, and the pgf of the delay of a random customer. In the numerical experiments, we focus on the impact of correlation between the classes of consecutive customers, and on the influence of different service time distributions on the system performance. (C) 2019 Elsevier B.V. All rights reserved