Detecting Markov Chain Instability: A Monte Carlo Approach

We devise a Monte Carlo based method for detecting whether a non-negative Markov chain is stable for a given set of parameter values. More precisely, for a given subset of the parameter space, we develop an algorithm that is capable of deciding whether the set has a subset of positive Lebesgue measure for which the Markov chain is unstable. The approach is based on a variant of simulated annealing, and consequently only mild assumptions are needed to obtain performance guarantees. The theoretical underpinnings of our algorithm are based on a result stating that the stability of a set of parameters can be phrased in terms of the stability of a single Markov chain that searches the set for unstable parameters. Our framework leads to a procedure that is capable of performing statistically rigorous tests for instability, which has been extensively tested using several examples of standard and non-standard queueing networks

A Correction Term for the Covariance of Renewal-Reward Processes with Multivariate Rewards

We consider a renewal-reward process with multivariate rewards. Such a process is constructed from an i.i.d.\ sequence of time periods, to each of which there is associated a multivariate reward vector. The rewards in each time period may depend on each other and on the period length, but not on the other time periods. Rewards are accumulated to form a vector valued process that exhibits jumps in all coordinates simultaneously, only at renewal epochs. We derive an asymptotically exact expression for the covariance function (over time) of the rewards, which is used to refine a central limit theorem for the vector of rewards. As illustrated by a numerical example, this refinement can yield improved accuracy, especially for moderate time-horizons

Analyzing large frequency disruptions in power systems using large deviations theory

We propose a method for determining the most likely cause, in terms of conventional generator outages and renewable fluctuations, of power system frequency reaching a predetermined level that is deemed unacceptable to the system operator. Our parsimonious model of system frequency incorporates primary and secondary control mechanisms, and supposes that conventional outages occur according to a Poisson process and renewable fluctuations follow a diffusion process. We utilize a large deviations theory based approach that outputs the most likely cause of a large excursion of frequency from its desired level. These results yield the insight that current levels of renewable power generation do not significantly increase system vulnerability in terms of frequency deviations relative to conventional failures. However, for a large range of model parameters it is possible that such vulnerabilities may arise as renewable penetration increases.Comment: Accepted to PMAPS 2020 (the 16th International Conference on Probabilistic Methods Applied to Power Systems

Ranking transmission lines by overload probability using the empirical rate function

We develop a non-parametric procedure for ranking transmission lines in a power system according to the probability that they will overload due to stochastic renewable generation or demand-side load fluctuations, and compare this procedure to several benchmark approaches. Using the IEEE 39-bus test network we provide evidence that our approach, which statistically estimates the rate function for each line, is highly promising relative to alternative methods which count overload events or use incorrect parametric assumptions.Comment: Accepted to PMAPS 2020 (the 16th International Conference on Probabilistic Methods Applied to Power Systems

Optimisation of stochastic networks with blocking: a functional-form approach

This paper introduces a class of stochastic networks with blocking, motivated by applications arising in cellular network planning, mobile cloud computing, and spare parts supply chains. Blocking results in lost revenue due to customers or jobs being permanently removed from the system. We are interested in striking a balance between mitigating blocking by increasing service capacity, and maintaining low costs for service capacity. This problem is further complicated by the stochastic nature of the system. Owing to the complexity of the system there are no analytical results available that formulate and solve the relevant optimization problem in closed form. Traditional simulation-based methods may work well for small instances, but the associated computational costs are prohibitive for networks of realistic size. We propose a hybrid functional-form based approach for finding the optimal resource allocation, combining the speed of an analytical approach with the accuracy of simulation-based optimisation. The key insight is to replace the computationally expensive gradient estimation in simulation optimisation with a closed-form analytical approximation that is calibrated using a single simulation run. We develop two implementations of this approach and conduct extensive computational experiments on complex examples to show that it is capable of substantially improving system performance. We also provide evidence that our approach has substantially lower computational costs compared to stochastic approximation

Transient provisioning and performance evaluation for cloud computing platforms: A capacity value approach

User demand on the computational resources of cloud computing platforms varies over time. These variations in demand can be predictable or unpredictable, resulting in ‘bursty’ fluctuations in demand. Furthermore, demand can arrive in batches, and users whose demands are not met can be impatient. We demonstrate how to compute the expected revenue loss over a finite time horizon in the presence of all these model characteristics through the use of matrix analytic methods. We then illustrate how to use this knowledge to make frequent short term provisioning decisions — transient provisioning. It is seen that taking each of the characteristics of fluctuating user demand (predictable, unpredictable, batchy) into account can result in a substantial reduction of losses. Moreover, our transient provisioning framework allows for a wide variety of system behaviors to be modeled and gives simple expressions for expected revenue loss which are straightforward to evaluate numerically

Modelling complex stochastic systems: approaches to management and stability

This thesis is about coping with variability in outcomes for complex stochastic systems. We focus on systems where jobs arrive randomly throughout time to utilise resources for a random amount of time before departure. The systems we investigate are primarily concerned with the communication and storage of data. The thesis is partitioned into two parts. The first part studies systems where congestion leads to jobs waiting for service (queueing systems) and the second part considers systems where congestion leads to losses due to departures before provision of service (loss systems).For queueing systems, we are mainly interested in the management objective of ensuring that the expected time a job must wait before entering is finite --- a property known as stability. Finite waiting times occur naturally for loss systems due to the balking behaviour of jobs in response to congestion and so our attention in this case turns to the more ambitious goal of managing systems in such a way that the number of lost jobs is minimised.Each part consists of an introductory chapter providing background knowledge, which is followed by three chapters containing original research. In both parts we progress through these chapters by first applying traditional analytical approaches to novel models and then developing novel simulation-based approaches for models which are out of reach of traditional approaches.We begin our research on queueing networks in Part 1 by considering a network of infinite-server queues with the special feature that, triggered by specific events, the network population vector may undergo a linear transformation. We use moment generating functions to obtain expressions for transient and stationary moments of the queue size vector and characterise the set of parameters for which the system is stable. A variety of systems fit in the framework developed, such as networks of retrial queues, networks in which jobs can be rerouted when links fail, and storage systems.In the next chapter of Part 1 we study the recently introduced Queue-Proportional Rate Allocation scheduling algorithm for multihop radio networks. The main contribution is a proof using fluid limit techniques to show that a natural generalisation of this policy to allow weighting of packets at each link, to reflect nonhomogeneous priorities, retains the maximal stability property. We also state a conjecture that in heavy traffic the diffusion-scaled workload process of the network converges weakly to a reflected Brownian motion and that in this weak limit the vector of queue lengths is always proportional to the traffic arrival rate vector.We conclude Part 1 by devising a simulation-based method for detecting whether a non-negative Markov chain is unstable for a given set of parameter values. More precisely, for a given subset of the parameter space, we develop an algorithm that can decide whether the set has a subset of positive Lebesgue measure for which the Markov chain is unstable. The approach is based on a variant of simulated annealing, and consequently only mild assumptions are needed to obtain rigorous performance guarantees. Our framework leads to a procedure that can perform statistically rigorous tests for instability, which has been extensively tested using several examples of standard and non-standard queueing networks.We begin our investigation of loss systems in Part 2 by considering a finite-capacity Erlang B model that alternates between active and inactive states according to a two-state modulating Markov process. Jobs arrives to the system as a Poisson process but are blocked from entry when the system is at capacity or inactive. We use Laplace transforms to derive expressions for the revenue lost during short term planning horizons. These expressions can be used to assess alternative system designs.In the next chapter of Part 2 we develop a sophisticated loss system type model for cloud computing systems. User demand on the computational resources of cloud computing platforms varies over time. These variations in the arrival process can be predictable or unpredictable, resulting in time-varying and `bursty' demand fluctuations. Furthermore, jobs can arrive in batches, and users whose demands are not met can be impatient. We demonstrate how to compute the expected revenue loss over a finite time horizon in the presence of all these model characteristics using matrix analytic methods. It is seen that taking these characteristics of fluctuating user demand into account can result in a substantial reduction of losses.We conclude Part 2 by developing an optimisation framework for a model applicable to mobile cloud edge computing systems. Our model is a stochastic network with blocking: jobs attempt to be processed sequentially at nodes in a network but are lost when they attempt to access a node that is at capacity. The problem is mathematically intractable in general and time consuming to solve using standard simulation methods. Our novel method combines simulation with analytical approximations to quickly obtain high quality solutions. We extensively test our approach using several complex models

Ranking transmission lines by overload probability using the empirical rate function

We develop a non-parametric procedure for ranking transmission lines in a power system according to the probability that they will overload due to stochastic renewable generation or demand-side load fluctuations, and compare this procedure to several benchmark approaches. Using the IEEE 39-bus test network we provide evidence that our approach

Analyzing large frequency disruptions in power systems using large deviations theory

We propose a method for determining the most likely cause, in terms of conventional generator outages and renewable fluctuations, of power system frequency reaching a predetermined level that is deemed unacceptable to the system operator. Our parsimonious model of system frequency incorporates primary and secondary control mechanisms, and suppo

Performance of Faulty Loss Systems with Persistent Connections

We consider a finite capacity Erlang loss system that alternates between active and inactive states according to a two state modulating Markov process. Work arrives to the system as a Poisson process but is blocked from entry when the system is at capacity or inactive. Blocked jobs cost the owner a fixed amount that depends on whether blockage was due to the system being at capacity or due to the system being inactive. Jobs which are present in the system when it becomes inactive pause processing until the system becomes active again. A Laplace transform expression for the expected undiscounted revenue lost in [0, t] due to blocking is found. Further, an expression for the total time discounted expected lost revenue in [0,?) is provided. We also derive a second order approximation to the former that can be used when the computing power to invert the Laplace transform is not available. These expressions can be used to ascribe a value to four alternatives for improving system performance: (i) increasing capacity, (ii) increasing the service rate, (iii) increasing the repair rate, or (iv) decreasing the failure rate